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Two closed orbits for non-degenerate Reeb flows

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

Published online by Cambridge University Press:  21 February 2020

MIGUEL ABREU ,
JEAN GUTT ,
JUNGSOO KANG  and
LEONARDO MACARINI
MIGUEL ABREU
Affiliation:
Center for Mathematical Analysis, Geometry and Dynamical Systems, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001Lisboa, Portugal. e-mail: mabreu@math.tecnico.ulisboa.pt
JEAN GUTT
Affiliation:
Institut de Mathématiques de Toulouse, Université Toulouse III Paul–Sabatier, 118, route de Narbonne, F-31062Toulouse, France. e-mail: jean.gutt@math.univ-toulouse.fr
JUNGSOO KANG
Affiliation:
Department of Mathematical Sciences, Research Institute in Mathematics, Seoul National University, Gwanak-Gu, Seoul08826, South Korea. e-mail: jungsoo.kang@snu.ac.kr
LEONARDO MACARINI
Affiliation:
Center for Mathematical Analysis, Geometry and Dynamical Systems, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001Lisboa, Portugal. e-mail: macarini@math.tecnico.ulisboa.pt
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Abstract

We prove that every non-degenerate Reeb flow on a closed contact manifold M admitting a strong symplectic filling W with vanishing first Chern class carries at least two geometrically distinct closed orbits provided that the positive equivariant symplectic homology of W satisfies a mild condition. Under further assumptions, we establish the existence of two geometrically distinct closed orbits on any contact finite quotient of M. Several examples of such contact manifolds are provided, like displaceable ones, unit cosphere bundles, prequantisation circle bundles, Brieskorn spheres and toric contact manifolds. We also show that this condition on the equivariant symplectic homology is preserved by boundary connected sums of Liouville domains. As a byproduct of one of our applications, we prove a sort of Lusternik–Fet theorem for Reeb flows on the unit cosphere bundle of not rationally aspherical manifolds satisfying suitable additional assumptions.

MSC classification

Primary: 53D40: Floer homology and cohomology, symplectic aspects
Secondary: 53D25: Geodesic flows 37J10: Symplectic mappings, fixed points 37J55: Contact systems
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 170 , Issue 3 , May 2021 , pp. 625 - 660
Copyright
© Cambridge Philosophical Society 2020

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