Geometry & Topology 9 (2005) 1775–1834

DOI: 10.2140/gt.2005.9.1775

arXiv: math.SG/0502448

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. Following an earlier paper, we also relate our refined capacity to the study of Hamiltonian paths with minimal Hofer length.

Keywords

Hofer–Zehnder capacity, symplectic submanifold, Floer homology

Mathematical Subject Classification

Primary: 53D40

Secondary: 37J45

References

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Publication

Received: 22 March 2005
Revised: 11 September 2005
Accepted: 12 August 2005
Published: 25 September 2005
Proposed: Leonid Polterovich
Seconded: Yasha Eliashberg, Eleny Ionel

Authors