As the title indicates, the setting is the world of contact structures on, and open book decompositions of, three-manifolds. Throughout, M will refer to a smooth, closed, orientable three-manifold, and ξ a contact structure on M ; that is to say that ξ = ker(α), where α is a globally defined 1-form satisfying the condition that α ∧ dα is a volume form on M . An open book decomposition of M is a...

We show that every open book decomposition of a contact 3–manifold can be represented (up to isotopy) by a smooth R–invariant family of pseudoholomorphic curves on its symplectization with respect to a suitable stable Hamiltonian structure. In the planar case, this family survives small perturbations, and thus gives a concrete construction of a stable finite energy foliation that has been used ...

In this note we observe that while all overtwisted contact structures on compact 3–manifolds are supported by planar open book decompositions, not all contact structures are. This has relevance to the Weinstein conjecture [1] and invariants of contact structures.

We determine parts of the contact homology of certain contact 3-manifolds in the framework of open book decompositions, due to Giroux. We study two cases: when the monodromy map of the compatible open book is periodic and when it is pseudo-Anosov. For an open book with periodic monodromy, we verify the Weinstein conjecture. In the case of an open book with pseudo-Anosov monodromy, suppose the b...

As authors, many of us have had less than satisfactory experiences in writing book chapters as part of a themed volume or textbook that, when published, are expensive, inaccessible, and cited and used infrequently through lack of availability [1]. It could even be argued we write them from some sense of obligation and need, but do not put our best science and efforts into them because we know t...