In this note we construct examples of closed connected Legendrian submanifolds in high dimensional contact vector space that admit an arbitrary finite number of topologically distinct infinite families of diffeomorphic, but not Hamiltonian isotopic exact Lagrangian fillings.

In this note we discuss the collection of statements known as Arnold conjecture for Hamiltonian diffeomorphisms of closed symplectic manifolds. We provide an overview of the homological, stable and strong versions of Arnold conjecture for non-degenerate Hamiltonian systems, a few versions of Arnold conjecture for possibly degenerate Hamiltonian systems, the degenerate version of Arnold conjecture for Hamiltonian homeomorphisms and Sandon’s version of Arnold conjecture for contactomorphisms.

The object of the present paper is to study weakly $\phi$-symmetric and weakly $\phi$-Ricci symmetric Kenmotsu manifolds. It is shown that weakly $\phi$-symmetric and weakly $\phi$-Ricci symmetric Kenmotsu manifolds are $\eta$-Einstein.

The structure of filtered algebras of Grothendieck's differential operators on a smooth fat point in a curve and graded Poisson algebras of their principal symbols is explicitly determined. A related infinitesimal-birational duality realized by a Springer type resolution of singularities and the Fourier transformation is presented. This algebro-geometrical duality is quantized in appropriate sense and its quantum origin is explained.

By using open book techniques we give an alternative proof of a theorem about the existence of contact structures on five-manifolds due to Geiges. The theorem asserts that simply-connected five-manifolds admit a contact structure in every homotopy class of almost contact structures.

We prove that the well-known Harder-Narsimhan filtration theory for bundles over a complex curve and the theory of optimal destabilizing $1$-parameter subgroups are the same thing when considered in the gauge theoretical framework.Indeed, the classical concepts of the GIT theory are still effective in this context and the Harder-Narasimhan filtration can be viewed as a limit object for the action of the gauge group, in the direction of an optimal destabilizing vector. This vector appears as an extremal...

Nous étudions les aspects infinitésimaux du problème suivant. Soit $H$ un hamiltonien de ${ℝ}^{2n}$ dont la surface d’énergie $\{H=1\}$ borde un domaine compact et étoilé de volume identique à celui de la boule unité de ${ℝ}^{2n}$. La surface d’énergie $\{H=1\}$ contient-elle une orbite périodique du système hamiltonien$$\left\{\begin{array}{cc}\dot{q}\hfill & =\frac{\partial H}{\partial p}\hfill \\ \dot{p}\hfill & =-\frac{\partial H}{\partial q}\hfill \end{array}\right.$$dont l’action soit au plus $\pi$ ?

Orbits of complete families of vector fields on a subcartesian space are shown to be smooth manifolds. This allows a description of the structure of the reduced phase space of a Hamiltonian system in terms of the reduced Poisson algebra. Moreover, one can give a global description of smooth geometric structures on a family of manifolds, which form a singular foliation of a subcartesian space, in terms of objects defined on the corresponding family of vector fields. Stratified...

Let $S$ be a Riemann surface. Let ${ℍ}^3$ be the $3$-dimensional hyperbolic space and let ${\partial }_{\infty }{ℍ}^3$ be its ideal boundary. In our context, a Plateau problem is a locally holomorphic mapping $\varphi :S\to {\partial }_{\infty }{ℍ}^3=\widehat{ℂ}$. If $i:S\to {ℍ}^3$ is a convex immersion, and if $N$ is its exterior normal vector field, we define the Gauss lifting, $\widehat{ı}$, of $i$ by $\widehat{ı}=N$. Let $\overrightarrow{n}:U{ℍ}^3\to {\partial }_{\infty }{ℍ}^3$ be the Gauss-Minkowski mapping. A solution to the Plateau problem $(S,\varphi )$ is a convex immersion $i$ of constant Gaussian curvature equal to $k\in (0,1)$ such that the Gauss lifting $(S,\widehat{ı})$ is complete and $\overrightarrow{n}\circ \widehat{ı}=\varphi$. In this paper, we show...