A Quantum Kirwan Map: Bubbling and Fredholm Theory for ... - KIAS

A Quantum Kirwan Map: Bubbling and FredholmTheory for Symplectic Vortices over the PlaneFabian ZiltenerAuthor address:Korea Institute for Advanced Study, 87 Hoegiro, Dongdaemun-gu,Seoul 130-722, Republic of KoreaE-mail address: fabian@kias.re.kr

ContentsChapter 1. Motivation and main results 11.1. Quantum deformations of the Kirwan map 11.2. Symplectic vortices, idea of the proof of existence of a quantumKirwan map 31.3. Bubbling for vortices over the plane 61.4. Fredholm theory for vortices over the plane 71.5. Remarks, related work, organization 11Chapter 2. Bubbling for vortices over the plane 172.1. Stable maps 172.2. Convergence to a stable map 212.3. An example: the Ginzburg-Landau setting 252.4. The action of the reparametrization group 292.5. Compactness modulo bubbling and gauge for rescaled vortices 312.6. Soft rescaling 432.7. Proof of the bubbling result 522.8. Proof of the result in Section 2.3 characterizing convergence 62Chapter 3. Fredholm theory for vortices over the plane 693.1. Equivariant homology, the Chern number, and the Maslov index 693.2. Proof of the Fredholm result 73Appendix A. Auxiliary results about vortices, weighted spaces, and othertopics 91A.1. Auxiliary results about vortices 91A.2. The invariant symplectic action 96A.3. Proofs of the results of Section 3.1 97A.4. Weighted Sobolev spaces and a Hardy-type inequality 103A.5. Smoothening a principal bundle 110A.6. Proof of the existence of a right inverse for d ∗ A 112A.7. Further auxiliary results 120Bibliography 127iii

AbstractConsider a Hamiltonian action of a compact connected Lie group on a symplecticmanifold (M,ω). Conjecturally, under suitable assumptions there exists a morphismof cohomological field theories from the equivariant Gromov-Witten theoryof (M,ω) to the Gromov-Witten theory of the symplectic quotient. The morphismshould be a deformation of the Kirwan map. The idea, due to D. A. Salamon, isto define such a deformation by counting gauge equivalence classes of symplecticvortices over the complex plane C.The present memoir is part of a project whose goal is to make this definitionrigorous. Its main results deal with the symplectically aspherical case. The firstone states that every sequence of equivalence classes of vortices over the plane hasa subsequence that converges to a new type of genus zero stable map, providedthat the energies of the vortices are uniformly bounded. Such a stable map consistsof equivalence classes of vortices over the plane and holomorphic spheres in thesymplectic quotient. The second main result is that the vertical differential of thevortex equations over the plane (at the level of gauge equivalence) is a Fredholmoperator of a specified index.Potentially the quantum Kirwan map can be used to compute the quantumcohomology of symplectic quotients.2000 Mathematics Subject Classification. Primary: 53D20, 53D45.Key words and phrases. Symplectic quotient, Kirwan map, (equivariant) Gromov-Witteninvariants, (equivariant) quantum cohomology, morphism of cohomological field theories, Yang-Mills-Higgs functional, stable map, compactification, weighted Sobolev spaces.The author acknowledges financial support by the Swiss National Science Foundation (fellowship200020-101611).iv

CHAPTER 1Motivation and main results1.1. Quantum deformations of the Kirwan mapLet (M,ω) be a symplectic manifold without boundary, and G a compact connectedLie group with Lie algebra g. We fix a Hamiltonian action of G on M andan (equivariant) momentum map 1 µ : M → g ∗ . Throughout this memoir, we makethe following standing assumption:Hypothesis (H): G acts freely on µ −1 (0) and the momentum map µ is proper.Then the symplectic quotient ( M := µ −1 (0)/G,ω ) is well-defined, smooth andclosed (i.e., compact and without boundary). The Kirwan map is a canonical ringhomomorphismκ : HG(M) ∗ → H ∗ (M).Here H ∗ and HG ∗ denote cohomology and equivariant cohomology with rationalcoefficients, and the product structures are the cup products. F. Kirwan proved[Kir] that this map is surjective. Based on this result, the cohomology ring H ∗ (M)was described in different ways by L. C. Jeffrey and F. Kirwan [JK, Theorem 8.1],S. Tolman and J. Weitsman [TW, Theorem 1], and many others.The present memoir is concerned with the problem of “quantizing” the Kirwanmap, which was first investigated by R. Gaio and D. A. Salamon. Assumingsymplectic asphericity and some other restrictive conditions, in [GS, Corollary A’]these authors constructed a ring homomorphism from HG ∗ (M) to the (small) quantumcohomology of (M,ω), which intertwines the Gromov-Witten invariants of thesymplectic quotient with the symplectic vortex invariants. Their result is based onan adiabatic limit in the symplectic vortex equations. It was used by K. Cieliebakand D. A. Salamon in [CS, Theorem 1.3] to prove that given a monotone linearsymplectic torus action on R 2n with minimal first equivariant Chern number atleast 2, the quantum cohomology of (M,ω) is isomorphic to the Batyrev ring.The result by Gaio and Salamon motivates the following conjecture. We denoteby QH ∗ G(M,ω) the equivariant quantum cohomology. By this we mean the Q-vectorspace of all maps α : H2 G (M, Z) → HG ∗ (M) satisfying an equivariant version of theNovikov condition, together with a product counting holomorphic maps from S 2to the fibers of the Borel construction for the action of G on M. 2 The Novikovcondition states that for every number C ∈ R there are only finitely many classesB ∈ H2 G (M, Z), such that α(B) ≠ 0 and 〈 [ω − µ],B 〉 ≤ C. Here [ω − µ] ∈ HG 2 (M)denotes the cohomology class of the two-cocycle ω − µ in the Cartan model.1 Momentum maps are often called moment maps by symplectic geometers. However, thefirst term seems more appropriate, since the notion generalizes the linear and angular momentaappearing in classical mechanics.2 For the definition of this product see [Gi, GiK, Kim, Lu, Ru].1

2 1. MOTIVATION AND MAIN RESULTSThe space QH ∗ G(M,ω) is naturally a module over the equivariant Novikov ringΛ µ ω. 3 We denote by QH ∗ (M,ω) the quantum cohomology of (M,ω) with coefficientsin this ring. A map(1.1) ϕ : H G 2 (M, Z) → Hom Q(H∗G (M),H ∗ (M) )satisfying the equivariant Novikov condition 4 , induces a Λ µ ω-module homomorphism(1.2) ϕ ∗ : QH ∗ G(M,ω) → QH ∗ (M,ω), (ϕ ∗ α)(B) := ∑ ϕ(B 1 )α(B 2 ),where the sum is over all pairs B 1 ,B 2 ∈ H2 G (M, Z) satisfying B 1 + B 2 = B. Wedenote by c G 1 (M,ω) ∈ HG 2 (M, Z) the first G-equivariant Chern class of (TM,ω),and by(1.3) N := inf ({〈 c G 1 (M,ω),B 〉 ∣ ∣ B ∈ HG2 (M, Z) : spherical } ∩ N ) ∈ N ∪ {∞} 5the minimal equivariant Chern number. We call (M,ω,µ) semipositive iff thereexists a constant c ∈ R such that〈[ω − µ],B〉= c〈cG1 (M,ω),B 〉 ,for every spherical class B ∈ H G 2 (M, Z), and if c < 0 then N ≥ 1 2 dimM.1. Conjecture (Quantum Kirwan map, semipositive case). Assume that (H)holds and that (M,ω,µ) is convex at ∞ 6 and semipositive. Then there existsa map ϕ as in (1.1), satisfying the equivariant Novikov condition, such that theinduced map ϕ ∗ as in (1.2) is a surjective ring homomorphism, and(1.4)ϕ(0) = κ,〈 〉(1.5) [ω − µ],B ≤ 0, B ≠ 0 =⇒ ϕ(B) = 0.Once proven, this conjecture will give rise to a recursion formula for QH ∗ (M,ω)in terms of QH ∗ G(M,ω) and ϕ. 7 As noticed in [NWZ], without the semipositivitycondition, the conjecture likely needs to be modified as follows:2. Conjecture (Quantum Kirwan map, general situation). Assume that (H)holds, and that (M,ω,µ) is convex at ∞. Then there exists a morphism of cohomologicalfield theories (CohFT’s) from the equivariant Gromov-Witten theory of(M,ω) to the Gromov-Witten theory of (M,ω).For the notion of a morphism between two CohFT’s V and W see [NWZ].Such a morphism consists of a sequence of S n -invariant multilinear mapsψ n : V ×n × H ∗ (M n,1 (A)) → W, n ∈ N 0 := N ∪ {0},3 This ring consists of all maps λ : H G2 (M, Z) → Q satisfying an equivariant version of theNovikov condition, analogous to the one above. The product is given by convolution.4 This condition is analogous to the one above.5 In this memoir N := {1, 2, . . .} does not include 0.6 This means that there exists an ω-compatible and G-invariant almost complex structure Jon M, such that the quadruple `M, ω, µ, J´ is convex at ∞ in the sense explained before Theorem3 below.7 The recursion is over the set(*+ )kX[ω − µ], B i k ∈ N ∪ {0}, B˛˛˛˛˛ i ∈ H2 G (M, Z) : ϕ(B i ) ≠ 0 or GW G B i ≠ 0, i = 1, . . . , k ,i=1where GW G B idenotes the 3-point genus 0 equivariant Gromov-Witten invariant of (M, ω) in theclass B i .

1.2. SYMPLECTIC VORTICES 3satisfying relations involving the composition maps of V and W. Here M n,1 (A)denotes the moduli space of stable n-marked scaled lines. Furthermore, the actionof the symmetric group S n is by permutations of the first n arguments of ψ n . Themap ψ 1 plays the role of ϕ ∗ as in Conjecture 1. The map ψ 0 measures how muchψ 1 fails to be a ring homomorphism. Once proven, Conjecture 2 will give rise toa recursion formula for the Gromov-Witten invariants of (M,ω) in terms of theequivariant Gromov-Witten invariants of (M,ω) and the morphism (ψ n ) n .The present memoir is part of a project whose goal is to prove Conjectures 1and 2. 8 The approach pursued here was suggested by D. A. Salamon. 9 The idea isto construct the maps of the conjectures by counting symplectic vortices over thecomplex plane C. In a first step, we will consider the (symplectically) asphericalcase. This means that∫(1.6)u ∗ ω = 0, ∀u ∈ C ∞ (S 2 ,M).S 2In this case the equivariant quantum cup product is induced by the ordinary cupproduct on HG ∗ (M).1.2. Symplectic vortices, idea of the proof of existence of a quantumKirwan mapTo explain the idea of the proofs of Conjectures 1 and 2, we recall the symplecticvortex equations: Let J be an ω-compatible and G-invariant almost complexstructure on M, 〈·, ·〉 g an invariant inner product on g, (Σ,j) a Riemann surface,and ω Σ a compatible area form on Σ. 10 For every smooth (principal) G-bundle Pover Σ we denote by A(P) the affine space of smooth connection one-forms on P,and by CG ∞ (P,M) the set of smooth equivariant maps from P to M. Consider theclass˜B := ˜B Σ := { w := (P,A,u) ∣ ∣ P smooth G-bundle over Σ,(1.7)A ∈ A(P), u ∈ C ∞ G (P,M) } .The symplectic vortex equations are the equations¯∂ J,A (u) = 0,(1.8)(1.9)F A + (µ ◦ u)ω Σ = 0for a triple (P,A,u) ∈ ˜B. To explain these conditions, note that the pullback bundleu ∗ TM → P descends to a complex vector bundle (u ∗ TM)/G → Σ. 11 For everyx ∈ M we denote by L x : g → T x M the infinitesimal action, corresponding tothe action of G on M. With this notation, ¯∂ J,A (u) means the complex antilinearpart of d A u := du + L u A, which we think of as a one-form on Σ with values in(u ∗ TM)/G → Σ. In (1.9) we view the curvature F A of A as a two-form on Σ withvalues in the adjoint bundle g P := (P × g)/G → Σ 12 . Furthermore, identifying g ∗with g via 〈·, ·〉 g , we view µ ◦ u as a section of g P . The vortex equations (1.8,1.9)were discovered by K. Cieliebak, A. R. Gaio, and D. A. Salamon [CGS], and8 Further relevant results will appear elsewhere, including [Zi4, Zi5].9 Private communication.10 This means that j and ωΣ determine the same orientation of Σ.11 The complex structure on this bundle is induced by the almost complex structure J.12 Here G acts on g in the adjoint way.

1.2. SYMPLECTIC VORTICES 5for B ∈ H G 2 (M, Z), α ∈ H ∗ G (M), and b ∈ H ∗(M). Under the hypotheses ofConjecture 1, this map is “morally” well-defined and satisfies the conditions ofthe conjecture: If J is chosen as in the definition of convexity below, then thereexists a compact subset of M/G containing the image of every finite energy vortexclass W ∈ B C whose image has compact closure. This ensures that for everyB ∈ H G 2 (M, Z), the space M B can be compactified by including holomorphicspheres in M and in the fibers of the Borel construction (M × EG)/G. In thetransverse case, it follows that the “boundary” of M B has codimension at least 2.This makes the map ϕ “well-defined”. It satisfies the equivariant Novikov conditionas a consequence of the compactification argument, conservation of the equivarianthomology class in the limit (see [Zi1, Zi4]), and the identityE(W) = 〈 [ω − µ],[W] 〉 .This holds for every vortex class W ∈ B C of finite energy, such that the image ofW has compact closure. 18 The identity also implies conditions (1.4,1.5).The ring homomorphism property for the induced map Qκ := ϕ ∗ follows froman argument involving two marked points on the plane C that either move togetheror infinitely apart. The semipositivity assumption ensures that in the limit thereis no bubbling of vortex classes over C without marked points. In contrast withholomorphic planes, such vortex classes may occur in stable maps in top dimensionalstrata, even in the transverse case. This is due to the fact that vortices over C“should not be rotated”, which is explained below.Surjectivity of Qκ will be a consequence of surjectivity of the Kirwan map κ,and the equivariant Novikov property. The idea of the proof of Conjecture 2 is todefine Qκ 1 := Qκ as above, and for general n ∈ N 0 , Qκ n in a similar way, using nmarked points. The map Qκ 0 counts vortex classes over C without marked points.The “quantum Kirwan morphism” (Qκ n ) n∈N0 will intertwine the genus 0 symplecticvortex invariants with the Gromov-Witten invariants of (M,ω). This willfollow from a bubbling argument for a sequence of vortex classes over the sphereS 2 , equipped with an area form that converges to ∞. 19The goal of the present memoir is to establish bubbling (i.e., “compactification”)and Fredholm results for vortices over C in the aspherical case. Togetherwith a transversality result (see [Zi5]), the Fredholm result will provide a naturalstructure of an oriented manifold on the set M B . Furthermore, the bubbling resultwill imply that the map(ev 0 ,ev ∞ ) : M B → (M × EG)/G × Mis a pseudocycle 20 .This will give a rigorous meaning to the integral (1.14). The ringhomomorphism property and the relations defining a morphism of CohFT’s will bea consequence of the bubbling result and a suitable gluing result.18 The equality follows from [CGS, Proposition 3.1] with Σ := S 2 ∼ = C∪{∞} and a smootheningargument at ∞.19 This corresponds to the adiabatic limit studied by Gaio and Salamon in [GS]. The newfeature here is that in the limit, vortex classes over C may bubble off.20 as defined in [MS2, Definition 6.5.1]

6 1. MOTIVATION AND MAIN RESULTS1.3. Bubbling for vortices over the planeTo explain the first main result of this memoir, we assume that (M,ω) isaspherical, i.e., condition (1.6) is satisfied. 21 We denote by(1.15) ˜M :={(P,A,u) ∈ ˜BC∣ ∣ (1.8,1.9)}, M := ˜M/∼the class of all vortices over C and the set of equivalence classes of such vortices. Thelatter is equipped with a natural topology. 22 Consider the subspace of all classesin M with fixed finite energy E > 0. There are three sources of non-compactnessof this space: Consider a sequence W ν ∈ M, ν ∈ N, of classes of energy E. In thelimit ν → ∞, the following scenarios (and combinations) may occur:Case 1. The energy density of W ν blows up at some point in C.Case 2. There exists a number r > 0 and a sequence of points z ν ∈ C that convergesto ∞, such that the energy density of W ν on the ball B r (z ν ) is boundedabove and below by some fixed positive constants.Case 3. The energy densities converge to 0, i.e., the energy is spread out more andmore.In case 1, by rescaling W ν around the bubbling point, in the limit ν → ∞,we obtain a non-constant J-holomorphic map from C to M. Using removal ofsingularity, this is excluded by the asphericity condition. In case 2, we pull W νback by the translation z ↦→ z + z ν , and in the limit ν → ∞, obtain a vortex classover C. Finally, in case 3, we “zoom out” more and more. In the limit ν → ∞ andafter removing the singularity at ∞, we obtain a pseudo-holomorphic map from S 2to the symplectic quotient M = µ −1 (0)/G.Hence in the limit, passing to a subsequence, we expect W ν to converge to a newsort of stable map, which consists of vortex classes over C and pseudo-holomorphicspheres in M. Here an important difference to Gromov-convergence for pseudoholomorphicmaps is the following: Although the vortex equations are invariantunder all orientation preserving isometries of Σ, only translations on C should beallowed as reparametrizations used to obtain a vortex class over C in the limit.Hence we should disregard some symmetries of the equations. The reasons are thatotherwise the reparametrization group does not always act with finite isotropy onthe set of simple stable maps, and that there is no suitable evaluation map on theset of vortex classes, which is invariant under rotation. 23We are now ready to formulate the first main result. Here we say that (M,ω,µ,J)is convex at ∞ iff there exists a proper G-invariant function f ∈ C ∞ (M,[0, ∞))and a constant C ∈ [0, ∞), such thatω(∇ v ∇f(x),Jv) − ω(∇ Jv ∇f(x),v) ≥ 0, df(x)JL x µ(x) ≥ 0,for every x ∈ f −1 ([C, ∞)) and 0 ≠ v ∈ T x M. Here ∇ denotes the Levi-Civitaconnection of the metric ω(·,J·). This condition reduces to the existence of a21 The general situation is discussed in Remark 17 in Section 2.1.22 It is induced by the C ∞ -topology on compact subsets of C.23 See Remarks 22 and 33 in Sections 2.2 and 2.4.

1.4. FREDHOLM THEORY FOR VORTICES OVER THE PLANE 7plurisubharmonic function in the case in which G is trivial. It is satisfied e.g. if Mis closed, and for linear actions on symplectic vector spaces. 243. Theorem (Bubbling). Assume that hypothesis (H) is satisfied, (M,ω) isaspherical, and (M,ω,µ,J) is convex at ∞. Let k ∈ N 0 , and for ν ∈ N let W ν ∈ Mbe a vortex class and z1,...,z ν k ν ∈ C be points. Suppose that the closure of theimage of each W ν is compact, andE(W ν ) > 0, ∀ν ∈ N,supE(W ν ) < ∞.ν∈NThen there exists a subsequence of ( W ν ,z ν 0 := ∞,z ν 1,...,z ν k)that converges tosome genus 0 stable map (W,z) consisting of vortex classes over C and pseudoholomorphicspheres in M, with k + 1 marked points. (See Definitions 15 and 20in Chapter 2. 25 )The proof of this result combines Gromov compactness for pseudo-holomorphicmaps with Uhlenbeck compactness. It relies on work [CGMS, GS] by K. Cieliebak,R. Gaio, I. Mundet i Riera, and D. A. Salamon. The idea is the following. In orderto capture all the energy, we “zoom out rapidly”, i.e., rescale the vortices so muchthat the energies of the rescaled vortices are concentrated near the origin in C.Now we “zoom back in” in such a way that we capture the first bubble, which mayeither be a vortex class over C or a J-holomorphic sphere in M. In the first casewe are done. In the second case we “zoom in” further, to obtain a finite number ofvortices and spheres that are attached to the first bubble. Iterating this procedure,we construct the limit stable map.The proof involves generalizations of results for pseudo-holomorphic maps tovortices: a bound on the energy density of a vortex, quantization of energy, compactnesswith bounded derivatives, and hard and soft rescaling. The proof thatthe bubbles connect and no energy is lost between them, uses an isoperimetric inequalityfor the invariant symplectic action functional, proved in [Zi2], based on aversion of the inequality by R. Gaio and D. A. Salamon [GS].Another crucial point is that when “zooming out”, no energy is lost locally inC in the limit. This relies on an upper bound on the “momentum map component”of a vortex, due to R. Gaio and D. A. Salamon.1.4. Fredholm theory for vortices over the planeThe space of gauge equivalence classes of symplectic vortices can be viewedas the zero set of a section of an infinite dimensional vector bundle. Formally,the second main result of this memoir states that in the case Σ = C the verticaldifferential of this section is Fredholm when seen as an operator between suitableweighted Sobolev spaces. We will first state the result and then interpret it in thisway.Statement of the Fredholm result. Consider the case Σ := C and ω C := ω 0 .Let p > 2 and λ be real numbers. 26 We define the set B p λas follows. For a24 See [CGMS, Example 2.8]. Here the standing assumption that µ is proper is used.25 The reasons for introducing the additional marked points z ν0 = ∞ are explained in Remark21 in Section 2.2.26 In this memoir, p and λ always refer to finite values, unless otherwise stated.

10 1. MOTIVATION AND MAIN RESULTSThe proof of Theorem 4 is based on a Fredholm result for the augmented verticaldifferential and the existence of a bounded right inverse for L ∗ w. (See Theorems 63and 64 in Section 3.2.1.) The proof of Theorem 63 has two main ingredients.The first one is the existence of a suitable complex trivialization of the bundleA 1 (g P ) ⊕ TM u . For R large, z ∈ C \ B R and p ∈ π −1 (z) ⊆ P such a trivializationrespects the splitting(1.28) T u(p) M = (imL C u(p) )⊥ ⊕ imL C u(p) ,where L C x : g ⊗ C → T x M denotes the complexified infinitesimal action, for x ∈M. The second ingredient are two propositions stating that the standard Cauchy-Riemann operator ∂¯z and a related matrix differential operator are Fredholm mapsbetween suitable weighted Sobolev spaces. These results are based on the analysisof weighted Sobolev spaces carried out by R. B. Lockhart and R. C. McOwen[Lo1, Lo2, Lo3, LM1, LM2, McO1, McO2, McO3]. A crucial analyticalingredient is a Hardy-type inequality (Proposition 91 in Appendix A.4).Motivation, a formal setting. To put the Fredholm result into context, let(Σ,j) be a connected smooth Riemann surface, equipped with a compatible areaform ω Σ . Recall the definitions (1.7,1.10) of ˜B, B. Consider the subclass ˜B ∗ ⊆ ˜B oftriples (P,A,u) for which there exists a point p ∈ P, such that the action of G atthe point u(p) ∈ M is free. We defineB ∗ := ˜B ∗ /∼,where ∼ is defined as before the definition (1.10). Formally, B ∗ may be viewedas an infinite dimensional manifold, since for every smooth G-bundle P over Σ,the natural action of the gauge group G P = CG ∞ (P,G) on the “infinite dimensionalmanifold”(1.29) ˜B∗ P := { (A,u) ∣ (P,A,u) ∈ ˜B∗}is free. Furthermore, the set of vortex classes may be viewed as the zero set ofa section of a vector bundle E over B ∗ , with infinite dimensional fiber, as follows.Consider the “vector bundle” Ẽ := ẼΣ over ˜B ∗ , whose fiber over a point w =(P,A,u) ∈ ˜B ∗ is given by(1.30) Ẽ w := Γ ( A 0,1 (TM u ) ⊕ A 2 (g P ) ) . 32The bundle E := E Σ over B ∗ is now defined to be the quotient of the bundle Ẽ → ˜B ∗under the natural equivalence relation lifting the relation ∼ on ˜B ∗ . Finally,S : B ∗ → Eis defined to be the section induced by˜S : ˜B ∗ → Ẽ, ˜S(A,u) :=(¯∂J,A (u),F A + (µ ◦ u)ω Σ).Heuristically, E is an infinite dimensional vector bundle over B ∗ , and S is a smoothsection of E. The zero set S −1 (0) ⊆ B ∗ consists of all vortex classes over Σ. Assumethat W ∈ S −1 (0). Then formally, there is a canonical map T : T (W,0) E → E W , whereE W ⊆ E denotes the fiber over W. We define the vertical differential of S at W tobe the map(1.31) d V S(W) = T dS(W) : T W B ∗ → E W .32 Here Γ(E) denotes the space of smooth sections of a vector bundle E → Σ.

1.5. REMARKS, RELATED WORK, ORGANIZATION 11Heuristically, if this map is Fredholm and surjective, for every W ∈ S −1 (0), thenthe zero set S −1 (0) is a smooth submanifold of B ∗ . The dimension of a connectedcomponent of this submanifold equals the Fredholm index of d V S(W), where W isany point in the connected component.At a formal level, in the case Σ = C, equipped with ω Σ = ω 0 , the verticaldifferential (1.31) coincides with the operator D p,λW, which was defined in (1.25,1.26)and occurred in the Fredholm result, Theorem 4. To see this, let W ∈ B ∗ . Weinterpret T W B ∗ as a quotient, as follows. Let P be a smooth G-bundle over Σ, and(A,u) ∈ ˜B P ∗ . Denoting w := (P,A,u), the infinitesimal action at the point (A,u),corresponding to the action of G P on ˜B P ∗ , is given byL w : Lie(G P ) = Γ(g P ) → T (A,u) ˜B∗ P = Γ ( A 1 (g P ) ⊕ TM u) ,where d A ξ := dξ + [A,ξ]. Defining(1.32) ˜Xw := T (A,u) ˜B∗ P /imL w ,we may identify( ∐(1.33) T W B ∗ = X W := ˜X w)/∼,w∈WL w ξ = (−d A ξ,L u ξ),where ∼ denotes the natural lift of the equivalence relation on ˜B ∗ . Assume formallythat ˜B ∗ P and Lie(G P) are equipped with a G P -invariant Riemannian metric and aG P -invariant inner product, respectively. For (A,u) ∈ ˜B ∗ P we denote by L∗ w :T (A,u) ˜B∗ P → Lie(G P ) the adjoint map of L w . Then by (1.32), we may identify˜X w = kerL ∗ w ⊆ Γ ( A 1 (g P ) ⊕ TM u) .Using this and (1.33,1.30), the vertical differential (1.31) at W ∈ S −1 (0) agreeswith the map( ) ( ∐∐ker L ∗ w /∼ → Γ ( A 0,1 (TM u ) ⊕ A 2 (g P ) )) /∼,w∈Ww∈Wgiven by (1.26), in the case Σ = C and ω Σ = ω 0 . Here on either side, ∼ denotes anatural lift of the equivalence relation on ˜B ∗ .Remarks.1.5. Remarks, related work, organization5. Remark (Vortices as triples). In some earlier work (e.g. [CGS] and [Zi1]),the G-bundle P was fixed and the vortex equations were seen as equations for apair (A,u) rather than a triple (P,A,u). 33 The motivation for making P part ofthe data is twofold:When formulating convergence for a sequence of vortex classes over C to a stablemap, one has to pull back the vortices by translations of C. (See Section 2.2.) If theprincipal bundle is fixed and vortices are defined as pairs (A,u) solving (1.8,1.9),then there is no natural such pullback. However, there is a natural pullback if the33 However, in [MT] I. Mundet i Riera and G. Tian took the viewpoint of the present memoir.

12 1. MOTIVATION AND MAIN RESULTSbundle is made part of the data for a vortex. 34 More generally, it is possible topull back vortex triples (P,A,u) by a Kähler transformation of a Riemann surfaceequipped with a compatible area form.Another motivation is the following: If the area form or the complex structureon the surface Σ vary, then in the limit we may obtain a surface Σ ′ with singularities.It does not make sense to consider P as a bundle over Σ ′ . One way of solving thisproblem is by decomposing Σ ′ into smooth surfaces, and constructing smooth G-bundles over these surfaces. Hence the G-bundle should be viewed as a varyingobject.Once P is made part of the data, it is natural to consider equivalence classes oftriples (P,A,u) ∈ ˜M (as defined in (1.15)), rather than the triples themselves. Onereason is that all important quantities, like energy density and energy, are invariant(or equivariant) under equivalence. Note also that the bubbling and Fredholmresults are more naturally stated for equivalence classes of vortices. Viewing theequivalence classes as the fundamental objects also matches the physical viewpointthat the “gauge field”, i.e., the connection A, is physically relevant only “up togauge”. ✷6. Remark. Let Σ be the plane C, equipped with the standard area formω 0 , and consider the trivial G-bundle P 0 := C × G. Then the solutions (A,u) ofthe vortex equations (1.8,1.9) on P 0 bijectively correspond to solutions (Φ,Ψ,f) ∈C ∞( C,g × g × M ) of the equations(1.34)(1.35)∂ s f + L f Φ + J(f)(∂ t f + L f Ψ) = 0,∂ s Ψ − ∂ t Φ + [Φ,Ψ] + µ(f) = 0.Here we denote by s and t the standard coordinates in C = R 2 , and in the secondequation we identify the Lie algebra g with its dual via the inner product 〈·, ·〉 g .The correspondence maps such a triple (Φ,Ψ,f) to ( A,u ) , where A denotes theconnection on P 0 defined byA (z,g) (ζ,gξ) := ( Φ(z)ds + Ψ(z)dt ) ζ + ξ, ∀ζ ∈ T z C, ξ ∈ g, z ∈ C, g ∈ G,and the map u : P 0 → M is given by u(z,g) := g −1 f(z). The group C ∞ (C,G) actson the set ˜M 0 of solutions of (1.34,1.35) by()h ∗ (Φ,Ψ,f) := h −1 ∂ s h + Ad h −1Φ,h −1 ∂ t h + Ad h −1Ψ,h −1 f ,where we denote the adjoint representation of an element g ∈ G by Ad g : g → g.This group naturally corresponds to the gauge group C ∞ G (P 0,G), and its action tothe actiong ∗ (A,u) := ( g −1 dg + Ad g −1A,g −1 u ) .Since every G-bundle over C is trivializable, it follows that the quotient of ˜M0by the action of C ∞ (C,G) bijectively corresponds to the quotient M = ˜M/ ∼,consisting of gauge equivalence classes of triples (P,A,u) of solutions of (1.8,1.9).Hence the results of the present memoir can alternatively be formulated in termsof solutions of the equations (1.34,1.35). However, the intrinsic approach usingequations (1.8,1.9) seems more natural. ✷34 Given a G-bundle P over C, we may of course choose a trivialization of P, and then definea pull back for pairs (A, u), using the trivialization. However, this approach is unnatural, since itdepends on the choice of a trivialization.

1.5. REMARKS, RELATED WORK, ORGANIZATION 137. Remark (Asphericity). Without the asphericity condition one needs to includeholomorphic spheres in the fibers of the Borel construction in the definitionof a stable map. In this situation, to compactify the space of vortices over C withan upper energy bound, one needs to combine the proof of Theorem 3 with theanalysis carried out by I. Mundet i Riera and G. Tian in [Mu1, MT], or by A. Ottin [Ott]. ✷8. Remark (Quotient spaces). The space X p,λWoccurring in Theorem 4 is aquotient of a disjoint union of normed vector spaces. It is canonically isomorphicp,λto the space ˜X w , for every representative w of W. Similar statements hold forY p,λp,λW. The description of the spaces XWand Yp,λWas such quotients may lookunconventional, however, it seems natural, since it does not involve any choice of arepresentative of W.p,λAlternatively, one could phrase the Fredholm result in terms of the spaces ˜X wand Ỹp,λ w . However, in view of the last part of Remark 5, this seems less naturalthan the present formulation. ✷9. Remark (Decay condition and vortices). The condition ‖ √ e w ‖ p,λ < ∞ inthe definition (1.16) of ˜B p λand the requirement 1 − 2/p < λ < 2 − 2/p in Theorem4(ii) capture the geometry of finite energy vortices over C, in the following sense.Let w = (P,A,u) ∈ ˜B p locbe a finite energy vortex such that u(P) has compactclosure. (Here ˜B p locis defined as at the beginning of Section 1.4.) Then for everyε > 0 there exists a constant C such that e w (z) ≤ C|z| −4+ε , for every z ∈ C \ B 1 .This follows from Theorem 99 in Appendix A.5 and [Zi2, Corollary 1.4].It follows that w ∈ ˜B p λif λ < 2 − 2/p. This bound is sharp. To see this,let λ ≥ 2 − 2/p and M := S 2 , equipped with the standard symplectic form ω st ,complex structure J := i, and the action of the trivial group G := {1}. Considerthe inclusion u : C × {1} ∼ = C → S 2 ∼ = C ∪ {∞}. Since this map is holomorphic, thetriple ( C × {1},0,u ) is a finite energy vortex whose image has compact closure. Itdoes not lie in ˜B p λ .On the other hand, every w ∈ ˜B p λhas finite energy whenever p > 2 and λ >1 − 2/p. 35 The latter condition is sharp. To see this, consider M := R 2 ,ω :=ω 0 ,G := {1},J := i. We choose a smooth map u : C × {1} ∼ = C → R 2 , such that( (√ ) (√ ))u(z) = cos log |z| ,sin log |z| , ∀z ∈ C \ B 2 .Then the triple ( C × {1},0,u ) lies in ˜B p λfor every p > 2 and λ ≤ 1 −2/p. However,it has infinite energy. 36 ✷10. Remark (Index). The condition λ < 2 − 2/p in part (ii) of Theorem 4 isneeded for the map D p,λWto have the right Fredholm index. Namely, let λ > 1 −2/p35 This follows from the estimates‖ √ e w‖ 2 ≤ ‚ ‚ √ e w〈·〉 λ‚ ‚ p‚ ‚ 〈·〉 −λ‚ ‚ q,‚‚〈·〉 −λ‚ ‚ q< ∞,where q := 2p/(p − 2). The first estimate is Hölder’s inequality and the second one follows froma calculation in radial coordinates.36 In the present setting, a simpler example of an infinite energy triple w = (P, A, u) satisfying√ ew ∈ L p λ for every p > 2 and λ ≤ 1 − 2/p, is w := `C“p ”× {1}, 0, u´, where u(z) := log |z|, 0 ,for every z ∈ C \ B 2 . However, the closure of the image of such a map u is non-compact, since itcontains the set [1, ∞) × {0}. Therefore, w does not lie in B ep λ for any p and λ.

14 1. MOTIVATION AND MAIN RESULTSbe such that λ + 2/p ∉ Z, and W ∈ B p λ. Then the proof of Theorem 4 shows thatis Fredholm with index equal toD p,λW(2 − k)(dim M − 2dim G) + 2 〈 c G 1 (M,ω),[W] 〉 ,where k is the largest integer less than λ + 2/p. In particular, the index changeswhen λ passes the value 2 − 2/p.Note also that the condition λ > 1 − 2/p is needed, in order for the homologyclass [W] to be well-defined. (See Remark 55 in Section 3.1.) ✷11. Remark (Weighted Sobolev spaces and energy density). The definition ofthe space X p,λWnaturally parallels the definition (1.17) of Bp λ. Namely, by linearizingwith respect to A and u the terms d A u,F A and µ◦u occurring in the energy densitye w , we obtain the terms ∇ A ζ,dµ(u)v and L u α. These expressions occur in ‖ζ‖ w,p,λ(defined as in (1.18)), except for the factor L u in L u α. 37 The expression ‖ζ‖ ∞ isneeded in order to make ‖ · ‖ w,p,λ non-degenerate. ✷12. Remark (Sobolev spaces and 0-th order terms). Consider the situation inwhich the norm (1.18) is replaced by the usual W 1,p -norm, and the norm ‖ · ‖ p,λ(defining Ỹp,λ w ) is replaced by the usual L p -norm. Then in general, the map definedby (1.26) does not have closed image, and hence it is not Fredholm. Note also thatthe 0-th order terms α ↦→ (L u α) 0,1 and v ↦→ ω 0 dµ(u)v in (1.26) are not compact(neither with respect to X p,λWand Yp,λW , nor with respect to the usual W 1,p - andL p -norms). The reason is that the embedding of W 1,p (C) (for p > 2) into the spaceof bounded continuous functions on C is not compact. Because of these terms, themap (1.26) is not well-defined between spaces that look like the standard weightedSobolev spaces in “logarithmic” coordinates τ + iϕ (with e τ+iϕ = z ∈ C \ {0}).This is in contrast with the situation in which Σ is the infinite cylinder R ×S 1 ,equipped with the standard complex structure and area form. In that situation thesplitting (1.28) is unnecessary, and standard weighted Sobolev spaces in “logarithmic”coordinates can be used. In the relative setting, with the cylinder replaced bythe infinite strip R × [0,1], this was worked out by U. Frauenfelder [Fr1, Proposition4.7]. The proof of the Fredholm result then relies on results [RoSa, Sa] byJ. Robbin and D. A. Salamon. 38 ✷Related work.Quantum Kirwan maps. The history of Conjectures 1 and 2 is as follows. Assumethat (H) holds, (M,ω) is aspherical, (M,ω,µ) is convex at ∞ and monotone,and HG ∗ (M) is generated by classes of degrees less than 2N, where N is the minimalequivariant Chern number (defined as in (1.3)). In this case R. Gaio andD. A. Salamon [GS] proved that there exists a ring homomorphism from HG ∗ (M)to the quantum cohomology of (M,ω) that agrees with the Kirwan map on classesof degrees less than 2N, see [GS, Corollary A’]. The idea of the proof of this resultis to fix an area form ω S 2 on S 2 and to relate symplectic vortices for the area formCω S 2 with pseudo-holomorphic spheres in M, for sufficiently large C > 0. Theauthors noticed that in general, this correspondence does not work, since in thelimit C → ∞, vortices over C may bubble off. Accordingly, the calculation of thequantum cohomology of monotone toric manifolds in [CS], which is based on [GS],37 It follows from hypothesis (H) and Lemma 84 in Appendix A.3 that this factor is irrelevant.38 In that setting, the index of the operator equals a certain spectral flow.

1.5. REMARKS, RELATED WORK, ORGANIZATION 15does not extend to the situation of a general toric manifold. This follows fromexamples by H. Spielberg [Sp1, Sp2].Based on these observations, Salamon suggested to construct a ring homomorphismfrom HG ∗ (M) to the quantum cohomology of (M,ω), by counting symplecticvortices over the plane C, provided that (M,ω) is aspherical and (M,ω,µ) is convexat ∞. This homomorphism should intertwine the symplectic vortex invariantsand the Gromov-Witten invariants of (M,ω). This gave rise to the Ph.D.-thesis[Zi1], which served as a basis for the present memoir. There it is observed that inthe definition of convergence for vortices over C to a stable map, only translationsshould be allowed as reparametrizations used to obtain a vortex component in thelimit. 39 C. Woodward realized that with this restriction, vortices over C withoutmarked points may appear in the bubbling argument used in the proof of the ringhomomorphism property of the quantum Kirwan map. (This may happen even inthe transverse case.) As a solution, he suggested to interpret the quantum Kirwanmap as a morphism of cohomological field theories. (See [NWZ] and Conjecture 2above.) On the other hand, under the semipositivity introduced above, the vorticeswithout marked points can be excluded in the transverse case. This gave rise toConjecture 1.In his recent article [Wo] C. Woodward developed these ideas in an algebraicgeometric setting. He defined a quantum Kirwan map in the case of a smoothprojective variety with an action of a complex reductive group. (See [Wo, Theorem1.3].) Theorem 3 of the present memoir is used in the proof of that result to showproperness of the Deligne-Mumford stack of stable scaled gauged maps in (M,ω)of genus 0. (See [Wo, Theorem 5.25].)In [GW1] E. Gonzalez and C. Woodward used Woodward’s definition to calculatethe quantum cohomology of a compact toric orbifold with projective coarsemoduli space. Furthermore, in [GW2] they used it to formulate a quantum versionof Kalkman’s wall-crossing formula.Bubbling and Fredholm results for vortices. Assume that Σ is closed, (H) holds,and M is symplectically aspherical and equivariantly convex at ∞. In this case,in [CGMS, Theorem 3.4], K. Cieliebak et al. proved compactness of the space ofvortex classes with energy bounded above by a fixed constant. In the case in whichM and Σ are closed, in [Mu1, Theorem 4.4.2] I. Mundet i Riera compactified thespace of bounded energy vortex classes with fixed complex structure on Σ. Assumingalso that G := S 1 , this was extended by I. Mundet i Riera and G. Tian in [MT,Theorem 1.4] to the situation of varying complex structure. That work is based ona version of Gromov-compactness for continuous almost complex structures, provedby S. Ivashkovich and V. Shevchishin in [IS].In [Ott, Theorem 1.8] A. Ott compactified the space of bounded energy vortexclasses in a different way, for a general Lie group and closed M and Σ, the lastwith fixed complex structure. He used the approach to Gromov-compactness byD. McDuff and D. A. Salamon in [MS2]. In the case in which Σ is an infinite strip,equipped with the standard area form and complex structure, the compactificationwas carried out in a relative setting by U. Frauenfelder in [Fr1, Theorem 4.12].(See also [Fr2].)39 As explained in Remark 22 in Section 2.2, the reason for this is that the evaluation of avortex class at a point in C is not invariant under rotations.

CHAPTER 2Bubbling for vortices over the planeIn this chapter, stable maps consisting of vortex classes over the plane C, holomorphicspheres in the symplectic quotient, and marked points, are defined, andthe first main result of this memoir, Theorem 3, is proven. This result states thatgiven a sequence of vortex classes over C, with uniformly bounded energies, andsequences of marked points, there exists a subsequence that converges to some stablemap. We also describe stable maps and convergence in the simplest interestingexample, the Ginzburg-Landau setting.2.1. Stable mapsLet M,ω,G,g, 〈·, ·〉 g ,µ,J be as in Chapter 1. Our standing hypothesis (H)implies that the symplectic quotient(M = µ −1 (0)/G,ω )is well-defined and closed. The structure J induces an ω-compatible almost complexstructure on M as follows. For every x ∈ M we denote by L x : g → T x M theinfinitesimal action at x. We define the horizontal distribution H ⊆ T(µ −1 (0)) byH x := kerdµ(x) ∩ imL ⊥ x , ∀x ∈ µ −1 (0).Here ⊥ denotes the orthogonal complement with respect to the metric ω(·,J·) onM. We denote by π : µ −1 (0) → M := µ −1 (0)/G the canonical projection. Wedefine ¯J to be the unique isomorphism of TM such that(2.1) ¯J dπ = dπJ on H.1We identify C ∪ {∞} with S 2 . The (Connectedness) condition in the definitionof a stable map below will involve the evaluation of a vortex class at the point∞ ∈ S 2 . In order to make sense of this, we need the following. We denote by Gxthe orbit of a point x ∈ M. Let P be a smooth (principal) G-bundle over C 2 andu ∈ CG ∞ (P,M) a map. We denote by M/G the orbit space of the action of G onM, and defineū : C → M/G, ū(z) := Gu(p),where p ∈ P is an arbitrary point in the fiber over z. For W ∈ B we define(2.2) ū W := ū,where w = (P,A,u) is any representative of W. This is well-defined, i.e., does notdepend on the choice of w. Recall the definition (1.15) of M, and that by the image1 Such a ¯J exists and is unique, since the map dπ is an isomorphism from H to TM, and Jpreserves H.2 Such a bundle is trivializable, but we do not fix a trivialization here.17

18 2. BUBBLING FOR VORTICES OVER THE PLANEof a class W ∈ M we mean the set of orbits of u(P), where (P,A,u) is any vortexrepresenting W. We define the set(2.3) M

. .2.1. STABLE MAPS 19.z 0α 0. ... .. .Figure 1. Stable map consisting of vortex classes over C andpseudo-holomorphic spheres in M.(iii) (Connectedness) Let α,β ∈ T 1 ∪ T ∞ be such that αEβ. Thenev zαβ (W α ) = ev zβα (W β ).Here ev is defined as in (2.4) and (2.5) and we set W α := ū α if α ∈ T ∞ .(iv) (Stability) Let α ∈ T.• If α ∈ T 1 and E(W α ) = 0 then the set{ } {(2.8)β ∈ T |αEβ ∪ i ∈ {0,...,k} |αi = α }contains at least two points.• If either α ∈ T ∞ and E(ū α ) = 0, or α ∈ T 0 , then the set (2.8) consists ofat least three points.This definition is modeled on the notion of a genus 0 pseudo-holomorphic stablemap, as introduced by Kontsevich in [Ko]. 4 Roughly speaking, a stable map in thesense of Definition 15 can be thought of as a collection of vortex classes over C,pseudo-holomorphic spheres in the symplectic quotient M, “ghost spheres of type0” corresponding to the vertices of T 0 , and marked and nodal points. A vortex classmay be connected to a sphere in M at the nodal point ∞, and to “ghost spheres oftype 0” at points in C. Furthermore, spheres of the same type may be connectedat nodal points. The “ghost spheres of type 0” should be thought of as constant4 For an exhaustive exposition of those stable maps see the book by D. McDuff and D. A. Salamon[MS2].

