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

Abstract

We use the minimal coupling procedure of Sternberg and Weinstein and our pseudo-symplectic capacity theory to prove that every closed symplectic submanifold in any symplectic manifold has an open neighborhood with finite (π1-sensitive) Hofer-Zehnder symplectic capacity. Consequently, the Weinstein conjecture holds near closed symplectic submanifolds in any symplectic manifold. I , W Γ I I ),π I , Π I , Γ I ,π I J , Π I J , λ I J J ⊂ I ∈ N (4.48)π I J : (π I J ) −1 M t I → Im(π I J ) ⊂ M t J

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.…”

“…Kerman's theorem holds when the ambient manifold P is symplectically aspherical while Schlenk's requires P to be only strongly semipositive. Furthermore, G. Lu, [Lu2], has proved the almost existence theorem for neighborhoods of symplectic submanifolds in any symplectic manifold by showing that the contractible Hofer-Zehnder capacity of such a neighborhood is finite using a deep and difficult result due to Liu and Tian, [LT].…”

Periodic orbits of twisted geodesic flows and the Weinstein–Moser theorem

“…In the setting of the generalized Weinstein-Moser theorem or of twisted geodesic flows, these problems are studied for low energy levels in, e.g., [CGK, Co, CIPP, FS, GG2, Gü, Ke3, Ma, Lu1, Lu2, Schl], following the original work of Hofer and Zehnder and of Struwe, [FHW,HZ1,HZ2,HZ3,St]. In particular, almost existence for periodic orbits near a symplectic extremum is established in [Lu2] under no restrictions on the ambient manifold P . When P is geometrically bounded and (stably) strongly semi-positive, almost existence is proved for almost all low energy levels in [Gü] under the assumption that ω| M does not vanish at any point, and in [Schl] when M has middle-dimension and ω| M = 0.…”

Symplectic capacities of classical domains

“…In [6] the first author defined the concept of pseudo symplectic capacities, and specially constructed a pseudo symplectic capacity of Hofer-Zehnder type. Using the latter he computed Gromov symplectic width and Hofer-Zehnder symplectic capacity for many symplectic manifolds, see [6,7,8]. For more detailed study history of symplectic capacities the reader may refer to [1,3,6] and references therein.…”