Preprint A148/2002
Hofer-Zehnder capacity and Hamiltonian circle actions
Leonardo Macarini
Keywords: Hofer-Zehnder capacity | periodic orbits | Marsden-Weinstein reduction | contact submanifolds | twisted symplectic structures | hypersurfaces of Donaldson type
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. 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 Káhler manifold ($n>2$) with $[\Om] \in H^2(M,\Z)$ and $\Sig$ a complex hypersurface representing the Poincaré dual of $k[\Om]$, for some $k \in \N$. Suppose either that $\Om$ vanishes on $\pi_2(\Sig)$ or that $k>2$. Then there exists a decomposition of $M\sm\Sig$ into an open dense subset with finite Hofer-Zehnder $G$-semicapacity and an isotropic CW-complex, where $G \subset \pi_1(M\sm\Sig)$ is the subgroup generated by the obvious circle action on the normal bundle of $\Sigma$. Moreover, we prove that if $(M,\Sig)$ is subcritical then $M\sm\Sig$ has finite Hofer-Zehnder $G$-semicapacity. 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 iff $k\leq 1/2$, where $G \subset \pi_1(TM\setminus M_0)$ is the subgroup generated by the fibers of $SM$. Finally, we will 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 $\Om$ on any compact Riemannian manifold $M$ then there exists a sequence of contractible periodic orbits of energy arbitrarily small, extending a previous result of L. Polterovich.