HOFER–ZEHNDER CAPACITY AND HAMILTONIAN CIRCLE ACTIONS

Abstract

We introduce the Hofer–Zehnder Γ-semicapacity [Formula: see text] of a symplectic manifold (M,ω) (or Γ-sensitive Hofer–Zehnder capacity) with respect to a subset Γ⊂π1(M)[Formula: see text] and prove that given a geometrically bounded symplectic manifold (M,ω) and an open subset N⊂M admitting a Hamiltonian free circle action with order greater than two then N has bounded Hofer–Zehnder Γ-semicapacity, where Γ⊂π1(N) is the subgroup generated by the orbits of the action. We give several applications of this result. Using Biran's decomposition theorem, we prove the following: let (M2n,Ω) be a closed Kähler manifold (n>2) with [Ω]∈H2(M,ℤ) and Σ a complex hypersurface representing the Poincaré dual of k[Ω], for some k∈ℕ. Suppose either that Ω vanishes on π2(M) or that k>2. Then there exists a decomposition of M into an open dense subset E such that E\Σ has finite Hofer–Zehnder Γ-semicapacity and an isotropic CW-complex, where Γ⊂π1(E\Σ) is the subgroup generated by the obvious circle action on the normal bundle of Σ. Moreover, we prove that if (M,Σ) is subcritical then M\Σ has finite Hofer–Zehnder Γ-semicapacity. We also show that given a hyperbolic surface M and TM endowed with the twisted symplectic form ω0+π*Ω, where Ω is the Kähler form on M, then the Hofer–Zehnder Γ-semicapacity of the domain Uk bounded by the hypersurface of kinetic energy k minus the zero section M0 is finite if k<1/2, where Γ⊂π1(Uk) is the subgroup generated by the fibers of SM. Finally, we consider the problem of the existence of periodic orbits on prescribed energy levels for magnetic flows. We prove that given any weakly exact magnetic field Ω on any compact Riemannian manifold M there exists a sequence of contractible periodic orbits of arbitrarily small energy, extending a previous result of Polterovich.