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Systolic ratio, index of closed orbits and convexity for tight contact forms on the three-sphere

Part of: Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems Symplectic geometry, contact geometry

Published online by Cambridge University Press:  06 November 2018

Alberto Abbondandolo ,
Barney Bramham ,
Umberto L. Hryniewicz  and
Pedro A. S. Salomão
Alberto Abbondandolo
Affiliation:
Ruhr Universität Bochum, Fakultät für Mathematik, Universitätsstrasse 150, D-44801 Bochum, Germany email [email protected]
Barney Bramham
Affiliation:
Ruhr Universität Bochum, Fakultät für Mathematik, Universitätsstrasse 150, D-44801 Bochum, Germany email [email protected]
Umberto L. Hryniewicz
Affiliation:
Universidade Federal do Rio de Janeiro, Departamento de Matemática Aplicada, Av. Athos da Silveira Ramos 149, Rio de Janeiro RJ, 21941-909, Brazil email [email protected]
Pedro A. S. Salomão
Affiliation:
Universidade de São Paulo, Instituto de Matemática e Estatística, Departamento de Matemática, Rua do Matão, 1010 – Cidade Universitária, São Paulo SP, 05508-090, Brazil email [email protected]
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Abstract

We construct a dynamically convex contact form on the three-sphere whose systolic ratio is arbitrarily close to 2. This example is related to a conjecture of Viterbo, whose validity would imply that the systolic ratio of a convex contact form does not exceed 1. We also construct, for every integer $n\geqslant 2$, a tight contact form with systolic ratio arbitrarily close to $n$ and with suitable bounds on the mean rotation number of all the closed orbits of the induced Reeb flow.

Keywords

Reeb dynamicssystolic geometryperiodic orbits

MSC classification

Primary: 53D10: Contact manifolds, general
Secondary: 37J05: General theory, relations with symplectic geometry and topology
Type
Research Article
Information
Compositio Mathematica , Volume 154 , Issue 12 , December 2018 , pp. 2643 - 2680
Copyright
© The Authors 2018 

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Footnotes

The research of A. Abbondandolo and B. Bramham is supported by the SFB/TRR 191 ‘Symplectic Structures in Geometry, Algebra and Dynamics’, funded by the Deutsche Forschungsgemeinschaft. P. A. S. Salomão is supported by the FAPESP grant 2016/25053-8 and the CNPq grant 306106/2016-7. U. L. Hryniewicz was supported by CNPq grant 309966/2016-7 and by the Humboldt Foundation; he also acknowledges the generous hospitality of the Mathematics Department of the Ruhr-Universität Bochum.

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