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

In the present article, we develop the analysis of the following nonlinear elliptic system of equations $$ \bar\partial^\pi w = 0, \, d(w^*\lambda \circ j) = 0 $$ first introduced by Hofer, associated to each given contact triad $(M,\lambda,J)$ on a contact manifold $(M,\xi)$. We directly work with this elliptic system on the contact manifold without involving the symplectization process. We establish the local a priori $C^k$ coercive pointwise estimates for all $k \geq 2$ in terms of $\|dw\|_{C^0}$ by doing tensorial calculations on contact manifold itself using the contact triad connection introduced by present the authors. Equipping the punctured Riemann surface $(\dot \Sigma,j)$ with a cylindrical K\"ahler metric and isothermal coordinates near every puncture, we prove the asymptotic (subsequence) convergence to the `spiraling' instantons along the `rotating' Reeb orbit for any solution $w$, not necessarily for $w^*\lambda \circ j$ being exact (i.e., allowing non-zero `charge' $Q \neq 0$), with bounded gradient $\|d w\|_{C^0} < C$ and finite $\pi$-harmonic energy. For nondegenerate contact forms, we employ the `three-interval method' to prove the exponential convergence to a closed Reeb orbit when $Q = 0$. (The Morse-Bott case using this method is treated in a sequel.) In particular, this also provides a new proof of the exponential convergence result even for the case of pseudo-holomorphic curves in symplectization, which was established by Hofer-Wyosocki-Zehnder and by Bourgeois.