20 2. BUBBLING FOR VORTICES OVER THE PLANEspheres in the Borel construction (M × EG)/G. They are needed for the bubblingresult (Theorem 3) to capture colliding marked points in C. 5Figure 1 shows an example of a stable map. Here the “teardrops” correspond tovortex classes over C, the solid and dashed spheres to pseudo-holomorphic spheresin M, and the dotted spheres to “ghost spheres of type 0”. The solid objects havepositive energy, and the dashed and dotted spheres are “ghosts”, i.e., their energyvanishes. Each “teardrop” is connected to a sphere (in M) via a nodal point at itsvertex, which corresponds to the point ∞ ∈ C ∪ {∞}.To explain the stability condition (iv), we fix α ∈ T. We define the set of nodalpoints on α to be(2.9) Z α := {z αβ∣ ∣ β ∈ T, αEβ} ⊆ S 2 ,the set of marked points on α to be{zi∣ ∣ αi = α, i ∈ {0,...,k} } ,and the set Y α of special points on α to be the union of these two sets. The stablemap of Figure 1 carries ten marked points, which are drawn as dots. The stabilitycondition says the following. Assume that α ∈ T is a “ghost component”, i.e.,α ∈ T 0 or W α carries zero energy (in the case α ∈ T 1 ∪ T ∞ ). Then the followingholds: If α ∈ T 1 then it carries at least one special point in C. 6 Furthermore, ifα ∈ T 0 ∪ T ∞ then α carries at least three special points.This condition ensures that the action of a natural reparametrization group onthe set of simple stable maps of a given type is free. 7 In a future definition of thequantum Kirwan map this will be needed in order to show that the evaluation mapon the set of non-trivial vortex classes (with marked points) is a pseudo-cycle.Remarks. Condition (i) implies that if T 1 is empty then so is T 0 , and hence astable map in the sense of Definition 15 is a genus 0 stable map of ¯J-holomorphicspheres in M.If α 0 ∈ T 1 then T 1 = {α 0 } and T ∞ = ∅. This follows from the second part ofcondition (i) for α ∈ T ∞ , using the last condition in (2.7). Hence in this case astable map consists of a single vortex class and special points.If α 0 ∈ T ∞ then the sets T ∞ and T 1 ∪ T ∞ are subtrees of T, and every elementof T 1 is adjacent to a unique vertex in T ∞ and to no vertex in T 1 . In particular,each element of T 1 is a leaf of the tree T 1 ∪ T ∞ . These statements follow fromcondition (i) and the fact that T does not contain any simple cycle.The vertices in T 0 are not adjacent to those in T ∞ . Furthermore, for eachconnected component of T 0 there exists a unique vertex in T 1 that is adjacent tosome element of the connected component. These assertions follow from condition(i) and the fact that T does not contain any simple cycle. ✷16. Remark. If 1 ≤ i ≤ k is such that α i ∈ T 1 then z i ≠ ∞. This follows fromcondition (ii) and the fact that either α 0 ∈ T 1 or every vertex in T 1 is adjacent tosome vertex in T ∞ . ✷5 As explained in [Zi1, Zi4] vortex classes over C evaluate to points in (M ×EG)/G at pointsin C. Therefore, identifying each “ghost sphere of type 0” with a point in (M × EG)/G, it makessense to ask that the sphere is connected to a vortex class over C at a given nodal point.6 It then also carries a special point at ∞.7 See Proposition 31 in Section 2.4 below.

2.2. CONVERGENCE TO A STABLE MAP 21. z0α 0Figure 2. This is the stable map described in Example 18 withl := 4 and a constant sphere ū 0 .17. Remark. Without the asphericity assumption, a stable map should alsoinclude holomorphic maps from the sphere S 2 to the fibers of the Borel construction(M ×EG)/G. These occur if in a sequence of vortices over C energy is concentratedaround some point in C. The necessary analysis was worked out by I. Mundet iRiera and G. Tian in [Mu1, MT], and by A. Ott in [Ott]. ✷Example. The simplest example of a stable map consists of the tree with onevertex T = T 1 = {α 0 }, a vortex class W ∈ M

22 2. BUBBLING FOR VORTICES OVER THE PLANELet α ∈ T := T 0∐T1∐T∞ and i = 0,...,k. We define z α,i ∈ S 2 as follows. Ifα = α i then we set(2.10) z α,i := z i .Otherwise we define z α,i to be the first special point encountered on a path of edgesfrom α to α i . To explain this, we denote by β ∈ T ∞ the unique vertex such thatthe chain of vertices of T running from α to α i is given by (α,β,...,α i ). (β = α iis also allowed.) We define(2.11) z α,i := z αβ .Let Σ be a compact connected smooth surface with non-empty boundary. Recallthe definition (1.10) of the set B Σ of equivalence classes of triples (P,A,u). Wedefine the C ∞ -topology τ Σ on this set as follows: We fix a (smooth) G-bundle Pover Σ, and denote by G P its gauge group. Since by hypothesis, G is connected,every G-bundle over Σ is trivializable. It follows that the map((2.12) A(P) × C∞G (P,M) ) /G P ∋ GP(A,u) ∗ ↦→ [P,A,u] ∈ B Σis a bijection. 819. Definition. We define τ Σ to be the pushforward of the quotient topologyof the C ∞ -topology on A(P) × CG ∞ (P,M) under the map (2.12).Let Σ be a smooth surface, W = [P,A,u] ∈ B Σ , and Ω ⊆ Σ an open subsetwith compact closure and smooth boundary. We define the restriction W | Ωto bethe equivalence class of the pullback of (P,A,u) under the inclusion map Ω → Σ.For W = [P,A,u] ∈ B Σ and ϕ a translation on C, we define the pullback of Wby ϕ to be(2.13) ϕ ∗ [P,A,u] := [ ϕ ∗ P,Φ ∗ (A,u) ] ,where Φ : ϕ ∗ P → P is defined by Φ(z,p) := p. 9 We define(2.14) M ∗ := { x ∈ M |if g ∈ G : gx = x ⇒ g = 1 } .Note that µ −1 (0) ⊆ M ∗ by our standing hypothesis (H).20. Definition (Convergence). The sequence (W ν ,z0 ν := ∞,z1,...,z ν k ν ) is saidto converge to (W,z), as ν → ∞, iff the limit E := lim ν→∞ E(W ν ) exists,(2.15) E = ∑ E(W α ) + ∑E(ū α ),α∈T 1 α∈T ∞and there exist Möbius transformations ϕ ν α : S 2 → S 2 , for α ∈ T, ν ∈ N, such thatthe following conditions hold.(i) • If α ∈ T 1 then ϕ ν α is a translation on C.• For every α ∈ T ∞ we have ϕ ν α(z α,0 ) = ∞, where z α,0 is defined as in(2.10), (2.11).• Let α ∈ T ∞ and ψ α be a Möbius transformation such that ψ α (∞) = z α,0 .Then the derivatives (ϕ ν α ◦ ψ α ) ′ (z) converge to ∞, for every z ∈ C.8 Recall here that A(P) denotes the affine space of smooth connection one-forms on P. Weuse the simplified notation [P, A, u] for the equivalence class [(P, A, u)].9 Recall here that a point in the pullback bundle ϕ ∗ P has the form (z, p), where z ∈ C and plies in the fiber of P over ϕ(z) ∈ C.

2.2. CONVERGENCE TO A STABLE MAP 23(ii) If α,β ∈ T are such that αEβ then (ϕ ν α) −1 ◦ϕ ν β → z αβ, uniformly on compactsubsets of S 2 \ {z βα }.(iii) • Let α ∈ T 1 and Ω ⊆ R 2 be an open connected subset with compactclosure and smooth boundary. Then the restriction (ϕ ν α) ∗ W ν | Ωconvergesto W α | Ω, with respect to the topology τ Ω(as in Definition 19).• Fix α ∈ T ∞ . Let Q be a compact subset of S 2 \ (Z α ∪ {z α,0 }). For νlarge enough, we haveū Wν ◦ ϕ ν α(Q) ⊆ M ∗ /G,and ū Wν ◦ϕ ν α converges to ū α in C 1 on Q. (Here Z α and ū Wν are definedas in (2.9,2.2).)(iv) We have (ϕ ν α i) −1 (z ν i ) → z i for every i = 1,...,k.The meaning of this definition is illustrated by Figure 3. It is based on thenotion of convergence of a sequence of pseudo-holomorphic spheres to a genus 0pseudo-holomorphic stable map. 10 An example in which it can be understood moreexplicitly, is discussed in the next section.Remark. The condition in the first part of (iii), that (ϕ ν α) ∗ W ν | Ω→ W α | Ωwith respect to τ Ω, is equivalent to the requirement that there exist representativesw ν of (ϕ ν α) ∗ W ν | Ω(for ν ∈ N) and w of W α | Ωsuch that w ν converges to w in theC k -topology, for every k ∈ N. This follows from a straight-forward argument, usingLemma 120 (Appendix A.7). ✷Remark. The last part of condition (i) and the second part of condition (iii)capture the idea of catching a pseudo-holomorphic sphere in M by “zooming out”:Fix α ∈ T ∞ , and consider the case z α,0 = ∞. Then there exist λ ν α ∈ C \ {0}and z ν α ∈ C such that ϕ ν α(z) = λ ν αz + z ν α. It follows from a direct calculation that(ϕ ν α) ∗ W ν is a vortex class with respect to the area form ω Σ = |λ ν α| 2 ω 0 , where ω 0denotes the standard area form on C. The last part of condition (i) means thatλ ν α → ∞, for ν → ∞. Hence in the limit ν → ∞ we obtain the equations¯∂ J,A (u) = 0, µ ◦ u = 0.These correspond to the ¯J-Cauchy-Riemann equations for a map from C to M. (SeeProposition 116 in Appendix A.7.) The second part of (iii) imposes that in fact thesequence of rescaled vortex classes converges (in a suitable sense) to a ¯J-holomorpicsphere and that this sphere equals ū α .It is unclear whether the bubbling result, Theorem 3, remains true if we replacethe C 1 -convergence in this part of condition (iii) by C ∞ -convergence. (Compare toRemark 39 in Section 2.5.) ✷Remark. The “energy-conservation” condition (2.15) has the important consequencethat the stable map (W,z) represents the same equivariant homologyclass as the vortex class W ν , for ν large enough. (See [Zi1, Proposition 5.4] and[Zi4].) ✷21. Remark. The purpose of the additional marked point (α 0 ,z 0 ) is to be ableto formulate the second part of condition (iii). For α ∈ T ∞ and ν ∈ N the mapGu ν ◦ ϕ ν α is only defined on the subset (ϕ ν α) −1 (C) ⊆ S 2 . Since by condition (i) we10 For that notion see for example [MS2].

24 2. BUBBLING FOR VORTICES OVER THE PLANE............. z0.α 0.Figure 3. Convergence of a sequence of vortex classes over C toa stable map.

2.3. AN EXAMPLE: THE GINZBURG-LANDAU SETTING 25have ϕ ν α(z α,0 ) = ∞, the composition ū Wν ◦ ϕ ν α : Q → M/G is well-defined for eachcompact subset Q ⊆ S 2 \ (Z α ∪ {z α,0 }). Hence the second part of condition (iii)makes sense.As another motivation, note that the bubbling result, Theorem 3, is in generalwrong, if we do not introduce the additional marked points z ν 0 := ∞ and (α 0 ,z 0 ).See Example 28 in Section 2.3 below. ✷22. Remark. One conceptual difficulty in defining the notion of convergenceis the following. Consider the group Isom + (Σ) of orientation preserving isometriesof Σ (with respect to the metric ω Σ (·,j·)). 11 This group acts on B Σ (defined as in(1.10)), as in (2.13). The set M

26 2. BUBBLING FOR VORTICES OVER THE PLANEthe energy of w. In the present situation the condition on the image in the definitionof M 0 be so large that v(z) ≠ 0 if |z| ≥ R. We denoteby r the radial coordinate in M = C and by γ the standard angular one-form on

2.3. AN EXAMPLE: THE GINZBURG-LANDAU SETTING 27R 2 \ {0}. 13 The one-form α := r22 γ is a primitive of ω 0, and therefore, by Stokes’theorem,∫ ∫(2.20)v ∗ ω 0 = v ∗ α.B RBy elementary arguments, we have∫(v ∗ γ = 2π deg SR 1 ∋ z ↦→ v(z) )SR1 |v(z)| ∈ S1 = 2π ∑deg u (z) = 2π deg(W).z∈B R(Here in the last equality we used the assumption that v(z) ≠ 0 if |z| ≥ R.) Itfollows that∫(2.21) π deg(W) min |v(z)| 2 ≤z∈SR1 v ∗ α ≤ π deg(W) max |v(z)| 2 .z∈SR1S 1 ROn the other hand, using the estimates E(w) < ∞ and |µ ◦u| ≤ √ e w , Lemma 72 inAppendix A.1 implies that |µ ◦ v(Rz)| = 1 2 (1 − |v(Rz)|2 ) converges to 0, uniformlyin z ∈ S 1 , as R → ∞. Combining this with (2.21,2.20,A.1), equality (2.19) follows.This proves Proposition 24.□This result has the following consequence.25. Corollary. Let w := (P,A,u) be a smooth vortex over C with positiveand finite energy. Then the image of u contains the open unit ball B 1 ⊆ C.S 1 RProof of Corollary 25. Consider the setX := { |u(p)| ∣ ∣ p ∈ P}.This set is connected, and hence an interval. Since E(w) < ∞, for every r < 1there exists a point p ∈ P such that |u(p)| ≥ r. On the other hand, positivity ofthe energy and Proposition 24 imply that u vanishes somewhere. It follows that Xcontains the interval [0,1). Since the image of u is invariant under the S 1 -action,it follows that it contains the ball B 1 . This proves Corollary 25.□Proposition 23 and Corollary 25 imply that the image of u equals B 1 , forevery smooth vortex over C with positive and finite energy. Fix now d ∈ N 0 . Wedenote by Sym d (C) the d-fold symmetric product. By definition this is the quotienttopological space for the action of the symmetric group S d on C d given byσ · (z 1 ,...,z d ) := ( z σ −1 (1),...,z σ −1 (d)).We identify Sym d (C) with the set ˜Sym d (C) of all maps m : C → N 0 such thatm(z) ≠ 0 for only finitely many points z ∈ C, and∑(2.22)m(z) = d,z∈Cby assigning to z := [z 1 ,...,z d ] ∈ Sym d (C) the multiplicity map m z : C → N 0 ,given bym z (z) := # { i ∈ {1,...,d} |z i = z } .We can now characterize vortex classes with energy dπ as follows.13 By our convention this form integrates to 2π over any circle centered at the origin.

28 2. BUBBLING FOR VORTICES OVER THE PLANE26. Proposition. The map(2.23) M d := { W ∈ M |E(W) = dπ } ∋ W ↦→ deg W ∈ ˜Sym d (C) = Sym d (C)is a bijection.Proof of Proposition 26. This follows from [JT, Chap. III, Theorem 1.1].□As a consequence of Proposition 26, we obtain a classification of stable mapsin the sense of Definition 15 in the present setting: Here the symplectic quotientM = µ −1 (0)/S 1 consists of a single point, the orbit S 1 ⊆ M = C. Hence everyholomorphic sphere in M is constant. Stable maps are thus classified in terms oftheir combinatorial structure ( T 0 ,T 1 ,T ∞ ,E ) , the location of the special points, andfor each α ∈ T 1 , a point in some symmetric product of C.Convergence to a stable map is explicitly described by the following result.Here we will use the inclusion(2.24)ι d : ∐ 0≤d ′ ≤d Symd′ (C) → Sym d (S 2 ),ι d ([z 1 ,...,z d ′]) := [ z 1 ,...,z d ′, ∞,...,∞ ] ,where we identify S 2 ∼ = C ∪ {∞}. We drop the constant maps to the symplecticquotient from the notation for a stable map, since no information gets lost.27. Proposition (Convergence in the Ginzburg-Landau setting). Let k ∈ N 0 ,for ν ∈ N let W ν ∈ M

2.4. THE ACTION OF THE REPARAMETRIZATION GROUP 29Then ( W ν ,z0 ν := ∞ ) converges to the stable map consisting of the sets T 0 := ∅,T 1 := {1,2}, T ∞ := {0}, the tree relation E := {(1,0),(0,1),(2,0),(0,2)}, thevortices W 1 and W 2 , the unique constant ¯J-holomorphic sphere ū 0 in M, the nodalpoints z 1 0 := z 2 0 := ∞, z 0 1 := 1, z 0 2 := 2 ∈ S 2 ∼ = C ∪ {∞}, and the marked point(α 0 ,z 0 ) := (0, ∞) ∈ T × S 2 . This follows from Proposition 27.It follows from this example that in general, the additional marked pointsz0 ν := ∞ are needed in the bubbling result, Theorem 3. Without these points, nosubsequence of W ν as above converges to a stable map. 14 This follows from anelementary argument. ✷2.4. The action of the reparametrization groupThis section covers an additional topic, which will not be used in this memoir,but will be relevant in a future definition of the quantum Kirwan map. Namely, weintroduce a natural group of reparametrizations and show that this group acts freelyon the set of simple stable maps consisting of vortex classes over C and pseudoholomorphicspheres in the symplectic quotient. The relevance of this result is thefollowing. For the definition of the quantum Kirwan map it will be necessary toshow that a certain natural evaluation map on the set of vortex classes over C (see[Zi1, Proposition 6.1] and [Zi4]) is a pseudo-cycle. This will rely on the fact thatits omega limit set has codimension at least two. In order to show this, one needsto cut down the dimension of each “boundary stratum” by dividing by the action ofthe reparametrization group. Heuristically, the freeness of the action of this groupimplies that the quotient is a smooth manifold, hence providing a meaning to thisprocedure.∐We∐fix finite sets T 0 ,T 1 ,T ∞ and a tree relation E on the disjoint union T :=T 0 T1 T∞ . We define the reparametrization group G T as follows. We defineAut(T) := Aut ( T 0 ,T 1 ,T ∞ ,E ) to be the subgroup of all automorphisms f of thetree (T,E), satisfying f(T i ) = T i , for i = 0,1, ∞. We denote by PSL(2, C) thegroup of Möbius transformations and by T C the group of translations of the planeC. We define{TC , if α ∈ TAut α :=1 ,PSL(2, C), if α ∈ T 0 ∪ T ∞ .We denote by Aut T the set of collections (ϕ α ) α∈T , such that ϕ α ∈ Aut α , for everyα ∈ T. The group Aut(T) acts on Aut T byf · (ϕ α ) α∈T := (ϕ f −1 (α)) α∈T .29. Definition. We define G T := G T0,T 1,T ∞,E to be the semi-direct productof Aut(T) and Aut T induced by this action.The group PSL(2, C) acts on the set of ¯J-holomorphic maps S 2 → M byϕ ∗ f := f ◦ ϕ.Furthermore, the group T C acts on the set M

30 2. BUBBLING FOR VORTICES OVER THE PLANEthe set of all stable maps of (combinatorial) type T. G T acts on M(T) as follows.For every (f,(ϕ α )) ∈ G T and (W,z) ∈ M(T) we defineW ′ α := ϕ ∗ f(α) W f(α), ∀α ∈ T 1 , ū ′ α := ū α ◦ ϕ f(α) , ∀α ∈ T ∞ ,z ′ αβ:= ϕ−1f(α) (z f(α)f(β)),∀αEβ,α i ′ := f(α i), z i ′ := ϕ−1α(z ′ α ′i i), ∀i = 0,...,k.(Here we set W α := ū α if α ∈ T ∞ . Furthermore, for ϕ ∈ T C we set ϕ(∞) := ∞.)30. Definition. We define)(f,(ϕ α )) ∗ (W,z) :=(T 0 ,T 1 ,T ∞ ,E,(W α) ′ α∈T1 ,(ū ′ α) α∈T∞ ,(z αβ) ′ αEβ ,(α i,z ′ i) ′ i=0,...,k .This defines an action of G T on M(T). Let now (M,J) be an almost complexmanifold. Recall that a J-holomorphic map u : S 2 → M is called multiply coverediff there exists a holomorphic map ϕ : S 2 → S 2 of degree at least two, and aJ-holomorphic map v : S 2 → M, such that u = v ◦ϕ. Otherwise, u is called simple.We call a stable map (W,z) simple iff the following conditions hold: For everyα ∈ T ∞ the ¯J-holomorphic map ū α is constant or simple. Furthermore, if α,β ∈ T 1are such that α ≠ β and E(W α ) ≠ 0, and ϕ ∈ T C , then ϕ ∗ W α ≠ W β . Moreover, ifα,β ∈ T ∞ are such that α ≠ β and ū α is nonconstant, and if ϕ ∈ PSL(2, C), thenū α ◦ ϕ ≠ ū β . We denote byM ∗ (T) := M ∗( T 0 ,T 1 ,T ∞ ,E ) ⊆ M(T)the subset of all simple stable maps. The action of G T on M(T) leaves M ∗ (T)invariant.31. Proposition. The action of G T on M ∗ (T) is free.The proof of this result uses the following lemma.32. Lemma. The action of T C on M

2.5. COMPACTNESS MODULO BUBBLING AND GAUGE FOR RESCALED VORTICES 31Note also that the action of Isom + (C) on the set of vortex classes of positiveenergy is not always free, as the next example shows. ✷34. Example. Consider the action of G := S 1 ⊆ C on M := C by multiplication.Let d ∈ N 0 be an integer. By Proposition 26 there exists a unique finiteenergy vortex class W over C such that{ d, if z = 0,deg W (z) =0, otherwise.For every rotation R ∈ SO(2) we havedeg R ∗ W = R ∗ deg W = deg W ,where SO(2) acts in a natural way on Sym d (C). Thus the action of Isom + (C) onthe set of vortex classes of positive energy is not free. ✷2.5. Compactness modulo bubbling and gauge for rescaled vorticesIn this section we formulate and prove a crucial ingredient (Proposition 37) ofthe proof of Theorem 3, which states the following. Consider a sequence of rescaledvortices over C with image in a fixed compact subset of M and uniformly boundedenergies. We assume that (M,ω) is aspherical. Then there exists a subsequencethat away from finitely many bubbling points and up to regauging, converges toa rescaled vortex over C. If the rescalings converge to ∞, then the limit objectcorresponds to a ¯J-holomorphic sphere in M.The proof of this result is based on compactness for rescaled vortices over thepunctured plane with uniformly bounded energy densities (Proposition 38 below).It also uses the fact that at each bubbling point at least the energy E min > 0 islost, which is the minimal energy of a vortex over C or pseudo-holomorphic spherein M. This is the content of Proposition 40 below, which is proved by a hardrescaling argument, using Proposition 38 and Hofer’s lemma. Another ingredientof the proof of Proposition 37 is Lemma 42 below, which says that the energydensities of a convergent sequence of rescaled vortices converge to the density ofthe limit.In order to explain the main result of this section, let M,ω,G,g, 〈·, ·〉 g ,µ,J,Σ,j, and ω Σ be as in Chapter 1. We fix a triple w = (P,A,u) ∈ ˜B Σ (defined as in(1.7)). Recall the definition (1.11) of the energy density ew ωΣ,j = e w .35. Remark. This density has the following transformation property: Let Σ ′be another surface, and ϕ : Σ ′ → Σ a smooth immersion. Consider the pullbackϕ ∗ w := ( ϕ ∗ P,Φ ∗ A,u ◦ Φ ) ,where the bundle isomorphism Φ : ϕ ∗ P → P is defined by Φ(z,p) := p. A straightforwardcalculation shows that(2.26) e ϕ∗ (ω Σ,j)ϕ ∗ w= e ωΣ,jw ◦ ϕ.Note also that w is a vortex with respect to (ω Σ ,j) if and only if ϕ ∗ w is a vortexwith respect to ϕ ∗ (ω Σ ,j). ✷36. Remark. If w is a vortex (with respect to (ω Σ ,j)) then(2.27) e ωΣ,jw = |∂ J,A u| 2 + |µ ◦ u| 2 ,

32 2. BUBBLING FOR VORTICES OVER THE PLANEwhere ∂ J,A u denotes the complex linear part of d A u = du + L u A, viewed as aone-form on Σ with values in the complex vector bundle (u ∗ TM)/G → Σ. 15 Thisfollows from the vortex equations (1.8,1.9). ✷Let R ∈ [0, ∞] and w ∈ ˜B Σ . Consider first the case 0 < R < ∞. Then wedefine the R-energy density of w to be(2.28) e R w := R 2 e R2 ω Σ,jw .This means that(2.29) e R w = 1 (|d A u| 2 ω2 Σ+ R −2 |F A | 2 ω Σ+ R 2 |µ ◦ u| 2) ,where the subscript “ω Σ ” means that the norms are taken with respect to the metricω Σ (·,j·).If R = 0 or ∞ then we definee R w := 1 2 |d Au| 2 ω Σ.We define the R-energy of w on a measurable subset X ⊆ Σ to be∫(2.30) E R (w,X) := e R wω Σ ∈ [0, ∞].XRemark. Consider the case (Σ,j) = C, equipped with the standard area formω 0 . Assume that 0 < R < ∞, and consider the map ϕ : C → C defined byϕ(z) := Rz. Then equality (2.26) implies thate R ϕ ∗ w = R 2 e ω0,iw ◦ ϕ.Hence in the present setting the R-energy transforms viaE R( ϕ ∗ w,ϕ −1 (X) ) = E(w,X) := E 1 (w,X).✷The (symplectic) R-vortex equations are the equations (1.8,1.9) with ω Σ replacedby R 2 ω Σ , i.e., the equations(2.31)¯∂ J,A (u) = 0,(2.32)F A + R 2 (µ ◦ u)ω Σ = 0.In the case R = ∞ we interpret the equation (2.32) asµ ◦ u = 0.We call a solution (A,u) ∈ ˜W p Σof equations (2.31,2.32) an R-vortex over Σ.Remarks. Consider the case (Σ,j) = C, equipped with ω 0 , and let 0 < R < ∞.We define the map ϕ : C → C by ϕ(z) := Rz. Then w ∈ ˜B C is a vortex if and onlyif ϕ ∗ w is an R-vortex.The rescaled energy density has the following important property. Let R ν ∈(0, ∞) be a sequence that converges to some R 0 ∈ [0, ∞], and for ν ∈ N 0 letw ν = (P ν ,A ν ,u ν ) be an R ν -vortex. If on compact sets A ν converges to A 0 in C 0and u ν converges to u 0 in C 1 thene Rνw ν→ e R0w 0in C 0 on compact sets. (See Lemma 42 below.) In the proof of Theorem 3, this willbe used in order to show that locally on C no energy is lost in the limit ν → ∞. ✷15 The complex structure on this bundle is induced by J.

2.5. COMPACTNESS MODULO BUBBLING AND GAUGE FOR RESCALED VORTICES 33We define the minimal energy E min as follows. Recall from (1.15) that ˜M denotesthe class consisting of smooth vortices, and that the energy of a ¯J-holomorphicmap f : S 2 → M is given by E(f) = ∫ S 2 f ∗ ω. We define 16(2.33) E 1 := inf ({ E(P,A,u) ∣ ∣ (P,A,u) ∈ ˜M : u(P) compact } ∩ (0, ∞) ) ,E ∞ := inf ({ E(f) ∣ ∣ f ∈ C ∞ (S 2 ,M) : ¯∂ ¯J(f) = 0 } ∩ (0, ∞) ) ,(2.34) E min := min{E 1 ,E ∞ }.Assume that M is equivariantly convex at ∞. Then Corollary 74 in Appendix A.1implies that E 1 > 0. Furthermore, our standing assumption (H) implies that M isclosed. It follows that E ∞ > 0. 17 Hence the number E min is positive.Remark. The infima (2.33) and (2.34) are attained, and hence the name “minimalenergy” for E min is justified. (This fact is not used anywhere in this memoir.)That (2.34) is attained follows from the fact that for every C ∈ R there existonly finitely many homotopy classes B ∈ π 2 (M) with 〈[ω],B〉 ≤ C that can berepresented by a J-holomorphic map S 2 → M. 18 That (2.33) is attained followsfrom the fact that for every C ∈ R there exist only finitely many homology classesB ∈ H G 2 (M, Z) with 〈 [ω − µ],B 〉 ≤ C that can be represented by a finite energyvortex whose image has compact closure. This is a consequence of Theorem 3 and[Zi1, Proposition 5.4] (Conservation of equivariant homology class). ✷The results of this and the next section are formulated for connections and mapsof Sobolev regularity. This is a natural setup for the relevant analysis. Furthermore,we restrict our attention to the trivial bundle Σ × G. 19We fix p > 2 20 and naturally identify the affine space of connections on Σ × Gof local Sobolev class W 1,plocclass W 1,plocwith the space of one-forms on Σ with values in g, of. Furthermore, we identify the space of G-equivariant maps from Σ × G1,pwith Wloc (Σ,M). Finally, we identify the gauge group21 on2,pwith Wloc (Σ,G). We denote˜W Σ := Ω 1 (Σ,g) × C ∞ (Σ,M),˜W p Σ := { one-form on Σ with values in g, of class W 1,p }loc × W1,ploc (Σ,M).to M of class W 1,plocΣ × G of class W 2,plocThe gauge group W 2,ploc (Σ,G) acts on ˜W p Σ byg ∗ (A,u) := ( Ad g −1A + g −1 dg,g −1 u ) ,where Ad g0 : g → g denotes the adjoint action of an element g 0 ∈ G. For r > 0 wedenote by B r ⊆ C the open ball of radius r, around 0.37. Proposition (Compactness modulo bubbling and gauge). Assume that(M,ω) is aspherical. Let R ν ∈ (0, ∞) be a sequence that converges to some R 0 ∈(0, ∞], r ν ∈ (0, ∞) a sequence that converges to ∞, and for every ν ∈ N letw ν = (A ν ,u ν ) ∈ ˜W p B rνbe an R ν -vortex (with respect to (ω 0 ,i)). Assume that there16 Here we use the convention that inf ∅ = ∞.17 See e.g. [MS2, Proposition 4.1.4].18 This is a corollary to Gromov compactness, see e.g. [MS2, Corollary 5.3.2].19 Since every smooth bundle over C is trivializable, this suffices for the proof of the mainresult.20 Recall that throughout this memoir, p < ∞, unless otherwise stated.21 i.e., the group of gauge transformations

34 2. BUBBLING FOR VORTICES OVER THE PLANEexists a compact subset K ⊆ M such that u ν (B rν ) ⊆ K, for every ν. Suppose alsothatsupE Rν (w ν ,B rν ) < ∞.νThen there exist a finite subset Z ⊆ C, an R 0 -vortex w 0 := (A 0 ,u 0 ) ∈ ˜W C\Z , and,passing to some subsequence, gauge transformations g ν ∈ W 2,ploc(C\Z,G), such thatthe following conditions hold.(i) If R 0 < ∞ then Z = ∅ and the sequence gν(A ∗ ν ,u ν ) converges to w 0 in C ∞on every compact subset of C.(ii) If R 0 = ∞ then on every compact subset of C\Z, the sequence gνA ∗ ν convergesto A 0 in C 0 , and the sequence gν −1 u ν converges to u 0 in C 1 .(iii) Fix a point z ∈ Z and a number ε 0 > 0 so small that B ε0 (z) ∩Z = {z}. Thenfor every 0 < ε < ε 0 the limitE z (ε) := lim (wν→∞ ERν ν ,B ε (z))exists andE z (ε) ≥ E min .Furthermore, the function (0,ε 0 ) ∋ ε ↦→ E z (ε) ∈ [E min , ∞) is continuous.Remark. Convergence in conditions (i,ii) should be understood as convergenceof the subsequence labelled by those indices ν for which B rν contains the givencompact set. ✷This proposition will be proved on page 42. The strategy of the proof is thefollowing. Assume that the energy densities e Rνw νare uniformly bounded on everycompact subset of C. Then the statement of Proposition 37 with Z = ∅ followsfrom an argument involving Uhlenbeck compactness, an estimate for ¯∂ J , ellipticbootstrapping (for statement (i)), and a patching argument.If the densities are not uniformly bounded then we rescale the maps w ν byzooming in near a bubbling point z 0 in a “hard way”, to obtain a positive energy˜R 0 -vortex in the limit, with ˜R 0 ∈ {0,1, ∞}. If R 0 < ∞ then ˜R 0 = 0, and we obtaina J-holomorphic sphere in M. This contradicts symplectic asphericity, and thusthis case is impossible.If R 0 = ∞ then either ˜R 0 = 1 or ˜R 0 = ∞, and hence either a vortex over Cor a pseudo-holomorphic sphere in M bubbles off. Therefore, at least the energyE min is lost at z 0 . Our assumption that the energies of w ν are uniformly boundedimplies that there can only be finitely many bubbling points. On the complementof these points a subsequence of w ν converges modulo gauge.The bubbling part of this argument is captured by Proposition 40 below,whereas the convergence part is the content of the following result.38. Proposition (Compactness with bounded energy densities). Let Z ⊆ C bea finite subset, R ν ≥ 0 be a sequence of numbers that converges to some R 0 ∈ [0, ∞],Ω 1 ⊆ Ω 2 ⊆ ... ⊆ C \ Z open subsets such that ⋃ ν Ω ν = C \ Z, and for ν ∈ N letw ν = (u ν ,A ν ) ∈ ˜W p Ω νbe an R ν -vortex. Assume that there exists a compact subsetK ⊆ M such that for ν large enough(2.35) u ν (Ω ν ) ⊆ K.Suppose also that for every compact subset Q ⊆ C \ Z, we have(2.36) sup { }‖e Rν ∣ ν ∈ N : Q ⊆ Ων < ∞.w ν‖ L ∞ (Q)

2.5. COMPACTNESS MODULO BUBBLING AND GAUGE FOR RESCALED VORTICES 35Then there exists an R 0 -vortex w 0 := (A 0 ,u 0 ) ∈ ˜W C\Z , and passing to somesubsequence, there exist gauge transformations g ν ∈ W 2,ploc(C \ Z,G), such that thefollowing conditions are satisfied.(i) If R 0 < ∞ then gνw ∗ ν converges to w 0 in C ∞ on every compact subset ofC \ Z.(ii) If R 0 = ∞ then on every compact subset of C \ Z, gνA ∗ ν converges to A 0 inC 0 , and gν −1 u ν converges to u 0 in C 1 .The proof of this result is an adaption of the argument of Step 5 in the proofof Theorem A in the paper by R. Gaio and D. A. Salamon [GS]. The proof ofstatement (i) is based on a compactness result for the case of a compact surface Σ(possibly with boundary). (See Theorem 78 in Appendix A.1. That result followsfrom an argument by K. Cieliebak et al. in [CGMS].) The proof also involves apatching argument for gauge transformations, which are defined on an exhaustingsequence of subsets of C \ Z.To prove statement (ii), we will show that curvatures of the connections A ν areuniformly bounded in W 1,p . This uses the second rescaled vortex equations anda uniform upper bound on µ ◦ u ν (Lemma 75 in Appendix A.1), due to R. Gaioand D. A. Salamon. The statement then follows from Uhlenbeck compactness withcompact base, compactness for ¯∂ J , and a patching argument.Proof of Proposition 38. We may assume w.l.o.g. that there exists a G-invariant compact subset K ⊆ M satisfying (2.35). (To see this, we choose acompact subset K satisfying this condition and consider the set GK.) We choosei 0 ∈ N so big that the balls ¯B 1/i0 (z), z ∈ Z, are disjoint and contained in B i0 . Forevery i ∈ N 0 we defineX i := ¯B i+i0 \ ⋃B 1 (z) ⊆ C.i+i 0z∈ZWe prove statement (i). Assume that R 0 < ∞. Using the hypotheses(2.35,2.36), it follows from Theorem 78 in Appendix A.1 (with Σ := X 2 ) thatthere exist an infinite subset I 1 ⊆ N and gauge transformations g 1 ν ∈ W 2,p (X 1 ,G)(ν ∈ I 1 ), such that X 1 ⊆ Ω ν andw 1 ν := (A 1 ν,u 1 ν) := (g 1 ν) ∗ (w ν |X 1 )is smooth, for every ν ∈ I 1 , and the sequence (w 1 ν) ν∈I 1 converges to some R 0 -vortexw 1 ∈ ˜W X 1, in C ∞ on X 1 .Iterating this argument, for every i ≥ 2 there exist an infinite subset I i ⊆ I i−1and gauge transformations g i ν ∈ W 2,p (X i ,G) (ν ∈ I i ), such that X i ⊆ Ω ν andw i ν := (A i ν,u i ν) := (g i ν) ∗ (w ν |X i )is smooth, for every ν ∈ I i , and the sequence (wν) i ν∈I i converges to some R 0 -vortexw i ∈ ˜W X 1, in C ∞ on X i .Let i ∈ N. For every ν ∈ I i we define h i ν := (gν i+1 |X i ) −1 gν. i We have(h i ν) ∗ (A i+1ν |X i ) = A i ν. Furthermore, the sequences (A i+1ν ) ν∈I i+1 and (A i ν) ν∈I i+1are bounded in W k,p on X i , for every k ∈ N. Hence it follows from Lemma 114(Appendix A.7) that the sequence (h i ν) ν∈I i+1 is bounded in W k,p on X i , for everyk ∈ N. Hence, using Morrey’s embedding theorem and the Arzelà-Ascoli theorem,

36 2. BUBBLING FOR VORTICES OVER THE PLANEit has a subsequence that converges to some gauge transformation h i ∈ C ∞ (X i ,G),in C ∞ on X i . Note that(2.37) (h i ) ∗ (w i+1 |X i ) = w i .We choose a map ρ i : X i+1 → X i such that ρ i = id on X i−1 . We define 22 k 1 := h 1 ,and recursively,(2.38) k i := h i (k i−1 ◦ ρ i−1 ) ∈ C ∞ (X i ,G), ∀i ≥ 2.Using (2.37) and the fact ρ i−1 = id on X i−2 , we have, for every i ≥ 2,(k i ) ∗ w i+1 = (k i−1 ◦ ρ i−1 ) ∗ w i = (k i−1 ) ∗ w i , on X i−2 .It follows that there exists a unique w ∈ ˜W C\Z that restricts to (k i+1 ) ∗ w i+2 onX i , for every i ∈ N. Let i ∈ N. We choose ν i ∈ I i+1 such that ν i ≥ i and a mapτ i : C \ Z → X i that is the identity on X i−1 . We defineg i := (gν i+1ik i ) ◦ τ i ∈ C ∞ (C \ Z,G).The sequence g ∗ i w ν iconverges to w, in C ∞ on every compact subset of C\Z. (Herewe use the C ∞ -convergence on X i of (w i ν) ν∈I i to w i and the facts X 1 ⊆ X 2 ⊆ · · ·and ⋃ i∈N X i = C \ Z.) Statement (i) follows.We prove statement (ii). Assume that R 0 = ∞.1. Claim. For every compact subset Q ⊆ C \ Z we have{ }(2.39) sup ‖FAν ‖ Lp∣(Q) ν ∈ N : Q ⊆ Ων < ∞.νProof of Claim 1. Let Ω ⊆ C be an open subset containing Q such that Ωis compact and contained in C \ Z. Hypothesis (2.36) implies that(2.40) sup ‖d Aν u ν ‖ Lν∞ (Ω) < ∞.It follows from our standing hypothesis (H) that there exists δ > 0 such that Gacts freely onK := { x ∈ M ∣ ∣ |µ(x)| ≤ δ}.Since µ is proper the set K is compact. Recall that L x : g → T x M denotes theinfinitesimal action at x. It follows that{ } |ξ|(2.41) sup|L x ξ| ∣ x ∈ K, 0 ≠ ξ ∈ g < ∞.Using the second R ν -vortex equation, we have√e Rνw ν|µ ◦ u ν | ≤ .R νHence by hypothesis (2.36) and the assumption R ν → ∞, we have ‖µ ◦u ν ‖ L ∞ (Ω)

2.5. COMPACTNESS MODULO BUBBLING AND GAUGE FOR RESCALED VORTICES 37Using Claim 1, Theorem 112 (Uhlenbeck compactness) in Appendix A.7 impliesthat there exist an infinite subset I 1 ⊆ N and, for each ν ∈ I 1 , a gauge transformationgν 1 ∈ W 2,p (X 1 ,G), such that X 1 ⊆ Ω ν and the sequence A 1 ν := (gν) 1 ∗ (A ν |X 1 )converges to some W 1,p -connection A 1 over X 1 , weakly in W 1,p on X 1 . By Morrey’sembedding theorem and the Arzelà-Ascoli theorem, shrinking I 1 , we may assumethat A 1 ν converges (strongly) in C 0 on X 1 .Iterating this argument, for every i ≥ 2 there exist an infinite subset I i ⊆ I i−1and gauge transformations gν i ∈ W 2,p (X 1 ,G), for ν ∈ I i , such that X i ⊆ Ω ν ,for every ν ∈ I i , and the sequence A i ν := (gν) i ∗ (A ν |X i ) converges to some W 1,p -connection A i over X i , weakly in W 1,p and in C 0 on X i .|X i ) −1 gν. i An argument as inthe proof of statement (i), using Lemma 114 (Appendix A.7), implies that thesequence (h i ν) ν∈I i has a subsequence that converges to some gauge transformationh i ∈ W 2,p (X i ,G), weakly in W 2,p on X i .Repeating the construction in the proof of statement (i) and using the weakW 1,p - and strong C 0 -convergence of A i ν on X i , we obtain an index ν i ≥ i+1 and agauge transformation g i ∈ W 2,p (C \ Z,G), for every i ∈ N, such that ν i ∈ I i+1 andgi ∗A ν iconverges to some W 1,p -connection A over C \ Z, weakly in W 1,p and in C 0on every compact subset of C\Z. Passing to the subsequence (ν i ) i , we may assumew.l.o.g. that A ν converges to A, weakly in W 1,p and in C 0 on every compact subsetof C \ Z.Let i ∈ N. For ν ∈ I i we define h i ν := (g i+1ν2. Claim. The hypotheses of Proposition 113 (Appendix A.7) with k = 1 aresatisfied.Proof of Claim 2. Let Ω ⊆ C \ Z be an open subset with compact closure,and ν 0 ∈ N be such that Ω ⊆ Ω ν0 . Since the sequence (A ν ) converges to A, weaklyin W 1,p (Ω), we have(2.42) supν≥ν 0‖A ν ‖ W 1,p (Ω) < ∞.Condition (A.56) is satisfied by assumption (2.35). We check condition (A.57):We denote by |Ω| the area of Ω and choose a constant C > 0 such that L x ξ ≤ C|ξ|,for every x ∈ K and ξ ∈ g. For ν ≥ ν 0 , we have(2.43)‖du ν ‖ L p (Ω) ≤ ‖d Aν u ν ‖ L p (Ω) + ‖L uν A ν ‖ L p (Ω)≤|Ω| 1 p ‖dAν u ν ‖ L ∞ (Ω) + C‖A ν ‖ L p (Ω).Here the second inequality uses the hypothesis (2.35). Combining this with (2.36)and (2.42), condition (A.57) follows.Condition (A.58) follows from the first vortex equation, (2.42), (A.57), andhypothesis (2.35). This proves Claim 2.□By Claim 2, we may apply Proposition 113, to conclude that, passing to somesubsequence, u ν converges to some map u ∈ W 2,p (C \ Z), weakly in W 2,p and inC 1 on every compact subset of C \ Z. The pair w := (A,u) solves the first vortexequation. Furthermore, multiplying the second R ν -vortex equation with Rν −2 , itfollows that µ ◦ u = 0. This means that w is an ∞-vortex.3. Claim. There exists a gauge transformation g ∈ W 2,p (C \ Z,G) such thatg ∗ (A,u) is smooth.

38 2. BUBBLING FOR VORTICES OVER THE PLANEProof of Claim 3. Since C \ Z continuously deformation retracts onto awedge of circles, and G is connected, there exists a continuous lift ṽ : C\Z → µ −1 (0)of Gu. By Proposition 116 in Appendix A.7 the map Gu : C \ Z → M is ¯Jholomorphic.Hence it is smooth. It follows that we may approximate ṽ by somesmooth lift v of Gu. We define g : C \ Z → G to be the unique solution ofg(z)v = u(z), for every z ∈ C\Z. Since the infinitesimal action at every x ∈ µ −1 (0)is injective, the equation ¯∂ J,g∗ A(g −1 u) = 0 and smoothness of v = g −1 u imply thatg ∗ A is smooth. This proves Claim 3.□We choose g as in Claim 3. Regauging A ν by g, statement (ii) follows. Thiscompletes the proof of Proposition 38.□Remark. One can try to circumvent the patching argument for the gaugetransformations in this proof by choosing an extension ˜g i ν of g i ν to C\Z, and definingg ν := ˜g ν ν. However, the sequence (g ν ) does not have the required properties, sinceg ∗ νw ν does not necessarily converge on compact subsets of C \Z. The reason is thatfor j > i the transformation g j ν does not need to restrict to g i ν on X i . ✷39. Remark. It is not clear if in the case R 0 = ∞ the g ν ’s can be chosen insuch a way that g ∗ νw ν converges in C ∞ on every compact subset of C \Z. To provethis, one approach is to fix an open subset of C with smooth boundary and compactclosure, which is contained in C \ Z. We can now try mimic the proof of [CGMS,Theorem 3.2]. In Step 3 of that proof the first and second vortex equations (andrelative Coulomb gauge) are used iteratively in an alternating way. This iterationfails in our setting, because of the factor R 2 ν in the second vortex equations, whichconverges to ∞ by assumption. ✷The second ingredient of the proof of Proposition 37 says that if the energydensities of a sequence of rescaled vortices are not uniformly bounded on somecompact subset Q, then at least the energy E min is lost in the limit, at some pointin Q. Here E min is defined as in (2.34).40. Proposition (Quantization of energy loss). Assume that (M,ω) is aspherical.Let Ω ⊆ C be an open subset, 0 < R ν < ∞ a sequence such that inf ν R ν > 0,and w ν ∈ ˜W p Ω an R ν-vortex, for ν ∈ N. Assume that there exists a compact subsetK ⊆ M such that u ν (Ω) ⊆ K for every ν and that sup ν E Rν (w ν ) < ∞. Then thefollowing conditions hold.(i) For every compact subset Q ⊆ Ω we havesupRν −2 ‖e Rνw ν‖ C 0 (Q) < ∞.ν(ii) If there exists a compact subset Q ⊆ Ω such that sup ν ||e Rνw ν|| C 0 (Q) = ∞ thenthere exists z 0 ∈ Q with the following property. For every ε > 0 so small thatB ε (z 0 ) ⊆ Ω we have(2.44) limsup E Rν (w ν ,B ε (z 0 )) ≥ E min .ν→∞The proof of Proposition 40 is built on a bubbling argument as in Step 5 in theproof of [GS, Theorem A]. The idea is that under the assumption of (ii) we mayconstruct either a ¯J-holomorphic sphere in M or a vortex over C, by rescaling thesequence w ν in a “hard way”. This means that after rescaling the energy densitiesare bounded. We need the following two lemmata.

