  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.
