Volume 9, issue 4 (2005)

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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Squeezing in Floer theory and refined Hofer–Zehnder capacities of sets near symplectic submanifolds

Ely Kerman

Geometry & Topology 9 (2005) 1775–1834

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 2000
Primary: 53D40
Secondary: 37J45
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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
Ely Kerman
Mathematics
University of Illinois at Urbana–Champaign
Urbana
Illinois 61801
USA
Institute of Science
Walailak University
Nakhon Si Thammarat
80160
Thailand