Exponential decay of $J$-holomorphic maps with finite energy

TL;DR

This paper proves that finite energy J-holomorphic maps in symplectic geometry exponentially converge to Reeb orbits, providing key insights into their asymptotic behavior in symplectic field theory.

Contribution

It establishes the exponential decay of finite energy J-holomorphic maps towards Reeb orbits, a significant result in symplectic and contact topology.

Findings

Finite energy J-holomorphic maps exponentially converge to Reeb orbits.

Provides rigorous proof of asymptotic behavior in symplectic field theory.

Enhances understanding of the structure of moduli spaces of holomorphic curves.

Abstract

We prove that every finite energy $J$-holomorphic map $u(s,t):\mathbb R\times S^1 \rightarrow {\mathbb R} \times \widetilde{M}$ exponentially converges to a periodic orbit of Reeb vector field of $\widetilde M,$ as $s\to \infty.$