Analysis of Contact Cauchy--Riemann maps I: a priori $C^k$ estimates and asymptotic convergence

Abstract

In the present article, we develop tensorial analysis for solutions $w$ of the following nonlinear elliptic system $$ {\overline \partial}^\pi w = 0, \, d(w^*\lambda \circ j) = 0, $$ associated to a contact triad $(M,\lambda,J)$. The novel aspect of this approach is that we work directly with this elliptic system on the contact manifold without involving the symplectization process. In particular, when restricted to the case where the one-form $w^*\lambda \circ j$ is exact, all a priori estimates for $w$-component can be written in terms of the map $w$ itself without involving the coordinate from the symplectization. We establish a priori $C^k$ coercive pointwise estimates for all $k \geq 2$ in terms of the energy density $\|dw\|^2$ by means of tensorial calculations on the contact manifold itself. Further, for any solution $w$ under the finite $\pi$-energy assumption and the derivative bound, we also establish the asymptotic subsequence convergence to `spiraling' instantons along the `rotating' Reeb orbit.