2002/05/02 by Macarini, Leonardo
#Differential Geometry (math.DG) #Dynamical Systems (math.DS) #FOS: Mathematics #Symplectic Geometry (math.SG)
paper · doi:10.48550/arxiv.math/0205030
We introduce the Hofer-Zehnder G-semicapacity cHZG(M,\om) of a symplectic manifold (M,\om) with respect to a subgroup G ⊂ π1(M) (cHZ(M,\om) ≤ cGHZ(M,\om)) and prove that if (M,\om) is tame and there exists an open subset U ⊂ M admitting a Hamiltonian free circle action with order greater than two then U has bounded Hofer-Zehnder G-semicapacity, where G ⊂ π1(M) is the subgroup generated by the orbits of the action, provided that the index of rationality of (M,\om) is sufficiently great (for instance, if [\om]|π2(M)=0). We give a lot of applications of this result. Using P. Biran's decomposition theorem, we prove the following: let (M2n,\Om) be a closed Kähler manifold (n>2) with [\Om] ∈ H2(M,\Z) and Σ a complex hypersurface representing the Poincaré dual of k[\Om], for some k ∈ \N. Suppose either that \Om vanishes on π2(Σ) or that k>2. Then there exists a decomposition of M∖Σ into an open dense connected subset with finite Hofer-Zehnder capacity and an isotropic CW-complex. Moreover, we prove that if (M,Σ) is subcritical then M∖Σ has finite Hofer-Zehnder capacity. We also show that given a hyperbolic surface M and TM endowed with the twisted symplectic form \om0 + π^*\Om, where \Om is the area form on M, then the Hofer-Zehnder G-semicapacity of the domain bounded by the hypersurface of kinetic energy k minus the zero section M0 is finite if k≤ 1/2, where G ⊂ π1(TM∖ M0) is the subgroup generated by the fibers of SM.