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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Fast finite-energy planes in symplectizations and applications


Author: Umberto Hryniewicz
Journal: Trans. Amer. Math. Soc. 364 (2012), 1859-1931
MSC (2010): Primary 53D35, 53D10; Secondary 53D25, 37J99
Published electronically: November 29, 2011
MathSciNet review: 2869194
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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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Additional Information

Umberto Hryniewicz
Affiliation: Departamento de Matemática Aplicada, IM-UFRJ, Rio de Janeiro, Brazil.
Address at time of publication: Institute for Advanced Study, Einstein Drive, Princeton, New Jersey 08540
Email: umberto@labma.ufrj.br, umbertolh@math.ias.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-2011-05387-0
PII: S 0002-9947(2011)05387-0
Keywords: Hamiltonian dynamics, pseudo-holomorphic curves, contact geometry
Received by editor(s): March 9, 2009
Received by editor(s) in revised form: March 13, 2009, June 19, 2009, February 16, 2010, and May 12, 2010
Published electronically: November 29, 2011
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.