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Rigid constellations of closed Reeb orbits

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

Published online by Cambridge University Press:  04 September 2017

Ely Kerman
Ely Kerman*
Affiliation:
Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA email [email protected]
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Abstract

We use Hamiltonian Floer theory to recover and generalize a classic rigidity theorem of Ekeland and Lasry. That theorem can be rephrased as an assertion about the existence of multiple closed Reeb orbits for certain tight contact forms on the sphere that are close, in a suitable sense, to the standard contact form. We first generalize this result to Reeb flows of contact forms on prequantization spaces that are suitably close to Boothby–Wang forms. We then establish, under an additional nondegeneracy assumption, the same rigidity phenomenon for Reeb flows on any closed contact manifold. A natural obstruction to obtaining sharp multiplicity results for closed Reeb orbits is the possible existence of fast closed orbits. To complement the existence results established here, we also show that the existence of such fast orbits cannot be precluded by any condition which is invariant under contactomorphisms, even for nearby contact forms.

Keywords

periodic orbitsReeb flowsFloer homology

MSC classification

Primary: 53D40: Floer homology and cohomology, symplectic aspects 53D10: Contact manifolds, general 37J45: Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods 70H12: Periodic and almost periodic solutions
Type
Research Article
Information
Compositio Mathematica , Volume 153 , Issue 11 , November 2017 , pp. 2394 - 2444
Copyright
© The Author 2017 

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