Embedded H-holomorphic maps and open book decompositions

TL;DR

This paper studies embedded H-holomorphic maps in stable Hamiltonian 3-manifolds, proving they form local foliations, satisfy no-intersection properties, and can be used to construct open book decompositions for contact structures, advancing the Weinstein conjecture.

Contribution

It introduces the concept of embedded H-holomorphic maps, proves their local foliation and intersection properties, and applies these results to construct open book decompositions for contact structures.

Findings

Embedded H-holomorphic maps locally foliate the manifold.

Such maps satisfy a no-first-intersection property.

Every contact structure admits an H-holomorphic open book decomposition.

Abstract

We investigate nicely embedded H--holomorphic maps into stable Hamiltonian three--manifolds. In particular we prove that such maps locally foliate and satisfy a no--first--intersection property. Using the compactness results of arXiv:0904.1603 we show that connected components of the space of such maps can be compactified if they contain a global surface of section. As an application we prove that any contact structure on a 3--manifold admits and H--holomorphic open book decomposition. This work is motivated by the program laid out by Abbas, Cieliebak and Hofer to give a proof to the Weinstein conjecture using holomorphic curves. The results in this paper, with the exception of the compactness statement, have been independently obtained by C. Abbas in arXiv:0907.3512.