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

Abstract:

Let $M$ be a three–dimensional contact manifold, and $\tilde {\psi }:D\setminus \{0\}\to M\times {\mathbb R}$ a finite–energy pseudoholomorphic map from the punctured disc in ${\mathbb C}$ that is asymptotic to a periodic orbit of the contact form. This article examines conditions under which smooth coordinates may be defined in a tubular neighbourhood of the orbit such that $\tilde {\psi }$ resembles a holomorphic curve, invoking comparison with the theory of topological linking of plane complex algebroid curves near a singular point. 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 (contact structure induced by the standard complex structure on ${\mathbb C}^{2}$), as well as contact structures arising from non-standard circle–fibrations of the three–sphere.