Related papers: Symplectic action around loops in $\text{Ham}(M)$

We prove the following to results: (1) A subgroup G of the isometry group of a Riemannian manifold M acts properly on M if and only if G is closed in the isometry group of M. (2) The orbits of an isometric action are closed if and only if…

Let $M$ be a compact orientable surface equipped with a volume form $\omega$, $P$ be either $\mathbb{R}$ or $S^1$, $f:M\to P$ be a $C^{\infty}$ Morse map, and $H$ be the Hamiltonian vector field of $f$ with respect to $\omega$. Let also…

While the Hamiltonian group actions on closed symplectic manifolds have been widely explored throughout the last couple of decades, the study on Hamiltonian group actions on symplectic manifolds with a contact type boundary has started only…

Let $(\bar{M}, \omega)$ be a compact symplectic manifold with convex boundary and $c_1(T\bar{M})=0$. Suppose that $(\bar{M}, \omega)$ is equipped with a convex Hamiltonian $G$-action for some connected, compact Lie group $G$. We construct…

We present a general framework for reduction of symplectic Q-manifolds via graded group actions. In this framework, the homological structure on the acting group is a multiplicative multivector field.

Let $G$ be a complex semisimple algebraic group and $X$ be a complex symmetric homogeneous $G$-variety. Assume that both $G$, $X$ as well as the $G$-action on $X$ are defined over real numbers. Then $G(\mathbb{R})$ acts on $X(\mathbb{R})$…

We prove the existence of at least $cl(M)$ periodic orbits for certain time dependant Hamiltonian systems on the cotangent bundle of an arbitrary compact manifold $M$. These Hamiltonians are not necessarily convex but they satisfy a certain…

In this paper, we compute contact homology of some quasi-regular contact structures, which admit Hamiltonian actions of Reeb type of Lie groups. We will discuss the toric contact case, (where the torus is of Reeb type), and the case of…

The main theme of this paper is the connection between the existence of infinitely many periodic orbits for a Hamiltonian system and the behavior of its action or index spectrum under iterations. We use the action and index spectra to show…

Let $(M,\omega)$ be a compact symplectic manifold with a Hamiltonian GKM action of a compact torus. We formulate a positive condition on the space; this condition is satisfied if the underlying symplectic manifold is monotone. The main…

We study hyperpolar actions on reducible symmetric spaces of the compact type. Our main result is that an indecomposable hyperpolar action on a symmetric space of the compact type is orbit equivalent to a Hermann action or of cohomogeneity…

Lalonde and McDuff showed that the natural action of the rational homology of the group of Hamiltonian diffeomorphisms of a closed symplectic manifold $(M, \omega)$ on the rational homology groups $H_*(M,{\mathbb Q})$ is trivial. In this…

We prove a convexity theorem for the image of the moment map of a Hamiltonian torus action on a $b^m$-symplectic manifold.

We show that if a Lie group acts properly on a co-oriented contact manifold preserving the contact structure, then the contact quotient is topologically a stratified space (in the sense that a neighborhood of a point in the quotient is a…

We classify symplectic actions of 2-tori on compact, connected symplectic 4-manifolds, up to equivariant symplectomorphisms. This extends results of Atiyah, Guillemin-Sternberg, Delzant and Benoist. The classification is in terms of a…

In this paper we define an action by the symplectomorphisms on a symplectic manifold on the space of real singular polarizations. It is then shown that under some topological conditions, this action preserves quantization by a fixed…

In systems where one coordinate undergoes periodic oscillation, the net displacement in any other coordinate over a single period is shown to be given by differentiation of the action integral associated with the oscillating coordinate.…

A study of symplectic actions of a finite group $G$ on smooth 4-manifolds is initiated. The central new idea is the use of $G$-equivariant Seiberg-Witten-Taubes theory in studying the structure of the fixed-point set of these symmetries.…

We prove several results on symplectic varieties with a Hamiltonian action of a reductive group having invariant Lagrangian subvarieties. Our main result states that the images of the moment maps of a Hamiltonian variety and of the…

In this paper, we study Hamiltonian R-actions on symplectic orbifolds [M/S], where R and S are tori. We prove an injectivity theorem and generalize Tolman-Weitsman's proof of the GKM theorem in this setting. The main example is the…