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

Abstract

© 2022 The Author(s).In [Y.-G. Oh and R. Wang, Analysis of contact Cauchy-Riemann maps I: A priori Ck 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 w = 0,d(w λ j) = 0 without involving symplectization for each given contact triad (Q,λ,J), and established the a priori Wk,2 elliptic estimates and proved the asymptotic (subsequence) convergence of the map w: ς˙ → Q for any solution, called a contact instanton, on ς˙ under the hypothesis w λ C0 < ∞ and d w L2 L4. The asymptotic limit of a contact instanton is a &apos;spiraling&apos; instanton along a &apos;rotating&apos; Reeb orbit near each puncture on a punctured Riemann surface ς˙. Each limiting Reeb orbit carries a &apos;charge&apos; arising from the integral of w λ j. In this paper, we further develop analysis of contact instantons, especially the W1,p estimate for p > 2 (or the C1-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 Eλ(j,w) for any pair (j,w) with a smooth map w satisfying d(w λ j) = 0, and develop the bubbling-off analysis and prove an -regularity result. We also develop the relevant Fredholm theory and carry out index calculations (for the case of vanishing charge).