Embedded H-holomorphic maps and open book decompositions

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.