2.5. COMPACTNESS MODULO BUBBLING AND GAUGE FOR RESCALED VORTICES 3941. Lemma (Hofer). Let (X,d) be a metric space, f : X → [0, ∞) a continuousfunction, x ∈ X, and δ > 0. Assume that the closed ball ¯B2δ (x) is complete. Thenthere exists ξ ∈ X and a number 0 < ε ≤ δ such thatd(x,ξ) < 2δ,Proof. See [MS2, Lemma 4.6.4].sup f ≤ 2f(ξ),B ε(ξ)εf(ξ) ≥ δf(x).The next lemma ensures that for a suitably convergent sequence of rescaledvortices in the limit ν → ∞ no energy gets lost on any compact set. Apart fromProposition 40, it will also be used in the proofs of Propositions 37 and 44, andTheorem 3.42. Lemma (Convergence of energy densities). Let (Σ,ω Σ ,j) be a surface withoutboundary, equipped with an area form and a compatible complex structure,R ν ∈ [0, ∞), ν ∈ N, a sequence of numbers that converges to some R 0 ∈ [0, ∞],and for ν ∈ N 0 let w ν := (A ν ,u ν ) ∈ ˜W p Σ be an R ν-vortex. Assume that on everycompact subset of Σ, A ν converges to A 0 in C 0 and u ν converges to u 0 in C 1 . Thenwe have(2.45) e Rνw ν→ e R0w 0in C 0 on every compact subset of Σ.Proof of Lemma 42. In the case R 0 < ∞ the statement of the lemma is aconsequence of equality (2.29) and the second rescaled vortex equation (2.32).Consider the case R 0 = ∞. It follows from our standing hypothesis (H) thatthere exists a constant δ > 0 such that G acts freely onK := {x ∈ M | |µ(x)| ≤ δ}.Properness of µ implies that K is compact.Let Q ⊆ Σ be a compact subset. The convergence of u ν and the fact µ ◦u 0 = 0imply that for ν large enough, we have u ν (Q) ⊆ K. Furthermore, our hypothesesabout the convergence of A ν and u ν imply that sup ν ‖d Aν u ν ‖ C0 (Q) < ∞. Finally,since K is compact and G acts freely on it, we have{ } |ξ|sup|L x ξ| ∣ x ∈ K, 0 ≠ ξ ∈ g < ∞.Therefore, we may apply Lemma 75 (Appendix A.1), to conclude thatsupR 2− 2 pν |µ ◦ u ν | < ∞.QSince p > 2,R ν → ∞, and e ∞ w 0= 1 2 |d A 0u 0 | 2 , the convergence (2.45) follows. Thiscompletes the proof of Lemma 42.□In the proof of Proposition 40 we will also use the following.43. Remark. Let (A,u) ∈ ˜W p Cbe an ∞-vortex, i.e., a solution of the equations¯∂ J,A (u) = 0 and µ ◦ u = 0. Then by Proposition 116 (Appendix A.7) the mapGu : C → M = µ −1 (0)/G is ¯J-holomorphic, and E ∞ (A,u) = E(ū). If this energyis finite, then by removal of singularities the map ū extends to a ¯J-holomorphicmap ū : S 2 → M. 23 It follows that E ∞ (w) ≥ E min , provided that E ∞ (w) > 0. ✷23 See e.g. [MS2, Theorem 4.1.2].□

40 2. BUBBLING FOR VORTICES OVER THE PLANEProof of Proposition 40. We write (A ν ,u ν ) := w ν . Consider the functionf ν := |d Aν u ν | + R ν |µ ◦ u ν | : Ω → R.Claim. Suppose that the hypotheses of Proposition 40 are satisfied and thatthere exists a sequence z ν ∈ Ω that converges to some z 0 ∈ Ω, such that f ν (z ν ) →∞. Then there exists a number(2.46) 0 < r 0 ≤ limsupν→∞and an r 0 -vortex w 0 ∈ ˜W C , such thatR ν(≤ ∞)f ν (z ν )(2.47) 0 < E r0 (w 0 ) ≤ limsup E Rν (w ν ,B ε (z 0 )),ν→∞for every ε > 0 so small that B ε (z 0 ) ⊆ Ω.Proof of the claim. Construction of r 0 : We define δ ν := f ν (z ν ) − 1 2 . Forν large enough we have ¯B 2δν (z ν ) ⊆ Ω. We pass to some subsequence such that thisholds for every ν. By Lemma 41, applied with (f,x,δ) := (f ν ,z ν ,δ ν ), there existζ ν ∈ B 2δν (z 0 ) and ε ν ≤ δ ν , such that(2.48)(2.49)(2.50)|ζ ν − z ν | < 2δ ν ,sup f ν ≤ 2f ν (ζ ν ),B εν (ζ ν)ε ν f ν (ζ ν ) ≥ f ν (z ν ) 1 2 .Since by assumption f ν (z ν ) → ∞, it follows from (2.48) that the sequence ζ νconverges to z 0 . We definec ν := f ν (ζ ν ), ˜Ων := { c ν (z − ζ ν ) ∣ ∣ z ∈ Ω } ,ϕ ν : ˜Ω ν → Ω,˜w ν := ϕ ∗ νw ν = (ϕ ∗ νA ν ,u ν ◦ ϕ ν ),ϕ ν (˜z) := c −1ν ˜z + ζ ν ,˜Rν := c −1ν R ν .Note that ˜w ν is an ˜R ν -vortex. Passing to some subsequence we may assume that˜R ν converges to some r 0 ∈ [0, ∞]. Since ε ν ≤ δ ν = f ν (z ν ) − 1 2 , it follows from (2.50)that f ν (z ν ) ≤ f ν (ζ ν ). It follows that the second inequality in (2.46) holds forthe original sequence.Construction of w 0 : We choose a sequence Ω 1 ⊆ Ω 2 ⊆ ... ⊆ C of open setssuch that ⋃ ν Ω ν = C and Ω ν ⊆ ˜Ω ν , for every ν ∈ N. We check the conditions ofProposition 38 with these sets, Z := ∅, and R ν ,w ν replaced by ˜R ν , ˜w ν : Condition(2.35) is satisfied by hypothesis.We check condition (2.36): A direct calculation involving (2.49) shows that(2.51) |d eAν ũ ν | + ˜R ν |µ ◦ ũ ν | = c −1ν f ν ◦ ϕ ν ≤ 2, on B ενc ν(0).It follows from (2.50) and the fact f ν (z ν ) → ∞, that ε ν c ν → ∞. Combining thiswith (2.51), condition (2.36) follows, for every compact subset Q ⊆ C.Therefore, applying Proposition 38, there exists an r 0 -vortex w 0 = (A 0 ,u 0 ) ∈˜W C and, passing to some subsequence, there exist gauge transformations g ν ∈W 2,p (C,G), with the following properties. For every compact subset Q ⊆ C, gνÃν∗converges to A 0 in C 0 on Q, and gν −1 ũ ν converges to u 0 in C 1 on Q.We prove the first inequality in (2.47): By Lemma 42 we haveR(2.52) e e ν Rew ν= e e νg→ν ∗ er0ewν w 0,

2.5. COMPACTNESS MODULO BUBBLING AND GAUGE FOR RESCALED VORTICES 41in C 0 (Q) for every compact subset Q ⊆ C. Sincee e R νew ν(0) = c −2ν e Rνw ν(ζ ν ) ≥ 1 2 ,it follows that e r0w 0(0) ≥ 1/2. This implies that E r0 (w 0 ) > 0. This proves the firstinequality in (2.47).We prove the second inequality in (2.47): Let ε > 0 be so small thatB ε (z 0 ) ⊆ Ω, and δ > 0. It follows from (2.52) that E r0 (w 0 ) ≤ sup ν E Rν (w ν ).By hypothesis this supremum is finite. Therefore, there exists R > 0 such thatE r0 (w 0 , C \ B R ) < δ. SinceE Rν( w ν ,B c−1ν R (ζ ν) ) = E e R ν( ˜w ν ,B R ),the convergence (2.52) implies that(2.53) limν→∞ ERν( w ν ,B c−1ν R (ζ ν) ) = E r0 (w 0 ,B R ) > E r0 (w 0 ) − δ.On the other hand, since c ν → ∞ and ζ ν → z 0 , for ν large enough the ball B c−1ν R (ζ ν)is contained in B ε (z 0 ). Combining this with (2.53), we obtainlimsup E Rν (w ν ,B ε (z 0 )) ≥ E r0 (w 0 ) − δ.ν→∞Since this holds for every δ > 0, the second inequality in (2.47) (for the originalsequence) follows.It remains to prove the first inequality in (2.46), i.e., that r 0 > 0. Assumeby contradiction that r 0 = 0. For a map u ∈ C ∞ (C,M) we denote byE(u) := 1 ∫|du| 22its (Dirichlet-)energy. 24 By the second R-vortex equation with R := 0 we haveF A0 = 0. Therefore, by Proposition 115 (Appendix A.7) there exists h ∈ C ∞ (C,G)such that h ∗ A 0 = 0. By the first vortex equation the map u ′ 0 := h −1 u 0 : C → M isJ-holomorphic. Let ε > 0 be such that B ε (z 0 ) ⊆ Ω. Using the second inequality in(2.47), we haveE(u ′ 0) = E 0 (w 0 ) ≤ limsup E Rν (w ν ,B ε (z 0 )).ν→∞Combining this with the hypothesis sup ν E Rν (w ν ,Ω) < ∞, it follows that theenergy E(u ′ 0) is finite. Hence by removal of singularities 25 , u ′ 0 extends to a smoothJ-holomorphic map v : S 2 → M. By the first inequality in (2.47) we have∫v ∗ ω = E(v) = E 0 (w 0 ) > 0.S 2This contradicts asphericity of (M,ω). Hence r 0 must be positive. This concludesthe proof of the claim.□Statement (i) of Proposition 40 follows from the claim, considering a sequencez ν ∈ Q, such that f ν (z ν ) = ‖f ν ‖ C0 (Q), and using (2.46).We prove statement (ii). Assume that there exists a compact subset Q ⊆ Ωsuch that sup ν ||e Rνw ν|| C0 (Q) = ∞. Let z ν ∈ Q be such that f ν (z ν ) → ∞. We choosea pair (r 0 ,w 0 ) as in the claim. Using the first inequality in (2.47) and Remark 4324 Here the norm is taken with respect to the metric ω(·, J·) on M.25 see e.g. [MS2, Theorem 4.1.2]C

42 2. BUBBLING FOR VORTICES OVER THE PLANE(in the case r 0 = ∞), we have E r0 (w 0 ) ≥ E min . Combining this with the secondinequality in (2.47), inequality (2.44) follows. This proves (ii) and concludes theproof of Proposition 40.□We are now ready to prove Proposition 37 (p. 33).Proof of Proposition 37. We abbreviate e ν := e Rνw ν.Claim. For every l ∈ N 0 there exists a finite subset Z l ⊆ C such that thefollowing holds. If R 0 < ∞ then we have Z l = ∅. Furthermore, if |Z l | < l then wehave(2.54) supν∈N{ }‖eν ‖ C0∣(Q) Q ⊆ Brν < ∞,for every compact subset Q ⊆ C \ Z l . Moreover, for every z 0 ∈ Z l and every ε > 0the inequality (2.44) holds.Proof of the claim. For l = 0 the assertion holds with Z 0 := ∅. We fixl ≥ 1 and assume by induction that there exists a finite subset Z l−1 ⊆ C such thatthe assertion with l replaced by l − 1 holds. If (2.54) is satisfied for every compactsubset Q ⊆ C \ Z l−1 , then the statement for l holds with Z l := Z l−1 .Assume that there exists a compact subset Q ⊆ C \Z l−1 , such that (2.54) doesnot hold. It follows from the induction hypothesis that(2.55) |Z l−1 | ≥ l − 1.By statement (ii) of Proposition 40 there exists a point z 0 ∈ Q such that inequality(2.44) holds, for every ε > 0. We set Z l := Z l−1 ∪ {z 0 }.It follows from the fact that (2.54) does not hold and statement (i) of Proposition40 that R 0 = lim ν→∞ R ν = ∞. Furthermore, since z 0 ∈ Q ⊆ C \ Z l−1 , (2.55)implies that |Z l | ≥ l. It follows that the statement of the claim for l is satisfied.By induction, the claim follows.□We fix an integerl > sup ν E Rν (w ν ,B rν )E minand a finite subset Z := Z l ⊆ C that satisfies the conditions of the claim. It followsfrom the inequality (2.44) that l > |Z|. Hence by the statement of the claim, thehypothesis (2.36) of Proposition 38 is satisfied with Ω ν := B rν \ Z. Applying thatresult and passing to some subsequence, there exist an R 0 -vortex w 0 ∈ ˜W C\Z andgauge transformations g ν ∈ W 2,ploc(C \ Z,G), such that the statements (i,ii) ofProposition 37 are satisfied. (Here we use that Z = ∅ if R 0 < ∞.)We prove( statement (iii). Passing to some “diagonal” subsequence, the limitlim ν→∞ E Rν wν ,B 1/i (z) ) exists, for every i ∈ N and z ∈ Z. Let now z ∈ Z andε > 0. We choose i ∈ N bigger than ε −1 . For 0 < r < R we denoteA(z,r,R) := ¯B R (z) \ B r (z).By Lemma 42 the limitlim ERν( w ν ,A(z,1/i,ε) )ν→∞exists and equals E R0 (w 0 ,A(z,1/i,ε)). It follows that the limitE z (ε) := limν→∞ ERν (w ν ,B ε (z))

2.6. SOFT RESCALING 43exists. Inequality (2.44) implies that E z (ε) ≥ E min . Since E R0 (w 0 ,A(z,1/i,ε))depends continuously on ε, the same holds for E z (ε). This proves statement (iii)and completes the proof of Proposition 37.□Remark. In the above proof the set of bubbling points Z is constructed by“terminating induction”. Intuitively, this is induction over the number of bubblingpoints. The “auxiliary index” l in the claim is needed to make this idea precise.Inequality (2.44) ensures that the “induction stops”. ✷2.6. Soft rescalingThe following proposition will be used inductively in the proof of Theorem 3 tofind the next bubble in the bubbling tree, at a bubbling point of a given sequence ofrescaled vortex classes. It is an adaption of [MS2, Proposition 4.7.1.] to vortices.44. Proposition (Soft rescaling). Assume that (M,ω) is aspherical. Let r > 0,z 0 ∈ C, R ν > 0 be a sequence that converges to ∞, p > 2, and for every ν ∈ N letw ν := (A ν ,u ν ) ∈ ˜W p B r(z 0) be an R ν-vortex, such that the following conditions aresatisfied.(a) There exists a compact subset K ⊆ M such that u ν (B r (z 0 )) ⊆ K for every ν.(b) For every 0 < ε ≤ r the limit(2.56) E(ε) := limν→∞ ERν (w ν ,B ε (z 0 ))exists and E min ≤ E(ε) < ∞. Furthermore, the function(2.57) (0,r] ∋ ε ↦→ E(ε) ∈ Ris continuous.Then there exist R 0 ∈ {1, ∞}, a finite subset Z ⊆ C, an R 0 -vortex w 0 := (A 0 ,u 0 ) ∈˜W C\Z , and, passing to some subsequence, there exist sequences ε ν > 0, z ν ∈ C,and g ν ∈ W 2,ploc(C \ Z,G), such that, definingϕ ν : C → C, ϕ ν (˜z) := ε ν˜z + z ν ,the following conditions hold.(i) If R 0 = 1 then Z = ∅ and E(w 0 ) > 0. If R 0 = ∞ and E ∞ (w 0 ) = 0 then|Z| ≥ 2.(ii) The sequence z ν converges to z 0 . Furthermore, if R 0 = 1 then ε ν = Rν−1 forevery ν, and if R 0 = ∞ then ε ν converges to 0 and ε ν R ν converges to ∞.(iii) If R 0 = 1 then the sequence gνϕ ∗ ∗ νw ν converges to w 0 in C ∞ on every compactsubset of C\Z. Furthermore, if R 0 = ∞ then on every compact subset of C\Z,the sequence gνϕ ∗ ∗ νA ν converges to A 0 in C 0 , and the sequence gν −1 (u ν ◦ ϕ ν )converges to u 0 in C 1 .(iv) Fix z ∈ Z and a number ε 0 > 0 such that B ε0 (z) ∩ Z = {z}. Then for every0 < ε < ε 0 the limitE z (ε) := limν→∞ EενRν( ϕ ∗ νw ν ,B ε (z) )exists and E min ≤ E z (ε) < ∞. Furthermore, the function (0,ε 0 ) ∋ ε ↦→E z (ε) ∈ R is continuous.(v) We have(2.58) lim limsup E Rν( w ν ,B R −1(z 0 ) \ B Rεν (z ν ) ) = 0.R→∞ν→∞

44 2. BUBBLING FOR VORTICES OVER THE PLANERoughly speaking, this result states that given a sequence of “zoomed out”vortices for which a positive amount of energy is concentrated around some pointz 0 , after “zooming back in”, some subsequence converges either to a vortex over Cor a holomorphic sphere in the symplectic quotient, up to bubbling at finitely manypoints. Furthermore, no energy is lost in the limit between the original sequenceand the “zoomed in” subsequence.To explain this, note that condition (b) in the hypothesis implies that given anarbitrarily small ball around the point z 0 , the energy of the vortex w ν on the ballis bounded below by a positive constant arbitrarily close to E min , provided that νis large enough. This means that in the limit ν → ∞, a positive amount of energybubbles off around the point z 0 . It follows that there exists a sequence of pointsz ν ∈ C converging to z 0 , such that the energy density of w ν at z ν converges to ∞.In the conclusion of the proposition, condition (ii) says that we are “zoomingin” around z 0 , and it specifies how fast we do so. Condition (iii) states that the“zoomed in” subsequence converges to either a vortex over C or a holomorphicsphere in M, depending on how fast we “zoom back in”. Condition (i) implies thatthe limit object is nontrivial or there are at least two new bubbling points. (Thelatter can only happen if the limit is a holomorphic sphere in M.) In the proof ofTheorem 3, this will guarantee that the new bubble is stable.Analogously to condition (b), condition (iv) implies that at each point in Z inthe limit ν → ∞, at least the energy E min bubbles off in the sequence of rescaledvortices ϕ ∗ νw ν . In the proof of Theorem 3, this will be used to prove that the inductiveconstruction of the bubble tree terminates after finitely many steps. Finally,condition (v) will ensure that at each step no energy is lost between the old andnew bubble.Remark. In condition (iii) the pullback ϕ ∗ νw ν is defined over the set ϕ −1ν (B r (z 0 )).This set contains any given compact subset Q ⊆ C \ Z, provided that ν is largeenough (depending on Q). Hence condition (iii) makes sense. ✷The proof of Proposition 44 is given on page 48. It is based on the followingresult, which states that the energy of a vortex over an annulus is concentratednear the ends, provided that it is small enough. For 0 ≤ r,R ≤ ∞ we denote theopen annulus around 0 with radii r,R byA(r,R) := B R \ ¯B r .Note that A(r, ∞) = C \ ¯B r , and A(r,R) = ∅ in the case r ≥ R. We defined : M × M → [0, ∞]to be the distance function induced by the Riemannian metric ω(·,J·). 26 We define(2.59) ¯d : M/G × M/G → [0, ∞], ¯d(¯x,ȳ) := minx∈¯x, y∈ȳ d(x,y).By Lemma 118 in Appendix A.7 this is a distance function on M/G which inducesthe quotient topology.45. Proposition (Energy concentration near ends). There exists a constantr 0 > 0 such that for every compact subset K ⊆ M and every ε > 0 there exists a26 If M is disconnected then d attains the value ∞.

2.6. SOFT RESCALING 45constant E 0 , such that the following holds. Assume that r ≥ r 0 ,R ≤ ∞, p > 2,and w := (A,u) ∈ ˜W p A(r,R) is a vortex (with respect to (ω 0,i)), such that(2.60)u(A(r,R)) ⊆ K,E(w) = E ( w,A(r,R) ) ≤ E 0 .Then we have 27 E ( w,A(ar,a −1 R) ) ≤ 4a −2+ε E(w), ∀a ≥ 2,(2.61)(2.62)sup z,z′ ∈A(ar,a −1 R) ¯d(Gu(z),Gu(z ′ )) ≤ 100a −1+ε√ E(w), ∀a ≥ 4.(Here Gx ∈ M/G denotes the orbit of a point x ∈ M.)The proof of this result is modeled on the proof of [Zi2, Theorem 1.3], which inturn is based on the proof of [GS, Proposition 11.1]. It is based on an isoperimetricinequality for the invariant symplectic action functional (Theorem 82 in AppendixA.2). It also relies on an identity relating the energy of a vortex over a compactcylinder with the actions of its end-loops (Proposition 83 in Appendix A.2). Theproof of (2.62) also uses the following remark.46. Remark. Let ( )M, 〈·, ·〉 M be a Riemannian manifold, G a compact Liegroup that acts on M by isometries, P a G-bundle over [0,1] 28 , A ∈ A(P) aconnection, and u ∈ CG ∞ (P,M) a map. We definel(A,u) :=∫ 10|d A u|dt,where d A u = du+L u A, and the norm is taken with respect to the standard metricon [0,1] and 〈·, ·〉 M . Furthermore, we defineū : [0,1] → M/G,ū(t) := Gu(p),where p ∈ P is any point over t. We denote by d the distance function induced by〈·, ·〉 M , and define ¯d as in (2.59). Then for every pair of points ¯x 0 , ¯x 1 ∈ M/G, wehave¯d(¯x 0 , ¯x 1 ) ≤ inf { l(A,u) ∣ ∣ (P,A,u) as above: ū(i) = ¯x i , i = 0,1 } .This follows from a straight-forward argument. ✷Proof of Proposition 45. For every subset X ⊆ M we definem X := inf { |L x ξ| ∣ ∣ x ∈ X, ξ ∈ g : |ξ| = 1 } ,where the norms are with respect to ω(·,J·) and 〈·, ·〉 g . We set(2.63) r 0 := m −1µ −1 (0) .Let K ⊆ M be a compact subset and ε > 0. Replacing K by GK, we mayassume w.l.o.g. that K is G-invariant. An elementary argument using our standinghypothesis (H) shows that there exists a number δ 0 > 0 such that G acts freely onK ′ := µ −1 ( ¯B δ0 ), and√(2.64) m K ′ ≥1 − ε 2 m µ −1 (0).p27 Note that for a > R/r we have A(ar, a −1 R) = ∅, hence (2.61,2.62) are non-trivial onlyfor a ≤ p R/r.28 Such a bundle is trivializable, but we do not fix a trivialization here.

46 2. BUBBLING FOR VORTICES OVER THE PLANEWe choose a constant δ as in Theorem 82 in Appendix A.2 (Isoperimetric inequality),corresponding to〈·, ·〉 M := ω(·,J·), K ′ , c := 12 − ε .Shrinking δ we may assume that it satisfies the condition of Proposition 83 inAppendix A.2 (Energy action identity) for K ′ . We choose a constant Ẽ0 > 0 as inLemma 72 in Appendix A.1 (called E 0 there), corresponding to K. We define(2.65) E 0 := min{Ẽ 0 , π 32 r2 0δ 2 0,δ 2128πAssume that r,R,p,w are as in the hypothesis. Without loss of generality, we mayassume that r < R.Consider first the case R < ∞, and assume that w extends to a smooth vortexover the compact annulus of radii r and R. We show that inequality (2.61) holds.We define the function(2.66) E : [0, ∞) → [0, ∞), E(s) := E ( w,A(re s ,Re −s ) ) .(2.67)Claim. For every s ∈ [ log 2,log(R/r)/2 ) we havedE(s) ≤ −(2 − ε)E(s).dsProof of the claim. Using the fact r ≥ r 0 and (2.60,2.65), it follows fromLemma 72 in Appendix A.1 (with r replaced by |z|/2) that{(2.68) e w (z) ≤ min δ0,2 δ 2 }4π 2 |z| 2 , ∀z ∈ A(2r,R/2).We define}.Σ s := ( s + log r, −s + log R ) × S 1 , ∀s ∈ R,ϕ : Σ 0 → C, ϕ(z) := e z , ˜w := (Ã,ũ) := ϕ∗ w.(Here we identify Σ 0∼ = C/ ∼, where z ∼ z + 2πin, for every n ∈ Z.) Let s0 ∈[log(2r),log(R/2)]. Combining (2.68) with the fact |µ ◦ u| ≤√ew and Remark 46,it follows that(2.69) ũ(s 0 ,t) ∈ K ′ = µ −1 ( ¯B δ0 ), ∀t ∈ S 1 , ¯l(Gũ(s0 , ·)) ≤ δ.Hence the hypotheses of Theorem 82 (Appendix A.2) are satisfied with K replacedby K ′ and c := 1/(2 − ε). By the statement of that result the loop ũ(s 0 , ·) isadmissible, and definingwe have(2.70)ι s0 : S 1 → Σ 0 ,ι s0 (t) := (s 0 ,t),∣ A(ι∗s0 ˜w )∣ ∣ ≤12 − ε ‖ι∗ s 0d eA ũ‖ 2 2 + 12m 2 K ′ ∥ ∥µ ◦ ũ ◦ ιs0∥ ∥22 .Here A denotes the invariant symplectic action, as defined in Appendix A.2. Furthermore,the L 2 -norms are with respect to the standard metric on S 1 ∼ = R/(2πZ),the metric ω(·,J·) on M, and the operator norm | · | op : g ∗ → R, induced by 〈·, ·〉 g .By (2.63,2.64) and the fact 2r ≤ e s0 , we have1(2.71)2 − ε |ι∗ s 0d eA ũ| 2 0(t) + 1 ∣ ∣ ∣µ ◦ ũ ◦2m 2 ιs0 2 1 (t) ≤ eK2 − ε e2s0 w (e s0+it ), ∀t ∈ S 1 .′

2.6. SOFT RESCALING 47Here the norm | · | 0 is with respect to the standard metric on S 1 ∼ = R/(2πZ), andwe used the fact that for every ϕ ∈ g ∗ ,|ϕ| op ≤ |ϕ| := √ ϕ(ξ)where ξ ∈ g is determined by 〈ξ, ·〉 g = ϕ. We fix s ∈ [ log 2,log(R/r)/2 ) . Recalling(2.66), we have∫E(s) = e 2s0 e w (e s0+it )dtds 0 .Σ sCombining this with (2.70,2.71), it follows that(2.72) −A ( ι ∗ −s+log R ˜w ) + A ( ι ∗ s+log r ˜w ) ≤ − 1 d2 − ε ds E(s).Using (2.69), the hypotheses of Proposition 83 are satisfied with K replaced by K ′ .Applying that result, we haveE(s) = −A ( ι ∗ −s+log R ˜w ) + A ( ι ∗ s+log r ˜w ) .Combining this with (2.72), inequality (2.67) follows. This proves the claim.By the claim the derivative of the function[log 2, log ( ))Rr∋ s ↦→ E(s)e (2−ε)s2is non-positive, and hence this function is non-increasing. Inequality (2.61) follows.We prove (2.62). Let z ∈ A(4r, √ rR). Using (2.60) and the fact E 0 ≤ Ẽ0, itfollows from Lemma 72 (with r replaced by |z|/2) that(2.73) e w (z) ≤ 32 ( )π|z| 2 E w,B |z| (z) .2We define a := |z|/(2r). Then a ≥ 2 and B |z|/2 (z) is contained in A(ar,a −1 R).Therefore, by (2.61) we haveE ( w,B |z| (z) ) ≤ 16r 2−ε |z| −2+ε E(w).2Combining this with (2.73) and the fact |d A u|(z) ≤ √ 2e w (z), it follows that(2.74) |d A u(z)v| ≤ Cr 1− ε 2 |z|−2+ ε 2√E(w)|v|, ∀z ∈ A(4r,√rR), v ∈ C.where C := 32/ √ π. A similar argument shows that(2.75) |d A u(z)v| ≤ CR −1+ ε 2 |z|− ε 2√E(w)|v|, ∀z ∈ A(√rR,R/4).Let now a ≥ 4 and z,z ′ ∈ A(ar,a −1 R). Assume that ε ≤ 1. (This is no realrestriction.) We define γ : [0,1] → C to be the radial path of constant speed,such that γ(0) = z and |γ(1)| = |z ′ |. Furthermore, we choose an angular pathγ ′ : [0,1] → C of constant speed, such that γ ′ (0) = γ(1), γ ′ (1) = z ′ , and γ ′ hasminimal length among such paths. (See Figure 4.)Consider the “twisted length” of γ ∗ (A,u), given by∫ 10∣ dA u ˙γ(t) ∣ dt.It follows from (2.74,2.75) and the fact ε ≤ 1, that this length is bounded aboveby 4C √ E(w)a −1+ε/2 . Similarly, it follows that the “twisted length” of γ ′∗ (A,u)□

48 2. BUBBLING FOR VORTICES OVER THE PLANEz’zγγ’Figure 4. The paths γ and γ ′ described in the proof of inequality(2.62) (Proposition 45).is bounded above by Cπ √ E(w)a −1+ε/2 . Therefore, using Remark 46, inequality(2.62) with ε replaced by ε/2 follows.Assume now that w is not smooth. By Theorem 76 in Appendix A.1 therestriction of w to any compact cylinder contained in A(r,R) is gauge equivalentto a smooth vortex. Hence the inequalities (2.61,2.62) follow from what we justproved, using the G-invariance of K.In the case R = ∞ the inequalities (2.61,2.62) follow from what we just proved,by considering an arbitrarily large radius R < ∞. This completes the proof ofProposition 45.□We are now ready for the proof of Proposition 44 (p. 43).Proof of Proposition 44. The function E as in (2.57) is increasing and, byhypothesis (b), bounded below by E min . Hence the limit(2.76) m 0 := limε→0E(ε)exists and is bounded below by E min . We fix a compact subset K ⊆ M as inhypothesis (a). We choose a constant E 0 > 0 as in Lemma 72 in Appendix A.1(Bound on energy density), depending on K. Considering the pullback of w ν bythe translation by z 0 , we may assume w.l.o.g. that z 0 = 0.1. Claim. We may assume w.l.o.g. that(2.77) ‖e Rνw ν‖ C0 ( ¯B r) = e Rνw ν(0).Proof of Claim 1. Suppose that we have already proved the propositionunder this additional assumption, and let r,z 0 = 0,R ν ,w ν be as in the hypothesesof the proposition. We choose 0 < ̂r ≤ r/4 so small that(2.78) E(4̂r) = limν→∞ ERν (w ν ,B 4br ) < m 0 + E 0 .

2.6. SOFT RESCALING 49For ν ∈ N we choose ˜z ν ∈ ¯B 2br such that(2.79) e Rνw ν(˜z ν ) = ‖e Rνw ν‖ C0 ( ¯B 2br ).2. Claim. The sequence ˜z ν converges to 0.Proof of Claim 2. Recall that A(r,R) denotes the open annulus of radii rand R. Let 0 < ε ≤ 2̂r. Inequality (2.78) implies that there exists ν(ε) ∈ N suchthatE Rν( w ν ,A(ε/2,4̂r) ) < E 0 ,for every ν ≥ ν(ε). Hence by Lemma 72 with r = ε/2 we have(2.80) e Rνw ν(z) < 32E 0πε 2 ,We define∀ν ≥ ν(ε), ∀z ∈ A(ε,2̂r).{ √ }m0δ 0 := min 2̂r,ε .64E 0Increasing ν(ε), we may assume that E Rν (w ν ,B δ0 ) > m 0 /2 for every ν ≥ ν(ε), andtherefore‖e Rνw ν‖ C0 ( ¯B δ0 ) > 32E 0πε 2 .Combining this with (2.79,2.80) and the fact δ 0 ≤ 2̂r, it follows that ˜z ν ∈ B ε , forevery ν ≥ ν(ε). This proves Claim 2.□By Claim 2 we may pass to some subsequence such that |˜z ν | < ̂r for every ν.We defineψ ν : B br → C, ψ ν (z) := z + z ν , ˜w ν := (Ã,ũ) := ψ∗ νw ν .Then (2.77) with w ν ,r replaced by ˜w ν , ̂r is satisfied. By elementary argumentsthe hypotheses of Proposition 44 are satisfied with (w ν ,r,z 0 ) replaced by ( ˜w ν , ̂r,0).Assuming that we have already proved the statement of the proposition for ˜w ν ,a straight-forward argument using Claim 2 shows that it also holds for w ν . Thisproves Claim 1.□So we assume w.l.o.g. that (2.77) holds.Construction of R 0 ,Z, and w 0 : Recall the definition (2.76) of m 0 , and thatwe have chosen E 0 > 0 as in Lemma 72. We choose constants r 0 and E 1 as inProposition 45, the latter (called E 0 there) corresponding to the compact set Kand ε := 1. We fix a constant{(2.81) 0 < δ < min m 0 , E 02 , E }1.2We pass to some subsequence such that(2.82) E Rν (w ν ,B r (0)) > m 0 − δ, ∀ν ∈ N.For every ν ∈ N, there exists 0 < ̂ε ν < r, such that(2.83) E Rν (w ν ,B bεν ) = m 0 − δ.

50 2. BUBBLING FOR VORTICES OVER THE PLANEIt follows from the definition of m 0 that(2.84) ̂ε ν → 0. 293. Claim. We have(2.85) infν ̂ε νR ν > 0.Proof of Claim 3. Equality (2.77) implies that(2.86) E Rν (w ν ,B bεν ) ≤ π̂ε 2 νe Rνw ν(0).The hypotheses R ν → ∞, (a), and (b) imply that the hypotheses of Proposition 40(Quantization of energy loss) are satisfied with Ω := B r . Thus by assertion (i) ofthat proposition with Q := {0}, we haveinfνR 2 νe Rνw ν(0) > 0.Combining this with (2.86,2.83) and the fact δ < m 0 , inequality (2.85) follows. Thisproves Claim 3.□Passing to some subsequence, we may assume that the limit(2.87) ̂R0 := limν→∞ ̂ε νR ν ∈ [0, ∞]exists. By Claim 3 we have ̂R 0 > 0. We define{(∞, ̂εν ), if(2.88) (R 0 ,ε ν ) :=̂R 0 = ∞,(1,Rν −1 ), otherwise,˜R ν := ε ν R ν ,ϕ ν : B ε−1ν r → B r, ϕ ν (z) := ε ν z, ˜w ν := (Ãν,ũ ν ) := ϕ ∗ νw ν .By Proposition 37 with R ν , w ν replaced by ˜R ν , ˜w ν and r ν := r/ε ν there exist afinite subset Z ⊆ C and an R 0 -vortex w 0 = (A 0 ,u 0 ) ∈ ˜W C\Z , and passing to somesubsequence, there exist gauge transformations g ν ∈ W 2,ploc(C \ Z,G), such that theconclusions of that proposition hold.We check the conditions of Proposition 44 with z ν := z 0 := 0: Condition44(ii) holds by (2.84,2.87,2.88). Condition 44(iii) follows from 37(i,ii), andcondition 44(iv) follows from 37(iii).We prove condition 44(v): We defineψ ν : B R−1ν r → B r, ψ ν (z) :=We choose 0 < ε ≤ r so small that̂R−1ν z, w ′ ν := ψ ∗ νw ν . 30lim (wν→∞ ERν ν ,B ε ) < m 0 + E 12 .Furthermore, we choose an integer ν 0 so large that for ν ≥ ν 0 , we have E Rν (w ν ,B ε ) 0, such that passing to some subsequence, we havebε ν ≥ ε. Combining this with (2.83) and considering the limit ν → ∞, we obtain a contradictionto (2.76).30 In the case R0 = 1 we have ψ ν = ϕ ν and hence w ′ ν = ew ν.

2.6. SOFT RESCALING 51It follows that the requirements of Proposition 45 are satisfied with r, R, w ν replacedby max{r 0 , ̂ε ν R ν }, εR ν , w ν. ′ Therefore, we may apply that result (with “ε” equalto 1), obtaining(E Rν w ν ,A ( amax{Rν −1 r 0 , ̂ε ν },a −1 ε )) ≤ 4a −1 E 1 , ∀a ≥ 2.Using (2.87,2.88) and the fact z ν = z 0 = 0, the equality (2.58) follows. This proves44(v).To see that condition 44(i) holds, assume first that R 0 = 1. Then Z = ∅ bystatement (i) of Proposition 37. The same statement and Lemma 42 imply thatE(w 0 ,B 2 b R0) = limν→∞ E( ˜w ν,B 2 b R0).It follows from the convergence ̂ε ν R ν → ̂R 0 > 0 and (2.83,2.81) that this limit ispositive. This proves condition 44(i) in the case R 0 = 1.Assume now that R 0 = ∞ and E ∞ (w 0 ) = 0. Then condition 44(i) is aconsequence of the following two claims.4. Claim. The set Z is not contained in the open ball B 1 .Proof of Claim 4. By 44(v) (which we proved above) there exists R > 0 sothat(2.89) limsup E Rν( w ν ,A(Rε ν ,R −1 ) ) < δ.ν→∞(Here we used that z 0 = z ν = 0.) Since R 0 = ∞, we have ̂ε ν = ε ν . Hence it followsfrom (2.83) and (2.56) that(2.90) limν→∞ ERν( w ν ,A(ε ν ,R −1 ) ) = E(R −1 ) − m 0 + δ.By the definition (2.76) of m 0 , the right hand side is bounded below by δ. Therefore,combining (2.90) with (2.89), it follows that(2.91) liminfν→∞ ERν (w ν ,A(ε ν ,ε ν R)) > 0.Suppose by contradiction that Z ⊆ B 1 . Then by 37(ii), the connection gνÃν∗converges to A 0 in C 0 on Ā(1,R) := B R \ B 1 , and the map gν −1 ũ ν converges to u 0in C 1 on Ā(1,R). Hence Lemma 42 implies thatE ∞( w 0 ,A(1,R) ) = limν→∞ E e R ν(˜wν ,A(1,R) ) .Combining this with (2.91), we arrive at a contradiction to our assumption E ∞ (w 0 ) =0. This proves Claim 4. □5. Claim. The set Z contains 0.Proof of Claim 5. By Claim 4 the set Z \ B 1 is nonempty. We choose a pointz ∈ Z \B 1 and a number ε 0 > 0 so small that B ε0 (z) ∩Z = {z}. We fix 0 < ε < ε 0 .Since ε ν → 0 (as ν → ∞), (2.77) implies thate e R νew ν(0) = ‖e e R νew ν‖ C 0 ( ¯B ε(z)),for ν large enough. Combining this with condition 44(iv) (which we proved above),it follows thatliminf e R e νν→∞ ew ν(0) ≥ E minπε 2 .

52 2. BUBBLING FOR VORTICES OVER THE PLANESince ε ∈ (0,ε 0 ) is arbitrary, it follows that(2.92) e e R νew ν(0) → ∞, as ν → ∞.RIf 0 did not belong to Z, then by 37(ii) and Lemma 42 the energy density e e νew ν(0)would converge to e ∞ w 0(0), a contradiction to (2.92). This proves Claim 5, andcompletes the proof of 44(i) and therefore of Proposition 44.□47. Remark. Assume that R 0 ,Z,w 0 are constructed as in the proof of condition(i) of Proposition 44, and that R 0 = ∞ and E ∞ (w 0 ) = 0. Then Z ⊆ ¯B 1 (andhence Z ∩ S 1 ≠ ∅ by Claim 4). This follows from the inequalitieslim E R e ν( ˜w ν ,A(1,R)) ≤ δ < E min , ∀R > 1.ν→∞Here the first inequality is a consequence of condition (2.83). ✷2.7. Proof of the bubbling resultBased on the results of the previous sections, we are now ready to prove thefirst main result of this memoir, Theorem 3. The proof is an adaption of the proofof [MS2, Theorem 5.3.1] to the present setting. The strategy is the following:Consider first the case k = 0, i.e., the only marked point is z ν 0 = ∞. We rescale thesequence W ν so rapidly that all the energy is concentrated around the origin in C.Then we “zoom back in” in a soft way, to capture the bubbles (spheres in M andvortices over C) in an inductive way. (See Claim 1 below.)Next we show that at each stage of this construction, the total energy of thecomponents of the tree plus the energy loss at the unresolved bubbling points equalsthe limit of the energies E(W ν ). Here we call a bubbling point “unresolved” if wehave not yet constructed any sphere or vortex which is attached to this point.(See Claim 2.) Furthermore, we prove that the number of vertices of the tree isuniformly bounded above. (See inequality (2.102).) This implies that the inductiveconstruction terminates at some point.We also show that the vortices over C and the bubbles in M have the requiredproperties and that the data fits together to a stable map, which is the limit of asubsequence of W ν . (See Claims 4 and 5.)For k ≥ 1 we then prove the statement of the theorem inductively, using thestatement for k = 0. At each induction step we need to handle one additionalmarked point in the sequence of vortex classes and marked points. In the limitthere are three possibilities for the location of this point:(I) It does not coincide with any special point.(II) It coincides with the marked point z i (lying on the vertex α i ), for some i.(III) It lies between two already constructed bubbles.In case (I) we just include the new marked point into the bubble tree. In case(II), we introduce either• a “ghost bubble”, which carries the two marked points and is connectedto α i , or• a “ghost vortex”, which is connected to α i , and a “ghost bubble of type0”, which is connected to the “ghost vortex” and carries the two markedpoints.

2.7. PROOF OF THE BUBBLING RESULT 5331=j 2=j 42=j 34Figure 5. This is a “partial stable map” as in Claim 1. It is apossible step in the construction of the stable map of Figure 1.The cross denotes a bubbling point that has not yet been resolved.When adding marked points, a “ghost sphere” in M will be introducedbetween the components 1 and 4.The second option will only occur if α i ∈ T ∞ . Which of the two options we choosedepends on the speed at which the two marked points come together. In case (III)we introduce a “ghost bubble” between the two bubbles, which carries the newmarked point.Proof of Theorem 3 (p. 7). We consider first the case k = 0. Let W νbe a sequence of vortex classes as in the hypothesis. For each ν ∈ N we choosea representative w ν := (P ν ,A ν ,u ν ) of W ν , such that P ν = C × G. Passing tosome subsequence we may assume that E(w ν ) converges to some constant E. Thehypothesis E(W ν ) > 0 (for every ν) implies that E ≥ E min . We choose a sequenceR ν ≥ 1 such that(2.93) E(W ν ,B Rν ) → E.We defineR ν 0 := νR ν , ϕ ν : C → C, ϕ ν (z) := R ν 0z, w ν 0 := ϕ ∗ νw ν ,j 1 := 0, z 1 := 0, Z 0 := {0}, z ν 0 := 0.The next claim provides an inductive construction of the bubble tree. (Some explanationsare given below. See also Figure 5.)1. Claim. For every integer l ∈ N, passing to some subsequence, there existan integer N := N(l) ∈ N and tuples(R i ,Z i ,w i ) i∈{1,...,N} , (j i ,z i ) i∈{2,...,N} , (R ν i ,z ν i ) i∈{1,...,N}, ν∈N ,where R i ∈ {1, ∞}, Z i ⊆ C is a finite subset, w i = (A i ,u i ) ∈ ˜W C\Zi is an R i -vortex,j i ∈ {1,...,i − 1}, z i ∈ C, Ri ν > 0, and zν i ∈ C, such that the following conditionshold.(i) For every i = 2,...,N we have z i ∈ Z ji . Moreover, if i,i ′ ∈ {2,...,N} aresuch that i ≠ i ′ and j i = j i ′ then z i ≠ z i ′.(ii) Let i = 1,...,N. If R i = 1 then Z i = ∅ and E(w i ) > 0. If R i = ∞ andE ∞ (w i ) = 0 then |Z i | ≥ 2.

54 2. BUBBLING FOR VORTICES OVER THE PLANE(iii) Fix i = 1,...,N. If R i = 1 then Ri ν = 1 for every ν, and if R i = ∞ thenRi ν → ∞. Furthermore,(2.94)R ν iR ν j i→ 0,z ν i − zν j iR ν j i→ z i .In the following we set ϕ ν i (z) := Rν i z + zν i , for i = 0,...,N and ν ∈ N.(iv) For every i = 1,...,N there exist gauge transformations gi ν ∈ W 2,ploc (C \Z i,G)such that the following holds. If R i = 1 then (gi ν)∗ (ϕ ν i )∗ w ν converges to w iin C ∞ on every compact subset of C. Furthermore, if R i = ∞ then on everycompact subset of C \ Z i the sequence (gi ν)∗ (ϕ ν i )∗ A ν converges to A i in C 0 ,and the sequence (gi ν)∗ (ϕ ν i )∗ u ν converges to u i in C 1 .(v) Let i = 1,...,N, z ∈ Z i and ε 0 > 0 be such that B ε0 (z) ∩ Z i = {z}. Then forevery 0 < ε < ε 0 the limit(E z (ε) := limν→∞ ERν i (ϕνi ) ∗ w ν ,B ε (z) )exists, and E min ≤ E z (ε) < ∞. Furthermore, the functionis continuous.(vi) For every i = 1,...,N, we have(0,ε 0 ) ∋ ε ↦→ E z (ε) ∈ [E min , ∞)lim limsup E ( w ν ,B R νR→∞ ji/R(zj ν i+ Rj ν iz i ) \ B RR νi(zi ν ) ) = 0.ν→∞(vii) If l > N then for every j = 1,...,N we have(2.95) Z j = { z i |j < i ≤ N,j i = j } .To understand this claim, note that the collection (j i ) i∈{2,...,N} describes a treewith vertices the numbers 1,...,N and unordered edges { (i,j i ),(j i ,i) } . Attachedto the vertices of this tree are vortex classes and ∞-vortex classes. (The latter willgive rise to holomorphic spheres in M.) Each pair (Ri ν,zν i ) defines a rescaling ϕν i ,which is used to obtain the i-th limit vortex or ∞-vortex. (See condition (iv).)The point z i is the nodal point on the j i -th vertex, at which the i-th vertexis attached. The corresponding nodal point on the i-th vertex is ∞. The set Z iconsists of the nodal points except ∞ (if i ≥ 2) on the i-th vertex together withthe bubbling points that have not yet been resolved.Condition (i) implies that the nodal points at a given vertex are distinct. Condition(ii) guarantees that once all bubbling points have been resolved, the i-thcomponent will be stable. 31Condition (iii) implies that the rescalings ϕ ν i “zoom out” less than the rescalingsϕ ν j i. A consequence of condition (v) is that at every nodal or unresolved bubblingpoint at least the energy E min concentrates in the limit. Condition (vi) meansthat no energy is lost in the limit between each pair of adjacent bubbles. Finally,condition (vii) means that in the case l > N all bubbling points have been resolved.Proof of Claim 1. We show that the statement holds for l := 1. We checkthe conditions of Proposition 44 (Soft rescaling) with z 0 := 0, r := 1 and R ν , w νreplaced by R ν 0, w ν 0. Condition 44(a) follows from Proposition 79 in Appendix A.1,31 Note that in the case i ≥ 2 there is another nodal point at ∞, and for i = 1 there will bea marked point at ∞, which is the limit of the sequence z ν 0 .

