Related papers: K-area, Hofer metric and geometry of conjugacy cla…

In this article we study the Hofer geometry of a compact Lie group $K$ which acts by Hamiltonian diffeomorphisms on a symplectic manifold $M$. Generalized Hofer norms on the Lie algebra of $K$ are introduced and analyzed with tools from…

We construct the vortex Floer homology group $VHF (M,\mu;H)$ for an aspherical Hamiltonian $G$-manifold $(M, \omega)$ with moment map $\mu$ and a class of $G$-invariant Hamiltonian loop $H_t$, following the proposal of [3]. This is a…

To every closed subset $X$ of a symplectic manifold $(M,\omega)$ we associate a natural group of Hamiltonian diffeomorphisms $Ham(X,\omega)$. We equip this group with a semi-norm $\Vert\cdot\Vert^{X,\omega}$, generalizing the Hofer norm. We…

We show that if (M,\omega) is a closed symplectic manifold which admits a nontrivial Hamiltonian vector field all of whose contractible closed orbits are constant, then Hofer's metric on the group of Hamiltonian diffeomorphisms of…

The commutator length of a Hamiltonian diffeomorphism $f\in \mathrm{Ham}(M, \omega)$ of a closed symplectic manifold $(M,\omega)$ is by definition the minimal $k$ such that $f$ can be written as a product of $k$ commutators in…

In this work we relate the known results about the homotopy type of classifying spaces for smooth foliations, with the homology and cohomology of the discrete group of diffeomorphisms of a smooth compact connected oriented manifold. The…

Using a "Hodge decomposition" of symplectic isotopies on a compact symplectic manifold $(M,\omega)$, we construct a norm on the identity component in the group of all symplectic diffeomorphisms of $(M,\omega)$ whose restriction to the group…

We show that if K: P \to R is an autonomous Hamiltonian on a symplectic manifold (P,\Omega) which attains 0 as a Morse-Bott nondegenerate minimum along a symplectic submanifold M, and if c_1(TP)|_M vanishes in real cohomology, then the…

In this paper, we construct a Hamiltonian Floer theory based invariant called relative symplectic cohomology, which assigns a module over the Novikov ring to compact subsets of closed symplectic manifolds. We show the existence of…

For a class of Riemannian manifolds that include products of arbitrary compact manifolds with manifolds of nonpositive sectional curvature on the one hand, or with certain positive-curvature examples such as spheres of dimension at least 3…

We construct Hamiltonian Floer complexes associated to continuous, and even lower semi-continuous, time dependent exhaustion functions on geometrically bounded symplectic manifolds. We further construct functorial continuation maps…

This is the first of two papers devoted to showing how the rich algebraic formalism of Eliashberg-Givental-Hofer's symplectic field theory (SFT) can be used to define higher algebraic structures on the symplectic cohomology of open…

This paper establishes robust obstructions to representing Hamiltonian diffeomorphisms as $k$-th powers ($k \geq 2$) or embedding them in flows for certain higher-dimensional symplectic manifolds $(M,\omega)$, including surface bundles. We…

In the case of a compact real analytic symplectic manifold M we describe an approach to the complexification of Hamiltonian flows [Se, Do1, Th1] and corresponding geodesics on the space of Kahler metrics. In this approach, motivated by…

The notion of linear K-system is introduced by the present authors as an abstract model arising from the structure of compactified moduli spaces of solutions to Floer's equation in the book [FOOO14]. The purpose of the present article is 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…

In this paper we find connection between the Hofer's metric of the group of Hamiltonian diffeomorphisms of a closed symplectic manifold, with an integral symplectic form, and the geometry, defined in a paper by Eliashberg and Polterovich,…

In the spirit of recent work of Harada-Kaveh and Nishinou-Nohara-Ueda, we study the symplectic geometry of Popov's horospherical degenerations of complex algebraic varieties with the action of a complex linearly reductive group. We…

We define Floer homology for a time-independent, or autonomous Hamiltonian on a symplectic manifold with contact type boundary, under the assumption that its 1-periodic orbits are transversally nondegenerate. Our construction is based on…

We prove that the rank of the cohomology of a closed symplectic manifold with coefficients in a field of characteristic $p$ is smaller than the number of periodic orbits of any non-degenerate Hamiltonian flow. Following Floer, the proof…