Noncontractible periodic orbits in cotangent bundles and Floer homology

2004/10/28 by Joa Weber · 4 citations
Mathematics · #Bounded function #Cotangent bundle #Floer homology #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry #Homology (biology) #Homotopy and Cohomology in Algebraic Topology #Hypersurface #Mathematical analysis #Mathematics #Pure mathematics #Symplectic geometry #Trigonometric functions #Unit sphere #math.DS #math.SG #msc:37Jxx #msc:53D40

paper · pdf · doi:10.1215/s0012-7094-06-13334-3

published as Duke Math. J. 133 no.3 (2006), 527--568 · 34 pages, 10 figures, submitted 15 April 2004

arxiv created 2004/10/28 · openalex publication_date 2006/06/14 · arxiv updated 2014/02/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

Let M be a closed connected Riemannian manifold, and let α be a homotopy class of free loops in M. Then, for every compactly supported time-dependent Hamiltonian on the open unit disk cotangent bundle which is sufficiently large over the zero section, we prove the existence of a 1-periodic orbit whose projection to M represents α. The proof shows that the Biran-Polterovich-Salamon capacity of the open unit disk cotangent bundle relative to the zero section is finite. If M is not simply connected, this leads to an existence result for noncontractible periodic orbits on level hypersurfaces corresponding to a dense set of values of any proper Hamiltonian on T*M bounded from below, whenever the levels enclose M. This implies a version of the Weinstein conjecture including multiplicities; we prove existence of closed characteristics—one associated to each nontrivial α—on every contact-type hypersurface in T*M enclosing M

Citations

• Non-contractible periodic orbits, Gromov invariants, and Floer-theoretic torsions
• Elliptic Partial Differential Equations of Second Order
• Floer Homology and the Heat Flow
• Perturbed closed geodesics are periodic orbits: Index and transversality
• Lectures on Morse theory, old and new
• Morse theory for periodic solutions of hamiltonian systems and the maslov index
• Functors and Computations in Floer homology with Applications Part II
• Riemannian Geometry and Geometric Analysis

Cited by

• Computation of annular capacity by Hamiltonian Floer theory of non-contractible periodic trajectories
• Open problems in billiards and quantitative symplectic geometry
• Geometry of contact transformations and domains: orderability versus squeezing
• Barcode entropy for Reeb flows on contact manifolds with Liouville fillings