Symplectic homology and periodic orbits near symplectic submanifolds

2002/10/30 by Kai Cieliebak, Cieliebak, Kai, Viktor L. Ginzburg +3 · 1 citation
Mathematics · #57R17 53D40 37J45 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric and Algebraic Topology #Geometry and complex manifolds #Homotopy and Cohomology in Algebraic Topology #Symplectic Geometry (math.SG) #math.DG #math.SG #msc:37J45 #msc:53D40 #msc:57R17

paper · pdf · doi:10.48550/arxiv.math/0210468

21 pages, 1 figure. reference added

openalex publication_date 2002/10/30 · arxiv created 2002/11/12 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that a small neighborhood of a closed symplectic submanifold in a geometrically bounded aspherical symplectic manifold has non-vanishing symplectic homology. As a consequence, we establish the existence of contractible closed characteristics on any thickening of the boundary of the neighborhood. When applied to twisted geodesic flows on compact symplectically aspherical manifolds, this implies the existence of contractible periodic orbits for a dense set of low energy values.

Citations

• Propagation in Hamiltonian dynamics and relative symplectic homology

Cited by

• On the Rigidity of Hamiltonians which are Zoll Near a Minimum, with an Application to Magnetic Systems and Almost-Kähler Manifolds

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