Symplectic homology and periodic orbits near symplectic submanifolds
By Kai Cieliebak, V. Ginzburg, E. Kerman. 2002
52 citations recorded by the source corpus.
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.
Subfields: Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/8f90ad3825720752359ff57a0e84c9e5aa09fa4d
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In Geometry and Topology
- V. Ginzburg
- Basak Z. Gurel
- Sheila Sandon
- Michael Usher
- L. Polterovich
- Urs Frauenfelder
- F. Schlenk
- Y. Eliashberg
In Mathematical Physics
- V. Ginzburg
- Basak Z. Gurel
- Sheila Sandon
- L. Polterovich
- Y. Eliashberg
- Urs Frauenfelder
- F. Schlenk
- Sang Seon Kim
Citations in the corpus
- Applications of Hofer's geometry to Hamiltonian dynamics (2003, background): “Note that if A is (stably) displaceable, then an entire neighbourhood of A is (stably) displaceable.”
- Contact Homology, Capacity and Non-Squeezing in R^2n x S^1 via Generating Functions (2009, methodology): “fiberwise starshaped domains in R2n× S1. This is a special instance of the Symplectic Field Theory, and is related to a version of the filtered symplectic homology of domains in R2n as used in [BPS03], [CGK04] and [GG04]. We will present here a proof of the same results using generating functions inst”
- Relative Hofer-Zehnder capacity and periodic orbits in twisted cotangent bundles (2003, methodology): “ submanifold of Wand let H: W→ R attain its absolute minimum H= 0 on M. Then the levels H= ǫcarry contractible in Wperiodic orbits for almost all small ǫ>0. This corollary generalizes a theorem of [CGK] where the existence of periodic orbits is established for a dense set of small ǫ>0. RELATIV”
- Erratum to “Geometry of contact transformations and domains: orderability versus squeezing” (2005, methodology): “We use a version of symplectic homology associated to negative compactly supported Hamiltonians as in [3; 10; 25], see Section 4 below for the definition and conventions about the grading.”
- Action and index spectra and periodic orbits in Hamiltonian dynamics (2008, background): “Throughout this section, we assume that M is weakly monotone and geometrically bounded and rational; see Audin and Lafontaine [2] and Cieliebak, Ginzburg and Kerman [6] for the definition and a detailed discussion of geometrically bounded manifolds.”