Squeezing in Floer theory and refined Hofer-Zehnder capacities of sets near symplectic submanifolds

Abstract

We use Floer homology to study the Hofer-Zehnder capacity of neighborhoods near a closed symplectic submanifold M of a geometrically bounded and symplectically aspherical ambient manifold. We prove that, when the unit normal bundle of M is homologically trivial in degree dim(M) (for example, if codim(M) > dim(M)), a refined version of the Hofer-Zehnder capacity is finite for all open sets close enough to M. We compute this capacity for certain tubular neighborhoods of M by using a squeezing argument in which the algebraic framework of Floer theory is used to detect nontrivial periodic orbits. As an application, we partially recover some existence results of Arnold for Hamiltonian flows which describe a charged particle moving in a nondegenerate magnetic field on a torus. We also relate our refined capacity to the study of Hamiltonian paths with minimal Hofer length.

Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol9/paper40.abs.html

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• Floer homology of families I
• Hofer-Zehnder capacity and length minimizing Hamiltonian paths
• Geometric variants of the Hofer norm
• Gromov-Witten invariants and pseudo symplectic capacities
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Cited by in corpus (3)

• Action and Index Spectra and Periodic Orbits in Hamiltonian Dynamics
• The contact property for nowhere vanishing magnetic fields on the two-sphere
• On symplectic capacities and their blind spots