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

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,