2.7. PROOF OF THE BUBBLING RESULT 55using the hypothesis that M is equivariantly convex at ∞. Condition 44(b) followsfrom the factslimν→∞ ERν 0 (wν0 ,B ε ) = E, ∀ε > 0, E ≥ E min .The first condition is a consequence of the facts R ν 0 = νR ν , E(w ν ) → E, and (2.93).Thus by Proposition 44, there exist R 0 ∈ {1, ∞}, a finite subset Z ⊆ C, andan R 0 -vortex w 0 ∈ ˜W p C\Z 1, and passing to some subsequence, there exist sequencesε ν > 0, z ν , and g ν , such that the conclusions of Proposition 44 with R ν ,w ν replacedby R ν 0,w ν 0 hold. We defineN := N(1) := 1, R 1 := R 0 , Z 1 := Z, w 1 := w 0 , R ν 1 := ε ν R ν 0, z ν 1 := R ν 0z ν .We check conditions (i)-(vii) of Claim 1 with l = 1: Conditions (i,vii) arevoid. Furthermore, conditions (ii)-(vi) follow from 44(i)-(v). This proves the statementof the Claim for l = 1.Let l ∈ N and assume, by induction, that we have already proved the statementof Claim 1 for l. We show that it holds for l + 1. By assumption there exists anumber N := N(l) and there exist collections(R i ,Z i ,w i ) i∈{1,...,N} , (j i ,z i ) i∈{2,...,N} , (R ν i ,z ν i ) i∈{1,...,N}, ν∈N ,such that conditions (i)-(vii) hold. IfZ j = { z i |j < i ≤ N, j i = j } ,∀j = 1,...,Nthen conditions (i)-(vii) hold with N(l + 1) := N, and we are done. Hence assumethat there exists a j 0 ∈ {1,...,N} such that(2.96) Z j0 ≠ { z i |j 0 < i ≤ N, j i = j 0}.We set N(l + 1) := N + 1 and choose an element(2.97) z N+1 ∈ Z j0 \ { z i |j < i ≤ N, j i = j 0}.We fix a number r > 0 so small that B r (z N+1 ) ∩ Z j0 = {z N+1 }. We apply Proposition44 with z 0 := z N+1 and R ν , w ν replaced by R ν j 0, (ϕ ν j 0) ∗ w ν . Condition 44(a)holds by hypothesis. Furthermore, by condition (v) for l, condition 44(b) is satisfied.Hence passing to some subsequence, there exist R 0 ∈ {1, ∞}, a finite subsetZ ⊆ C, an R 0 -vortex w 0 ∈ ˜W p C\Z , and sequences ε ν > 0, z ν , such that the conclusionof Proposition 44 holds. We defineR N+1 := R 0 , Z N+1 := Z, w N+1 := w 0 , j N+1 := j 0 ,R ν N+1 := ε νR ν j 0, z ν N+1 := Rν j 0z ν + z ν j 0.We check conditions (i)-(vii) of Claim 1 with l replaced by l + 1, i.e., Nreplaced by N + 1. Condition (i) follows from the induction hypothesis and (2.97).Conditions (ii)-(vi) follow from 44(i)-(v).We show that (vii) holds with N replaced by N+1: By the induction hypothesis,it holds for N. Hence (2.96) implies that N + 1 ≥ l + 1. So there is nothing tocheck. This completes the induction and the proof of Claim 1.□Let l ∈ N be an integer and N := N(l), (R i ,Z i ,w i ), (j i ,z i ), (Ri ν,zν i ) be asin Claim 1. Recall that Z 0 = {0} and z0 ν := 0. We fix i = 0,...,N. We defineϕ ν i (z) := Rν i z + zν i , for every measurable subset X ⊆ C we denoteE i (X) := E Ri (w i ,X), E i := E i (C \ Z i ), E ν i (X) := E Rν i ((ϕνi ) ∗ w ν ,X).

56 2. BUBBLING FOR VORTICES OVER THE PLANEFurthermore, for z ∈ Z i we define(2.98) m i (z) := limε→0limν→∞ Eν i (B ε (z)).For i = 0 it follows from (2.93) and R ν 0 = νR ν that the limit m 0 (0) exists andequals E. For i = 1,...,N it follows from condition (v) that the limit (2.98) existsand that m i (z) ≥ E min . For j,k = 0,...,N we defineZ j,k := Z j \ {z i |j < i ≤ k, j i = j}(This is the set of points on the j-th sphere that have not been resolved after theconstruction of the k-th bubble.) We define the function(2.99) f : {1,...,N} → [0, ∞), f(i) := E i + ∑z∈Z i,Nm i (z).2. Claim.N∑f(i) = E.i=1Proof of Claim 2. We show by induction that(k∑(2.100)E i + ∑ )m i (z) = E,i=1 z∈Z i,kfor every k = 1,...,N. Claim 2 is a consequence of this with k = N. For the proofof equality (2.100) we need the following.3. Claim. For every i = 1,...,N we have(2.101) m ji (z i ) = E i + ∑ z∈Z im i (z).Proof of Claim 3. Let i = 1,...,N. We choose a number ε > 0 so smallthat¯B ε (z i ) ∩ Z ji = {z i }, Z i ⊆ B ε −1 −ε,and if z ≠ z ′ are points in Z i then |z − z ′ | > 2ε. By condition (v) of Claim 1, foreach z ∈ Z i the limit lim ν→∞ Ei ν(B ε(z)) exists. Lemma 42 implies that(limν→∞ Eν i (B ε −1) = E i B ε −1 \ ⋃ )B ε (z) + ∑ limν→∞ Eν i (B ε (z)).z∈Z i z∈Z iCombining this with condition (vi) of Claim 1, equality (2.101) follows from astraight-forward argument. This proves Claim 3.□Since Z 1,1 = Z 1 , equality (2.100) for k = 1 follows from Claim 3 and the factm 0 (0) = E. Let now k = 1,...,N − 1 and assume that we have proved (2.100) fork. An elementary argument using Claim 3 with i := k + 1 shows (2.100) with kreplaced by k + 1. By induction, Claim 2 follows.□Consider the tree relation E on T := {1,...,N} defined by iEi ′ iff i = j i ′ ori ′ = j i . Lemma 117 in Appendix A.7 with f as in (2.99), k := 1, α 1 := 1 ∈ T, andE 0 := E min , implies that(2.102) N ≤ 2EE min+ 1.

2.7. PROOF OF THE BUBBLING RESULT 57(Hypothesis (A.61) follows from conditions (ii,v) of Claim 1.) Assume now thatwe have chosen l > 2E/E min +1. By (2.102) we have l > N, and therefore bycondition (vii) of Claim 1, equality (2.95) holds, for every j = 1,...,N. We defineT := {1,...,N}, T 0 := ∅, T 1 := {i ∈ T |R i = 1}, T ∞ := T \ T 1 ,and the tree relation E on T byiEi ′ ⇐⇒ i = j i ′ or i ′ = j i .Furthermore, for i,i ′ ∈ T such that iEi ′ we define the nodal points{∞, if iz ii ′ :=′ = j i ,z i ′, if i = j i ′.Moreover, we define the marked point(α 0 ,z 0 ) := (1, ∞) ∈ T × S 2 .4. Claim. Let i ∈ T. If i ∈ T 1 then E(w i ) < ∞ and u i (C × G) has compactclosure. Furthermore, if i ∈ T ∞ then the mapextends to a smooth ¯J-holomorphic mapGu i : C \ Z i → M = µ −1 (0)/Gū i : S 2 ∼ = C ∪ {∞} → M.Proof of Claim 4. We choose gauge transformations g ν i ∈ W 2,ploc (C \ Z i,G)as in condition (iv) of Claim 1, and define w ν i := (gν i )∗ (ϕ ν i )∗ w ν .Assume that i ∈ T 1 . It follows from Fatou’s lemma thatE(w i ) ≤ liminfν→∞ E(wν i ) = E < ∞.Furthermore, since by hypothesis M is equivariantly convex at ∞, by Proposition79 in Appendix A.1 there exists a G-invariant compact subset K 0 ⊆ M such thatu ν i (C) ⊆ K 0, for every ν ∈ N. Since u ν i converges to u i pointwise, it follows thatu i (C) ⊆ K 0 . Hence w i has the required properties.Assume now that i ∈ T ∞ . By Proposition 116 in Appendix A.7 the mapGu i : C \ Z i → M = µ −1 (0)/Gis ¯J-holomorphic, and e Gui = e ∞ w i. It follows from Fatou’s lemma thatE ∞ (w i , C \ Z i ) ≤ liminfν→∞ ERν i (wνi ) = E < ∞.Therefore, by removal of singularities, it follows that Gu i extends to a smooth¯J-holomorphic map ū i : S 2 → M. 32 This proves Claim 4.□5. Claim. The tuple()(W,z) := T 0 ,T 1 ,T ∞ ,E,([w i ]) i∈T1 ,(ū i ) i∈T∞ ,(z ii ′) iEi ′,(α 0 := 1,z 0 := ∞)is a stable map in the sense of Definition 15, and the sequence ([w ν ],z ν 0 := ∞)converges to (W,z) in the sense of Definition 20. (Here [w i ] denotes the gaugeequivalence class of w i .)32 See e.g. [MS2, Theorem 4.1.2].

58 2. BUBBLING FOR VORTICES OVER THE PLANEProof of Claim 5. We check the conditions of Definition 15. Condition(i) is a consequence of the definitions of T 0 ,T 1 and T ∞ . Condition (ii) followsfrom condition (i) of Claim 1 and the fact Z i = ∅, for i ∈ T 1 . (This follows fromcondition (ii) of Claim 1.)Condition (iii) follows from an elementary argument using Claim 1(iii,iv,vi)and Proposition 45. Condition (iv) follows from Claim 1(ii). Hence all conditionsof Definition 15 are satisfied.We check the conditions of Definition 20. Condition (2.15) follows fromClaim 2, using condition (vii) of Claim 1. Condition 20(i) follows from a straightforwardargument, using Claim 1(iii).Condition 20(ii) follows from Claim 1(iii) by an elementary argument. Condition20(iii) follows from Claim 1(iv). Finally, condition 20(iv) is void, sincek = 0. This proves Claim 5.□Thus we have proved Theorem 3 in the case k = 0.We prove that the theorem holds for every k ≥ 1: We prove by inductionthat for every k ∈ N 0 and every tuple ( W ν ,z ν 1,...,z ν k)as in the hypotheses ofTheorem 3, there exists a stable map (W,z) as in (2.6) and a collection (ϕ ν α) ofMöbius transformations, such that the conditions of Definition 20 hold,(2.103) ∀α ∈ T ∞ , Q ⊆ C \ Z α compact: limν→∞ E( W ν ,ϕ ν α(Q) ) = E(ū α ,Q),(2.104)(2.105)ϕ ν α(∞) = ∞,lim R→∞ limsup ν→∞ E ( W ν , C \ ϕ ν α 0(B R ) ) = 0,and for every edge αEβ such that α ∈ T 1 ∪ T ∞ and β lies in the chain of verticesfrom α to α 0 , we have(2.106) lim limsup E ( W ν ,ϕ ν β(B R −1(z βα )) \ ϕ ν α(B R ) ) = 0.R→∞ν→∞For k = 0 we proved this above. In that construction condition (2.103) follows fromstatement (iii) of Proposition 44. Furthermore, condition (2.105) is a consequence ofcondition (vi) of Claim 1 with i = 1, and the facts E(W ν ,B Rν ) → E and R ν 0 = νR ν .Finally, condition (2.106) follows from Claim 1(vi).Let now k ∈ N and assume that we have proved the statement for k−1. Passingto some subsequence, we may assume that for every i = 1,...,k − 1, the limit(2.107) z ki := limν→∞ (zν k − z ν i ) ∈ C ∪ {∞}exists. We setz k0 := ∞.Passing to a further subsequence, we may assume that the limitz αk := limν→∞ (ϕν α) −1 (z ν k) ∈ S 2exists, for every α ∈ T := T 0∐T1∐T∞ . There are three cases.Case (I) There exists a vertex α ∈ T, such that z αk is not a special point of (W,z)at α.Case (II) There exists an index i ∈ {0,...,k − 1} such that z αik = z i .

2.7. PROOF OF THE BUBBLING RESULT 59Case (III) There exists an edge αEβ such that z αk = z αβ and z βk = z βα .These three cases exclude each other. For the combination of the cases (II)and (III) this follows from the last part of condition (ii) (distinctness of the specialpoints) in Definition 15.6. Claim. One of the three cases always applies.Proof of Claim 6. This follows from an elementary argument, using that Tis finite and does not contain cycles.□Assume that Case (I) holds. We fix a vertex α ∈ T such that z αk is not aspecial point. 33 We define α k := α and introduce a new marked pointz k := z αk kon the α k -sphere. Then (W,z) augmented by (α k ,z k ) is again a stable map andthe sequence (W ν ,z ν 0,...,z ν k ) converges to this new stable map via (ϕν α) α∈T . Theinduction step in Case (I) follows.Assume that Case (II) holds. We fix an index 0 ≤ i ≤ k − 1 such thatz αik = z i . (It is unique.)Consider first Case (IIa): i ≠ 0 and the condition z ki ≠ 0 or α i ∈ T 0 ∪ T 1holds. In this case we extend the tree T as follows. We define⎧⎨ 0, if z ki = 0,j := 1, if z ki ≠ 0, ∞,⎩∞, if z ki = ∞,and introduce an additional vertex γ, which has type j and carries the new markedpoint. This vertex is adjacent to α i in the new tree. We move the i-th markedpoint from the vertex α i to the vertex γ and introduce an additional marked pointon γ. More precisely, we defineTj new ∐:= T j {γ}, Tnewj := T ′ j ′, j′ ∈ {0,1, ∞} \ {j},T new := T ∐ {γ}, E new := E ∐ { (α i ,γ),(γ,α i ) } ,α newi:= αk new := γ, zγα newi:= ∞, zα newiγ := z i ,{zi new := 0, zk new zki , if j = 1,:=1, if j = 0, ∞.If j = 1, i.e., γ ∈ T new1 , then we introduce the following “ghost vortex”: We choosea point x 0 in the orbit ū αi (z i ) ⊆ µ −1 (0), and define A γ := 0 ∈ Ω 1 (C,g) andu γ : C → M to be the map which is constantly equal to x 0 . We identify A γ with aconnection on C × G and u γ with a G-equivariant map C × G → M, and set(2.108) W γ := [ C × G,A γ ,u γ].If j = ∞, i.e., γ ∈ T∞new , then we define ū γ : S 2 → M to be the constant mapū γ ≡ ū αi (z i ).We denote by (W new ,z new ) the tuple obtained from (W,z) by making thechanges described above. By elementary arguments this is again a stable map.33 This vertex is unique, but we will not use this.

60 2. BUBBLING FOR VORTICES OVER THE PLANEWe define the sequence of Möbius transformations ϕ ν γ : S 2 → S 2 by{(2.109) ϕ ν z + zνγ(z) := i , if j = 1,(zk ν − zν i )z + zν i , if j = 0, ∞.7. Claim. There exists a subsequence of ( W ν ,z ν 0,...,z ν k)that converges to(W new ,z new ) via the Möbius transformations (ϕ ν α) α∈T new , ν∈N.Proof of Claim 7. Condition (2.15) (energy conservation) holds for everysubsequence, since the new component γ carries no energy. Conditions (i,ii)of Definition 20 hold (for the new collection of Möbius transformations), byelementary arguments.T new∞We check condition 20(iii) up to some subsequence. For every α ∈ T newwe write(2.110) W ν α := (ϕ ν α) ∗ W ν , ū ν α := ū W να: C → M/G,1 ∪where ū W ναis defined as in (2.2). In the case j = 0 condition 20(iii) does notcontain any new requirement.Assume that j = 1. It follows from Proposition 37 (Compactness modulobubbling and gauge) with R ν := 1, r ν := ν and W ν replaced by Wγ ν , that thereexists a vortex class ˜W γ over C such that passing to some subsequence, the sequenceW ν γ converges to ˜W γ , with respect to τ C (as in Definition 19), and the sequence ū ν γconverges to ū fWγ , uniformly on every compact subset of C.8. Claim. We have ˜W γ = W γ (defined as in (2.108)).Proof of Claim 8. A straight-forward argument implies that α i ∈ T new∞ .Therefore, condition 20(iii) for the sequence ( W ν ,z ν 0,...,z ν k−1)implies that thesequence ū ν α iconverges to ū αi , in C 1 on some neighborhood of z i . (Here we usedthat i ≠ 0.)Since ϕ ν α iγ := (ϕ ν α i) −1 ◦ϕ ν γ converges to z αiγ = z i , uniformly on every compactsubset of S 2 \ {z newγα i} = C, it follows thatū ν γ = ū ν α i◦ ϕ ν α iγ → ¯x 0 := ū αi (z i ),uniformly on every compact subset of C. It follows that ū fWγ ≡ ¯x 0 . We chooserepresentatives(Ãγ,ũ γ ) ∈ Ω 1 (C,G) × C ∞ (C,G)of ˜W γ and x 0 ∈ µ −1 (0) of ¯x 0 . By hypothesis (H) the action of G on µ −1 (0) isfree. Hence, after regauging, we may assume that ũ γ ≡ x 0 . (Here we use Lemma114 (Appendix A.7), which ensures that the gauge transformation is smooth.) Itfollows from the first vortex equation that Ãγ = 0. This shows that ˜W γ = W γ andhence proves Claim 8.□This proves condition 20(iii) in the case j = 1.Assume now that j = ∞. We have again α i ∈ T∞new . Hence condition 20(iii)for the sequence ( W ν ,z0,...,zk−1) ν ν implies that the sequence ūναiconverges to ū αi ,in C 1 on some neighborhood of z i . (Here we used that i ≠ 0.) Let Q ⊆ C = S 2 \Z γbe a compact subset. Since ϕ ν α iγ converges to z i , in C ∞ on Q, it follows thatū ν γ = ū ν α i◦ ϕ ν α iγ converges to ū γ ≡ ū αi (z i ) in C 1 on Q, as required. This provescondition 20(iii) in all cases.

2.7. PROOF OF THE BUBBLING RESULT 61Condition 20(iv) is a consequence of the definition (2.109) of ϕ ν γ. This provesClaim 7.□To see that condition (2.103) holds, observe that we need to prove this onlyin the case j = ∞ and then only for α = γ. In this case it follows from the samecondition for α = α i and 20(ii) with α = α i and β = γ. (Here we used that z i ≠ ∞and it does not coincide with any nodal point at α i .) The same conditions alsoimply (2.106) for α := γ and β := α i . (For the other pairs of adjacent vertices(2.106) follows from the induction hypothesis.)The induction step in Case (IIa) follows.Consider the Case (IIb): i = 0. We defineT∞ new ∐:= T ∞ {γ}, Tnewj := T j , j = 0,1, E new := E ∐ { (α 0 ,γ),(γ,α 0 ) } ,z newα0 new := αk new := γ, zγα new0:= 0, α 0γ := z 0 = ∞,ū γ ≡ ū α0 (z 0 ).z new0 := ∞, z newk:= 1,The tuple (W new ,z new ) obtained from (W,z) by making these changes, is again astable map. It follows from our assumption (2.104) that there exist λ ν ∈ C \ {0}and z ν ∈ C, such that ϕ ν α 0(z) = λ ν z + z ν . We define the sequence of Möbiustransformations ϕ ν γ byϕ ν γ(z) := (z ν k − z ν )z + z ν .We show that some subsequence of ( W ν ,z ν 0,...,z ν k)converges to (W new ,z new )via the Möbius transformations (ϕ ν α) α∈T new , ν∈N: Condition (2.15) holds with Treplaced by T new , since the map ū γ is constant. Conditions 20(i,ii,iv) follow fromstraight-forward arguments.We show that condition 20(iii) holds. Recall the definition (2.110) of ū ν α. Anelementary argument using condition 20(iii) with α := α 0 , our assumption (2.105),and Proposition 45, shows that for every ε > 0 there exist numbers R ≥ r 0 andν 0 ∈ N such that¯d ( ev z0=∞(W α0 ),ū ν α 0(z) ) < ε, ∀ν ≥ ν 0 , z ∈ C \ B R ,where W α0 := ū α0 if α 0 ∈ T ∞ , and ev ∞ is defined as in (2.4,2.5). The sequence ϕ ν α 0γconverges to zα new0γ = z 0 = ∞, uniformly on every compact subset of S 2 \{zγα new0= 0}.It follows that ū ν γ converges to the constant map ū γ ≡ ev ∞ (W α0 ), uniformly onevery compact subset of C \ {0} = S 2 \ {zγα new0,z0 new }. Condition 20(iii) with α = γis a consequence of the following.9. Claim. Passing to some subsequence the convergence is in C 1 on everycompact subset of C \ {0}.Proof of Claim 9. We denote R ν := |zk ν − z ν|, and choose a representativewγ ν of Wγ ν . This is an R ν -vortex. Our assumption z α0k = z 0 = ∞ implies thatR ν → R 0 := ∞. Hence by Proposition 37 there exists a finite subset Z ⊆ C andan ∞-vortex w γ := (A γ ,u γ ), and passing to some subsequence, there exist gauge(C \ Z,G), such that the assertions 37(ii,iii) hold withw ν := wγ. ν It follows that ū ν γ converges in C 1 on every compact subset of C \ Z.Furthermore, it follows from (2.105) and 37(iii) that Z ⊆ {0}. Claim 9 follows. □transformations g ν γ ∈ W 2,plocConditions (2.103,2.105) (for α new0 = γ) and (2.106) (for α := α 0 and β := γ)follow from condition (2.105) for α 0 , which holds by the induction hypothesis. This

62 2. BUBBLING FOR VORTICES OVER THE PLANEproves the induction step in Case (IIb).Consider now Case (IIc): z ki = 0 and α i ∈ T ∞ . Then we introduce a vertexγ ∈ T 1 adjacent to α i and a vertex δ ∈ T 0 adjacent to γ, such that δ carries thetwo new marked points. More precisely, we defineT new0 := T 0∐ {δ}, Tnew1 := T 1∐ {γ}, Tnew∞ := T ∞ ,T new := T ∐ {γ,δ}, E new := E ∐ { (α i ,γ),(γ,α i ),(γ,δ),(δ,γ) } ,zγα newi:= ∞, zα newiγ := z i , z γδ := 0, z δγ := ∞,α newi:= α newk:= δ, z newi := 0, z newk:= 1,and W γ as in case (IIa) with j = 1. We thus obtain a new stable map. The proofof convergence to this stable map and of conditions (2.103) (for γ) and (2.106) (forα = γ and β = α i ) is now analogous to the proof in Case (IIa). This proves theinduction step in Case (IIc).The Cases (IIa,b,c) cover all instances of Case (II), hence we have proved theinduction step in Case (II).Assume now that Case (III) holds. In this case we introduce a new vertexγ between α and β. Hence α and β are no longer adjacent, but are separated byγ. We define j := 0 if α or β lies in T 0 , and j := ∞, otherwise. 34 We may assumew.l.o.g. that β lies in the chain of vertices from α to α 0 . We definez newαγTj new ∐:= T j {γ}, Tnewj := T ′ j ′, j′ ∈ {0,1, ∞} \ {j},:= z αβ , z newβγ:= z βα , z newγα:= 0, z newγβ:= ∞, αnewk:= γ, zk new := 1.In the case j = ∞ we define ū γ : S 2 → M to be the constant map equal to ū β (z βα ).Again, we obtain a new stable map. By condition (2.104) there exist λ ν α ∈ C \ {0}and z ν α ∈ C, such that ϕ ν α(z) = λ ν αz + z ν α. We defineϕ ν γ(z) := (z ν k − z ν α)z + z ν α.The proof of convergence proceeds now analogously to the proof in Case (IIb),using (2.106) for the proof of condition 20(iii) rather than (2.105). Furthermore,conditions (2.103) with α = γ and (2.106) with (α,β) replaced by (α,γ) and (γ,β)follow from condition (2.106) for (α,β), which holds by the induction hypothesis.This proves the induction step in Case (III), and hence terminates the proof ofTheorem 3 in the case k ≥ 1.□Remark. In the above proof for k = 0 the stable map (W,z) is constructed by“terminating induction”. Intuitively, this is induction over the integer N occurringin Claim 1. The “auxiliary index” l in Claim 1 is needed to make this idea precise.Condition (vii) and the inequality (2.102) ensure that the “induction stops”. ✷2.8. Proof of the result in Section 2.3 characterizing convergenceUsing Compactness modulo bubbling and gauge for (rescaled) vortices, whichwas established in Section 2.5 (Proposition 37), in this section we prove Proposition27. The proof is based on the following result. Let G := S 1 ⊆ C act on M := C by34 In that case α or β lies in T∞.

2.8. PROOF OF THE RESULT IN SECTION 2.3 CHARACTERIZING CONVERGENCE 63multiplication, with momentum map µ : C → iR given by µ(z) := i 2 (1 − |z|2 ), andlet d ∈ N 0 . Recall the definition (2.24) of the mapι d : ∐Sym d′ (C) → Sym d (S 2 ),d ′ ≤dthe definitions (2.23,2.17) of the set M d of finite energy vortex classes of degree dand of the local degree map deg W : C → N 0 , and the Definition 19 of the “C ∞ -topology τ Σ on B Σ ”.48. Proposition. Let 0 ≤ d ′ ≤ d, W ∈ M d ′ and W ν ∈ M d , for ν ∈ N. Thenthe following conditions are equivalent.(i) For every open subset Ω ⊆ C with compact closure and smooth boundarythe restriction W ν | Ωconverges to W | Ωto Ω, as ν → ∞, with respect to thetopology τ Ω.(ii) The point in the symmetric productconverges to ι d (deg W ) ∈ Sym d (S 2 ).deg Wν∈ Sym d (C) ⊆ Sym d (S 2 )In the proof of this result we will use the following. Let X be a topologicalspace and d ∈ N 0 . We denote by Sym d (X) := X d /S d the d-fold symmetric productof X, and canonically identify it with the set ˜Sym d (X) of all maps m : X → N 0such that ∑ x∈X m(x) = d.35 We endow Sym d (X) with the quotient topology. Forevery subset Y ⊆ X and d 0 ∈ N 0 , we define}(2.111) V d0Y{m := ∈ ˜Sym d ∑(X) ∣ m(x) = d 0 , m(x) = 0, ∀x ∈ ∂Y ,x∈Ywhere ∂Y ⊆ X denotes the boundary of Y .49. Lemma. Let U be a basis for the topology of X. If X is Hausdorff then thesets V d0U , U ∈ U, d 0 ∈ N 0 , form a subbasis for the topology of ˜Sym d (X) ∼ = Sym d (X).Proof of Lemma 49. We denote by π : X d → Sym d (X) the canonical projection.For every U ∈ U and d 0 ∈ N 0 , we haveπ −1 (V d0U ) = S d · (U d0 × (X \ U) d−d0) ⊆ X d ,where · denotes the action of S d on X d . Since |S d | < ∞, this set is open, i.e., V d0Uis open. Now let V ⊆ ˜Sym d (X) be an open set, and m ∈ V . It suffices to show thatthere exists a finite set S ⊆ U ×N 0 , such that the intersection ⋂ (U,d V d00)∈S Ucontainsm and is contained in V . To see this, we denote by x 1 ,...,x k ∈ X the distinctpoints at which m does not vanish, and abbreviate d i := m(x i ). For i = 1,...,kwe choose a neighborhood U i ∈ U of x i , such that the sets U 1 ,...,U k are disjoint,and(2.112) U 1 × · · · × U 1 × · · · × U k × · · · × U k ⊆ π −1 (V ).Here the factor U i occurs d i times. (The sets exist by Hausdorffness of X, opennessof π −1 (V ), the definition of the product topology on X d , and the fact(x1 ,...,x 1 ,...,x k ,...,x k)∈ π −1 (m) ⊆ π −1 (V ).)35 Implicitly, here we require m(x) to vanish, except for finitely many points x ∈ X.

64 2. BUBBLING FOR VORTICES OVER THE PLANEWe define S := { (U i ,d i ) ∣ ∣ i = 1,...,k } . We havek⋂i=1)V diU i= π(U 1 × · · · × U 1 × · · · × U k × · · · × U k .Combining this with (2.112), it follows that ⋂ ki=1 V diU i⊆ V . On the other hand,since x i ∈ U i , we have m ∈ ⋂ ki=1 V diU i. Hence the set S has the required properties.This proves Lemma 49.□We will also use the following.50. Lemma. Let x ∈ C ( S 1 , C \ {0} ) be a loop. Then there exists a C 0 -neighborhood U ⊆ C ( S 1 , C \ {0} ) of x, such that every x ′ ∈ U is homotopic tox.Proof of Lemma 50. We define U to be the open C 0 -ball around x, withradius min z∈S 1 |x(z)|. Let x ′ ∈ U. We defineh : [0,1] × S 1 → C,h(t,z) := tx ′ (z) + (1 − t)x(z).This map does not vanish anywhere, and therefore is a homotopy between x andx ′ inside C \ {0}. This proves Lemma 50.□Proof of Proposition 48. We canonically identify the set B = B C of gaugeequivalence classes of smooth triples (P,A,u) (as defined in (1.10)) with the set ofgauge equivalence classes of smooth pairs(A,u) ∈ ˜W C := Ω 1 (C,g) × C ∞ (C,M).We show that (i) implies (ii). Assume that (i) holds. We denote by m : S 2 =C ∪ {∞} → N 0 the map given by deg W on C, and m(∞) := d −deg(W). We defineU := { B r (z) ∣ ∣ z ∈ C, r > 0 } ∪ { (C \ B r ) ∪ {∞} ∣ ∣ r ∈ [1, ∞) } .This is a basis for the topology of S 2 .Claim. Let U ⊆ U be such that deg W vanishes on ∂U. Then there existsν 0 ∈ N such that for every ν ≥ ν 0 we have deg Wν∈ V d0U , whered 0 := ∑ z∈Um(z).Proof of the claim. Let U be as in the hypothesis of the claim. Considerfirst the case U = B r (z), for some z ∈ C, r > 0. We choose a (smooth) representativew = (A,u) ∈ ˜W of W | Br(z) B . Since by assumption, deg r(z) W vanishes on∂U = Sr(z), 1 u is nonzero on Sr(z).1The C ∞ -topology on the set of smooth connections on B r (z) is second-countable.Furthermore, the action of the gauge group on this space is continuous. Therefore,by assumption (i) and Lemma 120 in Appendix A.7, for each ν ∈ N, there exists arepresentative w ν = (A ν ,u ν ) ∈ ˜W of W Br(z) ν, such that w ν converges to w in C ∞ ,as ν → ∞. Hence, using Lemma 50, there exists ν 0 ∈ N such that for ν ≥ ν 0 therestriction of u ν to Sr(z) 1 does not vanish anywhere, and is homotopic to u| S 1 r (z),as a map with target C \ {0}. It follows thatdeg(uν|u ν | : S1 r(z) → S 1 )= deg( u|u| : S1 r(z) → S 1 ),

2.8. PROOF OF THE RESULT IN SECTION 2.3 CHARACTERIZING CONVERGENCE 65for ν ≥ ν 0 . By elementary arguments, the left hand side of this equality agreeswith ∑ z ′ ∈B deg r(z) u ν(z ′ ), and the right hand side equals ∑ z ′ ∈B deg r(z) u(z ′ ). Sinceby definition, deg uν(z ′ ) = deg Wν(z ′ ) and deg u (z ′ ) = deg W (z ′ ), it follows that∑deg Wν(z ′ ) =∑deg W (z ′ ) = d 0 ,z ′ ∈B r(z)z ′ ∈B r(z)and therefore, deg Wν∈ V d0U . This proves the claim in the case U = B r(z) ⊆ C.Consider now the case U = (C \ B r ) ∪ {∞}, for some r ≥ 1. By what we justproved, there exists an index ν 0 , such that for every ν ≥ ν 0∑deg Wν(z ′ ) = ∑deg W (z ′ ),z ′ ∈B r z ′ ∈B rIt follows that∑z ′ ∈C\B rdeg Wν(z ′ ) =∑z ′ ∈(C\B r)∪{∞}m(z ′ ) = d 0 ,and therefore, deg Wν∈ V d0U , for every ν ≥ ν 0. This proves the claim in the caseU = (C \ B r ) ∪ {∞}.□The claim implies that if d 0 ∈ N and U ∈ U are such that m ∈ V d0Uthen thereexists ν 0 ∈ N such that for every ν ≥ ν 0 we have deg Wν∈ V d0U. Combining this withLemma 49 (with X := S 2 ), condition (ii) follows. This proves that (i) implies(ii).We show the opposite implication: Assume that (ii) is satisfied. We provethat given an open subset Ω ⊆ C with compact closure and smooth boundary, theclass W ν | Ωconverges to W | Ω, as ν → ∞, with respect to the topology τ Ω. ByLemma 119 (Appendix A.7) it suffices to show that for every such Ω and everysubsequence (ν i ) i∈N there exists a further subsequence (i j ) j∈N such that W νij | Ωconverges to W | Ω, as j → ∞. Let (ν i ) be a subsequence. Let i ∈ N. We choose arepresentative (A i ,u i ) ∈ ˜W C of W νi . It follows from Proposition 23 that the imageof u i is contained in the closed ball B 1 ⊆ C. 36 Furthermore, by Proposition 24,we have E(w i ) = dπ. Therefore, Proposition 37 (Compactness modulo bubblingand gauge) implies that there exist a smooth vortex w 0 := (A 0 ,u 0 ) over C, asubsequence (i j ) j∈N , and gauge transformations g j ∈ W 2,ploc(C,G) (for j ∈ N), suchthat the pair w j ′ := g∗ j (A i j,u ij ) is smooth and converges to w 0 , as j → ∞, in C ∞on every compact subset of C. We denote by W 0 the equivalence class of w 0 . Itfollows from Lemma 114 (Appendix A.7) that g j is smooth, and hence w j ′ representsW j := W νij , for every j. It follows that W j | Ωconverges to W 0 | Ωwith respect toτ Ω, as j → ∞, for every open subset Ω ⊆ C with compact closure and smoothboundary.It remains to show that W 0 = W. The energy densities e W j converge to e W0in C ∞ on compact subsets of C. Therefore, we haveE(W 0 ) ≤ liminf E(W j ) = dπ.j→∞By Proposition 24 we have E(W 0 ) = π deg(W 0 ), and therefore, d 0 := deg(W 0 ) ≤d. Hence ι d (deg W0) ∈ Sym d (S 2 ) is well-defined. It follows from the implication36 This is also true if E(Wνi ) = 0, since then the image of u i equals S 1 .

66 2. BUBBLING FOR VORTICES OVER THE PLANE(i)=⇒(ii) (which we proved above), that the pointdeg W j ∈ Sym d (C) ⊆ Sym d (S 2 )converges to ι d (deg W0), as j → ∞. By our assumption (ii), deg W j also convergesto ι d (deg W ). It follows that ι d (deg W0) = ι d (deg W ), hence deg W0= deg W , andtherefore, W 0 = W. (Here in the last step we used that the map (2.23) is injective.)This proves that (ii) implies (i) and concludes the proof of Proposition 48. □For the proof of Proposition 27 we also need the following lemma.51. Lemma. Let k be a positive integer, ϕ ν 0,...,ϕ ν kbe sequences of Möbiustransformations, and letz 0 ,...,z k−1 ,w 1 ,...,w k ∈ S 2be points. Assume that z 1 ≠ w 1 ,...,z k−1 ≠ w k−1 , and that(ϕ ν i ) −1 ◦ ϕ ν i+1 → z i ,uniformly on compact subsets of S 2 \{w i+1 }, for i = 0,...,k−1. Let Q ⊆ S 2 \{z 0 },Q ′ ⊆ S 2 \ {w k } be compact subsets. Then for ν large enough we have(2.113) ϕ ν 0(Q) ∩ ϕ ν k(Q ′ ) = ∅.Proof of Lemma 51. This follows from an elementary argument.Proof of Proposition 27. Assume that (i) holds. Then Proposition 24and (2.15) imply that equality (2.25) holds for ν large enough, as claimed. Thesecond part of condition (ii) follows from Proposition 48.Suppose now on the contrary that condition (ii) holds. Then Proposition24 and equality (2.25) imply (2.15). Let ϕ ν α (for α ∈ T) be as in condition (ii).We show 20(iii): The first part of this condition (concerning α ∈ T 1 ) follows fromProposition 48. We prove the second part of the condition: Let α ∈ T ∞ , andQ ⊆ S 2 \ (Z α ∪ {z α,0 }) be a compact subset. We denote by ¯x 0 the orbit S 1 ⊆ C.1. Claim. For every subsequence (ν i ) i∈N there exists a further subsequence(i j ) j∈N , such thatu νi j ◦ ϕν ijα (Q) ⊆ M ∗ = C \ {0}, ∀j ∈ N,Gu νi j ◦ ϕν ijα → ¯x 0 in C 1 on Q, as j → ∞.Proof of Claim 1. We fix a subsequence (ν i ) i∈N . We choose a Möbius transformationψ such that ψ(∞) = z α,0 . By condition 20(i) we haveϕ ν α ◦ ψ(∞) = ϕ ν α(z α,0 ) = ∞,and hence ϕ ν α ◦ ψ restricts to an automorphism of C. We defineW ν α := (ϕ ν α ◦ ψ) ∗ W ν ,R ν α := ∣ ∣ ϕνα ◦ ψ(1) − ϕ ν α ◦ ψ(0) ∣ ∣ ∈ (0, ∞).Then Wα ν is an Rα-vortex ν class over C. We choose a representative wα ν of Wα. ν Wecheck the hypotheses of Proposition 37 with the sequences R i := Rα νi , w i := wα νi ,and R 0 := ∞. By condition 20(i), Rα νi converges to ∞, as i → ∞. Furthermore,by Proposition 23 the image of u νiα is contained in the compact set K := B 1 ⊆ C.Finally, Proposition 24 implies that the energies of the vortices wανi (i ∈ N) areuniformly bounded. Hence the hypotheses of Proposition 37 are satisfied. Applyingthis result, there exist a finite subset Z ⊆ C, an ∞-vortex w 0 := (A 0 ,u 0 ) ∈ ˜W C\Z ,□

2.8. PROOF OF THE RESULT IN SECTION 2.3 CHARACTERIZING CONVERGENCE 67a subsequence (i j ), and gauge transformations g j ∈ W 2,p ( )loc C \ Z,S1, such thatassertions of the proposition hold, with the sequence (w ν ) replaced by (w νi jα ). Bycondition 37(ii) the map g −1 ( )j uνij ◦ ϕ νi jα converges to u0 ◦ ψ −1 in C 1 on everycompact subset of ψ(C \ Z) = S 2 \ ψ ( Z ∪ {∞} ) . Therefore, using the equalityψ(∞) = z α,0 , the statement of Claim 1 is a consequence of the following:2. Claim. The set ψ(Z) is contained in Z α .Proof of Claim 2. We choose a number R > 0 so large that∑(2.114)E ( ) πW β , C \ B R E(W β ,B R ) −π4|T 1 | .Let β ≠ γ ∈ T be vertices. We denote by ( β,β 1 ,...,β k−1 ,γ ) the chain of verticesconnecting β with γ, and write β 0 := β, β k := γ. By conditions 15(ii) and 20(ii)the hypothesis of Lemma 51 withϕ ν i := ϕ ν β i, z i := z βiβ i+1, w i := z βiβ i−1are satisfied. It follows that for every compact subset Q ⊆ S 2 \Z β and Q ′ ⊆ S 2 \Z γ ,for ν large enough, we have(2.116) ϕ ν β(Q) ∩ ϕ ν γ(Q ′ ) = ∅.Applying this several times with β,γ ∈ T 1 and Q := Q ′ := ¯B R , it follows that forν large enough the sets ϕ ν β (B R), β ∈ T 1 , are disjoint. Increasing j 0 we may assumew.l.o.g. that this holds for ν ≥ ν ij0 . Therefore, for j ≥ j 0 , we have()⋃E W νij , ϕ νi jβ (B R)β∈T 1= ∑ β∈T 1E ( W νij ,ϕ νi jβ (B R) )>( ∑β∈T 1E(W β ,B R ))− π 4(2.117)> ∑ β∈T 1E(W β , C) − π 2 .Here in the second line we used (2.115) and in the third line we used (2.114). Ourassumption (2.25) and Proposition 24 imply that∑β∈T 1E(W β , C) = E(W ν , C),

68 2. BUBBLING FOR VORTICES OVER THE PLANEfor ν large enough. Hence (2.117) implies that((2.118) E W νij , C \ ⋃ )β (B R) < π 2 ,β∈T 1ϕ νi jfor j large enough.Let now z ∈ S 2 \ (Z α ∪ {z α,0 }). We show that z does not belong to ψ(Z).We choose a number ε > 0 so small that ¯B ε (ψ −1 (z)) ⊆ C \ ψ −1 (Z α ). We defineQ := ψ ( ¯Bε (ψ −1 (z)) ) . By (2.116), we haveϕ ν α(Q) ∩ ϕ ν β( ¯B R ) = ∅,for ν large enough. Therefore, by (2.118) implies thatE ( W νij ,ϕ νi jα (Q) ) < π 2 ,for j large enough. On the other hand, if z belonged to ψ(Z), then by condition37(iii) we would havelim E( W νij ,ϕ νi jα (Q) ) = lim E( W νi jα , ¯B ε (ψ −1 (z)) ) ≥ E min = π.j→∞ j→∞This contradiction proves that z ∉ ψ(Z). It follows that ψ(Z) ⊆ Z α ∪ {z α,0 }. Sinceψ(∞) = z α,0 , Claim 2 follows. This concludes the proof of Claim 1. □Claim 1 and an elementary argument imply that u ν ◦ ϕ ν α(Q) ⊆ M ∗ for ν largeenough. Furthermore, it follows from the same claim and Lemma 119 in AppendixA.7 that Gu ν ◦ ϕ ν α → ¯x 0 in C 1 on Q, as ν → ∞. This proves the second part of20(iii). Hence all conditions of Definition 20 are satisfied, and therefore, condition27(i) holds. This concludes the proof of Proposition 27.□

CHAPTER 3Fredholm theory for vortices over the planeIn this chapter the equivariant homology class [W] of an equivalence class W oftriples (P,A,u) is defined, and the equivariant Chern number of [W] is interpretedas a certain Maslov index. This number entered the index formula (1.27). Furthermore,the second main result of this memoir, Theorem 4, is proven. It statesthe Fredholm property for the vertical differential of the vortex equations over theplane C, viewed as equations for equivalence classes W of triples (P,A,u). LetM,ω,G,g, 〈·, ·〉 g ,µ and J be as in Chapter 1 1 , and (Σ,j) := C, equipped with thestandard area form ω C = ω 0 .3.1. Equivariant homology, the Chern number, and the Maslov indexWe fix numbersp > 2, λ > 1 − 2 p .Then every equivalence class W ∈ B p λ of triples (P,A,u) ∈ ˜B p λ(defined as in (1.16))carries an equivariant homology class [W] ∈ H2 G (M, Z), whose definition is basedon the following lemma. Let P be a topological (principal) G-bundle over C 2 ,and u : P → M a continuous equivariant map. By an extension of the pair(P,u) to S 2 we mean a triple ( ˜P,ι,ũ), where ˜P is a topological G-bundle over S 2 ,ι : P → ˜P a morphism of topological G-bundles which descends to the inclusionC → C ∪ {∞} = S 2 , and ũ : ˜P → M a continuous G-equivariant map satisfyingũ ◦ ι = u.52. Lemma. The following statements hold.(i) For every triple (P,A,u) ∈ ˜B p λ there exists an extension ( ˜P,ι,ũ) of the pair(P,u), such that G acts freely on ũ( ˜P ∞ ) ⊆ M, where ˜P ∞ denotes the fiber of˜P over ∞.(ii) Let P be a topological G-bundle over C, u : P → M a continuous G-equivariant map, and ( ˜P,ι,ũ) an extension of (P,u), such that G acts freelyon ũ( ˜P ∞ ). Furthermore, let ( P ′ ,u ′ , ˜P ′ ,ι ′ ,ũ ′) be another such tuple, andΦ : P ′ → P an isomorphism of topological G-bundles, such that u ′ = u ◦ Φ.Then there exists a unique isomorphism of topological G-bundles ˜Φ : ˜P ′ → ˜P,satisfying(3.1) ˜Φ ◦ ι ′ = ι ◦ Φ, ũ ′ = ũ ◦ ˜Φ.The proof of this lemma is postponed to the appendix (page 99). For thedefinition of [W] we also need the following.1 As always, we assume that hypothesis (H) (see Chapter 1) is satisfied.2 Such a bundle is trivializable, but we do not fix a trivialization here.69

70 3. FREDHOLM THEORY FOR VORTICES OVER THE PLANE53. Remark. Let G be a topological group, X a closed 3 oriented topological(Hausdorff) manifold of dimension k, P → X a topological G-bundle, Y atopological space, equipped with a continuous action of G, and u : P → Y a G-equivariant map. Then u carries an equivariant homology class [u] G ∈ Hk G (Y, Z),defined as follows. We choose a universal G-bundle EG → BG and a continuousG-equivariant map θ : P → EG. 4 The map (u,θ) : P → Y ×EG descends to a mapu G : X → (Y × EG)/G. We define[u] G ∈ Hk G ( )(Y, Z) = H k (Y × EG)/G, Zto be the push-forward of the fundamental class of X, under the map u G . Thisclass does not depend on the choice of θ, nor on EG in the following sense. IfEG ′ → BG ′ is another universal G-bundle, then the corresponding class [u] ′ G ∈H k((Y × EG ′ )/G, Z ) is mapped to [u] G under the canonical isomorphism 5H k((Y × EG ′ )/G, Z ) → H k((Y × EG)/G, Z).✷54. Definition (Equivariant homology class). We define the equivariant homologyclass [W] ∈ H2 G (M, Z) of an element( )⋃W ∈/∼ pp>2, λ>1− 2 pto be the equivariant homology class of ũ : ˜P → M, where ( ˜P,ι,ũ) is an extensionas in Lemma 52, corresponding to any representative (P,A,u) of W. Here theequivalence relation ∼ p is defined as in Section 1.4.It follows from Lemma 52 that this class does not depend on the choices of( ˜P,ι,ũ) and w.55. Remark. The condition λ > 1 − 2/p is needed for [W] to be well-defined,for every W ∈ B p λ . Consider for example the case M := R2 ,ω := ω 0 ,J := i andG := {1}. Let p > 2. We choose a smooth map u : C × {1} ∼ = C → R 2 such thatu(z) =(sin(√log |z|),0˜B p λ), ∀z ∈ C \ B 1 .Then the equivalence class W of ( P := C×{1},0,u ) lies in B p λ, for every λ ≤ 1−2/p.However, there is no extension of (P,u) as in Lemma 52, since u(z) diverges, as|z| → ∞. Therefore, [W] is not well-defined. ✷56. Remark. Let p > 2 and W ∈ B p loc = ˜B p loc / ∼ p be a gauge equivalenceclass of triples (P,A,u) of Sobolev class. 6 Assume that every representative of Wsatisfies the vortex equations (1.8,1.9), the energy of W is finite, and the closure ofits image compact. As explained in Remark 9 in Section 1.5, we have W ∈ B p λ , forevery λ < 2 − 2/p. Therefore, the equivariant homology class of W is well-defined.✷3 i.e., compact and without boundary4 Such a map descends to a classifying map X → BG.5 This isomorphism is induced by an arbitrary continuous equivariant map from EG ′ to EG.6 Recall from Section 1.4 that each such triple consists of a G-bundle of class W2,ploc , and aconnection A and a G-equivariant map u : P → M of class W 1,ploc .

