2003/03/19 by Leonardo Macarini, Macarini, Leonardo
Mathematics · #Advanced Operator Algebra Research #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Symplectic Geometry (math.SG) #math.DS #math.SG
paper · pdf · doi:10.48550/arxiv.math/0303230
19 pages, 4 figures. Revised version
openalex publication_date 2003/03/19 · arxiv created 2003/08/18 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We introduce the concept of Hofer-Zehnder G-semicapacity (or G-sensitive Hofer-Zehnder capacity) and prove that given a geometrically bounded symplectic manifold (M,ω) and an open subset N ⊂ M endowed with a Hamiltonian free circle action ϕ then N has bounded Hofer-Zehnder Gϕ-semicapacity, where Gϕ⊂ π1(N) is the subgroup generated by the homotopy class of the orbits of ϕ. In particular, N has bounded Hofer-Zehnder capacity. We give two types of applications of the main result. Firstly, we prove that the cotangent bundle of a compact manifold endowed with a free circle action has bounded Hofer-Zehnder capacity. In particular, the cotangent bundle T^*G of any compact Lie group G has bounded Hofer-Zehnder capacity. Secondly, we consider Hamiltonian circle actions given by symplectic submanifolds. For instance, we prove the following generalization of a recent result of Ginzburg-Gürel: almost all low levels of a function on a geometrically bounded symplectic manifold carry contractible periodic orbits of the Hamiltonian flow, provided that the function attains its minimum along a closed symplectic submanifold.