Fast finite-energy planes in symplectizations and applications

Hryniewicz, Umberto

doi:10.1090/S0002-9947-2011-05387-0

Fast finite-energy planes in symplectizations and applications
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by Umberto Hryniewicz

Trans. Amer. Math. Soc. 364 (2012), 1859-1931

DOI: https://doi.org/10.1090/S0002-9947-2011-05387-0

Published electronically: November 29, 2011

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Abstract:

We define the notion of fast finite-energy planes in the symplectization of a closed $3$-dimensional energy level $M$ of contact type. We use them to construct special open book decompositions of $M$ when the contact structure is tight and induced by a (non-degenerate) dynamically convex contact form. The obtained open books have disk-like pages that are global surfaces of section for the Hamiltonian dynamics. Let $S \subset \mathbb {R}^4$ be the boundary of a smooth, strictly convex, non-degenerate and bounded domain. We show that a necessary and sufficient condition for a closed Hamiltonian orbit $P\subset S$ to be the boundary of a disk-like global surface of section for the Hamiltonian dynamics is that $P$ is unknotted and has self-linking number $-1$.

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