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

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