3.1. EQUIVARIANT HOMOLOGY, THE CHERN NUMBER, AND THE MASLOV INDEX 71The contraction appearing in formula (1.27) has a concrete geometric meaning.It can be interpreted as the Maslov index of a certain loop of linear symplectictransformations, as follows. Let (M,ω) be a symplectic manifold and G a connectedcompact Lie group, acting on M in a Hamiltonian way, with momentum map µ.Assume that G acts freely on µ −1 (0). Let Σ be a compact, connected, orientedtopological surface 7 with non-empty boundary. We associate to this data a mapm Σ,ω,µ , called “Maslov index”, as follows. The domain of this map consists of theweak (ω,µ)-homotopy classes of (ω,µ)-admissible maps from Σ → M, and it takesvalues in the even integers.Here we call a continuous map u : Σ → M (ω,µ)-admissible, iff for everyconnected component X of the boundary ∂Σ there exists a G-orbit which is containedin µ −1 (0), and contains the set u(X). We denote by C ( Σ,M;ω,µ ) the setof such maps. We call two maps u 0 ,u 1 ∈ C ( Σ,M;ω,µ ) weakly (ω,µ)-homotopic,iff they are homotopic through such maps. 8 This defines an equivalence relation onC ( Σ,M;ω,µ ) . We call the corresponding equivalence classes weak (ω,µ)-homotopyclasses.57. Example. Consider R 2n = C n , equipped with the standard symplecticform ω := ω 0 , and the action of S 1 ⊆ C given bywith momentum mapz · (z 1 ,...,z n ) := ( zz 1 ,...,zz n),(3.2) µ : C n → (Lie S 1 ) ∗ ∼ = Lie S 1 = iR, µ(z 1 ,...,z n ) := i 2(1 −n∑|z j |).2We have µ −1 (0) = S 2n−1 ⊆ C n , and the S 1 -orbits contained in the unit sphere arethe fibers of the Hopf fibration S 2n−1 → CP n−1 . Consider the case in which Σ isthe unit disk D ⊆ C. Then for each integer d ∈ Z and each vector v ∈ S 2n−1 themap(3.3) u d,v : D → C n , u d,v (z) := z d vis (ω 0 ,µ)-admissible. Furthermore, u d,v and u d ′ ,v ′ are weakly (ω 0,µ)-homotopicif and only if d = d ′ . To see this, note that if d = d ′ then a homotopy betweenthe two maps is given by [0,1] × D ∋ (t,z) ↦→ z d v(t), where v ∈ C ( [0,1],S 2n−1)is any path joining v and v ′ . Conversely, assume that u d,v and u d ′ ,v ′ are weakly(ω 0 ,µ)-homotopic. Then by an elementary argument, there exists a smooth weak(ω 0 ,µ)-homotopy h : [0,1] × D → C n between these maps. It follows that∫ ∫ ∫d ′ π = u ∗ d ′ ,v ′ω 0 = u ∗ d,vω 0 + h ∗ ω 0 = dπ + 0,DD[0,1]×∂Dwhere in the last equality we used the fact that for every v ∈ T∂D, dh(t, ·)v istangent to the characteristic distribution on S 2n−1 9 , and therefore ω 0(·,dh(t, ·)v)vanishes on vectors tangent to S 2n−1 . Therefore, we have d = d ′ . This proves theclaimed equivalence. ✷7 i.e., real two-dimensional topological manifold8 This means that there exists a continuous map u : [0, 1] × Σ → M, such that u(i, ·) = ui ,for i = 0, 1, and for every t ∈ [0, 1] we have u(t, ·) ∈ C`Σ, M; ω, µ´.9 This follows from the fact that by assumption, h(t, ∂D) is contained in a Hopf circle, forevery t ∈ [0, 1].j=1

72 3. FREDHOLM THEORY FOR VORTICES OVER THE PLANEThe definition of the map m Σ,ω,µ is based on the following Maslov index of aregular symplectic transport over a curve. Let X be an oriented, connected, closedtopological curve 10 , and (V,ω) a symplectic vector space. We denote by Autω thegroup of linear symplectic automorphisms of V . We define a regular symplectictransport over X to be a continuous map Φ : X × X → Autω satisfyingΦ(z,z) = 1, Φ(z ′′ ,z) = Φ(z ′′ ,z ′ )Φ(z ′ ,z), z,z ′ ,z ′′ ∈ X.We define the Maslov index of such a map to be(3.4) m X,ω (Φ) := 2deg ( X ∋ z ↦→ ρ ω ◦ Φ(z,z 0 ) ∈ S 1) .Here z 0 ∈ X is an arbitrary point, ρ ω : Aut(ω) → S 1 denotes the Salamon-Zehndermap (see [SZ, Theorem 3.1.]), and deg the degree of a map from X to S 1 . Thisdefinition does not depend on the choice of z 0 , by some homotopy argument.Let now M,ω,G,µ,Σ be as before, and a a weak (ω,µ)-homotopy class. Todefine m ω,µ (a), we choose a representative u : Σ → M of a, a symplectic vectorspace (V,Ω) of dimension dimM and a symplectic trivialization Ψ : Σ × V →u ∗ TM. 11 Let X be a connected component of ∂Σ. We define the mapΦ X : X × X → AutΩ, Φ X (z ′ ,z) := Ψ −1z g ′ z ′ ,z · Ψ z .Here g z ′ ,z ∈ G denotes the unique element satisfying u(z ′ ) = g z ′ ,zu(z), g z ′ ,z· denotesthe induced action of g z′ ,z on TM, and Ψ z := Ψ(z, ·). The map Φ X is a regularsymplectic transport.58. Definition (Maslov index). We define(3.5) m Σ,ω,µ (a) := ∑ m X,Ω (Φ X ),where the sum runs over all connected components X of ∂Σ.This definition does not depend on the choices of the symplectic vector space(V,Ω) and the trivialization Ψ. This follows from the fact that (3.5) agrees withthe Maslov index associated with a and the coisotropic submanifold µ −1 (0) ⊆ M,as defined in [Zi3]. See [Zi3, Lemma 45].Remark. This Maslov index is based on a definition by D. A. Salamon andR. Gaio (for Σ = D), see [GS]. ✷59. Example. Consider the situation of Example 57, with u d,v defined as in(3.3). This map carries the Maslov indexm D,ω0,µ([u d,v ]) = 2dn,where [· · · ] denotes the (ω 0 ,µ)-homotopy class. This follows from a straight-forwardcalculation. ✷We now return to the setting of the beginning of this section. We define Σ tobe the compact oriented topological surface obtained from C by “gluing a circleat ∞” to it. 12 The class W carries a weak (ω,µ)-homotopy class, whose definitionrelies on the following lemma.10 i.e., a real one-dimensional topological manifold11 Such a trivialization exists, since the group AutΩ is connected, and the surface Σ deformationretracts onto a wedge of circles. Here we use that Σ is connected and has non-emptyboundary.12 This surface is homeomorphic to the closed disk D ⊆ C.

3.2. PROOF OF THE FREDHOLM RESULT 7360. Lemma. Let p > 2, λ > 1 − 2/p, and (P,A,u) ∈ ˜B p λ. There exists a sectionσ : C → P of class W 1,ploc, such that the map u ◦σ : C → M continuously extends toΣ. Furthermore, for every such section σ the corresponding extension ũ : Σ → Mis (ω,µ)-admissible.The proof of this lemma is postponed to the appendix (page 99). Let( )⋃W ∈/∼ p .p>2, λ>1− 2 p61. Definition. We define the (ω,µ)-homotopy class of W to be the weak(ω,µ)-homotopy class of the continuous extension ũ : Σ → M of the map u ◦ σ :C → M. Here (P,A,u) is a representative of W, and σ a section of P as in theprevious lemma.By Lemma 85 in Appendix A.3 this homotopy class is independent of the choiceof the section σ. Furthermore, it follows from a straight-forward argument that it isindependent of the choice of the representative (P,A,u). The contraction appearingin formula (1.27) can now be expressed as follows.62. Proposition (Chern number and Maslov index). We have2 〈 c G 1 (M,ω),[W] 〉 = m Σ,ω,µ((ω,µ)-homotopy class of W).The proof of this result is postponed to the appendix (page 102).Example (Maslov index of a vortex). Let (M,ω,J,G) := (R 2 ,ω 0 ,i,S 1 ), 〈·, ·〉 gbe the standard inner product on g := Lie(S 1 ) = iR, the action of S 1 ⊆ C onR 2 = C be given by multiplication, with momentum map as in (3.2) with n = 1,Σ := C, equipped with the area form ω 0 , and W a gauge equivalence class of smoothfinite energy vortices. By Proposition 23 and Remark 56 the equivariant homologyclass [W] is well-defined. Furthermore, we have〈cG1 (M,ω),[W] 〉 = deg(W),where the degree deg(W) is defined as in (2.18). This follows from Proposition 62and a straight-forward calculation of the Maslov index of the (ω,µ)-homotopy classof W. ✷˜B p λ3.2. Proof of the Fredholm resultIn this section Theorem 4 is proved, based on a Fredholm theorem for theaugmented vertical differential and the existence of a bounded right inverse for L ∗ w,the formal adjoint of the infinitesimal action of the gauge group on pairs (A,u). Inthe present section we always assume that¯n := (dim M)/2 − dimG > 0.3.2.1. Reformulation of the Fredholm theorem. In this subsection westate the two results mentioned above and deduce Theorem 4 from them. Toformulate the first result, let p > 2, λ ∈ R, ˜Bpλbe defined as in (1.16), and w :=(P,A,u) ∈ ˜B p λ. We denoteimL := {(x,L x ξ) |x ∈ M, ξ ∈ g},

74 3. FREDHOLM THEORY FOR VORTICES OVER THE PLANEand by Pr : TM → TM the orthogonal projection onto imL. Pr induces anorthogonal projectionPr u : TM u := (u ∗ TM)/G → TM uonto (u ∗ imL)/G. Recall that A 1 (g P ) denotes the bundle of one-forms on R 2 withvalues in g P . We writePr u ζ := (α,Pr u v), ∀ζ = (α,v) ∈ A 1 (g P ) ⊕ TM u .Note that imL is in general not a subbundle of TM, since the dimension of imL xmay vary with x ∈ M. Recall that Γ 1,p1,ploc(E) denotes the space of Wloc -sections of avector bundle E. For ζ ∈ Γ 1,ploc(A 1 (g P ) ⊕ TM u) we define‖ζ̂‖ w,p,λ := ‖ζ‖ w,p,λ + ‖Pr u ζ‖ p,λ ,where ‖ζ‖ w,p,λ is defined as in (1.18). Recall the definition (1.21) of Ỹ p,λw . Wedenote by Γ p λ (g P) the space of L p λ -sections of g P. We define(3.6)(3.7)̂X w :=̂Xp,λw := { ζ ∈ Γ 1,ploc(A 1 (g P ) ⊕ TM u) ∣ ∣ ‖ζ̂‖w,p,λ < ∞ } ,Ŷ w := Ŷp,λ w:= Ỹp,λ w⊕ Γ p λ (g P),Recall the definition (1.19) of the formal adjoint map for the infinitesimal action ofthe gauge group on pairs (A,u). Restricting the domain and target, this becomesthe operatorL ∗ w : ̂Xp,λw → Γ p λ (g P), L ∗ w(α,v) := −d ∗ Aα + L ∗ uv.It follows from the fact L ∗ x = L ∗ x Pr x (for every x ∈ M) and compactness of u(P)that this operator is well-defined and bounded. We define the augmented verticaldifferential of the vortex equations over C at w to be the map(3.8)(3.9)⎛̂D w ζ := ⎝p,λ p,λ̂D w := ̂D w : ̂X w → Ŷp,λ w ,(∇ A v + L u α ) 0,1−12 J(∇ vJ)(d A u) 1,0d A α + ω 0 dµ(u)vL ∗ wζ63. Theorem (Fredholm property for the augmented vertical differential). Letp > 2 and λ > −2/p + 1 be real numbers, and w := (P,A,u) ∈ ˜B p λ. Then thefollowing statements hold.(i) The normed spaces ( p,λ ̂X w , ‖ · ̂‖)w,p,λ , Ỹwp,λ and Γ p λ (g P) are complete.(ii) Assume that −2/p+1 < λ < −2/p+2. Then the operator ̂D w (as in (3.8,3.9))is Fredholm of real indexind ̂D w = 2¯n + 2 〈 c G 1 (M,ω),[[w]] 〉 ,where [w] denotes the equivalence class of w, and [[w]] denotes the equivarianthomology class of [w].This theorem will be proved in Section 3.2.2, based on the existence of a suitabletrivialization of A 1 (g P ) ⊕ TM u in which the operator ̂D w becomes standard up toa compact perturbation.The second ingredient of the proof of Theorem 4 is the following.64. Theorem. Let p > 2, λ > 1 −2/p, and w := (P,A,u) ∈ ˜B p λ. Then the mapL ∗ p,λw : ̂X w → Γ p λ (g P) admits a bounded (linear) right inverse.⎞⎠ .

3.2. PROOF OF THE FREDHOLM RESULT 75The proof of Theorem 64 is postponed to Section 3.2.4 (page 86). It is based onthe existence of a bounded right inverse for the operator d ∗ A over a compact subsetof R n diffeomorphic to ¯B 1 (Proposition 69) and the existence of a neighborhoodU ⊆ M of µ −1 (0), such thatinf { |L x ξ| ∣ ∣ x ∈ U, ξ ∈ g : |ξ| = 1}> 0.Recall the definition (2.14) of the subset M ∗ of M. Theorem 4 can be reduced toTheorems 63 and 64, as follows:Proof of Theorem 4 (p. 9). Let p > 2, λ > 1 − 2/p, and W ∈ B p λ .We prove statement (i).Claim. We have˜X w :=˜Xp,λw= K := ker ( L ∗ w : ̂Xp,λw → Γ p λ (g P) ) ,and the restriction of the norm ‖ · ̂‖ w,p,λ to ˜X w is equivalent to ‖ · ‖ w,p,λ .Proof of the claim. It suffices to prove that ˜X w ⊆ K and this inclusion isbounded. Hypothesis (H) implies that there exists δ > 0 such that µ −1 ( ¯B δ ) ⊆ M ∗ .We havec := min { |L x ξ| ∣ ∣ x ∈ µ −1 ( ¯B δ ), ξ ∈ g : |ξ| = 1 } > 0.It follows from Lemma 84 in Appendix A.3 that there exists R > 0 such thatu(P | C\BR ) ⊆ µ −1 ( ¯B δ ). Let ζ = (α,v) ∈ ˜X w . Then L ∗ uv = d ∗ Aα, and thus, using thelast assertion of Remark 125 in Appendix A.7,‖Pr u v‖ p,λ ≤ c −1 ‖L ∗ uv‖ p,λ ≤ c −1 ‖∇ A α‖ p,λ ≤ c −1 ‖ζ‖ w,p,λ < ∞.Hence ˜X w ⊆ K, and this inclusion is bounded. This proves the claim.Statement (i) follows from Theorem 63(i) and the claim.Statement (ii) follows from Theorem 63(ii), Theorem 64 and Lemma 122(Appendix A.7) withX := ̂X w , Y := Ỹw, Z := Γ p λ (g P), T := L ∗ w, D ′ : ̂Xw → Ỹw,where D ′ ζ is defined to be the vector consisting of the first and second rows of ̂D w ζ(as in (3.9)). This proves Theorem 4.□3.2.2. Proof of Theorem 63 (augmented vertical differential). Thissubsection contains the core of the proof of Theorem 63. Here we introduce thenotion of a good complex trivialization, and state an existence result for such atrivialization (Proposition 66). Furthermore, we formulate a result saying thatevery good trivialization transforms D w into a compact perturbation of the directsum of copies of ∂¯z and a certain matrix operator (Proposition 67). The results ofthis subsection will be proved in Subsection 3.2.3.We denote by s and t the standard coordinates in R 2 = C. For v ∈ R n andd ∈ Z we denote〈v〉 := √ 1 + |v| 2 , p d : C → C, p d (z) := z d .We equip the bundle A 1 (g P ) with the (fiberwise) complex structure J P defined byJ P α := −α i. Furthermore, we denoteg C := g ⊗ R C, V := C¯n ⊕ g C ⊕ g C ,□

76 3. FREDHOLM THEORY FOR VORTICES OVER THE PLANEand for a ∈ C we use the notationa · ⊕id : V → V,(v 1 ,...,v¯n ,α,β ) ↦→ ( av 1 ,v 2 ,...,v¯n ,α,β ) .For x ∈ M we writeL C x : g C → T x Mfor the complex linear extension of L x . We defineH x := ker dµ(x) ∩ (imL x ) ⊥ , ∀x ∈ M.Note that in general, the union H of all the H x ’s is not a smooth subbundle ofTM, since the dimension of H x may depend on x. However, there exists an openneighborhood U ⊆ M of µ −1 (0) such that H| U is a subbundle of TM| U . Let p > 2,λ > −2/p + 1 and w := (P,A,u) ∈ ˜B p λ. We denote(3.10) d := 〈 c G 1 (M,ω),[W] 〉 .For z ∈ C we defineH u z := { G · (p,v) ∣ ∣ p ∈ π −1 (z) ⊆ P, v ∈ H u(p)}.Consider a complex trivialization (i.e, a bundle isomorphism descending to theidentity on the base C)Ψ : C × V → A 1 (g P ) ⊕ TM u .65. Definition. We call Ψ good, if the following properties are satisfied.(i) (Splitting) For every z ∈ C we have(3.11)(3.12)Ψ z (C¯n ⊕ g C ⊕ {0}) = {0} ⊕ TM u z ,Ψ z ({0} ⊕ {0} ⊕ g C ) =A 1 (g P ) ⊕ {0}.(3.13)Furthermore, there exists a number R > 0, a section σ of P → C \ B 1 , ofclass W 1,ploc , and a point x ∞ ∈ µ −1 (0), such that the following conditions aresatisfied. For every z ∈ C \ B R we haveΨ z (C¯n ⊕ {0} ⊕ {0}) = H u z ,u ◦ σ(re iϕ ) → x ∞ , uniformly in ϕ ∈ R, as r → ∞,σ ∗ A ∈ L p λ (C \ B 1,g),and for every z ∈ C \ B R and ( α,β = ϕ + iψ ) ∈ g C ⊕ g C , we have(3.14) Ψ z (0,α,β) = ( G · (σ(z),ϕds+ ψdt ) ,G · (u◦ σ(z),L C u◦σ(z) (α))) .(ii) There exists a number C > 0 such that for every (z,ζ) ∈ C × V(3.15) C −1 |ζ| ≤ ∣ ∣ Ψz (〈z〉 d · ⊕id)ζ ∣ ∣ ≤ C|ζ|.(iii) We have ∣ ∣ ∇A ( Ψ(p d · ⊕id) )∣ ∣ ∈ Lpλ (C \ B 1).The first ingredient of the proof of Theorem 63 is the following result.66. Proposition. If p > 2, λ > −2/p + 1, and w := (P,A,u) ∈ ˜B p λ, then thereexists a good complex trivialization of A 1 (g P ) ⊕ TM u .

3.2. PROOF OF THE FREDHOLM RESULT 77The proof of this proposition is postponed to subsection 3.2.3 (page 78). Thenext result shows that a good trivialization transforms ̂D w into a compact perturbationof some standard operator. We denote N 0 := N ∪ {0} andn∑|α| := α i , ∂ α := ∂ α11 · · · ∂αn n , ∀α = (α 1 ,...,α n ) ∈ N n 0.i=1Let 1 ≤ p ≤ ∞, n ∈ N, k ∈ N 0 , λ ∈ R, Ω ⊆ R n be an open subset, W a real orcomplex vector space, and u : Ω → W a k-times weakly differentiable map. Wedefine(3.16)(3.17)‖u‖ Lk,pλ (Ω,W) := ∑|α|≤k‖u‖ Wk,pλ (Ω,W) := ∑|α|≤k∥∥〈·〉 λ+|α| ∂ α u ∥ Lp∈ [0, ∞],(Ω,W)‖〈·〉 λ ∂ α u‖ Lp (Ω,W) ∈ [0, ∞],L k,pλ (Ω,W) := { u ∈ W k,ploc (Ω,W) | ‖u‖ L k,p (Ω,W)< ∞}W k,pλ (Ω,W) := { u ∈ W k,ploc (Ω,W) | ‖u‖ W k,pλ (Ω,W) < ∞} .If (X i , ‖ · ‖ i ), i = 1,...,k, are normed vector spaces then we endow X 1 ⊕ · · · ⊕ X kwith the norm ‖(x 1 ,...,x k )‖ := ∑ i ‖x i‖ i . Let d ∈ Z. If d < 0 then we chooseρ 0 ∈ C ∞ (C,[0,1]) such that ρ 0 (z) = 0 for |z| ≤ 1/2 and ρ 0 (z) = 1 for |z| ≥ 1.In the case d ≥ 0 we set ρ 0 := 1. The isomorphism of Lemma 89 (Appendix A.4)induces norms on the vector spaces(3.18) ̂X′p,λ,d := Cρ 0 p d + L 1,pλ−1−d (C, C), ̂X′′p,λ := C¯n−1 + L 1,pλ−1 (C, C¯n−1 ).We definêX d :=̂Xp,λd:= ̂X ′ p,λ,d ⊕̂X′′p,λ ⊕ W 1,pλ (C,gC ⊕ g C ),Ŷ d := Ŷp,λ d:= L p λ−d (C, C) ⊕ Lp λ(C, C¯n−1 ⊕ g C ⊕ g C) .For a complex vector space W we denote by ∂ W¯z (∂ W z ) the operator 1 2 (∂ s + i∂ t )( 1 2 (∂ s − i∂ t )) acting on functions from C to W. We denote by 〈·, ·〉 C g the hermitianinner product on g C (complex anti-linear in its first argument) extending 〈·, ·〉 g .Furthermore, we denote by A 0,1 (TM u ) the bundle of complex anti-linear one-formson C with values in TM u , and define the isomorphismsF 1 : TM u → A 0,1 (TM u ),F 2 : A 1 (g P ) → A 2 (g P ) ⊕ g P ,F : A 1 (g P ) ⊕ TM u → A 0,1 (TM u ) ⊕ A 2 (g P ) ⊕ g P ,F 1 (v) := (ds − Jdt)v,λF 2 (ϕds + ψdt) := (ψds ∧ dt,ϕ),F(α,v) := (F 1 v,F 2 α).We are now ready to formulate the second ingredient of the proof of Theorem 63:67. Proposition (Operator in good trivialization). Let p > 2, λ > −2/p + 1,w := (P,A,u) ∈ ˜B p λ , andΨ : C × V → A 1 (g P ) ⊕ TM ube a good trivialization. We define d as in (3.10). The following statements hold.(i) The following maps are well-defined isomorphisms of normed spaces:(3.19) ̂Xd ∋ ζ ↦→ Ψζ ∈ ̂X w , Ŷ d ∋ ζ ↦→ FΨζ ∈ Ŷw

78 3. FREDHOLM THEORY FOR VORTICES OVER THE PLANE(ii) There exists a positive 13 C-linear map S ∞ : g C → g C such that the followingoperator is compact:( )(3.20) S := (FΨ) −1 ̂Dw Ψ − ∂ C¯n ∂ gC¯z ⊕¯z id/2: ̂Xd →S ∞ 2∂ ŶdzgCThe proof of Proposition 67 is postponed to subsection 3.2.3 (page 81). It isbased on some inequalities and compactness properties for weighted Sobolev spacesand a Hardy-type inequality (Propositions 90 and 91 in Appendix A.4).Proof of Theorem 63 (p. 74). We fix p > 2, λ > −2/p + 1, and a triplew := (P,A,u) ∈ ˜B p λ. We prove part (i). The space (3.16) is complete, see [Lo1].The same holds for the space (3.17) by Proposition 90(ii) (Appendix A.4). Combiningthis with Propositions 66 and 67(i), part (i) follows.Part (ii) follows from Propositions 66 and 67(ii), Corollary 96 and Proposition97 (Appendix A.4). This proves Theorem 63. □68. Remark. An alternative approach to prove Theorem 63 is to switch to“logarithmic” coordinates τ + iϕ (defined by e τ+iϕ = s + it ∈ C \ {0}). In thesecoordinates and a suitable trivialization the operator ̂D w is of the form ∂ τ + A(τ).Hence one can try to apply the results of [RoSa]. However, this is not possible,since A(τ) contains the operator v ↦→ e 2τ dµ(u)v dτ ∧dϕ, which diverges for τ → ∞.✷3.2.3. Proofs of the results of subsection 3.2.2.Proof of Proposition 66 (p. 76). Let p,λ and w be as in the hypothesis.We choose a section σ of P | C\B1 and a point x ∞ ∈ µ −1 (0) as in Lemma 84 inAppendix A.3.1. Claim. There exists an open G-invariant neighborhood U ⊆ M of x ∞ suchthat H| U is a smooth subbundle of TM with the following property. There existsa smooth complex trivialization Ψ U : U × C¯n → H| U satisfyingΨ U gxv 0 = gΨ U x v 0 := gΨ U (x,v 0 ), ∀g ∈ G, x ∈ U, v 0 ∈ C¯n .Proof of Claim 1. By hypothesis (H) we have x ∞ ∈ M ∗ (where M ∗ is definedas in (2.14)). We choose a G-invariant neighborhood U 0 ⊆ M ∗ of x ∞ so smallthat kerdµ(x) and (imL x ) ⊥ intersect transversely, for every x ∈ U 0 . Then H| U0 isa smooth subbundle of TM| U0 . Furthermore, by the local slice theorem there existsa pair (U,N), where U ⊆ U 0 is a G-invariant neighborhood of x ∞ and N ⊆ U is asubmanifold of dimension dimM −dim G that intersects Gx transversely in exactlyone point, for every x ∈ U. We choose a complex trivialization of H| N and extendit in a G-equivariant way, thus obtaining a trivialization Ψ U of H| U . This provesClaim 1.□We choose U and Ψ U as in Claim 1. It follows from Lemma 84 that there existsR > 1 such that u(p) ∈ U, for p ∈ π −1 (z) ⊆ P, if z ∈ C \ B R . We define˜Ψ ∞ : (C \ B R ) × (C¯n ⊕ g C ) → TM u = (u ∗ TM)/G,)˜Ψ ∞ z (v 0 ,α) = G ·(u ◦ σ(z),Ψ U u◦σ(z) (z−d · ⊕id)v 0 + L C u◦σ(z) α .13 This means that 〈S∞v, v〉 C g > 0 for every 0 ≠ v ∈ g C .

3.2. PROOF OF THE FREDHOLM RESULT 79This is a complex trivialization of TM u | C\BR (of class W 1,ploc ).2. Claim. ˜Ψ ∞ | C\BR+1 extends to a complex trivialization of TM u .We define f : C \ {0} → S 1 by f(z) := z/|z|.Proof of Claim 2. We choose a complex trivializationof class W 1,ploc .14 We define(3.21)Ψ 0 : ¯B R × (C¯n ⊕ g C ) → TM u | ¯BRΦ : S 1 R := {z ∈ C | |z| = R} → Aut(C¯n ⊕ g C ),Φ z (v 0 ,α) := (Ψ 0 z) −1 (G · (u◦ σ(z),Ψ U u◦σ(z) v 0 + L C u◦σ(z) α)) .For a continuous map x : S 1 R → S1 we denote by deg(x) its degree.3. Claim. We have〈(3.22)cG1 (M,ω),[[w]] 〉 = deg(f ◦ det ◦Φ).Proof of Claim 3. We define ˜P to be the quotient of P ∐ ( (S 2 \ {0}) × G )under the equivalence relation generated by p ∼ (z,g), where g ∈ G is determinedby σ(z)g = p, for p ∈ π −1 (z) ⊆ P, z ∈ C \ {0}. Furthermore, we define{ũ : ˜P u(p), for p ∈ P,→ M, ũ([p]) :=ũ([∞,g]) := g −1 x ∞ , for g ∈ G.The statement of Lemma 84 implies that this map is continuous and extends u. Thefiberwise pullback form ũ ∗ ω on ˜P descends to a symplectic form ˜ω on the vectorbundle TM eu = (ũ ∗ TM)/G → S 2 . Similarly, the structure J induces an complexstructure ˜J on TM eu . The structures ˜ω and ˜J are compatible, and therefore, wehavec 1 (TM eu , ˜ω) = c 1(TM eu , ˜J ) .Using Lemma 86 in Appendix A.3, it follows that〈(3.23)cG1 (M,ω),[[w]] 〉 = 〈 (c 1 TM eu , ˜J ) ,[S 2 ] 〉 .We define the local complex trivializationΨ ∞ : (S 2 \ B R ) × (C¯n ⊕ g C ) → TM eu ,{ (G · [u ◦ σ(z)],ΨΨ ∞ Uz (v 0 ,α) :=u◦σ(z)v 0 + L C u◦σ(z) α) , if z ∈ C \ B R ,G · ([∞,1],Ψ U x ∞v 0 + L C x ∞α ) , if z = ∞.Recalling the definition (3.21), we haveΦ z = (Ψ 0 z) −1 Ψ ∞ z , ∀z ∈ S 1 R.Therefore, Φ is the transition map between Ψ 0 and Ψ ∞ . It follows that〈c1(TM eu , ˜J ) [S 2 ] 〉 = deg(f ◦ det ◦Φ).Combining this with (3.23), equality (3.22) follows. This proves Claim 3.□14 To see that such a trivialization exists, we first choose a continuous trivialization e Ψ 0 of thebundle. An argument using local trivializations of class W 1,ploc , shows that we may regularize e Ψ 0 ,so that it becomes of class W 1,ploc .

80 3. FREDHOLM THEORY FOR VORTICES OVER THE PLANEWe denote d := 〈 c G 1 (M,ω),[[w]] 〉 . By Claim 3 and Lemma 123 in AppendixA.7 the maps Φ andS 1 R ∋ z ↦→ (z d · ⊕id) ∈ Aut(C¯n ⊕ g C )are homotopic. Hence there exists a continuous maph : ¯B R \ B 1 → Aut(C¯n ⊕ g C ),such that h z := h(z) = (z d · ⊕id), if z ∈ S1, 1 and h z = Φ(z), if z ∈ SR 1 . We define⎧⎨ ˜Ψ ∞˜Ψ : C × (C¯n ⊕ g C ) → TM u z , for z ∈ C \ B R ,, ˜Ψz := Ψ⎩0 zh z (z −d · ⊕id), for z ∈ B R \ B 1 ,Ψ 0 z, for z ∈ B 1 .Regularizing ˜Ψ on the ball B R+1 , we obtain the required extension of ˜Ψ ∞ | C\BR+1 ,of class W 1,ploc. This proves Claim 2.□We define the trivialization(3.24) ̂Ψ ∞ : (C\B R )×g C → A 1( g P | C\BR), ̂Ψ∞ z (ϕ+iψ) := G·(σ(z),ϕds+ψdt ) .4. Claim. ̂Ψ ∞ | C\BR+1 extends to a complex trivialization of the bundle A 1 (g P )over C.Proof of Claim 4. We denote by Ad and Ad C the adjoint representationsof G on g and g C respectively. We havedet(Ad C g) = det(Ad g ) ∈ R, ∀g ∈ G.We choose a continuous section ˜σ of the restriction P | ¯BR . We define g : SR 1 →G to be the unique map such that σ(z) = ˜σ(z)g(z), for every z ∈ SR 1 . Sincef ◦ det(Ad C g) ≡ ±1, we have()deg SR 1 ∋ z ↦→ f ◦ det(Ad C g(z)) = 0.Hence Lemma 123 (Appendix A.7) implies that there exists a continuous mapΦ : ¯BR → Aut(g C ) satisfying Φ z := Φ(z) = Ad C g(z), for every z ∈ SR 1 . We definêΨ : C×g C → A 1 (g P ) to be the trivialization that equals ̂Ψ ∞ on C\B R , and satisfieŝΨ z α := G · (˜σ,ϕ ′ ds + ψ ′ dt ) ,where ϕ ′ +iψ ′ := Φ z α, for every z ∈ B R , α ∈ g C . Regularizing ̂Ψ on the ball B R+1 ,we obtain an extension of ̂Ψ ∞ | C\BR+1 , of class W 1,ploc. This proves Claim 4. □We choose extensions ˜Ψ and ̂Ψ of ˜Ψ ∞ and ̂Ψ ∞ as in Claims 2 and 4, and wedefineΨ : C × V → A 1 (g P ) ⊕ TM u , Ψ(z;v 0 ,α,β) := (̂Ψz β, ˜Ψ z (v 0 ,α) ) .5. Claim. The map Ψ is a good complex trivialization.Proof of Claim 5. Condition (i) of Definition 65 follows from the construction ofΨ. (The condition σ ∗ A ∈ L p λ (C \ B 1,g) follows from the statement of Lemma 84.)To prove (ii), note that for z ∈ C \ B R+1 and (v 0 ,α,β) ∈ V , we have(3.25) ∣ Ψz (z d · ⊕id)(v 0 ,α,β) ∣ 2 = |β| 2 + ∣ ∣ ΨUu◦σ(z) v 0 2 ∣+ ∣L Cu◦σ(z) α ∣ 2 .

3.2. PROOF OF THE FREDHOLM RESULT 81Here we used the fact H x = (imL C x) ⊥ , for every x ∈ M. By our choice of U,H| U ⊆ TM| U is a smooth subbundle of rank dimM − 2dim G. It follows thatimL C | U = H ⊥ | U is a smooth subbundle of TM| U of rank 2dim G. Hence L C x :g C → T x M is injective, for every x ∈ U. Since by assumption u(P) ⊆ M iscompact, the same holds for the set u(P | C\BR+1 ) ⊆ u(P). It follows that thereexists a constant C > 0 such thatC −1 |v 0 | ≤ ∣ ∣Ψ U u◦σ(z) v ∣0 ≤ C|v 0 |, C −1 |α| ≤ ∣ ∣L C u◦σ(z) α∣ ∣ ≤ C|α|,for every z ∈ C \ B R+1 , v 0 ∈ C¯n , and α ∈ g C . Combining this with equality (3.25),condition (ii) follows.We check condition (iii). Letζ := ( v 0 ,α,β = ϕ + iψ ) ∈ V, z ∈ C \ B R+1 , v ∈ T z C.We choose a point p ∈ π −1 (z) ⊆ P and a vector ṽ ∈ T p P such that π ∗ ṽ = v. Then∇ A v(˜Ψ(pd · ⊕id)(v 0 ,α) ) = G · (u(p), ˜∇ A ev (ΨU u v 0 + L C uα) ) ,where ˜∇ A evis defined as in (A.62). Furthermore, for every smooth vector field X onU we have˜∇ A ev X = (u∗ ∇) ev−p(Aev) X = ∇ dAu·evX.We defineC := max ∣ ∣ ∇v ′(Ψ U x v ′′ + L C xα) ∣ ∣ ,where the maximum is taken over all v ′ ∈ T x M, x ∈ u(P | C\BR+1 ) and (v ′′ ,α) ∈C¯n ⊕ g C such that |v ′ | ≤ 1, |(v ′′ ,α)| ≤ 1. Furthermore, we defineC ′ := ‖d A u‖ Lpλ (C\BR+1) .It follows that(3.26) ∥ ∇Av(˜Ψ(pd · ⊕id)(v 0 ,α) )∥ ∥ ≤ Lpλ (C\BR+1) CC′ |v||(v 0 ,α)|.We now define ˜ϕ, ˜ψ : P → g to be the unique equivariant maps such that ˜ϕ ◦σ ≡ ϕ,˜ψ ◦ σ ≡ ψ. We have d A ˜ϕ σ ∗ v = [(σ ∗ A)v,ϕ], and similarly for ˜ψ. Since˜∇ A σ ∗v(˜ϕds + ˜ψdt) = (d A ˜ϕσ ∗ v)ds + (d A ˜ψσ∗ v)dt,using (3.24), it follows that∣)∣∣∇ A v(̂Ψ(ϕds + ψdt) ∣ = ∣ ∣G · (σ(z), ˜∇ A σ ∗v(˜ϕds + ˜ψdt) )∣ ∣ = ∣ [ (σ ∗ A)v,β ]∣ ∣.Using the fact ‖σ ∗ A‖ p,λ < ∞ and inequality (3.26), condition (iii) follows. Thisproves Claim 5 and concludes the proof of Proposition 66.□Proof of Proposition 67 (p. 77). Let p,λ,w = (P,A,u) and Ψ be as inthe hypothesis. We choose ρ 0 ∈ C ∞ (C,[0,1]) such that ρ 0 (z) = 0 for z ∈ B 1/2 andρ 0 (z) = 1 for z ∈ C \ B 1 . We fix R ≥ 1, σ and x ∞ as in Definition 65(i).We prove statement (i). For every ζ ∈ W 1,1loc(C,V ) Leibnitz’ rule impliesthat(3.27) ∇ A (Ψζ) = ( ∇ A( Ψ(p d · ⊕id) )) (p −d · ⊕id)ζ + Ψ(p d · ⊕id)D ( (p −d · ⊕id)ζ ) .1. Claim. The first map in (3.19) is well-defined and bounded.

82 3. FREDHOLM THEORY FOR VORTICES OVER THE PLANEProof of Claim 1. Proposition 90(i) in Appendix A.4 and the fact λ >−2/p + 1 imply that there exists a constant C 1 such that(3.28) ∥ (〈·〉−d · ⊕id)ζ ∥ ∞≤ C 1 ‖ζ‖ bXd , ∀ζ ∈ ̂X d .We choose a constant C 2 := C as in part (ii) of Definition 65. Then by (3.15) and(3.28), we have(3.29) ‖Ψζ‖ ∞ ≤ C 1 C 2 ‖ζ‖ bXd ∀ζ ∈ ̂X d .It follows from (3.13) and (3.14), the definition H x := kerdµ(x) ∩ imL ⊥ x and thecompactness of u(P) that there exists C 3 ∈ R such that, for every ζ ∈ ̂X d ,(3.30) ∥ |dµ(u)v ′ | + |Pr u v ′ | + |α ′ | ∥ p,λ≤ C 3 ‖ζ‖ bXd ,where (v ′ ,α ′ ) := Ψζ. For r > 0 we denoteWe defineB C r := C \ B r ,‖ · ‖ p,λ;r := ‖ · ‖ Lpλ (BC r ) .C 4 := max {∥ ∥ ∇A ( Ψ(p d · ⊕id) )∥ ∥p,λ;1,C 2}.By condition (iii) of Definition 65 we have C 4 < ∞. Let ζ ∈ ̂X d . Then by (3.27)we have((3.31) ‖∇ A (Ψζ)‖ p,λ;1 ≤ C 4 ‖(p −d · ⊕id)ζ‖ L∞ (B1 C) + ∥ ( D (p−d · ⊕id)ζ )∥ ∥p,λ;1).DefiningC 5 := max { − d2 (−d+3)/2 ,2 } ,we have, by Proposition 90(iv),∥∥D ( (p −d · ⊕id)ζ )∥ ∥p,λ;1≤ C 5 ‖ζ‖ bXd .Combining this with (3.31) and (3.28), we get(3.32) ‖∇ A (Ψζ)‖ p,λ;1 ≤ C 4(2|d|2 C1 + C 5)‖ζ‖ bXd .By a direct calculation there exists a constant C 6 such that‖∇ A (Ψζ)‖ Lp (B 1) ≤ C 6 ‖ζ‖ bXd , ∀ζ ∈ ̂X d .Claim 1 follows from this and (3.29,3.30,3.32).□2. Claim. The map ̂X w ∋ ζ ′ ↦→ Ψ −1 ζ ′ ∈ ̂X d is well-defined and bounded.Proof of Claim 2. We choose a neighborhood U ⊆ M of µ −1 (0) as in Lemma126 (Appendix A.7), and define c as in (A.67), and C 1 := max{c −1 ,1}. Sinceu ◦ σ(re iϕ ) converges to x ∞ , uniformly in ϕ ∈ R, as r → ∞, there exists R ′ ≥ Rsuch that u(p) ∈ U, for every p ∈ π −1 (BR C ′) ⊆ P. Then (3.13,3.14) and (A.67)imply that(3.33)∥ (α,β)∥∥p,λ;R′ ≤ C 1∥ ∥Ψ(0,α,β)∥∥p,λ;R′ ≤ C 1 ‖ζ ′ ‖ w ,where (v 0 ,α,β) := Ψ −1 ζ ′ , for every ζ ′ ∈ ̂X d .3. Claim. There exists a constant C 2 such that for every ζ ′ ∈ ̂X w , we have((3.34) ∥ D (ρ0 p −d · ⊕id)Ψ −1 ζ ′)∥ ∥ ≤ C Lp 2‖ζ ′ ‖λ (C) w .

3.2. PROOF OF THE FREDHOLM RESULT 83Proof of Claim 3. It follows from equality (3.27) and conditions (ii) and(iii) of Definition 65 that there exist constants C and C ′ such that((3.35)∥ D (p−d · ⊕id)Ψ −1 ζ ′)∥ ∥p,λ;1≤ C(‖∇ A ζ ′ ‖ p,λ + ∥ ( ∇AΨ(p d · ⊕id) )∥ )∥p,λ‖ζ ′ ‖ ∞ ≤ C ′ ‖ζ ′ ‖ w , ∀ζ ′ ∈ ̂X w .On the other hand, Leibnitz’ rule implies thatD(Ψ −1 ζ ′ ) = Ψ −1( ∇ A ζ ′ − (∇ A Ψ)Ψ −1 ζ ′) .Hence by a short calculation, using Leibnitz’ rule again, it follows that there existsa constant C ′′ such that∥ D((ρ0 p −d · ⊕id)Ψ −1 ζ ′)∥ ∥L p (B 1) ≤ C′′ ‖ζ ′ ‖ w , ζ ′ ∈ ̂X w .Combining this with (3.35), Claim 3 follows.□Let ζ ′ ∈ ̂X w . We denote˜ζ := (ṽ 0 , ˜α, ˜β) := (ρ 0 p −d · ⊕id)Ψ −1 ζ ′ .By inequality (3.34) the hypotheses of Proposition 91 in Appendix A.4 with n := 2and λ replaced by λ − 1 are satisfied. It follows that there existsζ ∞ := ( v ∞ ,α ∞ ,β ∞)∈ V = C¯n ⊕ g C ⊕ g C ,such that ˜ζ(re iϕ ) → ζ ∞ , uniformly in ϕ ∈ R, as r → ∞, and(3.36) ‖(˜ζ − ζ ∞ )| · | λ−1∥ ∥ ≤ (dim M + 2dim G) p∥ ∥L p (C) D˜ζ| · |λ λ − 1 + 2 . L p (C)pSince λ > −2/p + 1, we have∫B C R ′ 〈·〉 pλ = ∞.Hence the convergence (˜α, ˜β) → (α ∞ ,β ∞ ) and the estimate (3.33) imply that(α ∞ ,β ∞ ) = (0,0). We choose a constant C > 0 as in part (ii) of Definition 65. Theconvergence ṽ 0 → v ∞ and the first inequality in (3.15) imply that(3.37) |v ∞ | ≤ ‖ṽ 0 ‖ ∞ ≤ 2 |d|2 C‖ζ ′ ‖ ∞ .We define(v 1 ,...,v¯n ,α,β ) := Ψ −1 ζ ′ − ( ρ 0 p d v 1 ∞,v 2 ∞,...,v¯n ∞,0,0 ) .Proposition 90(iv) in Appendix A.4 and inequalities (3.36) and (3.34) imply thatthere exists a constant C 6 (depending on p,λ,d and Ψ, but not on ζ ′ ) such that(3.38) ‖v 1 ‖ L1,pλ−1−d (BC 1 ) + ∥ ∥ ( v 2 ,...,v¯n ,α,β )∥ ∥L1,pλ−1 (BC 1 ) ≤ C 6‖ζ ′ ‖ w .Finally, by a straight-forward argument, there exists a constant C 7 (independentof ζ ′ ) such that‖Ψ −1 ζ ′ ‖ W 1,p (B R ′) ≤ C 7 ‖ζ ′ ‖ w .Combining this with (3.33,3.37,3.38), Claim 2 follows.□

84 3. FREDHOLM THEORY FOR VORTICES OVER THE PLANEClaims 1 and 2 imply that the first map in (3.19) is an isomorphism (of normedvector spaces). It follows from condition (ii) of Definition 65 that the second mapin (3.19) is an isomorphism. This completes the proof of (i).We prove statement (ii). Recall that we have chosen R > 0,σ and x ∞ asin Definition 65(i). We define S ∞ : g C → g C to be the complex linear extension ofL ∗ x ∞L x∞ : g → g. By our hypothesis (H) the Lie group G acts freely on µ −1 (0). Itfollows that L x∞ is injective. Therefore S ∞ is positive with respect to 〈·, ·〉 C g. By(3.11) and (3.12) there exist complex trivializationsΨ 1 : C × (C¯n ⊕ g C ) → TM u , Ψ 2 : C × g C → A 1 (g P ),such that Ψ z (v 0 ,α,β) = ( (Ψ 2 ) z β,(Ψ 1 ) z (v 0 ,α) ) . We denote byι : g C → C¯n ⊕ g C ,the canonical inclusion and projection. We definêX 1 dpr : C¯n ⊕ g C → g C:= L1,pλ−1−d(C, C) ⊕ L1,pλ−1 (C, C¯n−1 ) ⊕ W 1,pλ (C,gC 2), ̂X d:= W 1,pλ (C,gC ),̂Xd ′ := ̂X d 1 ⊕ ̂X d 2, 0 ̂X d:= Cρ 0 p d ⊕ C¯n−1 ⊕ {(0,0)} ⊆ ̂X d ,Ŷ 1 d := Lp λ−d (C, C) ⊕ Lp λ (C, C¯n−1 ⊕ g C ), Ŷ 2 d := Lp λ (C,gC ).Note that ̂X d = ̂X d 0 + ̂X d ′ and Ŷd = Ŷ1 d ⊕ Ŷ2 d . We define S : ̂Xd → Ŷd as in (3.20).Since ̂X d 0 is finite dimensional, S| X b0 is compact. Hence it suffices to prove that S| bX ′d dis compact. To see this, we denote(ds ∧ dtdµ(u)Q :=L ∗ u) (dA, T :=−d ∗ Aand we define S i j : ̂Xjd → Ŷi d (for i,j = 1,2) and ˜S 1 1 : ̂X1d→ Ŷ1 d byS 1 1v := (F 1 Ψ 1 ) −1 ((∇ A Ψ 1 )v) 0,1 , ˜S 1 1v := −(F 1 Ψ 1 ) −1( J(∇ Ψ1vJ)(d A u) 1,0 /2 ) ,S 1 2α := (F 1 Ψ 1 ) −1 (L u Ψ 2 α) 0,1 − ια/2, S 2 1v := ( (F 2 Ψ 2 ) −1 QΨ 1 − S ∞ pr ) v,S 2 2α := (F 2 Ψ 2 ) −1 (TΨ 2 )α.Here (TΨ 2 )α := T(Ψ 2 α), for α ∈ g C (viewed as a constant section of C×g C ). (Recallalso that S ∞ : g C → g C is the complex linear extension of L ∗ x ∞L x∞ : g → g.) Adirect calculation shows that),S(v,α) = ( S 1 1v + ˜S 1 1v + S 1 2α,S 2 1v + S 2 2α ) .For a subset X ⊆ C we denote by χ X : C → {0,1} its characteristic function. Itfollows that χ BR S| bX ′dis of 0-th order. Since it vanishes outside B R , it follows thatthis map is compact.4. Claim. The operators χ B CRS i j , i,j = 1,2, and χ B˜S 1 RC 1 are compact.Proof of Claim 4. To see that χ B CRS 1 1 is compact, note that Leibnitz’ rule andholomorphicity of p d imply that(∇ A Ψ 1 ) 0,1 = ( ∇ A( Ψ 1 (p d · ⊕id) )) 0,1(p−d · ⊕id), on C \ {0}.For a normed vector space V we denote by C b (C,V ) the space of bounded continuousmaps from C to V . Since λ > 1 − 2/p, assertions (iv) and (i) of Proposition90 imply that the map(ρ 0 p −d · ⊕id) : ̂X1d → C b(C, C¯n ⊕ g C)

3.2. PROOF OF THE FREDHOLM RESULT 85is well-defined and compact. By Definition 65(iii), the map(χ B CR∇A ( Ψ 1 (p d · ⊕id) )) 0,1 (: Cb C, C¯n ⊕ g C) → Γ p (λ A 0,1 (TM u ) )is bounded. Condition (ii) of Definition 65 implies boundedness of the mapCompactness of χ B CRS 1 1 follows.(F 1 Ψ 1 ) −1 : Γ p λ(A 0,1 (TM u ) ) → Ŷ1 d.By the definition of ˜B p λ , we have |d Au| ∈ L p λ(C). This together with Proposition90(iv) and (i) and Definition 65(ii) implies that the map χ B C˜S1R 1 is compact.Furthermore, it follows from Definition 65(i) that χ B CRS 1 2 = 0.To see that χ B CRS 2 1 is compact, we define f : BR C → End(gC ) by setting f(z) :g C → g C to be the complex linear extension of the mapL ∗ u◦σ(z) L u◦σ(z) − L ∗ x ∞L x∞ : g → g.Since u ◦ σ(re iϕ ) converges to x ∞ , uniformly in ϕ, as r → ∞, the map f(re iϕ )converges to 0, uniformly in ϕ, as r → ∞. Hence by Proposition 90(iii), the mapW 1,pλ (C,gC ) ∋ α ↦→ χ B CRfα ∈ L p λ (C,gC )is compact. Definition 65(i) implies that χ B CRS 2 1 = χ B CRfpr. It follows that thisoperator is compact.Finally, Proposition 90(i) and parts (iii) and (ii) of Definition 65 imply that themap χ B CRS 2 2 is compact. Claim 4 follows. It follows that the operator S : ̂Xd → Ŷd(as in (3.20)) is compact. This completes the proof of statement (ii) and hence ofProposition 67.□3.2.4. Proof of Theorem 64 (Right inverse for L ∗ w). For the proof ofthis result we need the following. Let n ∈ N, l ∈ N 0 , p > n/(l + 1), G be acompact Lie group, 〈·, ·〉 g an invariant inner product on g := Lie(G), X a manifold(possibly with boundary), and P → X a G-bundle of class W l+1,ploc. We denoteby g P := (P × g)/G → X the adjoint bundle, and by A l,ploc(P) the affine spaceof connections on P of class W l,ploc , i.e., of class W l,p on every compact subset ofX. If X is compact then we abbreviate A l,p (P) := A l,ploc (P). Let 〈·, ·〉 X be aRiemannian metric on X and A ∈ A l,ploc(P). The connection A induces a connectiond A on the adjoint bundle g P = (P × g)/G. For every i ∈ N this connection andthe Levi-Civita connection of 〈·, ·〉 X induce a connection ∇ A 〈·,·〉 Xon the bundle(T ∗ X) ⊗i ⊗ g P . We abbreviate these connections by ∇ A . Let k ∈ {0,...,l + 1}.For a vector bundle E over X we denote by Γ k,pk,ploc(E) the space of Wloc -sections ofE. For α ∈ Γ k,p)loc((T ∗ X) ⊗i ⊗ g P we define(3.39) ‖α‖ k,p,A := ‖α‖ k,p,X,〈·,·〉X,A := ∑∥ ∣ (∇ A ) j α ∣ ∥ ∥L ,〈·,·〉X,〈·,·〉 p g (X,〈·,·〉 X)j=0,...,kwhere | · | 〈·,·〉X,〈·,·〉 gdenotes the pointwise norm induced by 〈·, ·〉 X and 〈·, ·〉 g , and‖ · ‖ Lp (X,〈·,·〉 X) denotes the L p -norm of a function on X, induced by 〈·, ·〉 X . Wedenote by A i (g P ) the bundle of i-forms on X with values in g P , and(3.40)(3.41)Ω i k,p,A (g P) := { α ∈ Γ k,ploc(A i (g P ) ) ∣ ∣ ‖α‖ k,p,A < ∞ } ,Γ k,pA (g P) := Ω 0 k,p,A (g P), Γ p (g P ) := Γ 0,pA (g P).

