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 $SM$.

Abstract:
We introduce the concept of Hofer-Zehnder $G$-semicapacity (or $G$-sensitive Hofer-Zehnder capacity) and prove that given a geometrically bounded symplectic manifold $(M,\omega)$ and an open subset $N \subset M$ endowed with a Hamiltonian free circle action $\phi$ then $N$ has bounded Hofer-Zehnder $G_\phi$-semicapacity, where $G_\phi \subset \pi_1(N)$ is the subgroup generated by the homotopy class of the orbits of $\phi$. In particular, $N$ has bounded Hofer-Zehnder capacity. We give two types of applications of the main result. Firstly, we prove that the cotangent bundle of a compact manifold endowed with a free circle action has bounded Hofer-Zehnder capacity. In particular, the cotangent bundle $T^*G$ of any compact Lie group $G$ has bounded Hofer-Zehnder capacity. Secondly, we consider Hamiltonian circle actions given by symplectic submanifolds. For instance, we prove the following generalization of a recent result of Ginzburg-G\"urel: almost all low levels of a function on a geometrically bounded symplectic manifold carry contractible periodic orbits of the Hamiltonian flow, provided that the function attains its minimum along a closed symplectic submanifold.

Abstract:
The main theme of this paper is a relative version of the almost existence theorem for periodic orbits of autonomous Hamiltonian systems. We show that almost all low levels of a function on a geometrically bounded symplectically aspherical manifold carry contractible periodic orbits of the Hamiltonian flow, provided that the function attains its minimum along a closed symplectic submanifold. As an immediate consequence, we obtain the existence of contractible periodic orbits on almost all low energy levels for twisted geodesic flows with symplectic magnetic field. We give examples of functions with a sequence of regular levels without periodic orbits, converging to an isolated, but very degenerate, minimum. The proof of the relative almost existence theorem hinges on the notion of the relative Hofer-Zehnder capacity and on showing that this capacity of a small neighborhood of a symplectic submanifold is finite. The latter is carried out by proving that the flow of a Hamiltonian with sufficiently large variation has a non-trivial contractible one-periodic orbit, when the Hamiltonian is constant and equal to its maximum near a symplectic submanifold and supported in a neighborhood of the submanifold.

Abstract:
Let $M$ be a compact manifold with an effective semi-free circle action whose fixed point set has trivial normal bundle. We prove that its cotangent bundle endowed with the canonical symplectic form has bounded Hofer-Zehnder sensitive capacity. We give several examples like the product of any compact manifold with $\S^n$ or a connected sum $\CP^n # ... # \CP^n$.

Abstract:
We use Floer homology to study the Hofer-Zehnder capacity of neighborhoods near a closed symplectic submanifold M of a geometrically bounded and symplectically aspherical ambient manifold. We prove that, when the unit normal bundle of M is homologically trivial in degree dim(M) (for example, if codim(M) > dim(M)), a refined version of the Hofer-Zehnder capacity is finite for all open sets close enough to M. We compute this capacity for certain tubular neighborhoods of M by using a squeezing argument in which the algebraic framework of Floer theory is used to detect nontrivial periodic orbits. As an application, we partially recover some existence results of Arnold for Hamiltonian flows which describe a charged particle moving in a nondegenerate magnetic field on a torus. We also relate our refined capacity to the study of Hamiltonian paths with minimal Hofer length.

Abstract:
We use the criteria of Lalonde and McDuff to determine a new class of examples of length minimizing paths in the group $Ham(M)$. For a compact symplectic manifold $M$ of dimension two or four, we show that a path in $Ham(M)$, generated by an autonomous Hamiltonian and starting at the identity, which induces no non-constant closed trajectories of points in $M$, is length minimizing among homotopic paths. The major step in the proof involves determining an upper bound for the Hofer-Zehnder capacity for symplectic manifolds of the type $(M \times D(a))$ where $M$ is compact and has dimension two or four. In the appendix, we give an alternate proof of Polterovich's result that rotation in $CP^2$ and in the blow-up of $CP^2$ at one point is a length minimizing path with respect to the Hofer norm. Here we use the Gromov capacity and describe the necessary ball embeddings.

Abstract:
For every nontrivial free homotopy class $\alpha$ of loops in every closed connected Riemannian manifold $M$, we prove existence of a noncontractible 1-periodic orbit, for every compactly supported time-dependent Hamiltonian on the open unit cotangent bundle which is sufficiently large over the zero section. The proof shows that the Biran-Polterovich-Salamon capacity is finite for every closed connected Riemannian manifold and every free homotopy class of loops. This implies a dense existence theorem for periodic orbits on level hypersurfaces and, consequently, a refined version of the Weinstein conjecture: Existence of closed characteristics (one associated to each nontrivial $\alpha$) on hypersurfaces in $T^*M$ which are of contact type and contain the zero section.

Abstract:
We introduce a notion of symplectic capacity relative to a coisotropic submanifold of a symplectic manifold, and we define a modification of the Hofer-Zehnder capacity and show it provides an example. As a corollary, we obtain a non-squeezing theorem for symplectic embeddings relative to coisotropic constraints.

Abstract:
A Hamiltonian circle action on a compact symplectic manifold is known to be a closed geodesic with respect to the Hofer metric on the group of Hamiltonian diffeomorphisms. If the momentum map attains its minimum or maximum at an isolated fixed point with isotropy weights not all equal to plus or minus one, then this closed geodesic can be deformed into a loop of shorter Hofer length. In this paper we give a lower bound for the possible amount of shortening, and we give a lower bound for the index ("number of independent shortening directions"). If the minimum or maximum is attained along a submanifold B, then we deform the circle action into a loop of shorter Hofer length whenever the isotropy weights have sufficiently large absolute values and the normal bundle of B is sufficiently un-twisted.

Abstract:
We use the minimal coupling procedure of Sternberg and Weinstein and our pseudo-symplectic capacity theory to prove that every closed symplectic submanifold in any symplectic manifold has an open neighborhood with finite ($\pi_1$-sensitive) Hofer-Zehnder symplectic capacity. Consequently, the Weinstein conjecture holds near closed symplectic submanifolds in any symplectic manifold.