-rigidity of Lagrangian submanifolds and punctured holomorphic disks in the cotangent bundle

Our main result is the $\mathbb {\mathcal {C}}^{0}$ -rigidity of the area spectrum and the Maslov class of Lagrangian submanifolds. This relies on the existence of punctured pseudoholomorphic disks in cotangent bundles with boundary on the zero section, whose boundaries represent any integral homology class. We discuss further applications of these punctured disks in symplectic geometry.

Hyperkähler metrics near Lagrangian submanifolds and symplectic groupoids

The first part of this paper is a generalization of the Feix–Kaledin theorem on the existence of a hyperkähler metric on a neighborhood of the zero section of the cotangent bundle of a Kähler manifold. We show that the problem of constructing a hyperkähler structure on a neighborhood of a complex Lagrangian submanifold in a holomorphic symplectic manifold reduces to the existence of certain deformations of holomorphic symplectic structures. The Feix–Kaledin structure is recovered from the twisted cotangent bundle. We then show that every holomorphic symplectic groupoid over a compact holomorphic Poisson surface of Kähler type has a hyperkähler structure on a neighborhood of its identity section. More generally, we reduce the existence of a hyperkähler structure on a symplectic realization of a holomorphic Poisson manifold of any dimension to the existence of certain deformations of holomorphic Poisson structures adapted from Hitchin’s unobstructedness theorem.

Contact Dynamics: Legendrian and Lagrangian Submanifolds

We are proposing Tulczyjew’s triple for contact dynamics. The most important ingredients of the triple, namely symplectic diffeomorphisms, special symplectic manifolds, and Morse families, are generalized to the contact framework. These geometries permit us to determine so-called generating family (obtained by merging a special contact manifold and a Morse family) for a Legendrian submanifold. Contact Hamiltonian and Lagrangian Dynamics are recast as Legendrian submanifolds of the tangent contact manifold. In this picture, the Legendre transformation is determined to be a passage between two different generators of the same Legendrian submanifold. A variant of contact Tulczyjew’s triple is constructed for evolution contact dynamics.

Characterization of Lagrangian Submanifolds by Geometric Inequalities in Complex Space Forms

In this paper, we give an estimate of the first eigenvalue of the Laplace operator on a Lagrangian submanifold M n minimally immersed in a complex space form. We provide sufficient conditions for a Lagrangian minimal submanifold in a complex space form with Ricci curvature bound to be isometric to a standard sphere S n . We also obtain Simons-type inequality for same ambient space form.

Lagrangian submanifolds from tropical hypersurfaces

We prove that a smooth tropical hypersurface in [Formula: see text] can be lifted to a smooth embedded Lagrangian submanifold in [Formula: see text]. The idea of the proof is to use Lagrangian pairs of pants, which are the lifts of tropical hyperplanes introduced by the author in an earlier paper, as the main building blocks.

On forms, cohomology and BV Laplacians in odd symplectic geometry

We study the cohomology of the complexes of differential, integral and a particular class of pseudo-forms on odd symplectic manifolds taking the wedge product with the symplectic form as a differential. We thus extend the result of Ševera and the related results of Khudaverdian–Voronov on interpreting the BV odd Laplacian acting on half-densities on an odd symplectic supermanifold. We show that the cohomology classes are in correspondence with inequivalent Lagrangian submanifolds and that they all define semidensities on them. Further, we introduce new operators that move from one Lagragian submanifold to another and we investigate their relation with the so-called picture changing operators for the de Rham differential. Finally, we prove the isomorphism between the cohomology of the de Rham differential and the cohomology of BV Laplacian in the extended framework of differential, integral and a particular class of pseudo-forms.