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 $${\bar{\mathrm{\partial }}}^{\pi }w=0,d(w^{\ast }\lambda \circ j)=0$$ 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 $k\ge 2$ in terms of $\mathrm{\parallel }dw{\mathrm{\parallel }}_{C^0}$ by doing tensorial calculations on contact manifold itself using the contact triad connection introduced by present the authors. Equipping the punctured Riemann surface $(\dot{\mathrm{\Sigma }},j)$ 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 $w^{\ast }\lambda \circ j$ 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 (arXiv:1311.6196).)