86 3. FREDHOLM THEORY FOR VORTICES OVER THE PLANEWe denote by(3.42) d ∗ A = − ∗ d A ∗ : Ω 1 1,p,A(g P ) → Γ p (g P )the formal adjoint of d A , with respect to the L 2 -metrics induced by 〈·, ·〉 X and〈·, ·〉 g . Here ∗ denotes the Hodge star operator.69. Proposition (Right inverse for d ∗ A ). Let n,G and 〈·, ·〉 g be as above,p > n, and (X, 〈·, ·〉 X ) a Riemannian manifold. Assume that X is diffeomorphic toB 1 ⊆ R n . Then the following statements hold.(i) For every G-bundle P → X of class W 2,p and every connection one-form Aon P of class W 1,p there exists a bounded right inverse of the operator (3.42).(ii) There exist constants ε > 0 and C > 0 such that for every G-bundle P → Xof class W 2,p , and every A 1,p ∈ A(P) satisfying ‖F A ‖ p ≤ ε, there exists aright inverse R of the operator d ∗ A (as in (3.42)), satisfying‖R‖ := sup { ‖Rξ‖ 1,p,A∣ ∣ ξ ∈ Γ p (g P ) : ‖ξ‖ p ≤ 1 } ≤ C.We postpone the proof of Proposition 69 to the appendix (page 118).Proof of Theorem 64 (p. 74). Let p,λ and w = (P,A,u) be as in the hypothesis.We construct a map(3.43) R : L p loc (g P) → Γ p (loc A 1 (g P ) ⊕ TM u)such that L ∗ wR is well-defined and equals id, and we show that R restricts to abounded map from Γ p λ (g p,λP) to ̂X w . (See Claim 1 below.) It follows from hypothesis(H) that there exists δ > 0 such that µ −1 ( ¯B δ ) ⊆ M ∗ (defined as in (2.14)). Itfollows from Lemma 84 in Appendix 3.1 that there exists a number a > 0 suchthat u(p) ∈ µ −1 ( ¯B δ ), for every p ∈ π −1( C \ (−a,a) 2) ⊆ P. We choose constants ε 1and C 1 as in the second assertion of Proposition 69 (corresponding to ε and C, forn = 2). Furthermore, we choose constants C 2 and ε 2 as in Lemma 107 in AppendixA.6 (corresponding to C and ε). We define ε := min{ε 1 ,ε 2 }. By assumption wehave |F A | ∈ L p λ(C). Hence there exists an integer N > a such that(3.44) ‖F A ‖ )Lp < ε.λ(C\(−N,N) 2We choose a smooth function ρ : [−1,1] → [0,1] such that ρ = 0 on [−1, −3/4] ∪[3/4,1], ρ = 1 on [−1/4,1/4], and ρ(−t) = ρ(t) and ρ(t) + ρ(t − 1) = 1, for allt ∈ [0,1]. We choose a bijection(ϕ,ψ) : Z \ {0} → Z 2 \ {−N,...,N} 2 .We define ˜ρ : R → [0,1] by⎧⎨ 1, if |t| ≤ N,˜ρ(t) := ρ(|t| − N), if N ≤ |t| ≤ N + 1,⎩0, if |t| ≥ N + 1,and ρ 0 : C → [0,1] by ρ 0 (s,t) := ˜ρ(s)˜ρ(t). Furthermore, for i ∈ Z \ {0} we defineρ i : C → [0,1],ρ i (s,t) := ρ(s − ϕ(i))ρ(t − ψ(i)).We choose a compact subset K 0 ⊆ [−N −1,N +1] 2 diffeomorphic to ¯B 1 , such that[−N − 3/4,N + 3/4] 2 ⊆ intK 0 ,

3.2. PROOF OF THE FREDHOLM RESULT 87and we denote Ω 0 := intK 0 . Furthermore, we choose a compact subset K ⊆ [−1,1] 2diffeomorphic to ¯B 1 , such that [−3/4,3/4] 2 ⊆ intK. For i ∈ Z \ {0} we defineFor i ∈ Z we defineΩ i := intK + (ϕ(i),ψ(i)).T i := d ∗ A : Ω 1 1,p,A(gP | Ωi ) → Γ p (g P | Ωi ).By the first assertion of Proposition 69 there exists a bounded right inverse R 0 ofT 0 . We fix i ∈ Z \ {0}. Since λ > 1 − 2/p > 0, we have ‖F A ‖ L p (Ω i) ≤ ‖F A ‖ Lpλ (Ωi) ,and by inequality (3.44), the right hand side is bounded by ε. Hence it follows fromthe statement of Proposition 69 that there exists a right inverse R i of T i , satisfying(3.45) ‖R i ξ‖ 1,p,Ωi,A ≤ C 1 ‖ξ‖ Lp (Ω i), ∀ξ ∈ Γ 1,pA (g P | Ωi ).We definêR : L p loc (g P) → Γ 1,p (loc A 1 (g P ) ) ∑, ̂Rξ := ρ i · R i (ξ| Ωi ).Each section ξ : C → g P induces a section L u ξ : C → TM u . For every p ∈π −1( C \ (−N,N) 2) ⊆ P we have u(p) ∈ µ −1 ( ¯B δ ) ⊆ M ∗ , and therefore the mapL ∗ u(p) L u(p) : g → g is invertible. We define ˜R : L p loc (g P) → Γ p loc (TMu ) by{(3.46) ( ˜Rξ)(z) 0, for z ∈ (−N,N) 2 ,:=L u (L ∗ uL u ) −1( ξ − d ∗ ̂Rξ ) A (z), for z ∈ C \ (−N,N) 2 .Furthermore, we define the map (3.43) byRξ := (− ̂Rξ, ˜Rξ).The operator L ∗ wR is well-defined and equals id. The statement of Theorem 64 isnow a consequence of the following. We denote by Γ p λ (g P) the space of L p λ -sectionsof g P .1. Claim. The map R restricts to a bounded operator from Γ p λ (g P) to(defined as in (3.6)).Proof of Claim 1. We choose a constant C 3 so big that(3.47) supz∈Ω i〈z〉 pλ ≤ C 3 infz∈Ω i〈z〉 pλ , ∀i ∈ Z.i∈ZFor a weakly differentiable section ξ : C → g P we denote‖ξ‖ 1,p,λ,A := ‖ξ‖ p,λ + ‖d A ξ‖ p,λ .2. Claim. There exists a constant C 4 such that‖ξ − d ∗ A ̂Rξ‖ 1,p,λ,A ≤ C 4 ‖ξ‖ p,λ ,∀ξ ∈ Γ p λ (g P).Proof of Claim 2. Let ξ ∈ Γ p λ (g P). We denote α := ̂Rξ and α i := R i (ξ| Ωi ).Since ∑ i∈Z ρ i = 1, a straight-forward calculation shows that(3.48) d ∗ Aα = ξ − ∑ i∈Z∗ ( (dρ i ) ∧ ∗α i).̂Xp,λwFix z ∈ C. Then ∣ { i ∈ Z |ρ i (z) ≠ 0 }∣ ∣ ≤ 4. Hence equality (3.48) implies that∣ ( ξ − d ∗ Aα ) (z) ∣ p ∑≤ 4 p−1 ‖ρ ′ ‖ p ∞ |α i (z)| p .i∈Z

88 3. FREDHOLM THEORY FOR VORTICES OVER THE PLANECombining this with (3.45,3.47), we obtain‖ξ − d ∗ Aα‖ p p,λ ≤ 4p ‖ρ ′ ‖ p ∞ max { C p 1 , ‖R 0‖ p} ∑C 3i∈Zi∈Z‖ξ‖ p L p λ (Ωi).By (3.48), we have∣ (∣d A ξ − d∗A α ) (z) ∣ p ≤ 8 p−1 max { ‖ρ ′′ ‖ p ∞, ‖ρ ′ ‖ p } ∑ (∞ |αi | p + |∇ A α i | p) (z).Using again (3.45), Claim 2 follows.We choose C 4 as in Claim 2. The following will be used in the proofs of Claims3 and 4 below. Let ξ ∈ Γ p λ (g P). We abbreviate ˜ξ := ξ − d ∗ A ̂Rξ. By the fact˜ξ| (−N,N) 2 = 0, Lemma 107 in Appendix A.6 (Twisted Morrey’s inequality, using(3.44)), the fact λ > 1 − 2/p > 0 and Claim 2, we have(3.49) ‖˜ξ‖ ∞ ≤ C 2 ‖˜ξ‖ 1,p,λ,A ≤ C 2 C 4 ‖ξ‖ p,λ .Recall that we have chosen δ > 0 such that µ −1 ( ¯B δ ) ⊆ M ∗ . We define{ }|Lx ξ|(3.50) c := inf|ξ| ∣ x ∈ µ−1 ( ¯B δ ), 0 ≠ ξ ∈ g .Recall that Pr u : TM u → TM u denotes the orthogonal projection onto (u ∗ imL)/G.Claim 1 is now a consequence of the following three claims.3. Claim. We have∥∥sup{Rξ ∥ ∞+ ∥ ∥|dµ(u) ˜Rξ|∥ ∣ }+ |Pr u ˜Rξ| + | ̂Rξ| ∥p,λ ξ ∈ Γ p λ (g P) : ‖ξ‖ p,λ ≤ 1 < ∞.Proof of Claim 3. Let ξ ∈ Γ p λ (g P) be such that‖ξ‖ p,λ ≤ 1.We denote ˜ξ := ξ − d ∗ A ̂Rξ. Inequality (3.49), Remark 125, and the assumption‖ξ‖ p,λ ≤ 1 imply that(3.51) ‖ ˜Rξ‖ ∞ ≤ c −1 ‖˜ξ‖ L∞ (C\(−N,N) 2 ) ≤ c −1 C 2 C 4 ,where c is defined as in (3.50). We fix i ∈ Z and denote α i := R i (ξ| Ωi ). For i ≠ 0Lemma 107 in Appendix A.6, (3.45), the fact λ > 1 − 2/p > 0, and the assumption‖ξ‖ p,λ ≤ 1 imply thatFurthermore, we havewhere‖α i ‖ ∞ ≤ C 2 ‖α i ‖ 1,p,Ωi,A ≤ C 2 C 1 ‖ξ‖ L p (Ω i) ≤ C 2 C 1 .{C 5 := sup‖α 0 ‖ ∞ ≤ C 5 ‖R 0 ‖ ‖ξ| Ω0 ‖ p ≤ C 5 ‖R 0 ‖,‖α‖ ∞∣ ∣∣ α ∈ Γ1,pA(A 1 (g P | Ω0 ) ) : ‖α‖ 1,p,A ≤ 1}.An argument involving a finite cover of Ω 0 by small enough balls and Lemma 107,implies that C 5 < ∞. It follows that(3.52) ‖ ̂Rξ‖ ∞ ≤ sup ‖α i ‖ ∞ ≤ max { C 2 C 1 ,C 5 ‖R 0 ‖ } < ∞.iWe defineC := max{|dµ(x)| ∣ }x ∈ u(P) .□

3.2. PROOF OF THE FREDHOLM RESULT 89The definition (3.46), the second inequality in (A.66) (Remark 125 in AppendixA.7), the statement of Claim 2, and the assumption ‖ξ‖ p,λ ≤ 1 imply that∥(3.53) ∥ dµ(u) ˜Rξ ∥p,λ ≤ Cc −1 ‖˜ξ‖ Lpλ (C\(−N,N)2 ) ≤ Cc −1 C 4 .By the last equality in (A.66), we havePr u ˜Rξ = Lu (L ∗ uL u ) −1˜ξ.Hence using again the second inequality in (A.66) and the statement of Claim 2,we obtain∥ ∥(3.54)∥Pr u ˜Rξ ∥p,λ ≤ c −1 C 4 .For every z ∈ C there are at most four indices i ∈ Z for which ρ i (z) ≠ 0. Therefore,we have‖ ̂Rξ‖ p p,λ ≤ 4p ∑ i‖α i ‖ p L p λ (Ωi).Using (3.45,3.47) and the assumption ‖ξ‖ p,λ ≤ 1, it follows that‖ ̂Rξ‖ p p,λ ≤ 4p max{C p 1 , ‖R 0‖ p }C 3 .Combining this with (3.51,3.52,3.53,3.54), Claim 3 follows.□4. Claim. We havesup { ‖∇ A ( ˜Rξ)‖ p,λ∣ ∣ ξ ∈ Γ p λ (g P) : ‖ξ‖ p,λ ≤ 1 } < ∞.Proof of Claim 4. Let ξ ∈ Γ p λ (g P). We define˜ξ := ξ − d ∗ A ̂Rξ,η := (L ∗ uL u ) −1˜ξ,and ρ ∈ Ω 2 (M,g) as in (A.65) in Appendix A.7. By Lemma 124 in the sameappendix, we have(3.55) ∇ A (L u η) = L u d A η + ∇ dAuX η ,where X ξ0 denotes the vector field on M generated by an element ξ 0 ∈ g. Usingthe second part of Lemma 124 (with v := L u η), it follows that(3.56) L ∗ uL u d A η = d A˜ξ − ρ(dA u,L u η) − L ∗ u∇ dAuX η .We choose a constant C so big that|ρ(v,v ′ )| ≤ C|v‖v ′ |, |∇ v X ξ0 | ≤ C|v‖ξ 0 |, ∀x ∈ µ −1 ( ¯B δ ), v,v ′ ∈ T x M, ξ 0 ∈ g.We define C 0 := max { c −1 ,3Cc −2} . Since ˜Rξ = L u η, equalities (3.55,3.56) andRemark 125 imply that‖∇ A ( ˜Rξ)‖ p,λ ≤ C 0(∥ ∥dA˜ξ∥ ∥p,λ + ‖d A u‖ p,λ∥ ∥˜ξ∥ ∥∞).Since ‖d A u‖ p,λ < ∞, Claim 2 and (3.49) now imply Claim 4.□5. Claim. We havesup { ‖∇ A ( ̂Rξ)‖ p,λ∣ ∣ ξ ∈ Γ p λ (g P) : ‖ξ‖ p,λ ≤ 1 } < ∞.

90 3. FREDHOLM THEORY FOR VORTICES OVER THE PLANEProof of Claim 5. Let ξ ∈ Γ p λ (g P) be such that ‖ξ‖ p,λ ≤ 1. We write α i := R i (ξ| Ωi ).Then we have∇ A ( ̂Rξ) = ∑ (ρi ∇ A )α i + dρ i ⊗ α i .iSettingC := 8 p ‖ρ ′ ‖ p ∞C 3 max{C p 1 , ‖R 0‖ p },it follows that‖∇ A ( ̂Rξ)‖ p p,λ ≤ ∑ ()8p−1 ‖∇ A α i ‖ p p,λ + ‖ρ′ ‖ p ∞‖α i ‖ p p,λ≤ C.iHere in the second inequality we used the fact ‖ρ ′ ‖ ∞ ≥ 1, and (3.45). This provesClaim 5, and completes the proof of Claim 1 and hence of Theorem 64. □

APPENDIX AAuxiliary results about vortices, weighted spaces,and other topicsIn the appendix some additional results are recollected, which were used in theproofs of the main theorems of this memoir. As always, we denote N 0 := N ∪ {0},by B r ⊆ C the open ball of radius r, by Sr 1 ⊆ C the circle of radius r, centeredat 0, and by A(P) the affine space of smooth connection one-forms on a smoothprincipal bundle P. Let M,ω,G,g, 〈·, ·〉 g ,µ,J be as in Chapter 1, and (Σ,j) aRiemann surface, equipped with a compatible area form ω Σ . For ξ ∈ g and x ∈ Mwe denote by L x ξ ∈ T x M the (infinitesimal) action of ξ at x. Let p > 2, P be a(principal) G-bundle over Σ of class W 2,ploc, A a connection one-form on P of classW 1,p1,ploc, and u : P → M a G-equivariant map of class Wloc . We denote w := (P,A,u),define the energy density e w as in (1.11), and denote by∫E(w) := e w ω Σthe energy of w.A.1. Auxiliary results about vorticesLet M,ω,G,g, 〈·, ·〉 g ,µ,J be as in Chapter 1. 1 The following result was used inthe proof of Proposition 24 in Section 2.3.70. Proposition. Let w := (P,A,u) be a smooth vortex over C with finiteenergy, such that the closure of the image u(P) ⊆ M is compact. Then there existsa smooth section σ : C → P, such that∫(A.1)(u ◦ σ) ∗ ω → E(w), as R → ∞.B RThis result is a consequence of [GS, Proposition 11.1]. For the convenience ofthe reader we include a proof here. 2 We need the following. We denote by γ thestandard angular one-form on R 2 \ {0}. 371. Lemma. Let P be a smooth G-bundle over R 2 and A ∈ A(P). Then thereexists a smooth section σ : R 2 → P, such that∫ ∫∫(A.2)|σ ∗ A|Rγ ≤ |F A |d 2 x + |σ ∗ A|γ,B R\B 1S 1 R1 As always, we assume that hypothesis (H) is satisfied.2 This proof is similar to the one of [GS, Proposition 11.1], however, it relies on the isoperimetricinequality for the invariant symplectic action functional proved in [Zi2] (Theorem 1.2)rather than the earlier inequality [GS, Lemma 11.3].3 By our convention this form integrates to 2π over any circle centered at the origin.ΣS 1 191

92 A. AUXILIARY RESULTSwhere the norms are with respect to the standard metric on R 2 .Proof of Lemma 71. We choose σ such that(A.3) (σ ∗ A) x x = 0, ∀x ∈ R 2 \ B 1 .(This means that σ ∗ A is in radial gauge on R 2 \B 1 . Such a section exists, since (A.3)corresponds to a family of ordinary differential equations, one for each directionx ∈ S 1 .) We identify R/(2πZ) with S 1 and defineϕ : Σ := [0, ∞) × S 1 → C = R 2 , ϕ(s,t) := e s+it , Ψ := ( (σ ◦ ϕ) ∗ A ) t : Σ → g,where the subscript “t” refers to the t-component of the one-form (σ ◦ ϕ) ∗ A. Itfollows from (A.3) that the s-component of this form vanishes, and therefore,Using the estimate(σ ◦ ϕ) ∗ A = Ψdt, (σ ◦ ϕ) ∗ F A = ∂ s Ψdsdt.|Ψ(s,t)| ≤∫ s0|∂ s Ψ(s ′ ,t)|ds ′ + |Ψ(0,t)|,it follows that∫∣ (σ ◦ ϕ) ∗ A ∣ ∫∫∣ ∣ ∣ Σdt =(σ ◦ ϕ) ∗ F ∣Σ A ds ′ dt +(σ ◦ ϕ) ∗ A ∣ Σdt,{s}×S 1 [0,s]×S 1 {0}×S 1for every s ≥ 0, where the subscript “Σ” indicates that the norms are taken withrespect to the standard metric on Σ. Inequality (A.2) follows from this by a straightforwardcalculation. This proves Lemma 71.□Proof of Proposition 70. Let w := (P,A,u) be as in the hypothesis andR > 0. We choose a section σ : C → P as in Lemma 71. Denoting by E(w,B R )the energy of w over B R , we have, by [CGS, Proposition 3.1],∫(E(w,B R ) = (u ◦ σ) ∗ ω − d 〈 µ ◦ u ◦ σ,σ ∗ A 〉 )gB∫R∫(A.4)= (u ◦ σ) ∗ 〈ω − µ ◦ u ◦ σ,σ ∗ A 〉 . gB RLet ε > 0. By [Zi2, Corollary 1.4] there exists a constant C 1 such that√(A.5)ew (z) ≤ C 1 |z| −2+ε , ∀z ∈ C \ B 1 .Combining this with the estimate |F A | ≤ √ e w , and using (A.2), it follows that∫|σ ∗ A|Rγ ≤ C 2 R ε + C 3 , where C 2 := 2πC ∫1, C 3 := |σ ∗ A|γ.εS 1 RCombining this with the inequality |µ ◦ u| ≤ √ e w and (A.5), it follows that∫〈 µ ◦ u ◦ σ,σ ∗ A 〉 ∣g∣ ≤ C 1C 2 R −2+2ε + C 1 C 3 R −2+ε .S 1 RBy choosing ε ∈ (0,1) and using (A.4), the convergence (A.1) follows. This provesProposition 70.□The next result was used in the proofs of Propositions 44 and 45 (Section 2.6).S 1 RS 1 1

A.1. AUXILIARY RESULTS ABOUT VORTICES 9372. Lemma (Bound on energy density). Let K ⊆ M be a compact subset.Then there exists a constant E 0 > 0 such that the following holds. Let r > 0, P bea smooth G-bundle over B r , p > 2, and (A,u) a vortex on P of class W 1,ploc , suchthatu(P) ⊆ K,E(w,B r ) ≤ E 0 .(where E(w,B r ) denotes the energy of w over the ball B r ). Then we havee w (0) ≤ 8πr 2 E(w,B r).For the proof of Lemma 72 we need the following lemma.73. Lemma (Heinz). Let r > 0 and c ≥ 0. Then for every function f ∈C 2 (B r , R) satisfying the inequalities∫f ≥ 0, ∆f ≥ −cf 2 , f < πB r8cwe havef(0) ≤ 8πr 2 ∫B rf.Proof of Lemma 73. This is [MS2, Lemma 4.3.2].Proof of Lemma 72. Since G is compact, we may assume w.l.o.g. that Kis G-invariant. The result then follows from Theorem 76 below, the calculation inStep 1 of the proof of [GS, Proposition 11.1.], and Lemma 73.□Lemma 72 has the following consequence.74. Corollary (Quantization of energy). If M is equivariantly convex at ∞4 , then we haveinf E(w) > 0,wwhere w = (P,A,u) ranges over all vortices over C with P smooth and (A,u) ofclass W 1,plocfor some p > 2, such that E(w) > 0 and ū(P) is compact.Proof of Corollary 74. This is an immediate consequence of Proposition79 below and Lemma 72. □This corollary implies that the minimal energy E 1 of a vortex over C (definedas in (2.33)) is positive, and therefore E min > 0 (defined as in (2.34)).The next lemma was used in the proofs of Proposition 38 and Lemma 42 (Section2.5). It is a consequence of [GS, Lemma 9.1]. Let (Σ,ω Σ ,j) be a surface withan area form and a compatible complex structure.75. Lemma (Bounds on the momentum map component). Let c > 0, Q ⊆ Σ\∂Σand K ⊆ M be compact subsets, and p > 2. Then there exist positive constants□4 as defined in Chapter 1 on p. 6

94 A. AUXILIARY RESULTSR 0 and C p such that the following holds. Let R ≥ R 0 , P be a smooth G-bundleover Σ, and (A,u) an R-vortex on P of class W 1,ploc, such thatThen∫Qu(P) ⊆ K,‖d A u‖ L∞ (Σ) ≤ c,|ξ| ≤ c|L u(p) ξ|, ∀p ∈ P, ∀ξ ∈ g.|µ ◦ u| p ω Σ ≤ C p R −2p , sup |µ ◦ u| ≤ C p R 2 p −2 ,Qwhere we view |µ ◦ u| as a function from Σ to R.Proof of Lemma 75. This follows from the proof of [GS, Lemma 9.1], usingTheorem 76 below.□The next result was also used in the proofs of Proposition 45 (Section 2.6) andLemma 72, and will be used in the proof of Proposition 77.76. Theorem (Regularity modulo gauge over a compact surface). Assume thatΣ is compact. Let P be a smooth G-bundle over Σ, p > 2, and (A,u) a vortex onP of class W 1,p . Then there exists a gauge transformation g ∈ W 2,p (Σ,G) suchthat g ∗ w is smooth over Σ \ ∂Σ.Proof of Theorem 76. This follows from the proof of [CGMS, Theorem3.1], using a version of the local slice theorem allowing for boundary (see [We,Theorem 8.1]).□The next result will be used in the proof of Proposition 79 below.77. Proposition (Regularity modulo gauge over C). Let R ≥ 0 be a number,P a smooth G-bundle over C, p > 2, and w := (A,u) an R-vortex on P of classW 1,p2,ploc. Then there exists a gauge transformation g on P of class Wloc such thatg ∗ w is smooth.Proof of Proposition 77. 5Claim. There exists a collection (g j ) j∈N , where g j is a gauge transformationover B j+1 of class W 2,p , such that for every j ∈ N, we have(A.6)(A.7)g ∗ j w is smooth over B j+1,g j+1 = g j over B j .Proof of the claim. By Theorem 76 there exists a gauge transformationg 1 ∈ W 2,p (B 2 ,G) such that g ∗ 1w is smooth. Let l ∈ N be an integer and assumeby induction that there exist gauge transformations g j ∈ W 2,p (B j+1 ,G), for j =1,...,l, such that (A.6) holds for j = 1,...,l, and (A.7) holds for j = 1,...,l − 1.We show that there exists a gauge transformation g l+1 ∈ W 2,p (B l+2 ,G) such that(A.8)(A.9)g ∗ l+1 w smooth over B l+2,g l+1 = g l over B l .5 This proof follows the lines of the proofs of [Fr1, Theorems 3.6 and A.3].

A.1. AUXILIARY RESULTS ABOUT VORTICES 95We choose a smooth function ρ : ¯B l+2 → B l+1 such that ρ(z) = z for z ∈ B l . ByTheorem 76 there exists a gauge transformation h ∈ W 2,p (B l+2 ,G) such that h ∗ wis smooth over ¯B l+2 . We defineg l+1 := h ( (h −1 g l ) ◦ ρ ) .Then g l+1 is of class W 2,p over B l+2 , and (A.9) is satisfied. Furthermore, g ∗ l w =(h −1 g l ) ∗ h ∗ w is smooth over B l+1 . Therefore, using smoothness of h ∗ w over B l+2 ,Lemma 114(ii) below implies that h −1 g l is smooth over B l+1 . It follows thatg ∗ l+1w = ( (h −1 g l ) ◦ ρ ) ∗h ∗ wis smooth over B l+2 . This proves (A.8), terminates the induction, and concludesthe proof of the claim.□We choose a collection (g j ) as in the claim, and define g to be the uniquegauge transformation on P that restricts to g j over B j . This makes sense by (A.7).Furthermore, (A.6) implies that g ∗ w is smooth. This proves Proposition 77. □The next result was used in the proof of Proposition 38 (Section 2.5).78. Theorem (Compactness for vortex classes over compact surface). Let Σbe a compact surface (possibly with boundary), ω Σ an area form, j a compatiblecomplex structure on Σ, P a G-bundle over Σ, K ⊆ M a compact subset, R ν ∈[0, ∞), p > 2, and (A ν ,u ν ) an R ν -vortex on P of class W 1,p , for every ν ∈ N.Assume that R ν converges to some R 0 ∈ [0, ∞), andu ν (P) ⊆ K,sup ‖d Aν u ν ‖ L p (Σ) < ∞.νThen there exist a smooth R 0 -vortex (A 0 ,u 0 ) on P |(Σ \ ∂Σ) and gauge transformationsg ν on P of class W 2,p , such that g ∗ ν(A ν ,u ν ) converges to (A 0 ,u 0 ), in C ∞on every compact subset of Σ \ ∂Σ.Proof of Theorem 78. This follows from a modified version of the proof of[CGMS, Theorem 3.2]: We use a version of Uhlenbeck compactness for a compactbase with boundary, see Theorem 112 below, and a version of the local slice theoremallowing for boundary, see [We, Theorem 8.1]. Note that the proof carries over tothe case in which R ν = 0 for some ν ∈ N, or R 0 = 0.□The following result was used in the proofs of Theorem 3 (Section 2.7) andCorollary 74.79. Proposition (Boundedness of image). Assume that M is equivariantlyconvex at ∞. Then there exists a G-invariant compact subset K 0 ⊆ M such thatthe following holds. Let p > 2, P a G-bundle over C, and (A,u) a vortex on P ofclass W 1,ploc , such that E(w) < ∞ and u(P) is compact. Then we have u(P) ⊆ K 0.Proof of Proposition 79. Let P be a G-bundle over C. By an elementaryargument every smooth vortex on P is smoothly gauge equivalent to a smoothvortex that is in radial gauge outside B 1 . Using Proposition 77, it follows thatevery vortex on P of class W 1,plocis gauge equivalent to a smooth vortex that is inradial gauge outside B 1 . Hence the statement of Proposition 79 follows from [GS,Proposition 11.1].□

96 A. AUXILIARY RESULTSA.2. The invariant symplectic actionThe proof of Proposition 45 (Energy concentration near ends) in Section 2.6was based on an isoperimetric inequality and an energy action identity for theinvariant action functional (Theorem 82 and Proposition 83 below). Building onwork by D. A. Salamon and R. Gaio [GS], we define this functional as follows. 6We first review the usual symplectic action functional: Let (M,ω) be a symplecticmanifold without boundary. We fix a Riemannian metric 〈·, ·〉 M on M, and denoteby d,exp, |v|,ι x > 0, and ι X := inf x∈X ι x ≥ 0 the distance function, the exponentialmap, the norm of a vector v ∈ TM, and the injectivity radii of a point x ∈ M and asubset X ⊆ M, respectively. We define the symplectic action of a loop x : S 1 → Mof length l(x) < 2ι x(S1 ) to be∫A(x) := − u ∗ ω.DHere D ⊆ C denotes the (closed) unit disk, and u : D → M is any smooth mapsuch thatu(e it ) = x(t), ∀t ∈ R/(2πZ) ∼ = S 1 , d ( u(z),u(z ′ ) ) < ι x(S1 ), ∀z,z ′ ∈ D.80. Lemma. The action A(x) is well-defined, i.e., a map u as above exists, andA(x) does not depend on the choice of u.Proof. The lemma follows from an elementary argument, using the exponentialmap exp x(0+Z) : T x(0+Z) M → M.□Let now G be a compact connected Lie group with Lie algebra g. Suppose thatG acts on M in a Hamiltonian way, with (equivariant) momentum map µ : M → g ∗ ,and that 〈·, ·〉 M is G-invariant. We denote by 〈·, ·〉 : g ∗ × g → R the naturalcontraction. Let P be a smooth G-bundle over S 1 and x ∈ CG ∞ (P,M). We call(P,x) admissible iff there exists a section s : S 1 → P such that l(x ◦ s) < 2ι x(P) ,and∫〈A(g · (x ◦ s)) − A(x ◦ s) = µ ◦ x ◦ s,g −1 dg 〉 ,S 1for every g ∈ C ∞ (S 1 ,G) satisfying l(g · (x ◦ s)) ≤ l(x ◦ s).81. Definition. Let (P,x) be an admissible pair, and A be a connection onP. We define the invariant symplectic action of (P,A,x) to be∫〈 〉A(P,A,x) := A(x ◦ s) + µ ◦ x ◦ s,Ads ,S 1where s : S 1 → P is a section as above.To formulate the isoperimetric inequality, we need the following. If X is amanifold, P a G-bundle over X and u ∈ CG ∞ (P,M), then we define ū : X → Mby ū(y) := Gu(p), where p ∈ P is any point in the fiber over y. We define M ∗as in (2.14). For a loop ¯x : S 1 → M ∗ /G we denote by ¯l(¯x) its length w.r.t. theRiemannian metric on M ∗ /G induced by 〈·, ·〉 M . Furthermore, for each subsetX ⊆ M we definem X := inf { |L x ξ| ∣ ∣ x ∈ X, ξ ∈ g : |ξ| = 1}.The first ingredient of the proof of Proposition 45 is the following.6 This is the definition from [Zi2], written in a more intrinsic way.

A.3. PROOFS OF THE RESULTS OF SECTION 3.1 9782. Theorem (Isoperimetric inequality). Assume that there exists a G-invariantω-compatible almost complex structure J such that 〈·, ·〉 M = ω(·,J·). Then for everycompact subset K ⊆ M ∗ and every constant c > 1 2there exists a constant δ > 0with the following property. Let P be a G-bundle over S 1 and x ∈ CG ∞ (P,M), suchthat x(P) ⊆ K and ¯l(¯x) ≤ δ. Then (P,x) is admissible, and for every connectionA on P we have|A(P,A,x)| ≤ c‖d A x‖ 2 2 + 12m 2 ‖µ ◦ x‖ 2 2.KHere we view d A x as a one-form on S 1 with values in the bundle (x ∗ TM)/G →S 1 , and µ ◦x as a section of the co-adjoint bundle (P ×g ∗ )/G → S 1 . Furthermore,S 1 is identified with R/(2πZ), and the norms are taken with respect to the standardmetric on R/(2πZ), the metric 〈·, ·〉 M on M, and the operator norm on g ∗ inducedby 〈·, ·〉 g . 7Proof of Theorem 82. This is [Zi2, Theorem 1.2].□The second ingredient of the proof of Proposition 45 is the following. Fors ∈ R we denote by ι s : S 1 → R × S 1 the map given by ι s (t) := (s,t). Let X,X ′be manifolds, f ∈ C ∞ (X ′ ,X), P a G-bundle over X, A a connection on P, andu ∈ C ∞ G (P,M). Then the pullback triple f ∗ (P,A,u) consists of a G-bundle P ′ overX ′ , a connection on P ′ , and an equivariant map from P ′ to M.83. Proposition (Energy action identity). For every compact subset K ⊆ M ∗there exists a constant δ > 0 with the following property. Let s − ≤ s + be numbers,Σ := [s − ,s + ] × S 1 , ω Σ an area form on Σ, j a compatible complex structure, andw := (A,u) a smooth vortex over Σ, such that u(P) ⊆ K and ¯l(ū ◦ ι s ) < δ, forevery s ∈ [s − ,s + ]. Then the pairs ι ∗ s ±(P,u) are admissible, andE(w) = −A ( ι ∗ s +(P,A,u) ) + A ( ι ∗ s −(P,A,u) ) .Proof of Proposition 83. This follows from [Zi2, Proposition 3.1].□A.3. Proofs of the results of Section 3.1This section contains the proofs of Lemmas 52,60 and Proposition 62, whichwere stated in Section 3.1. We also state and prove Lemma 85, which was usedin Definition 61 in that section. Let M,ω,G,,g, 〈·, ·〉 g ,µ and J be as in Chapter1, Σ := C, ω Σ the standard area form ω 0 , p > 2, and λ > 1 − 2/p. Assume thathypothesis (H) holds.Lemma 52 was used in the definition of the equivariant homology class of anequivalence class of triples (P,A,u) (Definition 54). Its proof is based on the followingresult, which was also used in the proofs of Theorem 4 (Section 3.2.1), 64(Section 3.2.4) and Proposition 66 (Section 3.2.2).84. Lemma. For every (P,A,u) ∈ ˜B p λthere exists a section σ of the restrictionof the bundle P to B1 C := C\B 1 , of class W 1,ploc , and a point x ∞ ∈ µ −1 (0), such thatu ◦ σ(re iϕ ) converges to x ∞ , uniformly in ϕ ∈ R, as r → ∞, and σ ∗ A ∈ L p λ (BC 1 ).Proof of Lemma 84.7 Note that in [Zi2, Theorem 1.2] S 1 was identified with R/Z instead. Note also that hypothesis(H) is not needed for Theorem 82.

98 A. AUXILIARY RESULTS1. Claim. The expression |µ ◦ u|(re iϕ ) converges to 0, uniformly in ϕ ∈ R, asr → ∞.Proof of Claim 1. We define the function f := |µ ◦ u| 2 : M → R. It followsfrom the ad-invariance of 〈·, ·〉 g that(A.10) df = 2 〈 d A (µ ◦ u),µ ◦ u 〉 g = 2〈 dµ(u)d A u,µ ◦ u 〉 g .Since u(P) ⊆ M is compact, we have sup C |dµ(u)| < ∞ and sup C |µ ◦ u| < ∞.Furthermore, |d A u| ≤ √ e w ∈ L p λ. Combining this with (A.10), it follows that df ∈L p λ. Therefore, by Proposition 91 in the next section (Hardy-type inequality, appliedwith u replaced by f and λ replaced by λ − 1) the expression f(re iϕ ) converges tosome number y ∞ ∈ R, as r → ∞, uniformly in ϕ ∈ R. Since |µ ◦ u| ≤ √ e w ∈ L p λ , itfollows that y ∞ = 0. This proves Claim 1.□It follows from hypothesis (H) that there exists a δ > 0 such that µ −1 ( ¯B δ ) ⊆ M ∗(defined as in (2.14)). We choose R > 0 so big that |µ ◦ u|(z) ≤ δ if z ∈ BR−1 C =C \ B R−1 . Since G is compact, the action of it on M is proper. Hence the localslice theorem implies that M ∗ /G carries a unique manifold structure such thatthe canonical projection π M ∗: M ∗ → M ∗ /G is a submersion. Consider the mapū : BR−1 C → M ∗ /G defined by ū(z) := Gu(p), where p ∈ π −1 (z) ⊆ P is arbitrary.2. Claim. The point ū(re iϕ ) converges to some point ¯x ∞ ∈ µ −1 (0)/G, uniformlyin ϕ ∈ R, as r → ∞.Proof of Claim 2. We choose n ∈ N and an embedding ι : M ∗ /G → R n .Furthermore, we choose a smooth function ρ : C → R that vanishes on B R−1 andequals 1 on B C R . We define f : C → Rn to be the map given by ρ · ι ◦ ū on B C R−1and by 0 on B R−1 . It follows that(A.11) ‖df‖ Lpλ (BC R ) ≤ ∥ ∥ dι(ū)dū∥L pλ (BC R ) + ‖ι ◦ ū dρ‖ L p λ (BR\BR−1) .A short calculation shows that |dū| ≤ |d A u|, and therefore,∥(A.12) ∥dι(ū)dū ∥ ≤ ‖dι(ū)‖ Lpλ (BC R ) L ∞ (BR C) ‖d A u‖ Lpλ (BC R ) .Our assumption w = (P,A,u) ∈ ˜B p λ implies that ‖d Au‖ Lpλ (BC R ) < ∞. Furthermore,µ is proper by the hypothesis (H), hence the set µ −1 ( ¯B δ ) is compact. Thus thesame holds for the set π M ∗ (µ −1 ( ¯B δ )). This set contains the image of ū. It followsthat ‖dι(ū)‖ L∞ (B C R ) < ∞. Combining this with (A.11,A.12), we obtain the estimate‖df‖ Lpλ (C) ≤ ‖df‖ Lpλ (BR) + ‖df‖ Lpλ (BC R ) < ∞.Hence the hypotheses of Proposition 91 (with λ replaced by λ − 1) are satisfied.It follows that the point f(re iϕ ) converges to some point y ∞ ∈ R n , uniformly inϕ ∈ R, as r → ∞. Claim 2 follows.□Let ¯x ∞ be as in Claim 2. We choose a local slice around ¯x ∞ , i.e., a pair (Ū, ˜σ),where Ū ⊆ M ∗ /G is an open neighborhood of ¯x ∞ , and ˜σ : Ū → M ∗ is a smoothmap satisfying π M ∗ ◦ ˜σ = id Ū . Then there exists a unique section σ ′ of P | B CR, ofclass W 1,ploc , such that ˜σ ◦ ū = u ◦ σ′ . By the continuous homotopy lifting propertyof P we may extend this to a continuous section σ ′′ of P | B C1. Regularizing σ ′′ onB R+1 \B 1 , we obtain a section σ of P | B C1, of class W 1,ploc .8 We define x ∞ := ˜σ(¯x ∞ ).8 For this we regularize σ ′′ suitably in local trivializations.

