A proof of Weinstein's conjecture in \(\mathbb R^{2n}\)
type: Item
instance of: scholarly article
title: A proof of Weinstein's conjecture in \(\mathbb R^{2n}\)
zbMATH Open document ID: 0631.58013
DOI: 10.1016/S0294-1449(16)30363-8
author: Claude Viterbo
published in: Annales de l'Institut Henri Poincaré. Analyse Non Linéaire
publication date: 1987-01-01T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/78135
review text: Let \((M^{2n},\omega)\) be a symplectic manifold, \(\Sigma\) a hypersurface of \(M^{2n}\). \(\Sigma\) is said to be of contact type if there is a 1- form \(\theta\) on \(\Sigma\), such that \(d\theta =j^*\omega\) (j: \(\Sigma\to W\) is the inclusion map), and \(\theta \wedge (d\theta)^{n- 1}\) is a volume form on \(\Sigma\). A characteristic is a curve everywhere tangent to the line field ker \(\omega|_{\Sigma}\). Theorem. If \(\Sigma \subset ({\mathbb{R}}^{2n},\omega_ 0)\) is a compact hypersurface of contact type, then \(\Sigma\) has a closed characteristic (here \(\omega_ 0=\sum^{n}_{i=1}dx_ i\wedge dy^ i).\) The starting point of the investigation was Seifert's result (1948) on the existence of closed characteristics for some special class of convex hypersurfaces. Weinstein extended this result to general \(C^ 2\) convex hypersurfaces and made the following conjecture: if \(\Sigma\subset (M,\omega)\) is a compact hypersurface of contact type and \(H^ 1(\Sigma,{\mathbb{R}})=0\), then \(\Sigma\) has a closed characteristic.
Mathematics Subject Classification ID: 53D05
Mathematics Subject Classification ID: 53D10
Mathematics Subject Classification ID: 53C15
Mathematics Subject Classification ID: 53C40
zbMATH DE Number: 4026387
zbMATH Keywords: symplectic manifold
zbMATH Keywords: characteristic
zbMATH Keywords: existence of closed characteristics
Wikidata QID: Q60304902
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4026387