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

Abstract

In the present article, we develop the analysis of the following nonlinear elliptic system of equations $${\bar{\partial }}^{\pi }w=0,d(w^{\ast }\lambda \circ j)=0$$ 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 $\Vert dw{\Vert }_{C^0}$ 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 Kähler 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' $Q\ne 0$), with bounded gradient $\Vert dw{\Vert }_{C^0}<C$ 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).)

v3): change of the title, largely re-written, much simplification and improvement of the presentation of the a priori estimates, new proof of exponential convergence via the 3-interval method, 3 appendices added, v4): 31 pages, The final version to appear in Osaka J. Math with exponetial convergence part removed, Full details of (local) $W^{2,k}$-estimates for general $k\ge 2$ are added

References in corpus (3)

• Analysis of contact Cauchy-Riemann maps II: canonical neighborhoods and exponential convergence for the Morse-Bott case
• Compactness results for H-holomorphic maps
• Embedded H-holomorphic maps and open book decompositions

Cited by in corpus (3)

• Analysis of contact Cauchy-Riemann maps II: canonical neighborhoods and exponential convergence for the Morse-Bott case
• Contact Hamiltonian dynamics and perturbed contact instantons with Legendrian boundary condition
• Pseudoholomoprhic curves on the $\mathfrak{LCS}$-fication of contact manifolds