Perturbation theory of asymptotic operators of contact instantons and pseudoholomorphic curves on symplectization

Abstract

In this paper, we first provide precise tensorial formulae for the asymptotic operators of contact instantons $w:\dot \Sigma \to Q$ and of pseudoholomorphic curves $u:(\dot\Sigma,j) \to (Q \times \mathbb R, \widetilde J)$ on the symplectization of contact manifold $(Q,\lambda)$. The formula exhibits explicit dependence on the compatible pair $(\lambda,J)$ of the given contact triad $(Q,\lambda, J)$. Then based on this, we present a perturbation theory of the asymptotic operator under the change of compatible CR almost complex structure $J$ for given contact form $\lambda$. This perturbation theory has been missing in the literature on the study of pseudoholomorphic curves on symplectization. Through this systematic tensorial approach, we also provide the study of finer asymptotic behavior at the punctures, such as convergence of tangent plane of contact instantons, in terms of the eigenvalues and eigenvectors of the asymptotic operator, which also simplifies the corresponding study of pseudoholomorphic curves on symplectization in the literature.