Geometry & Topology

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

Ely Kerman

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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.

Article information

Geom. Topol., Volume 9, Number 4 (2005), 1775-1834.

Received: 22 March 2005
Revised: 11 September 2005
Accepted: 12 August 2005
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53D40: Floer homology and cohomology, symplectic aspects
Secondary: 37J45: Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods

Hofer–Zehnder capacity symplectic submanifold Floer homology


Kerman, Ely. Squeezing in Floer theory and refined Hofer–Zehnder capacities of sets near symplectic submanifolds. Geom. Topol. 9 (2005), no. 4, 1775--1834. doi:10.2140/gt.2005.9.1775.

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