We extend the constructions and results of Damian to get topological obstructions to the existence of closed monotone Lagrangian embeddings into the cotangent bundle of a space which is the total space of a fibration over the circle.

We show that a Lagrangian submanifold of a complex space form attaining equality in the inequality obtained by Oprea in [8], must be totally geodesic.

In one of his papers, C. Viterbo defined a distance on the set of Hamiltonian diffeomorphisms of ${ℝ}^{2n}$ endowed with the standard symplectic form ${\omega }_0=dp\wedge dq$. We study the completions of this space for the topology induced by Viterbo’s distance and some others derived from it, we study their different inclusions and give some of their properties. In particular, we give a convergence criterion for these distances that allows us to prove that the completions contain non-ordinary elements, as for example, discontinuous...

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.

We construct a version of rational Symplectic Field Theory for pairs $(X,L)$, where $X$ is an exact symplectic manifold, where $L\subset X$ is an exact Lagrangian submanifold with components subdivided into $k$ subsets, and where both $X$ and $L$ have cylindrical ends. The theory associates to $(X,L)$ a $\mathbb{Z}$-graded chain complex of vector spaces over ${\mathbb{Z}}_2$, filtered with $k$ filtration levels. The corresponding $k$-level spectral sequence is invariant under deformations of $(X,L)$ and has the following property: if $(X,L)$ is obtained by joining a...

We prove the existence of stationary discs in the ball for small almost complex deformations of the standard structure. We define a local analogue of the Riemann map and establish its main properties. These constructions are applied to study the local geometry of almost complex manifolds and their morphisms.

All second order scalar differential invariants of symplectic hyperbolic and elliptic Monge-Ampère equations with respect to symplectomorphisms are explicitly computed. In particular, it is shown that the number of independent second order invariants is equal to 7, in sharp contrast with general Monge-Ampère equations for which this number is equal to 2. We also introduce a series of invariant differential forms and vector fields which allow us to construct numerous scalar differential invariants...

The notes consist of a study of special Lagrangian linear subspaces. We will give a condition for the graph of a linear symplectomorphism $f:({ℝ}^{2n},\sigma ={\sum }_{i=1}^{n}d{x}_i\wedge d{y}_i)\to ({ℝ}^{2n},\sigma )$ to be a special Lagrangian linear subspace in $({ℝ}^{2n}\times {ℝ}^{2n},\omega =\pi *₂\sigma -\pi *₁\sigma )$. This way a special symplectic subset in the symplectic group is introduced. A stratification of special Lagrangian Grassmannian $S{\Lambda }_{2n}\simeq SU\left(2n\right)/SO\left(2n\right)$ is defined.

We construct a family of Lagrangian submanifolds in the complex sphere which are foliated by $(n-1)$-dimensional spheres. Among them we find those which are special Lagrangian with respect to the Calabi-Yau structure induced by the Stenzel metric.

We give a canonical construction of an “isotropic average” of given $C^1$-close isotropic submanifolds of a symplectic manifold. For this purpose we use an improvement (obtained in collaboration with H. Karcher) of Weinstein’s submanifold averaging theorem and apply “Moser’s trick”. We also present an application to Hamiltonian group actions.

In [Ch00], Chekanov showed that the Hofer norm on the Hamiltonian diffeomorphism group of a geometrically bounded symplectic manifold induces a nondegenerate metric on the orbit of any compact Lagrangian submanifold under the group. In this paper we consider the orbits of more general submanifolds. We show that, for the Chekanov–Hofer pseudometric on the orbit of a closed submanifold to be a genuine metric, it is necessary for the submanifold to be coisotropic, and we show that this condition is...