Analysis of pseudoholomorphic curves on symplectization: Revisit via contact instantons

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.