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 [Formula: see text] 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 [Formula: see text] without involving symplectization for each given contact triad [Formula: see text], and established the a priori [Formula: see text] elliptic estimates and proved the asymptotic (subsequence) convergence of the map [Formula: see text] for any solution, called a contact instanton, on [Formula: see text] under the hypothesis [Formula: see text] and [Formula: see text]. The asymptotic limit of a contact instanton is a ‘spiraling’ instanton along a ‘rotating’ Reeb orbit near each puncture on a punctured Riemann surface [Formula: see text]. Each limiting Reeb orbit carries a ‘charge’ arising from the integral of [Formula: see text]. In this paper, we further develop analysis of contact instantons, especially the [Formula: see text] estimate for [Formula: see text] (or the [Formula: see text]-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 [Formula: see text] for any pair [Formula: see text] with a smooth map [Formula: see text] satisfying [Formula: see text], and develop the bubbling-off analysis and prove an [Formula: see text]-regularity result. We also develop the relevant Fredholm theory and carry out index calculations (for the case of vanishing charge).