Hofer-Zehnder capacity and Hamiltonian circle actions

Abstract

We introduce the Hofer-Zehnder $G$-semicapacity $c_{HZ}^G(M,\om)$ of a symplectic manifold $(M,\om)$ with respect to a subgroup $G \subset \pi_1(M)$ ($c_{HZ}(M,\om) \leq c^G_{HZ}(M,\om)$) and prove that if $(M,\om)$ is tame and there exists an open subset $U \subset M$ admitting a Hamiltonian free circle action with order greater than two then $U$ has bounded Hofer-Zehnder $G$-semicapacity, where $G \subset \pi_1(M)$ 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 $[\om]|_{\pi_2(M)}=0$). We give a lot of applications of this result. Using P. Biran's decomposition theorem, we prove the following: let $(M^{2n},\Om)$ be a closed K\"ahler manifold ($n>2$) with $[\Om] \in H^2(M,\Z)$ and $\Sigma$ a complex hypersurface representing the Poincar\'e dual of $k[\Om]$, for some $k \in \N$. Suppose either that $\Om$ vanishes on $\pi_2(\Sigma)$ or that $k>2$. Then there exists a decomposition of $M\setminus\Sigma$ into an open dense connected subset with finite Hofer-Zehnder capacity and an isotropic CW-complex. Moreover, we prove that if $(M,\Sigma)$ is subcritical then $M\setminus\Sigma$ has finite Hofer-Zehnder capacity. We also show that given a hyperbolic surface $M$ and $TM$ endowed with the twisted symplectic form $\om_0 + \pi^*\Om$, 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 $M_0$ is finite if $k\leq 1/2$, where $G \subset \pi_1(TM\setminus M_0)$ is the subgroup generated by the fibers of $SM$.