Analysis of contact Cauchy-Riemann maps III: Energy, bubbling and Fredholm theory

Abstract

In [Y.-G. Oh and R. Wang, Analysis of contact Cauchy-Riemann maps I: A priori C-k estimates and asymptotic convergence, Osaka J. Math. 55(4) (2018) 647-679; Y.-G. Oh and R. Wang, Analysis of contact Cauchy-Riemann maps II: Canonical neighborhoods and exponential convergence for the Morse-Bott case, Nagoya Math. J. 231 (2018) 128-223], the authors studied the nonlinear elliptic system (& part;)over bar(pi)w = 0, d(w & lowast;lambda o j) = 0 without involving symplectization for each given contact triad (Q, lambda, J), and established the a priori W-k,W-2 elliptic estimates and proved the asymptotic (subsequence) convergence of the map w : (sigma)over dot -> Q for any solution, called a contact instanton, on (sigma)over dot under the hypothesis llw & lowast;lambda ll(C0) < infinity and d(pi)w is an element of L-2 & cap; L-4. The asymptotic limit of a contact instanton is a "spiraling&apos; instanton along a "rotating&apos; Reeb orbit near each puncture on a punctured Riemann surface (sigma)over dot. Each limiting Reeb orbit carries a "charge&apos; arising from the integral of w & lowast;lambda o j. In this paper, we further develop analysis of contact instantons, especially the W-1,W-p estimate for p > 2 (or the C-1-estimate), which is essential for the study of compactification of the moduli space and the relevant Fredholm theory for contact instantons. In particular, we define a Hofer-type off-shell energy E-lambda(j, w) for any pair (j, w) with a smooth map w satisfying d(w & lowast;lambda o j) = 0, and develop the bubbling-off analysis and prove an epsilon-regularity result. We also develop the relevant Fredholm theory and carry out index calculations (for the case of vanishing charge).