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Hofer-Zehnder capacity and Hamiltonian circle actions | | Leonardo Macarini
; | | Date: |
3 May 2002 | | Subject: | Symplectic Geometry; Differential Geometry; Dynamical Systems | math.SG math.DG math.DS | | 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 Kähler manifold ($n>2$) with $[Om] in H^2(M,)$ and $Sigma$ a complex hypersurface representing the Poincaré 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 $MsetminusSigma$ 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 $MsetminusSigma$ 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 $kleq 1/2$, where $G subset pi_1(TMsetminus M_0)$ is the subgroup generated by the fibers of $SM$. | | Source: | arXiv, math.SG/0205030 | | Services: | Forum | Review | PDF | Favorites | |
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