Trapped Reeb orbits do not imply periodic ones
type: Item
instance of: scholarly article
title: Trapped Reeb orbits do not imply periodic ones
zbMATH Open document ID: 1304.37019
author: Hansjörg Geiges
author: Nena Röttgen
author: Kai Zehmisch
published in: Inventiones Mathematicae
publication date: 2014-10-30T00:00:00Z
full work available at URL: https://arxiv.org/abs/1309.4982
review text: A theorem by \textit{Y. Eliashberg} and \textit{H. Hofer} [Differ. Integral Equ. 7, No. 5--6, 1303--1324 (1994; Zbl 0803.58045)] states that any contact form \(\alpha\) on \(\mathbb R^3\) which equals the standard form outside a compact set and whose Reeb vector field does not have any periodic orbits, is diffeomorphic to the standard one. In terms of the Reeb vector field, this is equivalent to say that if the Reeb vector field of \(\alpha\) has a trapped orbit, then it also has a periodic orbit. H. Hofer conjectured the higher-dimensional analogue of this result, but in this paper the authors disprove that conjecture by an example.
reviewed by: Manuel de León
Mathematics Subject Classification ID: 37C27
Mathematics Subject Classification ID: 37C70
Mathematics Subject Classification ID: 53D10
zbMATH DE Number: 6362528
zbMATH Keywords: contact manifolds
zbMATH Keywords: Reeb vector field
MaRDI profile type: Publication
OpenAlex ID: W2079445093
Wikidata QID: Q60228720
arXiv ID: 1309.4982
cites work: Isoperimetric inequalities for minimal submanifolds in Riemannian manifolds: a counterexample in higher codimension
cites work: First Steps Towards a Symplectic Dynamics
cites work: A Hamiltonian characterization of the three-ball
cites work: An Introduction to Contact Topology
DOI: 10.1007/S00222-014-0500-9
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description: scientific article; zbMATH DE number 6362528