Pseudoholomorphic curves in symplectizations with applications to the Weinstein conjecture in dimension three
type: Item
instance of: scholarly article
title: Pseudoholomorphic curves in symplectizations with applications to the Weinstein conjecture in dimension three
zbMATH Open document ID: 0797.58023
DOI: 10.1007/BF01232679
published in: Inventiones Mathematicae
publication date: 1994-10-11T00:00:00Z
full work available at URL: https://eudml.org/doc/144157
review text: A contact form on the smooth closed orientable \((2n + 1)\)-dimensional manifold \(M\) is a 1-form \(\lambda\) such that \(\lambda \wedge (d \lambda)^ n\) is a volume form on \(M\). Associated to \(\lambda\) there are two important structures: the Reeb vector field \(X = X_ \lambda\) defined by \(i_ X \lambda \equiv 1\), \(i_ X d\lambda \equiv 0\) and the contact structure \(\xi = \xi_ \lambda \to M\) given by \(\xi_ \lambda = \text{Kern}(\lambda) \subset TM\). The contact structure is stable [see \textit{J. W. Gray}, Ann. Math., II. Ser. 69, 421-450 (1959; Zbl 0092.39301)], but in contrast the dynamics of the Reeb vector field changes drastically under small perturbation. It is known the Weinstein conjecture [\textit{A. Weinstein}, J. Differ. Equations 33, 353-358 (1979; Zbl 0388.58020)]: for every contact form on a closed manifold \(M\) the Reeb vector field has at least one periodic orbit, provided \(H^ 1(M;R) = 0\). The Weinstein conjecture holds for \(M\) provided for every contact form \(\lambda\) on \(M\) the associated Reeb vector field has a closed orbit. The author is restricted to the case \(\dim M = 3\), since in three dimensions his method seems to be most powerful and gives a more complete picture of the Weinstein conjecture. One of the main results is the existence of periodic orbits for the Reeb vector field associated to overtwisted contact forms. For such contact forms there does not exist any compact symplectic manifold \((W,\omega)\) with \(M = \partial W\) and \(d \lambda = \omega\mid M\). There are for example many overtwisted 1-forms on \(S^ 3\). As a corollary \(S^ 3\) with such structure can never be embedded into \(\mathbb{R}^ 4\) in such a way that the Hamiltonian flow on the embedded \(S^ 3\) is the conjugate to the abstract Reeb flow. The problem for higher dimensions and some applications will be given in a future paper.
Mathematics Subject Classification ID: 37J10
Mathematics Subject Classification ID: 53D35
Mathematics Subject Classification ID: 58E05
Mathematics Subject Classification ID: 53D10
Mathematics Subject Classification ID: 32Q65
zbMATH DE Number: 549637
zbMATH Keywords: Reeb vector field
zbMATH Keywords: Weinstein conjecture
zbMATH Keywords: contact form
Wikidata QID: Q29037381
author: Helmut Hofer
reviewed by: Boris V. Loginov
MaRDI profile type: Publication
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OpenAlex ID: W2091362639
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description: scientific article; zbMATH DE number 549637