Hofer-Zehnder semicapacity of cotangent bundles and symplectic submanifolds

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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.