Abstract
These notes provide an introduction to Giroux’s theory of convex surfaces in contact 3-manifolds and its simplest applications. The first goal is to explain why all the information about a contact structure in a neighborhood of a generic surface is encoded by finitely many curves on the surface. Then we will describe the bifurcations that happen in generic families of surfaces (with one or sometimes two parameters). We sketch a proof of the fact that the standard contact structure on S 3 is tight (due to Bennequin) and that all tight contact structures on S 3 are isotopic to it (due to Eliashberg).
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Notes
- 1.
One may worry about the fact that \(S \times \mathbb {R}\) is non-compact but here the vector field constructed during the proof of this theorem is tangent to S t which is compact for all t hence its flow is well defined for all times.
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Massot, P. (2014). Topological Methods in 3-Dimensional Contact Geometry. In: Bourgeois, F., Colin, V., Stipsicz, A. (eds) Contact and Symplectic Topology. Bolyai Society Mathematical Studies, vol 26. Springer, Cham. https://doi.org/10.1007/978-3-319-02036-5_2
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