Geometry & Topology, Vol. 9 (2005) Paper no. 40, pages 1775--1834.

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

Ely Kerman

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.

Keywords. Hofer-Zehnder capacity, symplectic submanifold, Floer homology

AMS subject classification. Primary: 53D40. Secondary: 37J45.

E-print: arXiv:math.SG/0502448

DOI: 10.2140/gt.2005.9.1775

Submitted to GT on 22 March 2005. (Revised 11 September 2005.) Paper accepted 12 August 2005. Paper published 25 September 2005.

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Ely Kerman
Mathematics, University of Illinois at Urbana-Champaign
Urbana, IL 61801, USA
Institute of Science, Walailak University
Nakhon Si Thammarat, 80160, Thailand

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