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

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 [18, 19,26]. As an application, we partially recover some existence results of Arnold [1] for Hamiltonian flows which describe a charged particle moving in a nondegenerate magnetic field on a torus. Following [43], we also relate our refined capacity to the study of Hamiltonian paths with minimal Hofer length.

Totally Non-Coisotropic Displacement and Its Applications to Hamiltonian Dynamics

“…The almost existence theorem due to Hofer and Zehnder and to Struwe,[HZ1,HZ2,St], asserts that almost all regular level sets of a proper autonomous Hamiltonian on R 2n carry periodic orbits. A similar result has also been proved for CP n , symplectic vector bundles, subcritical Stein manifolds, and certain other symplectic manifolds; see, e.g., [FS,GG,HV,Ke2,Lu2,Sc] and also the survey [Gi3] and references therein. Here, similarly to [CGK,FS,Gi1,GG,GK1,GK2,Ke1,Ke2,Lu1,Mac,Pol2,Schl], we focus on these theorems for Hamiltonians supported in a neighborhood of a closed submanifold.…”

“…In this setting, as a particular case of Theorem 1.5, we obtain the existence of contractible twisted geodesics on almost all low energy levels, provided that the magnetic field is nowhere zero -a result complementing numerous other theorems on the existence of twisted geodesics; see, e.g., [CGK,Gi1,GG,GK1,GK2,Ke1,Ke2,Lu1,Mac,Pol2,Schl]. Note that the assumption that η is nowhere zero ensures that M is nowhere coisotropic.…”

“…Note in this connection that almost existence in a neighborhood of a symplectic submanifold satisfying certain additional hypotheses was proved by Kerman,[Ke2]. On the other hand, almost existence in a neighborhood of a non-Lagrangian submanifold of middle dimension was established by Schlenk, [Schl].…”

Finiteness of the Hofer-Zehnder capacity of neighborhoods of symplectic submanifolds

“…Very recently, motivated by these and a very useful formula for the Hofer-Zehnder symplectic capacity of the product of a symplectic manifold and the standard symplectic ball, (cf. [4,8,15,16,23,30]), Kerman [11] proposed the following. [29, page 69]).…”

Coisotropic intersections

“…Proposition 5.1 has a counter-part asserting the existence of homotopy connecting trajectories "transferring" action selectors; cf. [CGK,Ke2]. This is a much more standard result and we treat it in lesser detail.…”