A.3. PROOFS OF THE RESULTS OF SECTION 3.1 99It follows from Claim 2 that u◦σ(re iϕ ) converges to x ∞ , uniformly in ϕ, for r → ∞.Furthermore,Since‖σ ∗ L u A‖ Lpλ (BC R+1 ) ≤ ‖σ ∗ du‖ Lpλ (BC R+1 ) + ‖σ ∗ d A u‖ Lpλ (BC R+1 ) ,σ ∗ du = du dσ = d˜σ dū, on BR+1 C , |dū| ≤ |d Au|.{inf |L u (p)ξ| ∣ }p ∈ P |B CR+1, ξ ∈ g : |ξ| = 1 > 0, ‖d A u‖ p,λ < ∞,it follows that σ ∗ A ∈ L p λ (BC 1 ). This proves Lemma 84.Proof of Lemma 52. We prove statement (i): We denote by π : P → Cthe bundle projection, and identify S 2 ∼ = C ∪ {∞}. We choose σ and x ∞ as inLemma 84, and we define ˜P to be the quotient of P ∐ ( (S 2 \ {0}) × G ) underthe equivalence relation generated by p ∼ (π(p),g), where g ∈ G is determined by(σ ◦ π(p))g = p, for p ∈ P. We define ι to be the canonical map from P to ˜P, andũ : ˜P → M to be the unique map satisfyingũ ◦ ι = u, ũ([(∞,g)]) := g −1 x ∞ , ∀g ∈ G.The statement of Lemma 84 implies that this map is continuous. This proves (i).We prove statement (ii). Uniqueness of ˜Φ follows from the condition ˜Φ ◦ ι ′ =ι ◦Φ and continuity of ˜Φ. We prove existence: We define the map ϕ : ˜P ∞ ′ → ˜P ∞ asfollows. The map ũ descends to a continuous map ˜f : S 2 → M/G. Recall that M ∗denotes the set of points in M where G acts freely. We denote Ũ := ˜f −1 (M ∗ /G).Since M ∗ is open, the set Ũ is, as well. Since G is compact, the canonical mapM ∗ → M ∗ /G defines a smooth G-bundle. We denote by ˜π : ˜P → S 2 the projectionmap. We define(A.13) ˜Ψ : ˜π −1 (Ũ) → ˜f| ∗ e UM ∗ , ˜Ψ(˜p) :=(˜π(˜p),ũ(˜p)).Furthermore, we define ˜f ′ ,Ũ ′ , ˜π ′ , ˜Ψ ′ in an analogous way, using ˜P ′ and ũ ′ . Since Φdescends to the identity on C, and u ′ = u ◦ Φ, the maps u and u ′ descend to thesame map C → M/G. Since ũ ◦ ι = u, ũ ′ ◦ ι ′ = u ′ , and ũ and ũ ′ are continuous,it follows that ˜f = ˜f ′ . We claim that there exists a unique map ˜Φ : ˜P′→ ˜P,satisfying(A.14) ˜Φ = ˜Ψ−1 ◦ ˜Ψ ′ on (˜π ′ ) −1 (Ũ), ˜Φ ◦ ι ′ = ι ◦ Φon P ′ .To see uniqueness of this map, note that the hypothesis ũ ′ ( ˜P ′ ∞) ⊆ M ∗ impliesthat ˜P ′ ∞ ⊆ (˜π ′ ) −1 (Ũ). Hence conditions (A.14) determine ˜Φ on the whole of ˜P ′ .Existence of this map follows from the equality ˜Ψ ′ ◦ι ′ = ˜Ψ◦ι◦Φ on π ′ −1 (Ũ ∩C) ⊆ P ′ ,which follows from the assumptions ũ ′ ◦ ι ′ = u ′ , ũ ◦ ι = u, and u ◦ Φ = u ′ . The map˜Φ has the required properties. This proves (ii) and completes the proof of Lemma52. □We now prove Lemma 60 (p. 73), which was used in Section 3.1, in order todefine the (ω,µ)-homotopy class of an equivalence class W of triples (P,A,u). (SeeDefinition 61.)Proof of Lemma 60. To prove the first statement, we choose a section σ 0of the restriction of P to the disk D, of class W 2,ploc. By Lemma 84 there exists asection ˜σ of the restriction of the bundle P to B1 C := C \ B 1 , of class W 1,ploc , and a□

100 A. AUXILIARY RESULTSpoint x ∞ ∈ µ −1 (0), such that u ◦ ˜σ(re iϕ ) converges to x ∞ , uniformly in ϕ ∈ R, asr → ∞. We define g ∞ : S 1 → G to be the unique map satisfying ˜σ = σ 0 g ∞ , on S 1 ,and σ to be the continuous section of P that agrees with σ 0 on B 1 , and satisfiesσ(z) = ˜σ(z)g ∞ (z/|z|) −1 , ∀z ∈ B C 1 .By regularizing σ, we may assume that it is of class W 1,ploc. This section satisfies therequirements of the first part of the lemma.To prove the second statement, let σ be a section of P of class W 1,ploc, such thatthe map u ◦ σ : C → M continuously extends to a map ũ : Σ → M. It follows fromClaim 2 in the proof of Lemma 84 that there exists a point ¯x ∞ ∈ M = µ −1 (0)/G,such that Gũ(z) = ¯x ∞ , for every z ∈ ∂Σ. The second statement follows from this.This proves Lemma 60.□The next lemma was used in Definition 61.85. Lemma. Let p > 2, λ > 1 − 2/p, (P,A,u) ∈ ˜B p λbe a triple, and σ,σ′sections of P as in Lemma 60. Then the continuous extensions ũ,ũ ′ : Σ → M ofu ◦ σ,u ◦ σ ′ are weakly (ω,µ)-homotopic.Proof of Lemma 85. Let R ∈ (0, ∞). We denote by B R and B R the openand closed balls in C, of radius R, centered around 0.Claim. There exists a continuous map h : [0,1] × Σ → M such that(A.15)h(0, ·) = ũ, h(1,z) = ũ ′ (z), ∀z ∈ B R , h(t,z) = h(0,z), ∀z ∈ ∂Σ,t ∈ [0,1].Proof of the claim. We defineg 0 : C → G,g 0 (z)σ ′ (z) := σ(z).There exists a continuous map g : [0,1] × C → G such that(A.16) g(0, ·) = g 0 , g(1,z) = 1, ∀z ∈ B R , g(t,z) = g 0 (z), ∀z ∈ C \ B R+1 .To see this, observe that we may assume without loss of generality that g 0 (0) = 1.(Here we use the assumption that G is connected.) We choose a continuous mapf : [0,1] × C → C such thatf(0, ·) = id, f(1,z) = 0, ∀z ∈ B R , f(t,z) = z, ∀z ∈ C \ B R+1 .We define g := g 0 ◦ f. This map satisfies (A.16). We now define{ ( )u g(t,z)σ(z) , if z ∈ C,h(t,z) :=ũ(z), if z ∈ ∂Σ.This map satisfies the conditions (A.15). This proves the claim.It follows from hypothesis (H) that there exists a number R > 0 such thatũ(Σ \ B R ) ⊆ M ∗ . By the claim, we may assume without loss of generality thatũ = ũ ′ on B R . Since G is compact, the canonical projection π : M ∗ → M ∗ /Gnaturally defines a smooth G-bundle. It follows that the map[0,1] × (Σ \ B R ) ∋ (t,z) ↦→ π ◦ ũ(z) ∈ M ∗ /Ghas a continuous lift h : [0,1] × (Σ \ B R ) → M ∗ that agrees with the map (0,z) ↦→ũ(z) on {0} × (Σ \ B R ), with the map (1,z) ↦→ ũ ′ (z) on {1} × (Σ \ B R ), and with□

A.3. PROOFS OF THE RESULTS OF SECTION 3.1 101the map (t,z) ↦→ ũ(z) on [0,1] × SR 1 , where S1 R ⊆ C denotes the circle of radius R,around 0. The map{ũ(z), if z ∈ BR ,[0,1] × Σ ∋ (t,z) ↦→h(t,z), otherwise,is a weak (ω,µ)-homotopy from ũ to ũ ′ . This proves Lemma 85.□We now prove Proposition 62. We need the following. Let w be a representativeof W. Let ( ˜P,ι,ũ) be an extension as in Lemma 52 (p. 69). Then ω induces afiberwise symplectic form ˜ω on the topological vector bundle TM eu = (ũ ∗ TM)/Gover S 2 . We denote by c 1 (TM eu , ˜ω) its first Chern class.86. Lemma (Chern number). We have〈(A.17)cG1 (M,ω),[W] 〉 = 〈 c 1 (TM eu , ˜ω),[S 2 ] 〉 .For the proof of this lemma we need the following remark.87. Remark. Let G be a topological group, X and X ′ topological spaces, P →X and P ′ → X ′ topological G-bundles, M a topological G-space, E a G-equivariantsymplectic vector bundle over M, and u : P → M, θ : P → P ′ continuous G-equivariant maps. We define f : X → (M ×P ′ )/G to be the map induced by u andθ. Then the symplectic vector bundles(A.18) (u ∗ E)/G, f ∗( (E × P ′ )/G )are isomorphic. Here we denote by (u ∗ E)/G the symplectic vector bundle over Xobtained from the pullback bundle u ∗ E → P as the quotient by the G-action, byE × P ′ the natural symplectic vector bundle over M × P ′ , and by (E × P ′ )/G theinduced symplectic vector bundle over (M × P ′ )/G. An isomorphism between thebundles in (A.18) is given by the mapG(p,v) ↦→ ( π(p),G(v,θ(p)) ) .✷Proof of Lemma 86. We choose a continuous G-equivariant map θ : ˜P →EG. We denote byf : S 2 ∼ = C∐{∞} → (M × EG)/Gthe map induced by (ũ,θ) : ˜P → M × EG. By definition, cG1 (M,ω) is the firstChern class of the vector bundle(TM × EG)/G → (M × EG)/G,equipped with the fiberwise symplectic form induced by ω. Furthermore, we have[W] = f ∗ [S 2 ]. Therefore, equality (A.17) follows from naturality of the first Chernclass under pullback by the map f, and Remark 87, with X := S 2 , (P,u) replacedby ( ˜P,ũ), and (E,P ′ ) := (TM,EG). This proves Lemma 86.□For the proof of Proposition 62 we also need the following.88. Remark. Let Σ be an oriented topological surface homeomorphic to theclosed disk. We denote by Σ ′ the surface obtained from Σ by collapsing its boundaryto a point 9 , and by f : Σ → Σ ′ the canonical “collapsing” map. Let (E,ω) be a9 The surface Σ ′ is homeomorphic to S 2 .

102 A. AUXILIARY RESULTSsymplectic vector bundle over Σ ′ , (V,Ω) a symplectic vector space of dimension therank of E, and Ψ : Σ × V → f ∗ E a symplectic trivialization. Then we have〈(A.19) c1 (E,ω),[Σ ′ ] 〉 )= m ∂Σ,Ω(∂Σ × ∂Σ ∋ (z,z ′ ) ↦→ Ψ −1z Ψ ′ z ∈ AutΩ .Here [Σ ′ ] denotes the fundamental class of Σ ′ and AutΩ the group of linear symplecticautomorphisms of V , and m ∂Σ,Ω is defined as in (3.4). Furthermore, wedenote by π : Σ → Σ ′ the canonical map, and we use the canonical identification(f ∗ E) z = E π(z) , for z ∈ ∂Σ. Equality (A.19) follows from an elementary argument(e.g. an argument as in the proof of [MS1, Theorem 2.69].) ✷Proof of Proposition 62 (p. 73). We choose an extension ( ˜P,ι,ũ) as inLemma 52(i), such that ũ( ˜P ∞ ) ⊆ µ −1 (0). (It follows from the proof of Lemma52(i) that ũ may be chosen to satisfy this condition.) By Lemma 86, equality (A.17)holds. We denote by ˜π : ˜P → S 2 the canonical projection, by Σ the compact surfaceobtained from C by “gluing a circle at ∞”, and byf : Σ → C ∐ {∞} ∼ = S 2the map that is the identity on the interior C = intΣ, and maps the boundary∂Σ ∼ = S 1 to ∞. We choose a continuous map ˜σ : Σ → ˜P, such that ˜π ◦ ˜σ = f. Wedefine v := ũ ◦ ˜σ : Σ → M. This map continuously extends the map u ◦ σ, whereσ := ˜σ| C . Hence, by definition, the (ω,µ)-homotopy class of W equals the (ω,µ)-homotopy class of v. We choose a symplectic trivialization Ψ : Σ × V → v ∗ TM.For z,z ′ ∈ ∂Σ we define g z′ ,z ∈ G to be the unique element satisfying(A.20)˜σ(z ′ )g z ′ ,z = ˜σ(z).It follows that v(z ′ ) = g z′ ,zv(z), and hence, using the definition of the Maslov index,( )m ω,µ (ω,µ)-homotopy class of W()(A.21) = m ∂Σ,Ω ∂Σ × ∂Σ ∋ (z ′ ,z) ↦→ Ψ −1 g z ′ ,z · Ψ z ∈ AutΩ .The statement of Proposition 62 is now a consequence of the following claim.Claim. The number (A.21) agrees with 〈 c 1(TM eu , ˜ω ) ,[S 2 ] 〉 .Proof of the claim. We define the map˜Ψ : Σ × V → f ∗ TM eu , ˜Ψz w := ˜Ψ(z,w) := ( z,G (˜σ(z),Ψ z w )) .This is a continuous symplectic trivialization. We denote by Σ ′ the surface obtainedfrom Σ by collapsing its boundary ∂Σ to a point. There is a canonical homeomorphismC ∐ {∞} → Σ ′ , and the composition of f with this map agrees with thecollapsing map. Therefore, applying Remark 88 with E := TM eu and ω,Ψ replacedby ˜ω, ˜Ψ, we have(A.22)〈c1(TM eu , ˜ω ) ,[S 2 ] 〉 = m ∂Σ,Ω(∂Σ × ∂Σ ∋ (z,z ′ ) ↦→Equality (A.20) implies that˜Ψ −1z˜Ψz = Ψ −1′ z g ′ z ′ ,z · Ψ z , ∀z,z ′ ∈ ∂Σ.z ′˜Ψ−1z ′ ˜Ψz ∈ AutΩ ) .Combining this with (A.22), the claim follows. This proves Proposition 62.□

A.4. WEIGHTED SOBOLEV SPACES AND A HARDY-TYPE INEQUALITY 103A.4. Weighted Sobolev spaces and a Hardy-type inequalityLet d ∈ Z. The following lemma was used in Section 3.2.2 in order to definenorms on the vector spaces ̂X p,λ,d ′ ′′and ̂Xp,λdefined in (3.18). If d < 0 then letρ 0 ∈ C ∞ (C,[0,1]) be such that ρ 0 (z) = 0 for |z| ≤ 1/2 and ρ 0 (z) = 1 for |z| ≥ 1.In the case d ≥ 0 we set ρ 0 := 1. Recall the definitionsp d : C → C, p d (z) := z d , 〈x〉 := √ 1 + |v| 2 , ∀v ∈ R n .89. Lemma. For every 1 < p < ∞ and λ > −2/p the mapC ⊕ L 1,pλ−d (C, C) → C · ρ 0p d + L 1,pλ−d(C, C), (vis an isomorphism of vector spaces.∞,v) ↦→ v ∞ ρ 0 p d + vProof of Lemma 89. This follows from a straight-forward argument.The following proposition was used in the proofs of Theorem 63 (Section 3.2.2)and Proposition 67 (Section 3.2.3). For every normed vector space V we denoteby C b (R n ,V ) the space of bounded continuous maps from R n to V . We denote byB r the ball of radius r in R n , and by X C the complement of a subset X ⊆ R n .Recall the definitions (3.16,3.17) of the weighted Sobolev spaces L k,pλ (Ω,W).W k,pλ□(Ω,W) and90. Proposition (Weighted Sobolev spaces). Let n ∈ N. Then the followingstatements hold.(i) Let n < p < ∞. Then for every λ ∈ R there exists C > 0 such that(A.23) ‖u〈·〉 λ+ n 1,1p ‖L ∞ (R n ) ≤ C‖u‖ L1,pλ (Rn ), ∀u ∈ Wloc (Rn ).If λ > −n/p then L 1,pλ (Rn ) is compactly contained in C b (R n ).(ii) For every k ∈ N 0 , 1 < p < ∞ and λ ∈ R the mapW k,pλ (Rn ) ∋ u ↦→ 〈·〉 λ u ∈ W k,p (R n )is a well-defined isomorphism (of normed spaces).(iii) Let p > 1, λ ∈ R, and f ∈ L ∞ (R n ) be such that ‖f‖ L ∞ (R n \B i) → 0, fori → ∞. Then the operatorW 1,pλ (Rn ) ∋ u ↦→ fu ∈ L p λ (Rn )is compact.(iv) For every 1 < p < ∞, λ ∈ R, d ∈ Z, and u ∈ L 1,pλ (BC 1 ) the following inequalityholds:‖p d u‖ L1,pλ−d (BC 1 ) ≤ max { − d2 (−d+3)/2 ,2 } ‖u‖ L1,pλ (BC 1 ).Proof of Proposition 90. Proof of statement (i): Inequality (A.23) followsfrom inequality (1.11) in Theorem 1.2 in the paper [Ba] by R. Bartnik. Assumenow that λ > −n/p. Then it follows from Morrey’s embedding theorem that thereexists a canonical bounded inclusion L 1,pλ (Rn ) ֒→ C b (R n ). In order to show thatthis inclusion is compact, let u ν ∈ L 1,pλ (Rn ) be a sequence such that(A.24)C := sup ‖u ν ‖ L1,pνλ (Rn ) < ∞.By Morrey’s embedding theorem and the Arzelà-Ascoli theorem on ¯B j (for j ∈ N)and a diagonal subsequence argument, there exists a subsequence u νj of u ν that

104 A. AUXILIARY RESULTSconverges to some map u ∈ W 1,ploc (Rn ), weakly in W 1,p (B j ), and strongly in C( ¯B j ),for every j ∈ N.Claim. We have u ∈ C b (R n ) and u νj converges to u in C b (R n ).Proof of the claim. We choose a constant C ′ as in the first part of (i). For everyR > 0 we have‖u‖ L1,pλ (BR)≤ limsup ‖u νj ‖ L1,pλ (BR)≤ C.jHence u ∈ L 1,pλ (Rn ). Since λ > −n/p, by inequality (A.23), this implies u ∈ C b (R n ).To see the second statement, we choose a smooth function ρ : R n → [0,1] such thatρ(x) = 0 for x ∈ B 1 , ρ(x) = 1 for x ∈ B C 3 , and |Dρ| ≤ 1. Let R ≥ 1 and j ∈ N. Wedefine ρ R := ρ(·/R) : R n → [0,1]. Abbreviating v j := u νj − u, we have(A.25) ‖v j ‖ ∞ ≤ ∥ ∥ vj (1 − ρ R ) ∥ ∥∞+ ∥ ∥ vj ρ R∥∥∞ .Inequality (A.23) and the fact ρ R = 0 on B R imply that∥(A.26)∥ vj ρ ∥∞ R ≤ C ′ R −λ− n p ‖vj ρ R ‖ 1,p,λ .Furthermore, using (A.24), we have‖v j ρ R ‖ 1,p,λ ≤ 2‖v j ‖ 1,p,λ ≤ 4C.Combining this with (A.25) and (A.26), and the fact lim j→∞ ‖v j ‖ L ∞ (B 3R) = 0, itfollows thatlimsup ‖v j ‖ ∞ ≤ 4CC ′ R −λ− n p .j→∞Since λ > −n/p and R ≥ 1 is arbitrary, it follows that u νj converges to u in C b (R n ).This proves the claim and completes the proof of statement (i).Statement (ii) follows from a straight-forward calculation.Proof of statement (iii): Let f ∈ L ∞ (R n ) be as in the hypothesis. Letu ν ∈ W 1,pλ(Rn ) be a sequence such thatC := sup ‖u ν ‖ W1,pνλ (Rn ) < ∞.By the Rellich-Kondrashov compactness theorem on ¯B j (for j ∈ N) and a diagonalsubsequence argument there exists a subsequence (ν j ) and a map v ∈ L p loc (Rn ),such that fu νj converges to v, strongly in L p (K), as j → ∞, for every compactsubset K ⊆ R n . Elementary arguments show that v ∈ L p λ (Rn ) and fu νj convergesto v in L p λ (Rn ). (For the latter we use the hypothesis that ‖f‖ L ∞ (R n \B i) → 0, asi → ∞.) This proves (iii).Statement (iv) follows from a straight-forward calculation. This completesthe proof of Proposition 90.□The next result was used in the proofs of Proposition 67 (Section 3.2.3) andLemma 84 (Appendix A.3).91. Proposition (Hardy-type inequality). Let n ∈ N, p > n, λ > −n/p andu ∈ W 1,1loc (Rn , R) be such that ‖Du| · | λ+1 ‖ L p (R n ) < ∞. Then u(rx) converges tosome y ∞ ∈ R, uniformly in x ∈ S n−1 , as r → ∞, and(A.27) ∥ (u − y∞ )| · | λ∥ ∥ ≤ p∥ ∥Lp (R n ) Du| · |λ+1 λ + n .Lp (R n )p

A.4. WEIGHTED SOBOLEV SPACES AND A HARDY-TYPE INEQUALITY 105For the proof of this proposition we need the following. We denote by B r theball of radius r in R n , and by X C the complement of a subset X ⊆ R n .92. Lemma (Hardy’s inequality). Let n ∈ N, 1 < p < ∞, λ > −n/p andu ∈ W 1,1loc (Rn , R). If there exists R > 0 such that u| B CR= 0 then‖u| · | λ ‖ L p (R n ) ≤p∥ ∥ Du| · |λ+1 λ + n Lp(∈ [0, ∞]).(R n )pProof of Lemma 92. If u is smooth then the stated inequality follows from[Kav, Chapter 6, Exercise 21]. The general case can be reduced to this case bymollifying the function u. This proves the lemma.□Proof of Proposition 91. Let n,p,λ be as in the hypothesis. We defineε := λ + n p .Claim. There exists a constant C 1 such that for every weakly differentiablefunction u : R n → R and x,y ∈ R n satisfying 0 < |x| ≤ |y|, we have|u(x) − u(y)| ≤ C 1 |x| −ε∥ ∥ Du| · |λ+1 ∥ ∥L p (B C |x| ).Proof of the claim. By Morrey’s theorem there is a constant C such that|u(0) − u(x)| ≤ Cr 1− n p ‖Du‖Lp (B r),for every r > 0, weakly differentiable function u : B r → R, and x ∈ B r . Let u,xand y be as in the hypothesis of the claim. Let N ∈ N be such that 2 N−1 |x| ≤|y| ≤ 2 N |x|. For i = 0,...,N we define x i := 2 i x ∈ R n . Furthermore, we setx N+7 := 2 N |x| y|y| , x N+8 := y,and we choose pointssuch thatx i ∈ S n−12 N |x| := { y ∈ R n ∣ ∣ |y| = 2 N |x| } , i = N + 1,...,N + 6,|x i − x i−1 | ≤ 2 N−1 |x|, ∀i = N + 1,...,N + 7.For i = 0,...,N −1 we have x i ∈ ¯B 2 i |x|(x i+1 ). Hence it follows from the statementof Morrey’s theorem that|u(x i+1 ) − u(x i )| ≤ C(2 i |x|) −ε∥ ∥Du| · | λ+1∥ ∥Lp (B C |x| ).Moreover, for i = N,...,N +7 we have x i+1 ∈ ¯B 2 N−1 |x|(x i ), and hence analogously,Using the inequalitythe claim follows.|u(x i+1 ) − u(x i )| ≤ C(2 N−1 |x|) −ε ‖Du| · | λ+1 ‖ Lp (B C |x| ) .|u(y) − u(x)| ≤∑i=0,...,N+7|u(x i+1 ) − u(x i )|,Let u ∈ W 1,1loc (Rn , R) be such that ‖Du| · | λ+1 ‖ Lp (R n ) < ∞. It follows fromthe claim that there exists y ∞ ∈ R such that u(rx) converges to y ∞ , as r → ∞,uniformly in x ∈ S n−1 . To prove inequality (A.27), we choose a smooth map□

106 A. AUXILIARY RESULTSρ : [0, ∞) → [0,1] such that ρ(t) = 1 for 0 ≤ t ≤ 1, ρ(t) = 0 for t ≥ 2 and|ρ ′ (t)| ≤ 2. We fix a number R > 0 and defineρ R : R → [0,1],ρ R (x) := ρ(|x|/R).We abbreviate v := u − y ∞ . Using Lemma 92 with u replaced by ρ R v, we have‖v| · | λ ‖ L p (B R) ≤ ∥ ρR v| · | λ∥ ∥ ≤ p∥L p (R n )λ + n D(ρR v)| · | λ+1∥ ∥ . L p (R n )pCombining this with a calculation using Leibnitz’ rule, it follows that(A.28) ‖v| · | λ ‖ Lp (B R) ≤ p (λ + n 4‖v| · | λ ‖ Lp (B 2R\B R) + ∥ ∥ )Du| · |λ+1 Lp.(R n )pThe above claim implies thatUsing the equalities|v(x)| ≤ C 1 |x| −ε ‖Du| · | λ+1 ‖ L p (B C R ) , ∀x ∈ B C R.∫B 2R\B R|x| −n dx = log 2|S n−1 |and ε = λ + n/p, it follows that‖v| · | λ ‖ p L p (B 2R\B R) ≤ Cp 1 log 2|Sn−1 | ∥ ∥Du| · | λ+1∥ ∥ p L p (B C R ).Inequality (A.27) follows by inserting this into the right hand side of (A.28) andsending R to ∞. This proves Proposition 91.□The next result will be used to prove Corollary 96 below, which was used inthe proof of Theorem 63 (Section 3.2.2). For every d ∈ Z we define P d and ¯P d tobe the spaces of polynomials in z ∈ C and ¯z of degree less than d. 10 We abbreviateL 1,pλ:= L1,pλ (C, C), Lp λ := Lp λ (C, C), ∂¯z := ∂ C¯z , ∂ z := ∂z C .Let X be a normed vector space and Y ⊆ X a closed subspace. We denote by X ∗the dual space of X and equip X/Y with the quotient norm.93. Proposition (Fredholm property for ∂¯z ). For every d ∈ Z, 1 < p < ∞and −2/p + 1 < λ < −2/p + 2 the following conditions hold.(i) The operator T := ∂¯z : L 1,pλ−1−d → Lp λ−dis Fredholm.(ii) We have kerT = P d .(iii) The map(A.29) ¯P−d → ( ∫L p λ−d /imT) ∗, u ↦→(v + imT ↦→is well-defined and a C-linear isomorphism.C)uv dsdtThe proof of this proposition is based on the following result, which is due toR. B. Lockhart.94. Theorem. Let n,k,m ∈ N 0 be such that n ≥ 2, k ≥ m, λ ∈ R, andT : L k,pλ (Rn , C) → L k−m,pλ+m (Rn , C)a constant coefficient homogeneous elliptic linear operator of order m on R n . Ifλ + n/p ∉ Z then T is Fredholm.10 Hence if d ≤ 0 we have Pd = {0}.

A.4. WEIGHTED SOBOLEV SPACES AND A HARDY-TYPE INEQUALITY 107Proof of Theorem 94. This is an immediate consequence of R. B. Lockhart’sresult [Lo2, Theorem 4.3], using that a bounded linear operator betweenBanach spaces is Fredholm if its adjoint operator is Fredholm.□95. Remark. Let X be a normed vector space and Y ⊆ X a closed subspace.We equip X/Y with the quotient norm. The mapY ⊥ := { ϕ ∈ X ∗ ∣ ∣ ϕ(x) = 0, ∀x ∈ Y}→ (X/Y ) ∗ , ϕ ↦→ ( x + Y ↦→ ϕ(x) ) ,is well-defined and an isometric isomorphism. This follows from a straight-forwardargument. ✷We denote by S the space of Schwartz functions on C and by S ′ the space oftemperate distributions. By ̂ : S ′ → S ′ we denote the Fourier transform, and by∨ : S ′ → S ′ the inverse transform.Proof of Proposition 93. Let d,p,λ, and T be as in the hypothesis.Statement (i) follows from Theorem 94, observing that ∂¯z is elliptic, i.e., itsprincipal symbolσ T : R 2 = C → C, σ T (ζ) = ζ 2does not vanish on R 2 \ {0}.We prove statement (ii). A calculation in polar coordinates shows that forevery polynomial u in z we have(A.30)u ∈ L 1,pλ−1−d ⇐⇒ deg u < d − λ + 1 − 2 p .Hence our assumption λ < −2/p+2 implies that ker T ⊇ P d . Therefore, statement(ii) is a consequence of the following claim.1. Claim. We have ker T ⊆ P d .Proof of Claim 1. Let u ∈ ker T. Then 0 = ̂∂¯z u(ζ) = i 2ζû (as temperatedistributions). It follows that the support of û is either empty or consists of thepoint 0 ∈ C. Hence the Paley-Wiener theorem implies that u is real analyticin the variables s and t, where z = s + it, and there exists N ∈ N such thatsup z∈C |u(z)|〈z〉 N < ∞. 11 Therefore, by Liouville’s Theorem u is a polynomial inthe variable z. Since by our assumption λ > −2/p + 1, it follows from (A.30) thatu ∈ P d . This proves Claim 1.□To prove statement (iii), we define p ′ := p/(p − 1). Consider the isometricisomorphism( ∫ )Φ : L p′−λ+d → (Lp λ−d )∗ , Φ(u) := v ↦→ uv .CDenoting by T ∗ the adjoint operator of T, we haveT ∗ Φ = ∂ z : L p′−λ+d → (L1,p λ−1−d )∗ ,where the derivatives are taken in the sense of distributions.2. Claim. We have ker(T ∗ Φ) = ¯P −d .11 See e.g. [ReSi, Theorem IX.12].

108 A. AUXILIARY RESULTSProof of Claim 2. For every polynomial u in ¯z we have(A.31) u ∈ L p′−λ+d ⇐⇒ deg u < −d + λ − 2 p ′ = −d + λ − 2 + 2 p .Our assumption λ > −2/p + 1 and (A.31) imply that kerT ∗ ⊇ ¯P −d . Furthermore,the inclusion kerT ∗ ⊆ ¯P −d is proved analogously to the inclusion kerT ⊆ P d , usingλ < −2/p + 2 and (A.31). This proves Claim 2.□It follows from Claim 2 that the map Φ restricts to a C-linear isomorphismbetween ¯P −d and kerT ∗ = (imT) ⊥ . The composition of this map with the canonicalisomorphism (imT) ⊥ → ( L p λ−d /imT) ∗described in Remark 95, equals the map(A.29). Statement (iii) follows. This completes the proof of Proposition 93. □Let d ∈ Z, 1 < p < ∞, −2/p+1 < λ < −2/p+2, and ρ 0 : C → [0,1] be a smoothfunction that vanishes on B 1/2 and equals 1 on B1 C . We equip Cρ 0 p d + L 1,pλ−1−dwith the norm induced by the isomorphism of Lemma 89. This norm is complete.(See e.g. [Lo1].)96. Corollary. The map ∂¯z : Cρ 0 p d + L 1,preal index 2 + 2d.λ−1−d → Lp λ−dis Fredholm, withProof of Corollary 96. The composition of the isomorphism of Lemma89 with the above map is given byT + S : C ⊕ L 1,pλ−1−d → Lp λ−d , T(x ∞,u) := ∂¯z u, S(x ∞ ,u) := x ∞ (∂¯z ρ 0 )p d .The map T is the composition of the canonical projection pr : C⊕L 1,pλ−1−d → L1,p λ−1−dwith the operator ∂¯z : L 1,pλ−1−d → Lp λ−d. Using Proposition 93, it follows that Tis Fredholm of real index 2 + 2d. Furthermore, S is compact, since it equals thecomposition of the canonical projection C ⊕ L 1,pλ−1−d→ C (which is compact) witha bounded operator. Corollary 96 follows.□The next result was used in the proof of Theorem 63 (Fredholm property forthe augmented vertical differential) in Section 3.2.2. Let (V, 〈·, ·〉) be a finite dimensionalhermitian vector space, A,B : V → V positive linear maps, λ ∈ R and1 < p < ∞. We define( )T λ :=∂¯z A: W 1,pB ∂λ (C,V ⊕ V ) → Lp λ(C,V ⊕ V ).z97. Proposition. The operator T λ is Fredholm of index 0.For the proof of Proposition 97 we need the following result.98. Proposition. Let (V, 〈·, ·〉),p and A be as above, and n ∈ N. Then themap−∆ + A : W 2,p (R n ,V ) → L p (R n ,V )is an isomorphism (of Banach spaces).Proof of Proposition 98. Consider first the case dim C V = 1 and A = 1.We defineG := (2π) n 2(〈·〉−2 )∨ ∈ S ′ .The map S ∋ u ↦→ G ∗ u ∈ S is well-defined. By Calderón’s Theorem this mapextends uniquely to an isomorphism(A.32)L p (R n , C) ∋ u ↦→ G ∗ u ∈ W 2,p (R n , C).

A.4. WEIGHTED SOBOLEV SPACES AND A HARDY-TYPE INEQUALITY 109(See [Ad, Theorem 1.2.3.].) Note that(−∆ + 1)(G ∗ u) = ( 〈·〉 2 (G ∗ u)̂)∨ = u,for every u ∈ S. It follows that the inverse of (A.32) is given by−∆ + 1 : W 2,p (R n , C) → L p (R n , C).Hence this is an isomorphism.The general case can be reduced to the above case by diagonalizing the mapA. This proves Proposition 98. □Proof of Proposition 97. We abbreviate L p := L p (C,V ⊕ V ), etc.Assume first that λ = 0. We denote by A 1/2 ,B 1/2 : V → V the uniquepositive linear maps satisfying (A 1 2 ) 2 = A, (B 1 2 ) 2 = B. We define( )L :=∂¯z A 1 2 B 1 2: W 1,p → L p .B 1 2 A 1 2 ∂ zA short calculation shows that(A.33) T 0 = ( A 1 2 ⊕ B12)L(A− 1 2 ⊕ B− 1 2).Claim. The operator L is an isomorphism.Proof of the claim. We define( )L ′ −∂z A 1 2 B 1 2:=: W 2,p → W 1,p .B 1 2 A 1 2 −∂¯zBy a short calculation we haveLL ′ = ( − ∆ 4 + A 1 1 ) ( ∆2 BA 2 ⊕ −4 + B 1 1 )2 AB 2 : W 2,p → L p .Since the linear mapsA 1 1 1 12 BA 2 ,B 2 AB 2 : V → Vare positive, Proposition 98 implies that LL ′ is an isomorphism. We denote by(LL ′ ) −1 : L p → W 2,p its inverse and defineR := L ′ (LL ′ ) −1 : L p → W 1,p .Then R is bounded and LR = id L p.By a short calculation, we have LL ′ (u,v) = L ′ L(u,v), for every Schwartz function(u,v) ∈ S. This implies that(LL ′ ) −1 L| S = L(LL ′ ) −1 | S ,and therefore RL| S = id S . Since RL : W 1,p → W 1,p is continuous and S ⊆ W 1,pis dense, it follows that RL = id W 1,p. The claim follows.□The mapsA 1 2 ⊕ B12 : L p → L p , A − 1 2 ⊕ B− 1 2 : W 1,p → W 1,pare automorphisms. Therefore, (A.33) and the claim imply that T 0 is an isomorphism.Consider now the general case λ ∈ R. The mapL p ∋ (u,v) ↦→ 〈·〉 −λ (u,v) ∈ L p λ

110 A. AUXILIARY RESULTSis an isometric isomorphism. Furthermore, by Proposition 90(ii) the mapW 1,pλ∋ (u,v) ↦→ 〈·〉 λ (u,v) ∈ W 1,pis well-defined and an isomorphism. We defineDirect calculations show thatS := 〈·〉 λ (∂¯z 〈·〉 −λ ) ⊕ 〈·〉 λ (∂ z 〈·〉 −λ ) : W 1,p → L p .T λ = 〈·〉 −λ (T 0 + S)〈·〉 λ , |∂¯z 〈·〉 −λ | ≤ |λ|〈·〉 −λ−1 /2, |∂ z 〈·〉 −λ | ≤ |λ|〈·〉 −λ−1 /2.Therefore, Proposition 90(iii) implies that the operator S is compact. Since weproved that T 0 is an isomorphism, it follows that T λ is a Fredholm map of index 0.This proves Proposition 97 in the general case.□A.5. Smoothening a principal bundleThe main result of this section states that a principal bundle of Sobolev classW 2,plocis Sobolev isomorphic to a smooth bundle, if p is large enough. This will beused in the proofs of Propositions 69, 103, and 106 in the next section. Let n ∈ N bean integer, p > n/2 a real number, X a smooth manifold (possibly with boundary)of dimension n, G a compact Lie group, and P a G-bundle over X of class W 2,ploc .1299. Theorem (Smoothening a principal bundle). If p > n 2and P is as abovethen there exists an isomorphism of principal G-bundles of class W 2,ploc, from P toa smooth bundle over X.The proof of this result relies on the facts that there exists a smooth G-bundlethat is “C 0 -close” to P, and that if two “Sobolev bundles” are “C 0 -close” then theyare “Sobolev-isomorphic”. In order to explain this, let U be an open cover of X.We call a collection of functionscompatible iff it satisfiesg U ′ ,U : U ∩ U ′ → G (U,U ′ ∈ U)(A.34) g U,U = 1, g U ′′ ,U ′g U ′ ,U = g U ′′ ,U on U ∩ U ′ ∩ U ′′ , ∀U,U ′ ,U ′′ ∈ U.Remark. The second condition means that the collection (g U ′ ,U) is a Čech1-cocycle. ✷Recall that a cover U of X is called locally finite iff every point x ∈ X possessesa neighborhood which intersects only finitely many sets in U. By a refinement ofthe cover U we mean a cover V of X, together with a map V ∋ V ↦→ U V ∈ U, suchthat V ⊆ U V , for every V ∈ V. The first ingredient of the proof of Theorem 99 isthe following.100. Proposition (Smoothening a compatible collection of maps). Letg U,U ′ ∈ W 2,ploc (U ∩ U ′ ,G) (U,U ′ ∈ U)12 By definition, this means the following. Let U, U ′ ⊆ X be open subsets, and Φ : U ×G → Pand Φ ′ : U ′ × G → P local trivializations. Then the corresponding transition function g U ′ ,U :U ∩ U ′ → G is bounded in W 2,p on every compact subset K ⊆ U ∩ U ′ . Note here that K mayintersect the boundary ∂X.

A.5. SMOOTHENING A PRINCIPAL BUNDLE 111be a compatible collection, and W a neighborhood of the diagonal in G × G. Thenthere exists a locally finite refinement V ∋ V ↦→ U V ∈ U consisting of open precompactsets, and a compatible collection of smooth mapssuch that(A.35)h V,V ′ : V ∩ V ′ → G (V,V ′ ∈ V),(hV,V ′(x),g UV ,U V ′(x) ) ∈ W, ∀V,V ′ ∈ V, x ∈ V ∩ V ′ .Remark. This result says that there exists a smooth Čech 1-cocycle, whichis arbitrarily close in the uniform sense to a given Čech 1-cocycle of Sobolev class.Condition (A.35) may look unconventional, but it is a natural way of stating thecloseness condition. An alternative would be to introduce a Riemannian metric onG and formulate the condition in terms of the induced distance function on G. ✷Proof of Proposition 100. We may assume w.l.o.g. that ∂X = ∅, by consideringthe double X#X. If X is compact then the statement follows from [Is,Theorem 2.1], using the fact that W 2,p -maps are continuous, since p > n/2. In thegeneral case, it follows from an adaption of that proof: We first choose a locally finiterefinement V ∋ V ↦→ U V ∈ U by precompact sets, and define g V,V ′ := g ′ U V ,U V ′ | V ∩V ′.Then we follow the argument of the proof of [Is, Theorem 2.1], for the cover V. 13This proves Proposition 100.□The next lemma will also be used in the proof of Theorem 99.101. Lemma. There exists a neighborhood N of the diagonal G × G with thefollowing property. Let n ∈ N be an integer, p > n 2a real number, X a smoothmanifold of dimension n, and U a locally finite cover of X by precompact sets.Then there exists a refinement V ∋ V ↦→ U V ∈ U such that the following holds. Letg U,U ′ ∈ W 2,p( U ∩ U ′ ,G ) , h U,U ′ ∈ W 2,p( U ∩ U ′ ,G ) (U,U ′ ∈ U)be compatible collections of maps satisfying(gU,U ′(x),h U,U ′(x) ) ∈ N, ∀x ∈ U ∩ U ′ , U,U ′ ∈ U.Then there exists a collection of maps k V ∈ W 2,p (V,G) (V ∈ V), such that(A.36)k −1V ′ h V ′ ,V k V = g V ′ ,V on V ∩ V ′ .Proof of Lemma 101. This is a direct consequence of [We, Lemma 7.2].For the proof of Theorem 99, we also need the following.102. Remark. Let U be an open cover of X, and g U ′ ,U (U,U ′ ∈ U) be acompatible collection of maps. We define the setP (gU ′ ,U ) := { (U,x,g) ∣ ∣ U ∈ U, x ∈ U, g ∈ G}/ ∼,where the equivalence relation ∼ is defined by(U,x,g) ∼ (U ′ ,x ′ ,g ′ ) iff x = x ′ ∈ U ∩ U ′ and g ′ = gg U ′ ,U(x).If the maps g U ′ ,U are smooth, then this set naturally is a smooth G-bundle, and ifthe maps g U ′ ,U are of Sobolev class W k,plocwith kp > n := dimX, then it naturally13 Note that we can choose the sets in the refinement of V as in that proof to be precompact,since the sets in V are precompact.□

112 A. AUXILIARY RESULTSis a G-bundle of class W k,ploc. In either case, a system of local trivializations is givenby(A.37) Φ U : U × G → P, (Φ U ) x (g) := Φ U (x,g) := [U,x,g],where [U,x,g] denotes the equivalence class of (U,x,g). The map g U ′ ,U is thetransition map from Φ U to Φ U ′. This means that(Φ U ′) −1x (Φ U ) x (g) = g g U ′ ,U(x), ∀x ∈ U ∩ U ′ , g ∈ G.Let kp > n, and (g U ′ ,U) and (h U ′ ,U) be compatible collections of maps of classW k,ploc . Then the bundles P (g U ′ ,U ) and P (hU ′ ,U ) are W k,ploc-isomorphic, if there existsa collection of maps k U ∈ W k,ploc(U,G), satisfying the equation(A.38) k U ′(x) −1 h U ′ ,U(x)k U (x) = g U ′ ,U(x), ∀x ∈ U ∩ U ′ , ∀U,U ′ ∈ U.Defining Φ U as in (A.37) and Ψ U similarly, with g U ′ ,U replaced by h U ′ ,U, an isomorphismP (gU ′ ,U ) → P (hU ′ ,U ) is given by[U,x,g] ↦→ (Ψ U ) x(kU (x)(Φ U ) −1x (g) ) .(It follows from the compatibility condition (A.34) and (A.38) that this map iswell-defined, i.e., the right hand side above does not depend on the choice of therepresentative (U,x,g).) ✷Proof of Theorem 99 (p. 110). We choose a cover U of X, and a systemof local trivializations Φ U : U × G → P of class W 2,ploc (U ∈ U). For U,U ′ ∈ U wedenote by g U ′ ,U : U ∩ U ′ → G the corresponding transition map, defined by·g U ′ ,U(x) := (Φ U ′) −1x (Φ U ) x ,where for g ∈ G, ·g : G → G denotes right multiplication. We choose a neighborhoodN of the diagonal in G × G as in Lemma 101, and a refinement V and acollection of smooth maps h V ′ ,V (V,V ′ ∈ V) as in Proposition 100. Using (A.35),we may apply Lemma 101 with U,g U ′ ,U,h U ′ ,U replaced by V,g UV ′,U V,h V ′ ,V , toconclude that there exists a refinement W ∋ W ↦→ V W ∈ V, and a collection ofmaps k W ∈ W 2,p (W,G) (W ∈ W), such thatk −1W ′h V W ′,V Wk W = g UVW′ ,UV W on W ∩ W ′ , ∀W,W ′ ∈ W.Therefore, the statement of Theorem 99 follows from Remark 102.A.6. Proof of the existence of a right inverse for d ∗ AIn this section we prove Proposition 69 (Section 3.2.4). We need the followingresults. Let n ∈ N, G be a compact Lie group with Lie algebra g, 〈·, ·〉 gan invariant inner product on g, and (X, 〈·, ·〉 X ) a Riemannian manifold (possiblywith boundary) of dimension n. Recall the definitions (3.39,3.40,3.41) of‖α‖ k,p,A ,Ω i k,p,A (g P),Γ k,pA (g P),Γ p (g P ). If X is compact, p > n/2, and P is a G-bundle over X of Sobolev class W 2,p , then we denote by A 1,p (P) the affine spaceof connection one-forms on P of class W 1,p .103. Proposition (Uhlenbeck gauge). Let n/2 < p < ∞. Assume that X iscompact and diffeomorphic to the closed ball ¯B 1 ⊆ R n . Then there exist constantsε > 0 and C with the following property. Let P 0 and P be G-bundles over X ofclass W 2,p , and A 0 ,A ∈ A 1,p (P) connections, such that A 0 is flat and‖F A ‖ p ≤ ε.□

A.6. PROOF OF THE EXISTENCE OF A RIGHT INVERSE FOR d ∗ A 113Then there exists an isomorphism of G-bundles Φ : P 0 → P, of class W 2,p , suchthat(A.39) ‖Φ ∗ A − A 0 ‖ 1,p,A0 ≤ C‖F A ‖ p .The proof of this result is based on the following.104. Lemma (Uhlenbeck gauge with trivial connection). Let n/2 < p < ∞.Assume that X = B 1 ⊆ R n , equipped with the standard metric 〈·, ·〉 0 . We denoteby P 0 the trivial G-bundle X ×G, and by A 0 the trivial connection on this bundle.Then there exist constants ε > 0 and C > 0, such that for every connectionA ∈ A 1,p (P 0 ), satisfying ‖F A ‖ p ≤ ε, there exists a gauge transformation g on P 0 ofclass W 2,p , such that‖g ∗ A − A 0 ‖ 1,p,A0,〈·,·〉 0≤ C‖F A ‖ p .Proof of Lemma 104. This follows e.g. from [We, Theorem 6.3].In the proof of Proposition 103 we also use the following.105. Remark. Letl ∈ N 0 ,p ∈( nl + 1 , ∞ ),and 〈·, ·〉 ′ X be a Riemannian metric on X. Assume that X is compact. Then thereexists a constant C > 0 such thatC −1 ‖α‖ k,p,〈·,·〉X,A ≤ ‖α‖ k,p,〈·,·〉 ′X ,A ≤ C‖α‖ k,p,〈·,·〉X,A,for every G-bundle P over X, of class W l+1,p , connection A ∈ A l,p (P), integer k ∈{0,...,l + 1}, and every differential form α on X with values in g P , of class W k,p .(Here the degree of α is arbitrary.) This follows from an elementary argument,using induction over k. ✷Proof of Proposition 103. Consider first the case X = B 1 together withthe standard metric. We choose constants ε,C as in Lemma 104. Let P 0 ,P,A 0 ,Abe as in the hypothesis. We show that the required isomorphism Φ exists. It followsfrom Theorem 99 that we may assume without loss of generality that P 0 and Pare smooth. Since X is smoothly retractible to a point, P 0 and P are smoothlyisomorphic to the trivial bundle over X. Hence we may assume without loss ofgenerality that they are the trivial bundle. Since A 0 is flat, it follows from Lemma104 that A 0 is W 2,p -gauge equivalent to the trivial connection on X ×G. Therefore,the statement of Proposition 103 is a consequence of Lemma 104.The situation in which 〈·, ·〉 X is a general metric, can be reduced to the abovecase, using Remark 105. This proves Proposition 103.□The proof of Proposition 69 is based on the following. Recall from (3.42) thatd ∗ A = − ∗ d A ∗ : Ω 1 1,p,A(g P ) → Γ p (g P )denotes the formal adjoint of the operator d A . If (X, ‖ · ‖ X ) and (Y, ‖ · ‖ Y ) arenormed vector spaces and T : X → Y a bounded linear map then we denote by‖T ‖ := sup { ‖Tx‖ Y∣ ∣ x ∈ X : ‖x‖X ≤ 1 }the operator norm of T.□

114 A. AUXILIARY RESULTS106. Proposition. Let p > n. Assume that X is diffeomorphic to B 1 ⊆ R n ,and (X, 〈·, ·〉 X ) can be embedded (as a Riemannian manifold) into R n , togetherwith the standard metric. Then there exist constants ε > 0 and C > 0, suchthat for every G-bundle P → X of class W 2,p , and every connection A ∈ A 1,p (P)satisfying ‖F A ‖ p ≤ ε, there exists a right inverse R of the operator(A.40) d ∗ Ad A : Γ 2,pA (g P) → Γ p (g P ),with operator norm ‖R‖ bounded above by C.For the proof of this result, we need the following three lemmas.107. Lemma (Twisted Morrey’s inequality). Let p > n. Assume that X isdiffeomorphic to B 1 . Then there exist constants C and ε > 0 such that for everyG-bundle P → X of class W 2,p , and A ∈ A 1,p (P), the following holds. If ‖F A ‖ p ≤ εthen(A.41) ‖α‖ ∞ ≤ C‖α‖ 1,p,A , ∀α ∈ Ω i 1,p,A(g P ), i ∈ {0,...,n}.Proof of Lemma 107. We denote by A 0 the trivial connection on the trivialbundle P 0 := X × G.Claim. There exists a constant C 1 > 0 such that the inequality (A.41) holdswith C = C 1 and A = A 0 .Proof of the claim. This follows from Morrey’s theorem, using the hypothesisp > n.□We choose constants ε > 0 and C 2 := C as in Proposition 103. Let P be aG-bundle over X, of class W 2,p , and A ∈ A 1,p (P), such that ‖F A ‖ p ≤ ε. By thestatement of Proposition 103, there exists an isomorphism Φ : P 0 → P of classW 2,p , such that(A.42) ‖Φ ∗ A − A 0 ‖ 1,p,A0 ≤ C 2 ‖F A ‖ p .Let i ∈ {0,...,n} and α ∈ Ω i 1,p,A (g P). We setA ′ := Φ ∗ A, α ′ := Φ ∗ α, C 3 := max { |[ξ,η]| ∣ ∣ ξ,η ∈ g : |ξ| ≤ 1, |η| ≤ 1},where the norm | · | is with respect to 〈·, ·〉 g . A direct calculation shows that(∇ A0 − ∇ A′ )α ′ = [(A ′ − A 0 ) ⊗ α ′ ],where [·] : g P ⊗ g P → g P denotes the map induced by the Lie bracket on g. Usingthe above claim, it follows that(A.43) ‖α‖ ∞ = ‖α ′ ‖ ∞ ≤ C 1 ‖α ′ ‖ 1,p,A0 ≤ C 1(‖α ′ ‖ 1,p,A ′ + C 3 ‖A ′ − A 0 ‖ ∞ ‖α ′ ‖ p).The above claim, inequality (A.42), and the assumption ‖F A ‖ p ≤ ε imply theestimate ‖A ′ − A 0 ‖ ∞ ≤ C 1 C 2 ε. Combining this with (A.43), the statement ofLemma 107 follows.□108. Lemma. Let p > n and ε > 0. Assume that X is diffeomorphic to B 1 .Then there exists a constant δ > 0 with the following property. Let P → X be aG-bundle of class W 2,p , A 0 ,A ∈ A 1,p (P), and ξ ∈ Γ 2,pA 0(g P ), such that A 0 is flat(A.44) ‖A − A 0 ‖ 1,p,A0 ≤ δ.

