Analysis of Pseudoholomorphic Curves on Symplectization: Revisit via Contact Instantons

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 \(\overline{\delta }^\pi w = 0,{\mkern 1mu} d(w^* \lambda \circ j) = 0\) on the contact manifold \((Q,\lambda )\) , which has been carried out by a coordinate-free covariant tensorial calculus. The latter was initiated by Wang and the first author of the present survey in [OW18a, OW18b] for the closed string case. More recently the first author has extended the machinery to the open string case and applied it to the problems of quantitative contact topology and contact dynamics [Oh21a], [Oh22a], [OY23]. When the analysis is applied to that of pseudoholomorphic curves \(u \, = \, \left( {w,f} \right)\) with \(w = \pi \circ u\) , \(f = s \circ u\) 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, especially \(W^{2,2}\) -estimate, are largely consequences of various Weitzenböck-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.