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

Abstract

Let M M be a three–dimensional contact manifold, and ψ ~ : D ∖ { 0 } → M × R \tilde {\psi }:D\setminus \{0\}\to M\times {\mathbb R} a finite–energy pseudoholomorphic map from the punctured disc in C {\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 ψ ~ \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 E p , q × R {\mathbb E}_{p,q}\times {\mathbb R} , where E p , q {\mathbb E}_{p,q} denotes a rational ellipsoid (contact structure induced by the standard complex structure on C 2 {\mathbb C}^{2} ), as well as contact structures arising from non-standard circle–fibrations of the three–sphere.