Related papers: Lagrangian Circle Actions

On any closed symplectic manifold we construct a path-connected neighborhood of the identity in the Hamiltonian diffeomorphism group with the property that each Hamiltonian diffeomorphism in this neighborhood admits a Hofer and spectral…

We consider a connected symplectic manifold $M$ acted on properly and in a Hamiltonian fashion by a connected Lie group $G$. Inspired to the recent paper \cite{gb2}, see also \cite{ch} and \cite{pacini}, we study Lagrangian orbits of…

Given a Lagrangian submanifold $L$ in a symplectic manifold $X$, the homological Lagrangian monodromy group $\mathcal{H}_L$ describes how Hamiltonian diffeomorphisms of $X$ preserving $L$ setwise act on $H_*(L)$. We begin a systematic study…

Given a closed, orientable Lagrangian submanifold $L$ in a symplectic manifold $(X, \omega)$, we show that if $L$ is relatively exact then any Hamiltonian diffeomorphism preserving $L$ setwise must preserve its orientation. In contrast to…

We prove that when Hodge theory survives on non-compact symplectic manifolds, a compact symplectic Lie group action having fixed points is necessarily Hamiltonian, provided the associated almost complex structure preserves the space of…

This note describes some recent results about the homotopy properties of Hamiltonian loops in various manifolds, including toric manifolds and one point blow ups. We describe conditions under which a circle action does not contract in the…

We study the dynamics of Hamiltonian diffeomorphisms on convex symplectic manifolds. To this end we first establish the Piunikhin-Salamon-Schwarz isomorphism between the Floer homology and the Morse homology of such a manifold, and then use…

We study the extension of homologically trivial symplectic or Hamiltonian cyclic actions to Hamiltonian circle actions on irrational ruled symplectic $4$-manifolds. On one hand, we construct symplectic involutions on minimal irrational…

Let $(M,\omega_M)$ be a six dimensional closed monotone symplectic manifold admitting an effective semifree Hamiltonian $S^1$-action. We show that if the maximal and the minimal fixed component are both two dimensional, then $(M,\omega_M)$…

We compare Hofer's geometries on two spaces associated with a closed symplectic manifold M. The first space is the group of Hamiltonian diffeomorphisms. The second space L consists of all Lagrangian submanifolds of $M \times M$ which are…

We address the problem of computing the fundamental group of a symplectic $S^1$-manifold for non-Hamiltonian actions on compact manifolds, and for Hamiltonian actions on non-compact manifolds with a proper moment map. We generalize known…

In this paper, we complete the classification of six-dimensional closed monotone symplectic manifolds admitting semifree Hamiltonian $S^1$-actions. We also show that every such manifold is $S^1$-equivariantly symplectomorphic to some…

We explore the topology of real Lagrangian submanifolds in a toric symplectic manifold which come from involutive symmetries on its moment polytope. We establish a real analog of the Delzant construction for those real Lagrangians, which…

Let $\text{Ham(M)}$ be the group of Hamiltonian symplectomorphisms of a quantizable, compact, symplectic manifold $(M,\omega)$. We prove the existence of an action integral around loops in $\text{Ham(M)}$, and determine the value of this…

Let $(M, \omega)$ be a connected, compact symplectic manifold equipped with a Hamiltonian $G$ action, where $G$ is a connected compact Lie group. Let $\phi$ be the moment map. In \cite{L}, we proved the following result for $G=S^1$ action:…

In this paper we provide a criterion for the quasi-autonomous Hamiltonian path (``Hofer's geodesic'') on arbitrary closed symplectic manifolds $(M,\omega)$ to be length minimizing in its homotopy class in terms of the spectral invariants…

Let $(M,\omega_M)$ be a six dimensional closed monotone symplectic manifold admitting an effective semifree Hamiltonian $S^1$-action. We show that if the minimal (or maximal) fixed component of the action is an isolated point, then…

In this article, we study the behavior of iterations of symplectomorphisms and Hamiltonian diffeomorphisms on symplectic manifolds. We prove that symplectomorphisms and Hamiltonian diffeomorphisms do not have $C^1$-recurrence on negatively…

We study the action of Hamiltonian diffeomorphisms of a compact symplectic manifold ($X,\omega$) on $C^\infty(X)$ and on functions $C^\infty(X)\to \mathbb R$. We describe various properties of invariant convex functions on $C^\infty(X)$.…

This paper studies Hamiltonian circle actions, i.e. circle subgroups of the group Ham(M,\om) of Hamiltonian symplectomorphisms of a closed symplectic manifold (M,\om). Our main tool is the Seidel representation of \pi_1(\Ham(M,\om)) in the…