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

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→∞