Analysis of pseudoholomorphic curves on symplectization: Revisit via contact instantons

TL;DR

This paper revisits the analysis of pseudoholomorphic curves on symplectization through contact instantons, providing stronger estimates and explicit tensorial formulas, thereby advancing the mathematical understanding of these structures.

Contribution

It introduces a coordinate-free tensorial approach to analyze pseudoholomorphic curves and contact instantons, offering improved estimates and explicit formulas for linearized and asymptotic operators.

Findings

A priori estimates depend solely on the contact instanton component w.

Derived explicit tensorial formulas for linearized and asymptotic operators.

Established a perturbation theory for these operators with respect to almost complex structures.

Abstract

In this survey article, we present the analysis of pseudoholomorphic curves $u:(\dot \Sigma,j) \to (Q \times \mathbb{R}, \widetilde J)$ on the symplectization of contact manifold $(Q,\lambda)$ as a subcase of the analysis of contact instantons $w:\dot \Sigma \to Q$, i.e., of the maps $w$ satisfying the equation $$ {\bar{\partial}}^\pi w = 0, \, d(w^*\lambda \circ j) = 0 $$ on the contact manifold $(Q,\lambda)$, which has been carried out by a coordinate-free covariant tensorial calculus. When the analysis is applied to that of pseudoholomorphic curves $u = (w,f)$ with $w = \pi_Q \circ u$, $f = s\circ u$ on symplectization, the outcome is generally stronger and more accurate than the common results on the regularity presented in the literature in that all of our a priori estimates can be written purely in terms $w$ not involving $f$. The a priori elliptic estimates for $w$ are largely consequences of various Weitzenb\"ock-type formulae with respect to the contact triad connection introduced by Wang and the first author in [OW14], and the estimate for $f$ is a consequence thereof by simple integration of the equation $df = w^*\lambda \circ j$. We also derive a simple precise tensorial formulae for the linearized operator and for the asymptotic operator that admit a perturbation theory of the operators with respect to (adapted) almost complex structures: The latter has been missing in the analysis of pseudoholomorphic curves on symplectization in the existing literature.