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The action spectrum characterizes closed contact 3-manifolds all of whose Reeb orbits are closed

Authors :
Marco Mazzucchelli
Daniel Cristofaro-Gardiner
University of California [Santa Cruz] (UCSC)
University of California
Unité de Mathématiques Pures et Appliquées (UMPA-ENSL)
École normale supérieure - Lyon (ENS Lyon)-Centre National de la Recherche Scientifique (CNRS)
University of California [Santa Cruz] (UC Santa Cruz)
University of California (UC)
École normale supérieure de Lyon (ENS de Lyon)-Centre National de la Recherche Scientifique (CNRS)
Source :
Commentarii Mathematici Helvetici, Commentarii Mathematici Helvetici, European Mathematical Society, 2020, 95 (3), pp.461-481. ⟨10.4171/CMH/493⟩, Commentarii Mathematici Helvetici, 2020, 95 (3), pp.461-481. ⟨10.4171/CMH/493⟩
Publication Year :
2020
Publisher :
HAL CCSD, 2020.

Abstract

A classical theorem due to Wadsley implies that, on a connected contact manifold all of whose Reeb orbits are closed, there is a common period for the Reeb orbits. In this paper we show that, for any Reeb flow on a closed connected 3-manifold, the following conditions are actually equivalent: (1) every Reeb orbit is closed; (2) all closed Reeb orbits have a common period; (3) the action spectrum has rank 1. We also show that, on a fixed closed connected 3-manifold, a contact form with an action spectrum of rank 1 is determined (up to pull-back by diffeomorphisms) by the set of minimal periods of its closed Reeb orbits.<br />Comment: 18 pages; version 3: we specified that the contact manifolds are required to be connected. To appear in Commentarii Mathematici Helvetici

Details

Language :
English
ISSN :
00102571 and 14208946
Database :
OpenAIRE
Journal :
Commentarii Mathematici Helvetici, Commentarii Mathematici Helvetici, European Mathematical Society, 2020, 95 (3), pp.461-481. ⟨10.4171/CMH/493⟩, Commentarii Mathematici Helvetici, 2020, 95 (3), pp.461-481. ⟨10.4171/CMH/493⟩
Accession number :
edsair.doi.dedup.....dd3b4d9288ddb1c37a10859d8885d35c
Full Text :
https://doi.org/10.4171/CMH/493⟩