A.6. PROOF OF THE EXISTENCE OF A RIGHT INVERSE FOR d ∗ A 115Then the following inequalities hold:∥ ( d ∗ A d ) ∥(A.45)A − d ∗ A 0d A0 ξ∥p ≤ ε‖ξ‖ 1,p,A0 ,(A.46)(A.47)(A.48)Proof of Lemma 108. We have‖ξ‖ 2,p,A ≤ (1 + ε)‖ξ‖ 2,p,A0 .d ∗ A d A − d ∗ A 0d A0= (d A − d A0 ) ∗ (d A − d A0 ) + d ∗ A 0(d A − d A0 ) + (d A − d A0 ) ∗ d A0 ,d A − d A0 = [ (A − A 0 ) ∧ ·].It follows that there exists a constant C > 0 such that‖ ( d ∗ A d )A − d ∗ A 0d A0 ξ‖p≤ C(‖A − A 0 ‖ 2 ∞‖ξ‖ p + ∥ ∇A 0(A − A 0 ) ∥ )p‖ξ‖ ∞ + ‖A − A 0 ‖ ∞ ‖d A0 ξ‖ p ,for every G-bundle P → X of class W 2,p , A 0 ,A ∈ A 1,p (P), and ξ ∈ Γ 2,pA 0(g P ). 14Using Lemma 107 and flatness of A 0 , it follows that there exists a constant δ > 0such that inequality (A.45) holds for every P,A 0 ,A,ξ as in the hypothesis.Inequality (A.46) follows from an analogous argument, involving formulas for∇ A ∇ A − ∇ A0 ∇ A0 and ∇ A − ∇ A0 similar to (A.47,A.48). This proves Lemma108. □109. Lemma. Let X and Y be Banach spaces, and T 0 ,S : X → Y and R 0 :Y → X bounded linear maps, such thatT 0 R 0 = id, ‖S‖ ≤ 12‖R 0 ‖ .Then there exists a right inverse R of T 0 + S, satisfying ‖R‖ ≤ 2‖R 0 ‖.The proof of this lemma will use the following remark.110. Remark. Let X be a Banach space, and T : X → X a linear mapsatisfying ‖T ‖ < 1. Then id + T is invertible, and‖(id + T) −1 ‖ ≤11 − ‖T ‖ .This follows from a standard argument, using the Neumann series ∑ n∈N 0(−1) n T n .✷Proof of Lemma 109. By hypothesis, we have‖SR 0 ‖ ≤ ‖S‖ ‖R 0 ‖ ≤ 1 2 .Hence using Remark 110, it follows that the mapR := R 0 (id + SR 0 ) −1 : Y → Xis well-defined and has the desired properties. This proves Lemma 109.□Proof of Proposition 106 (p. 114).14 The constant C depends on the Lie bracket on g and the metric on X, but not on thebundle P.

116 A. AUXILIARY RESULTS1. Claim. Let 1 < p < ∞. There exists a constant C > 0 such that for everyG-bundle P → X of class W 2,p , and every flat connection A on P of class W 1,p ,there exists a right inverse R of the operator (A.40), satisfying ‖R‖ ≤ C.Proof of Claim 1. By Theorem 99 we may assume without loss of generalitythat P is smooth. We choose a smooth flat connection A 0 on P. By Proposition103 there exists an automorphism Φ of P, of class W 2,p , such that inequality (A.39)holds. Since A is flat, it follows that Φ ∗ A = A 0 , and thus Φ ∗ A is smooth. Hencewe may assume w.l.o.g. that A is smooth.For an open subset U ⊆ R n we denote by C0 ∞ (U) the compactly supportedsmooth real valued functions on U. We define the operator ˜T : C0 ∞ (B 1 ) → C ∞ (B 1 )as follows. We denote by Φ : R n \{0} → R the fundamental solution for the Laplaceequation. It is given byΦ(x) ={ 12πlog |x|, if n = 2,Γ(n/2)2(2−n)π n |x|2−n , if n ≠ 2,2where Γ denotes the gamma function. (See e.g. [Ev], p. 22.) Let f ∈ C ∞ 0 (B 1 ).We define ˜f : R n → R to be the extension of f by 0 outside B 1 . We denote by ∗convolution in R n and define˜Tf := (Φ ∗ ˜f)| B1 .Note that Φ is locally integrable, hence the convolution is well-defined. Furthermore,˜Tf is smooth, and ∆˜Tf = f. 152. Claim. There exists a constant C such that‖ ˜Tf‖ W 2,p (B 1) ≤ C‖f‖ L p (B 1), ∀f ∈ C ∞ 0 (B 1 ).Proof of Claim 2. Young’s inequality states that‖ ˜Tf‖ L p (B 1) ≤ ‖Φ‖ L 1 (B 2)‖f‖ L p (B 1), ∀f ∈ C ∞ 0 (B 1 ).Furthermore, the Calderón-Zygmund inequality states that there exists a constantC such that for every f ∈ C0 ∞ (R n ) we have ‖D 2 (Φ∗f)‖ p ≤ C‖f‖ p . 16 The statementof Claim 2 follows from this.□We fix a constant C as in Claim 2. By this claim the map ˜T uniquely extendsto a bounded linear mapT : L p (B 1 ) → W 2,p (B 1 ).Since ∆ ˜Tf = f, for every f ∈ C ∞ 0 (B 1 ), a density argument shows that ∆Tf = f,for every f ∈ W 2,p (B 1 ), i.e., T is a right inverse for ∆ : W 2,p (B 1 ) → L p (B 1 ).Let now G be a compact Lie group, 〈·, ·〉 g an invariant inner product on g =Lie G, and (X, 〈·, ·〉 X ) a Riemannian manifold as in the hypothesis of Proposition106. Without loss of generality we may assume that X is a submanifold (withboundary) of R n , and that 〈·, ·〉 X is the standard metric. Let π : P → X be aG-bundle, and A ∈ A(P) be a flat connection. We fix a point p 0 ∈ P, and denoteby σ : X → P the A-horizontal section through p 0 . This is the unique smooth15 The first assertion follows from differentiation under the integral, and for the second seee.g. [Ev, Chap. 2, Theorem 1].16 This follows e.g. [MS2, Theorem B.2.7], using (∂j Φ) ∗ f = ∂ j (Φ ∗ f).

A.6. PROOF OF THE EXISTENCE OF A RIGHT INVERSE FOR d ∗ A 117section of P satisfying Adσ = 0 and σ(0) = p 0 . (Such a section exists, since A isflat, and X is diffeomorphic to B 1 .) For k ≥ 0 we define the mapΨ k : W k,p (B 1 ,g) → Γ k,pA (g P),Ψ k ξ := G · (σ,ξ).This is an isometric isomorphism. We define R := −Ψ 2 TΨ −10 . It follows that∥∥R ∥ ≤ ‖Ψ 2 ‖‖T ‖‖Ψ −10 ‖ = ‖T ‖ ≤ C.A straight-forward calculation shows thatd ∗ Ad A Ψ 2 = Ψ 0 d ∗ d = −Ψ 0 ∆ : W 2,p (B 1 ,g) → Γ p (g P ).It follows that R is a right inverse for d ∗ A d A. This proves Claim 1.We choose constants C 1 := C as in Claim 1, δ as in Lemma 108, correspondingto ε = min { 1/(2C 1 ),1 } , (ε 1 ,C 2 ) := (ε,C) as in Proposition 103, and (ε 2 ,C 3 ) :=(ε,C) as in Lemma 107. We define{ } δε := min ,ε 1 ,ε 2 .C 2Let P → X be a G-bundle of class W 2,p , and A ∈ A 1,p (P), such that ‖F A ‖ p ≤ ε.The statement of Proposition 106 is a consequence of the following claim.3. Claim. There exists a right inverse R of d ∗ A d A, satisfying ‖R‖ ≤ 4C 1 .Proof of Claim 3. We choose a flat connection Ã0 on P of class W 1,p . 17 Since‖F A ‖ p < ε ≤ ε 1 , by the statement of Proposition 103 with P 0 = P there exists anautomorphism Φ of P of class W 2,p , such that(A.49) ‖Φ ∗ A − Ã0‖ 1,p, e A0≤ C 2 ‖F A ‖ p .We define A 0 := Φ ∗ Ã 0 . By the statement of Claim 1 there exists a right inverse R 0of the operatord ∗ A 0d A0 : Γ 2,pA 0(g P ) → Γ p (g P ),satisfying ‖R 0 ‖ A0 ≤ C 1 , where ‖ · ‖ A0 denotes the operator-norm. 18 The assumption‖F A ‖ p ≤ ε ≤ δ/C 2 and (A.49) imply that the condition (A.44) is satisfied.Therefore, by (A.45) with “ε”= 1/(2C 1 ), we have∥ d∗A d A − d ∗ A 0d A0∥∥A0≤ 12C 1.Therefore, applying Lemma 109, there exists a right inverse R of the operatord ∗ Ad A : Γ 2,pA 0(g P ) → Γ p (g P ),satisfying ‖R‖ A0 ≤ 2C 1 . Combining this with inequality (A.46) (with “ε”= 1),it follows that ‖R‖ A ≤ 4C 1 . This proves Claim 3 and completes the proof ofProposition 106.□In the proof of Proposition 69 we will also use the following lemma.□17 It follows from an argument involving Theorem 99 that such a connection exists.18 Here we use the subscript “A0 ” to indicate that the definition of this norm involves theconnection A 0 .

118 A. AUXILIARY RESULTS111. Lemma. Let n ∈ N, p ∈ [1, ∞], and f ∈ L p (B n 1 ). Then the functionlies in L p .B n 1 ∋ x ↦→∫ x10f ( t,x 2 ,...,x n)dtProof of Lemma 111. Consider first the case p = 1 and n = 1, and letf ∈ L 1 ((0,1)). Defining χ(t,x) := 1 if t ≤ x, and 0, otherwise, Fubini’s theoremimplies that∫ 1 ∫ x ∫∫ 1|f(t)|dtdx = χ(t,x)|f(t)|dtdx = (1 − t)|f(t)|dt < ∞.00[0,1]×[0,1]The statement of the lemma in the case p = 1 and n = 1 follows. For p = 1 and ageneral n the proof is similar.For a general exponent p, Hölder’s inequality implies that∣∫ x10f ( ∣) ∣∣∣p ∫ x1∣t,x 2 ,...,x n dt ≤ |x 1 | p−1 ∣f ( )∣t,x 2 ,...,x n ∣pdt.Hence in general, the statement of the lemma follows from what we have alreadyproved, by considering the function |f| p .□We are now ready to prove that d ∗ A admits a right inverse:Proof of Proposition 69 (p. 86). We prove statement (i).Let n,G, 〈·, ·〉 g ,p,X, 〈·, ·〉 X ,P,A be as in the hypothesis. We show that the operatord ∗ A admits a bounded right inverse. By Theorem 99 we may assume w.l.o.g. that Pis smooth. Since by assumption, X is diffeomorphic to B 1 , we may assume withoutloss of generality that X = B 1 . The Hodge ∗-operator induces an isomorphismbetween Ω i k,p,A,〈·,·〉 X(g P ) and Ω n−ik,p,A,〈·,·〉 X(g P ). Using the equality d ∗ A = − ∗ d A∗, itfollows that the operatord ∗ A : Ω 1 1,p,A,〈·,·〉 X(g P ) → Γ p 〈·,·〉 X(g P )admits a bounded right inverse, if and only if the operatord A : Ω n−11,p,A,〈·,·〉 X(g P ) → Ω n 0,p,A,〈·,·〉 X(g P )does so. Since X = B 1 is compact, using Remark 105, it follows that the conditionthat d ∗ A admits a bounded right inverse, is independent of the metric 〈·, ·〉 X. Hencewe may assume without loss of generality that 〈·, ·〉 X is the standard metric onB 1 ⊆ R n .1. Claim. There exists a bounded linear map(A.50) T : Γ 1,pA (g P) → Ω 1 1,p,A(g P ),such that d ∗ A T = id.Proof of Claim 1. We define Ω ⊆ R × P to be the subset consisting of all(t,p) such that(A.51) |t + x 1 | 2 + x 2 2 + · · · + x 2 n < 1,00

120 A. AUXILIARY RESULTSSince π ∗ ṗ(0) = e i and π ◦ Ψ(s,p(t)) = x + se 1 + te i , it follows that|d A η(x)e i | ≤∫ 0−x 1∣ ∣dA ξ(x + se 1 )e i∣ ∣ds.By Lemma 111 with f := |d A ξ e i |, using the assumption ξ ∈ Γ 1,pA (g P), it followsthat |d A η e i | ∈ L p (B n 1 ). Therefore, η lies in Ω 1 1,p,A (g P). To prove equality (A.54),observe thatd ∗ A(ηdx 1 ) = − ∗ d A(η dx 2 ∧ ... ∧ dx n) = − ∗ ( ξ dx 1 ∧ ... ∧ dx n) = −ξ,where in the second step we used (A.55). This proves Claim 2 and hence Claim 1.□We choose a map T as in Claim 1, and a flat connection A 0 ∈ A(P). ByProposition 106 there exists a bounded right inverse R 0 ofd ∗ A 0d A0 : Γ 2,pA 0(g P ) → Γ p (g P ).Statement (i) is now a consequence of the following.3. Claim. The operatorR := d A R 0 + T ( id − d ∗ Ad A R 0): L p (g P ) → Ω 1 1,p,A(g P )is well-defined and bounded, and d ∗ A R = id.Proof of Claim 3. A short calculation shows that S := d ∗ A d A − d ∗ A 0d A0 isof first or zeroth order. Hence it is bounded as a map from Γ 2,pA (g P) to Γ 1,pA (g P).Furthermore, the equalityid − d ∗ Ad A R 0 = −SR 0holds on smooth sections of g P . This implies that R is well-defined and bounded.A short calculation shows that d ∗ AR = id. This proves Claim 3, and completes theproof of (i).□To prove statement (ii), we choose constants ε and C as in Proposition 106.Let P → X be a G-bundle of class W 2,p , and A ∈ A 1,p (P), such that ‖F A ‖ p ≤ ε.By the statement of Proposition 106, there exists a right inverse R of the operatord ∗ A d A : Γ 2,pA (g P) → Γ p (g P ), satisfying ‖R‖ ≤ C. The operator d A R is a rightinverse for d ∗ A , satisfying ‖d AR‖ ≤ ‖d A ‖ ‖R‖ ≤ C. Here in the last inequality weused the fact ‖d A ‖ ≤ 1, whered A : Γ 2,pA (g P) → Ω 1 1,p,A(g P ).This proves (ii), and completes the proof of Proposition 69.A.7. Further auxiliary resultsThe next two results were used in the proofs of Proposition 38 (Section 2.5)and Theorem 78 (Appendix A.1).112. Theorem (Uhlenbeck compactness). Let n ∈ N, G be a compact Liegroup, X a compact smooth Riemannian n-manifold (possibly with boundary), Pa smooth G-bundle over X, p > n/2 a number, and A ν a sequence of connectionson P of class W 1,p . Assume thatsup ‖F Aν ‖ L p (X) < ∞.ν∈N□

A.7. FURTHER AUXILIARY RESULTS 121Then passing to some subsequence there exist gauge transformations g ν of classW 2,p , such that g ∗ νA ν converges weakly in W 1,p .Proof of Theorem 112. This is [We, Theorem A]. See also [Uh, Theorem1.5]. □113. Proposition (Compactness for ¯∂ J ). Let M be a manifold without boundary,k ∈ N, p > 2 numbers, J an almost complex structure on M of class C k ,Ω 1 ⊆ Ω 2 ⊆ ... ⊆ C open subsets, and u ν : Ω ν → M a sequence of functions ofclass W 1,ploc . Assume that ¯∂ J u ν is of class W k,ploc, for every ν, and that for every opensubset Ω ⊆ ⋃ ν Ω ν with compact closure the following holds. If ν 0 ∈ N is so largethat Ω ⊆ Ω ν0 then(A.56)(A.57)(A.58)∃K ⊆ M compact: u ν (Ω) ⊆ K, ∀ν ≥ ν 0 ,sup ν≥ν0 ‖du ν ‖ Lp (Ω) < ∞,sup ν≥ν0 ‖¯∂ J u ν ‖ W k,p (Ω) < ∞.Then there exists a subsequence of u ν that converges weakly in W k+1,p and in C kon every compact subset of ⋃ ν Ω ν.Proof of Proposition 113. The proof goes along the lines of the proof of[MS2, Proposition B.4.2].□The next lemma was used in the proofs of Propositions 38 (Section 2.5) and 77(Appendix A.1), and of Theorem 3.114. Lemma (Regularity of the gauge transformation). Let X be a smoothmanifold, G a compact Lie group, P a G-bundle over X, k ∈ N 0 , and p > dim X.Then the following assertions hold.(i) Let g be a gauge transformation of class W 1,plocand A a connection on P ofclass C k , such that g ∗ A is of class C k . Then g is of class C k+1 .(ii) Assume that X is compact (possibly with boundary). Let U be a subset ofthe space of W k,p -connections on P that is bounded in W k,p . Then thereexists a W k+1,p -bounded subset V of the set of W k+1,p -gauge transformationson P, such that the following holds. Let A ∈ U and g be a W 1,p -gaugetransformation, such that g ∗ A ∈ U. Then g ∈ V.Proof of Lemma 114. This follows by induction over k, using the equalitydg = g ( g ∗ A) − Ag and Morrey’s inequality (for (ii)). (For details see [We, LemmaA.8].)□The next proposition was used in the proof of Proposition 40 (Quantization ofenergy loss) in Section 2.5.115. Proposition. Let n ∈ N, G be a compact Lie group, P a G-bundle overR n , and A,A ′ smooth flat connections on P. Then there exists a smooth gaugetransformation g such that A ′ = g ∗ A.Proof of Proposition 115. 21 In the case n = 1 such a g exists, since thenthe condition A ′ = g ∗ A can be viewed as an ordinary differential equation for g.Let n ∈ N and assume by induction that we have already proved the statement forn. Let P be a G-bundle over R n+1 , and A,A ′ smooth flat connections on P. We21 In the case n = 2, see also [Fr1, Corollary 3.7].

122 A. AUXILIARY RESULTSdefine ι : R n → R n+1 by ι(x) := (x,0). By the induction hypothesis there exists asmooth gauge transformation g 0 on ι ∗ P → R n , such that(A.59) g ∗ 0ι ∗ A = ι ∗ A ′ .Since P is trivializable, there exists a smooth gauge transformation ˜g 0 on P suchthat ι ∗˜g 0 = g 0 .Let x ∈ R n . We define ι x : R → R n+1 by ι x (t) := (x,t). There exists a uniquesmooth gauge transformation h x on ι ∗ xP → R, such that(A.60) h ∗ xι ∗ x˜g ∗ 0A = ι ∗ xA ′ , h x (p) = 1, ∀p ∈ fiber of ι ∗ xP over 0 ∈ R.To see this, note that these conditions can be viewed as an ordinary differentialequation for h x with prescribed initial value. Since this solution depends smoothlyon x, there exists a unique smooth gauge transformation h on P such that ι ∗ xh = h x ,for every x ∈ R n . The gauge transformation g := ˜g 0 h on P satisfies the equationA ′ = g ∗ A. This follows from (A.59,A.60) and flatness of A and A ′ . This provesProposition 115.□The next result was used in the proofs of Proposition 38, Remark 43 (Section2.5), and Theorem 3. Let M,ω,G,g, 〈·, ·〉 g ,µ,J,Σ,ω Σ ,j be as in Chapter 1. Wedefine the almost complex structure ¯J on M as in (2.1). The energy density of amap f ∈ W 1,p (Σ,M) is given bye f (z) := 1 2 |df|2 ,where the norm is with respect to the metrics ω Σ (·,j·) on Σ and ω(·, ¯J·) on M.Let P be a smooth G-bundle over Σ, A a connection on P, and u : P → M anequivariant map. We definee ∞ A,u := 1 2 |d Au| 2 ,where the norm is taken with respect to the metrics ω Σ (·,j·) on Σ and ω(·,J·) onM. Furthermore, we defineū : Σ → M,ū(z) := Gu(p),where p ∈ P is an arbitrary point in the fiber over z.116. Proposition (Pseudo-holomorphic curves in the symplectic quotient).Let P be a smooth G-bundle over Σ, p > 2, A a W 1,ploc-connection on P, andu : P → M a G-equivariant map of class W 1,ploc, such that µ ◦ u = 0. Then we haveeū = e ∞ A,u.If (A,u) also solves the equation ¯∂ J,A (u) = 0 then¯∂ ¯Jū = 0.Proof of Proposition 116. This follows from an elementary argument. Forthe second part see also [Ga, Section 1.5].□In the proof of Theorem 3 we used the following lemma.117. Lemma (Bound for tree). Let k ∈ N 0 be a number, (T,E) a finite tree,α 1 ,...,α k ∈ T vertices, f : T → [0, ∞) a function, and E 0 > 0 a number. Assumethat for every vertex α ∈ T we have(A.61) f(α) ≥ E 0 or # { β ∈ T |αEβ } + # { i ∈ {1,...,k} |α i = α } ≥ 3.

A.7. FURTHER AUXILIARY RESULTS 123Then#T ≤ 2∑ α∈T f(α)E 0+ k.Proof of Lemma 117. This follows from an elementary argument. 22The next result was used in the proof of Proposition 44 (Section 2.6). Let(X,d) be a metric space 23 , G a topological group, and ρ : G ×X → X a continuousaction by isometries. By π : X → X/G we denote the canonical projection. Thetopology on X, determined by d, induces a topology on the quotient X/G.118. Lemma (Induced metric on the quotient). Assume that G is compact.Then the map ¯d : X/G × X/G → [0, ∞] defined by¯d(¯x,ȳ) := min d(x,y)x∈¯x, y∈ȳis a metric on X/G that induces the quotient topology on X/G.Proof of Lemma 118. This follows from an elementary argument.The following two lemmas were used in the proof of Proposition 48 (Section2.8).119. Lemma. Let X be a topological space, x ∈ X, and x ν ∈ X be a sequence.Then x ν converges to x, as ν → ∞, if and only if for every subsequence (ν i ) i∈Nthere exists a further subsequence (i j ) j∈N , such that x νij converges to x, as j → ∞.Proof of Lemma 119. This follows from an elementary argument.Let X be a topological space and G a group. We fix an action of G on X.120. Lemma (Convergence in the quotient). Assume that X is first-countableand that the action of every g ∈ G is a continuous self-map of X. Let y ν ∈ X/G,ν ∈ N, be a sequence that converges to a point y ∈ X/G, and x be a representativeof y. Then there exists a representative x ν of y ν , for each ν ∈ N, such that x νconverges to x.Proof of Lemma 120. This follows from an elementary argument.The connection ∇ A . Next we explain the twisted connection ∇ A , which appearedin the definition of the space X p,λW, occurring in Theorem 4. Let E → Mbe a real (smooth) vector bundle. We denote by C(E) the affine space of (smoothlinear) connections on E. Let ∇ E ∈ C(E). Let N be a smooth manifold, andu : N → M be a smooth map. We denote by u ∗ E → N the pullback bundle. Thepullback connection u ∗ ∇ E ∈ C(u ∗ E) is uniquely determined by the equality(u ∗ ∇ E ) v (s ◦ u) = ∇ E u ∗vs, ∀v ∈ TN, s ∈ Γ(E).Let G be a Lie group, π : P → X a (right-)G-bundle, and E → P a G-equivariantvector bundle. Then the quotient E/G has a natural structure of a vector bundleover X. Assume that G acts on a manifold M, and let E → M be a G-equivariantvector bundle. We denote by C G (E) the space of G-invariant connections on E.We fix A ∈ A(P), ∇ E ∈ C G (E), and u ∈ CG ∞ (P,M). We define(A.62) ˜∇A ∈ C G (u ∗ E), ˜∇A ev ˜s := (u ∗ ∇ E ) ev−p(Aev)˜s,22 See e.g. [MS2, Exercise 5.1.2.].23 d is allowed to attain the value ∞.□□□□

124 A. AUXILIARY RESULTSfor ˜s ∈ Γ(u ∗ E), p ∈ P, and ṽ ∈ T p P. We denote by E u the quotient bundle(u ∗ E)/G → X. We define the connection ∇ A ∈ C(E u ) by(A.63)∇ A v s := G · (p 0 , ˜∇ A ev ˜s),for s ∈ Γ(E u ) and v ∈ TX, where (p 0 ,ṽ) ∈ TP is an arbitrary vector such thatπ ∗ ṽ = v, and ˜s ∈ Γ(u ∗ E) is the G-invariant section defined by s◦π(p) = G·(p, ˜s(p)),for every p ∈ P. This definition is independent of the choice of (p 0 ,ṽ), since theconnection ˜∇ A is basic (i.e., G-invariant and horizontal).The following lemma was mentioned in Chapter 1. Let M,ω,G,g, 〈·, ·〉 g ,µ,Jbe as in that Chapter. Let p > 2, λ ∈ R, and P → C be a G-bundle of class W 2,ploc .Recall the definition (1.24) of ˜Bpλ(P). We denote by G2,ploc(P) the group of gaugetransformations on P of class W 2,ploc .121. Lemma. If λ > 1 − 2/p then the group G 2,ploc(P) acts freely on the set˜B p λ (P).Proof of Lemma 121. Assume that λ > 1 − 2/p. Let w := (A,u) ∈ ˜B p λ (P)and g ∈ G 2,ploc (P) be such that g ∗w = w. Let p 1 ∈ P. We show that g(p 1 ) = 1.It follows from hypothesis (H) that there exists δ > 0 such that µ −1 (B δ ) ⊆ M ∗(defined as in (2.14)). Furthermore, Lemma 84 (Appendix A.3) implies that u(p 0 ) ∈µ −1 (B δ ), for every p 0 ∈ P, for which |π(p 0 )| is large enough. We fix such a pointp 0 . Our hypothesis p > 2 implies that P is a C 1 -bundle. Hence we may choosea path p ∈ C 1 ([0,1],P) such that p(i) = p i , for i = 0,1. Consider the maph := g ◦p : [0,1] → G. By assumption, we have g ∗ u = g ◦u = u. Since u(p 0 ) ∈ M ∗ ,it follows that h(0) = 1. Furthermore, the assumption g ∗ A = A implies that hsolves the ordinary differential equation(A.64)ḣ = hAṗ − (Aṗ)h.The hypothesis p > 2 implies that the map Aṗ : [0,1] → g is continuous. Hencethe equation (A.64) is of the form ḣ(t) = f(t,h(t)), where f is continuous in tand Lipschitz continuous in h. Therefore, by the Picard-Lindelöf theorem, we haveh ≡ 1. In particular, we have g(p 1 ) = h(1) = 1. It follows that g ≡ 1. This provesLemma 121.□The next lemma was used in the proof of Theorem 4 (Section 3.2.1). Here fora linear map D : X → Y we denote coker D := Y/imD.122. Lemma. Let X,Y,Z be vector spaces and D ′ : X → Y and T : X → Zbe linear maps. We define D := D ′ | ker T . Then the following holds.(i) ker D = ker(D ′ ,T).(ii) The map Φ : coker D → coker(D ′ ,T), Φ(y + imD) := (y,0) + im(D ′ ,T), iswell-defined and injective. If T : X → Z is surjective then Φ is also surjective.(iii) Let ‖ · ‖ Y , ‖ · ‖ Z be norms on Y and Z and assume that im(D ′ ,T) is closedin Y ⊕ Z. Then imD is closed in Y .The proof of Lemma 122 is straight-forward and left to the reader.The following result was used in the proof of Proposition 66 in Section 3.2.3.We define the map f : C \ {0} → S 1 by f(z) := z/|z|. For two topological spacesX and Y we denote by C(X,Y ) the set of all continuous maps from X to Y ,

A.7. FURTHER AUXILIARY RESULTS 125and by [X,Y ] the set of all (free) homotopy classes of such maps. Let V be afinite dimensional complex vector space. We denote by End(V ) the space of its(complex) endomorphisms of V , by det : End(V ) → C the determinant map, andby Aut(V ) ⊆ End(V ) the group of automorphisms of V .123. Lemma. The map C(S 1 ,Aut(V )) → Z given by Φ ↦→ deg ( f ◦ det ◦Φ )descends to a bijection [ S 1 ,Aut(V ) ] → Z.Proof of Lemma 123. We choose a hermitian inner product V and denoteby U(V ) the corresponding group of unitary automorphisms of V . The mapdet : U(V ) → S 1 induces an isomorphism of fundamental groups, see e.g. [MS1,Proposition 2.23]. Furthermore, the space Aut(V ) strongly deformation retractsonto U(V ). 24 Let Φ 0 ∈ Aut(V ). It follows that the map{Φ ∈ C(S 1 ,Aut(V )) ∣ } ( )Φ(1) = Φ0 → Z, Φ ↦→ deg f ◦ det ◦Φdescends to an isomorphism between the fundamental group π 1 (Aut(V ),Φ 0 ) andZ. Since this group is abelian, the mapπ 1 (Aut(V ),Φ 0 ) → [ S 1 ,Aut(V ) ]that forgets the base point Φ 0 , is a bijection. The statement of Lemma 123 followsfrom this.□The next lemma was used in the proof of Theorem 64 (Section 3.2.4). Let Xand M be manifolds, G a Lie group with Lie algebra g, 〈·, ·〉 g an invariant innerproduct on g, 〈·, ·〉 M a G-invariant Riemannian metric on M, and ∇ its Levi-Civitaconnection. For ξ ∈ g we denote by X ξ the vector field on M generated by ξ. Wedefine the tensor ρ : TM ⊕ TM → g by(A.65) 〈ξ,ρ(v,v ′ )〉 g := 〈∇ v X ξ ,v ′ 〉 M .A short calculation shows that ρ is skew-symmetric. This two-form was introducedin [Ga, p. 181]. The next lemma corresponds to [Ga, Proposition 7.1.3(a,b)]. LetP → X be a G-bundle, A ∈ A(P), u ∈ C ∞( X,(P × M)/G ) , v ∈ Γ(TM u ), andξ ∈ Γ(g P ). We define the connection ∇ A on TM u → X as on page 123.124. Lemma. ∇ A L u ξ − L u d A ξ = ∇ dAuX ξ , d A L ∗ uv − L ∗ u∇ A v = ρ(d A u,v).Proof of Lemma 124. This follows from short calculations.Let M,ω,G,g, 〈·, ·〉 g ,µ and J be as in Chapter 1, and 〈·, ·〉 M := ω(·,J·). Thefollowing remark was used in the proofs of Theorems 4 (Section 3.2.1) and 64(Section 3.2.4). Recall the definition (2.14) of M ∗ ⊆ M, and that Pr : TM → TMdenotes the orthogonal projection onto imL.125. Remark. Let K ⊆ M ∗ be compact. We define{ }|Lx ξ|c := inf|ξ| ∣ x ∈ K, 0 ≠ ξ ∈ g .Then c > 0. Let x ∈ K. Then L ∗ xL x is invertible, and(A.66) |(L ∗ xL x ) −1 | ≤ c −2 , ∣ Lx (L ∗ xL x ) −1∣ ∣ ≤ c −1 , L x (L ∗ xL x ) −1 L ∗ x = Pr x ,where the | · |’s denote operator norms. Furthermore, |Pr x v| ≤ c −1 |L ∗ xv|, for everyv ∈ T x M. These assertions follow from short calculations. ✷24 This follows from the Gram-Schmidt orthonormalization procedure.□

126 A. AUXILIARY RESULTSAssume that hypothesis (H) holds. The following lemma was used in the proofof Proposition 67 in Section 3.2.3. For x ∈ M we denote by L C x : g C → T x M thecomplex linear extension of the infinitesimal action.126. Lemma. There exists a neighborhood U ⊆ M of µ −1 (0), such that(A.67) c := inf {∣ ∣ dµ(x)LCx α ∣ ∣ + |Pr LCx α| ∣ ∣ x ∈ U, α ∈ g C : |α| = 1 } > 0.Proof of Lemma 126. It follows from hypothesis (H) that there exists δ 0 > 0such that µ −1 ( ¯B δ0 ) ⊆ M ∗ . We defineC := sup { |[ξ,η]| ∣ } ξ,η ∈ g : |ξ| ≤ 1, |η| ≤ 1 ,{ }|Lx ξ|c 0 := inf|ξ| ∣ x ∈ µ−1 ( ¯B δ0 ), 0 ≠ ξ ∈ g .Since the action of G on M ∗ is free, it follows that L x : g → T x M is injective, forx ∈ M ∗ . Furthermore, by hypothesis (H) the set µ −1 ( ¯B δ0 ) is compact. It followsthat c 0 > 0. We choose a positive number δ < min{δ 0 ,c 0 /C,c 3 0/C}, and we defineU := µ −1 (B δ ).Claim. Inequality (A.67) holds.Proof of the claim. Let x ∈ U and α = ξ + iη ∈ g C . Then(A.68) dµ(x)L C xα = [µ(x),ξ] + L ∗ xL x η.Using the last assertion in (A.66), we have(A.69)Pr x L C xα = L x ξ − L x (L ∗ xL x ) −1 [µ(x),η].By the first assertion in (A.66), we have |L ∗ xL x η| ≥ c 2 0|η|. Combining this with(A.68,A.69) and the second assertion in (A.66), we obtain∣ dµ(x)LCx α ∣ + |Pr LCx α| ≥ −Cδ|ξ| + c 2 0|η| + c 0 |ξ| − c −10 Cδ|η|.Inequality (A.67) follows now from our choice of δ. This proves the claim andcompletes the proof of Lemma 126.□

Bibliography[Ad] D. R. Adams, L. I. Hedberg, Function Spaces and Potential Theory, Grundlehren derMathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences), 314,Springer-Verlag, Berlin, 1996, xii+366 pp.[Ba] R. Bartnik, The Mass of an Asymptotically Flat Manifold, Comm. Pure Appl. Math. 39(1986), no. 5, 661–693.[CGS] K. Cieliebak, A.R. Gaio and D.A. Salamon, J-holomorphic curves, moment maps, andinvariants of Hamiltonian group actions, Internat. Math. Res. Notices 2000, no. 16, 831–882.[CGMS] K. Cieliebak, A. R. Gaio and I. Mundet i Riera, D. A. Salamon, The symplectic vortexequations and invariants of Hamiltonian group actions, Journal of Symplectic Geometry 1(2002), no. 3, 543–645.[CS] K. Cieliebak and D.A. Salamon, Wall crossing for symplectic vortices and quantum cohomology,Math. Ann. 335 (2006), no. 1, 133–192.[Ev] L. C. Evans, Partial differential equations, Graduate Studies in Mathematics, 19, AmericanMathematical Society, Providence, RI, 1998, xviii+662 pp.[Fr1] U. Frauenfelder, Floer Homology of Symplectic Quotients and the Arnold-Givental Conjecture,Ph.D.-thesis, Eidgenössische Technische Hochschule Zürich (Switzerland), 2003, 235pp.[Fr2] U. Frauenfelder, The Arnold-Givental Conjecture and Moment Floer Homology,Int. Math. Res. Not. 42 (2004), 2179–2269.[Ga] A. R. Gaio, J-Holomorphic Curves and Moment Maps, Ph.D.-thesis, University of Warwick(U.K.), 2000, 219 pp.[GS] A.R. Gaio and D.A. Salamon, Gromov-Witten invariants of symplectic quotients and adiabaticlimits, J. Symplectic Geom. 3 (2005), no. 1, 55–159.[GL] V. L. Ginzburg and L. D. Landau, On the theory of superconductors,Zh. Eksp. Teor. Fiz. (Sov. Phys. JETP) 20 (1950), 1064–1082. English translation in:L. D. Landau, Collected Papers, Oxford, Pergamon Press, 1965, p. 546.[Gi] A. Givental, Equivariant Gromov-Witten Invariants, Int. Math. Res. Not., 1996, no. 13,613–663.[GiK] A. Givental and B. Kim, Quantum cohomology of flag manifolds and Toda lattices,Comm. Math. Phys. 168 (1995), no. 3, 609–641.[GW1] E. Gonzalez and C. Woodward, Quantum Cohomology of Toric Orbifolds,arXiv:1207.3253.[GW2] E. Gonzalez and C. Woodward, A wall-crossing formula for Gromov-Witten invariantsunder variation of git quotient, arXiv:1208.1727.[Is] T. Isobe, Topological and analytical properties of Sobolev bundles, I: The critical case,Ann. Global Anal. Geom. 35 (2009), no. 3, 277–337.[IS] S. Ivashkovich, V. Shevchishin, Gromov compactness theorem for J-complex curves withboundary, Internat. Math. Res. Notices 2000, no. 22, 1167–1206.[JT] A. Jaffe and C. Taubes, Vortices and monopoles, Structure of static gauge theories, Progressin Physics, 2, Birkhäuser, Boston, Mass., 1980, v+287 pp.[JK] L. C. Jeffrey, F. Kirwan, Localization for nonabelian group actions, Topology 34 (1995),no. 2, 291–327.[Kav] O. Kavian, Introduction à la théorie des points critiques et applications aux problèmeselliptiques (Introduction to critical point theory and applications to elliptic problems),Mathématiques & Applications (Berlin), 13, Springer, 1993, viii+325 pp.127

128 BIBLIOGRAPHY[Kir] F. Kirwan, Cohomology of Quotients in Symplectic and Algebraic Geometry, MathematicalNotes, 31, Princeton University Press, Princeton, NJ, 1984, i+211 pp.[Kim] B. Kim, On Equivariant Quantum Cohomology, Internat. Math. Res. Notices, 1996, no. 17,841–851.[Ko] M. Kontsevich, Enumeration of rational curves via torus actions, 335–368 in R. Dijkgraaf,C. Faber, G. van der Geer, eds., The moduli space of curves, Progress in Mathematics 129,Birkhäuser Boston, 1995.[Lo1] R. B. Lockhart, Fredholm properties of a class of elliptic operators on non-compact manifolds,Ph.D.-thesis, University of Illinois at Urbana-Champaign (U.S.A.), 1979, 91 pp.[Lo2] R. B. Lockhart, Fredholm properties of a class of elliptic operators on non-compact manifolds,Duke Math. J. 48 (1981), no. 1, 289–312.[Lo3] R. B. Lockhart, Fredholm, Hodge and Liouville Theorems on Noncompact Manifolds,Trans. Amer. Math. Soc. 301 (1987), no. 1, 1–35.[LM1] R. B. Lockhart and R. C. McOwen, On elliptic systems in R n , Acta Math. 150 (1983),125–135.[LM2] R. B. Lockhart and R. C. McOwen, Elliptic Differential Operators on Noncompact Manifolds,Ann. Sc. Norm. Sup. Pisa 12 (1985), 409–447.[Lu] P. Lu, A rigorous definition of fiberwise quantum cohomology and equivariant quantum cohomology,Comm. Anal. Geom. 6 (1998), no. 3, 511–588.[McO1] R. C. McOwen, The Behavior of the Laplacian on Weighted Sobolev Spaces, Comm. PureAppl. Math. 32 (1979), 783–795.[McO2] R. C. McOwen, Fredholm theory of partial differential equations on complete Riemannianmanifolds, Pacific J. Math. 87, no. 1 (1980), 169–185.[McO3] R. C. McOwen, On elliptic operators in R n , Comm. Part. Diff. Eq. 5 (1980), 913–933.[MS1] D. McDuff and D.A. Salamon, Introduction to Symplectic Topology, 2nd edition, OxfordMathematical Monographs, The Clarendon Press, Oxford University Press, 1998, x+486 pp.[MS2] D. McDuff and D. A. Salamon, J-Holomorphic Curves and Symplectic Topology, AMSColloquium Publications, Vol. 52, American Mathematical Society, Providence, RI, 2004,xii+669 pp.[Mu1] I. Mundet i Riera, Yang-Mills-Higgs theory for symplectic fibrations, Ph.D.-thesis, UniversidadAutónoma de Madrid (Spain), 1999, 159 pp.[Mu2] I. Mundet i Riera, Hamiltonian Gromov-Witten invariants, Topology 42 (2003), 525–553.[MT] I. Mundet i Riera and G. Tian, A Compactification of the Moduli Space of Twisted HolomorphicMaps, Adv. Math. 222 (2009), no. 4, 1117–1196.[NWZ] K. L. Nguyen, C. Woodward, F. Ziltener, Morphisms of CohFT Algebras and Quantizationof the Kirwan Map, arXiv:0903.4459, to appear in Proceedings of the Hayashibara Forum.[Ott] A. Ott, Removal of singularities and Gromov compactness for symplectic vortices,arXiv:0912.2500, to appear in J. Symplectic Geom.[ReSi] M. Reed and B. Simon, Methods of Modern Mathematical Physics, I, Functional analysis,Academic Press, New York, 1980, xv+400 pp.[RoSa] J. Robbin and D. A. Salamon, The spectral flow and the Maslov index, Bull. LondonMath. Soc. 27 (1995), no. 1, 1–33.[Ru] Y. Ruan, Virtual neighborhoods and pseudo-holomorphic curves, Proceedings of 6th GökovaGeometry-Topology Conference. Turkish J. Math. 23 (1999), no. 1, 161–231.[Sa] D. A. Salamon, Lectures on Floer Homology, in Symplectic Geometry and Topology, editedby Y. Eliashberg and L. Traynor, IAS/Park City Mathematics Series, Vol. 7 (1999), 143–230.[SZ] D. A. Salamon, E. Zehnder, Morse theory for periodic solutions of Hamiltonian systems andthe Maslov index, Comm. Pure Appl. Math. 45 (1992), 1303–1360.[Sp1] H. Spielberg, The Gromov-Witten invariants of symplectic toric manifolds, and their quantumcohomology ring, C. R. Acad. Sci. Paris Sér. I Math. 329 (1999), no. 8, 699–704.[Sp2] H. Spielberg, Counting generic genus-0 curves on Hirzebruch surfaces,Proc. Amer. Math. Soc. 130 (2002), no. 5, 1257–1264 (electronic).[VW] S. Venugopalan and C. Woodward, Hitchin-Kobayashi correspondence for affine vortices,in preparation.[TW] S. Tolman and J. Weitsman, The Cohomology Rings of Symplectic Quotients,Comm. Anal. Geom. 11 (2003), no. 4, 751–773.[Uh] K. Uhlenbeck, Connections with L p Bounds on Curvature, Comm. Math. Phys. 83 (1982),31–42.

BIBLIOGRAPHY 129[We] K. Wehrheim, Uhlenbeck Compactness, EMS Series of Lectures in Mathematics, EuropeanMathematical Society, Zürich, 2004, viii+212 pp.[Wi] E. Witten, The Verlinde algebra and the cohomology of the Grassmannian, Geometry, topology,& physics, 357–422, Conf. Proc. Lect. Notes Geom. Top., IV, Int. Press, Cambridge,MA, 1995.[Wo] C. Woodward, Quantum Kirwan Morphism and Gromov-Witten Invariants of Quotients,arXiv:1204.1765.[Zi1] F. Ziltener, Symplectic Vortices on the Complex Plane and Quantum Cohomology, Ph.D.-thesis, Eidgenössische Technische Hochschule Zürich (Switzerland), 2006, 258 pp.[Zi2] F. Ziltener, The Invariant Symplectic Action and Decay for Vortices, J. Symplectic Geom. 7(2009), no. 3, 357–376.[Zi3] F. Ziltener, A Maslov Map for Coisotropic Submanifolds, Leaf-wise Fixed Points andPresymplectic Non-Embeddings, arXiv:0911.1460.[Zi4] F. Ziltener, A Quantum Kirwan Map: Conservation of Homology and an Evaluation Map,in preparation.[Zi5] F. Ziltener, A Quantum Kirwan Map: Transversality, in preparation.