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.
Notes on file formats
Ely Kerman
Mathematics, University of Illinois at Urbana-Champaign
Urbana, IL 61801, USA
and
Institute of Science, Walailak University
Nakhon Si Thammarat, 80160, Thailand
Email: ekerman@math.uiuc.edu
URL: http://www.math.uiuc.edu/~ekerman/
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