Branch structure of J-holomorphic curves near periodic orbits of a contact manifold

Abstract

Let $M$ be a three-dimensional contact manifold and $\psi :D\setminus \{0\}\to M\times \mathbb{R}$ a finite-energy pseudoholomorphic map from a punctured disc in $\mathbb{C}$, that is asymptotic to a periodic orbit of the Reeb vector field. This article examines conditions under which smooth coordinates may be defined in a tubular neighbourhood of the orbit such that $\psi$ resembles a holomorphic curve, invoking comparison with the theory of topological linking of plane complex algebroid curves near a singularity. Examples of this behaviour which are studied in some detail include pseudoholomorphic maps into ${\mathbb{E}}_{p,q}\times \mathbb{R}$, where ${\mathbb{E}}_{p,q}$ denotes a rational ellipsoid with contact structure induced by the complex structure of the ambient ${\mathbb{C}}^2$. Contact structures arising from non-standard circle-fibrations of the three-sphere are also examined.

28 pages