Related papers: Characteristic Laplacian in sub-Riemannian geometr…

Consider the complex linear space C^n endowed with the canonical pseudo-Hermitian form of signature (2p,2(n-p)). This yields both a pseudo-Riemannian and a symplectic structure on C^n. We prove that those submanifolds which are both…

Submanifolds of a manifold are described as sections of a certain fiber bundle that enables one to consider their Lagrangian and (polysymplectic) Hamiltonian dynamics as that of a particular classical field theory. In particular, their…

We show that complex symplectic structures need not be preserved under small deformations, and we find sufficient conditions for this to happen. We study various cohomologies of compact complex symplectic manifolds, obtaining some…

We consider Laplacians on $\Z^2$-periodic discrete graphs. The following results are obtained: 1) The Floquet-Bloch decomposition is constructed and basic properties are derived. 2) The estimates of the Lebesgue measure of the spectrum in…

We discuss symplectic manifolds where, locally, the structure is that encountered in Lagrangian dynamics. Exemples and characteristic properties are given. Then, we refer to the computation of the Maslov classes of a Lagrangian submanifold.…

We mainly investigate some properties of quasiconformal mappings between smooth 2-dimensional surfaces with boundary in the Euclidean space, satisfying certain partial differential equations (inequalities) concerning Laplacian, and in…

We investigate properties of pseudodifferential operators on $L^2$ space on manifold with ends including asymptotically conical or hyperbolic ends. Our pseudodifferential operators are a generalization of the canonical quantization which…

New isoperimetric inequalities for lower order eigenvalues of the Laplacian on closed hypersurfaces, of the biharmonic Steklov problems and of the Wentzell-Laplace on bounded domains in a Euclidean space are proven. Some open questions for…

We show that the exterior derivative operator on a symplectic manifold has a natural decomposition into two linear differential operators, analogous to the Dolbeault operators in complex geometry. These operators map primitive forms into…

We introduce a Hopf algebroid associated to a proper Lie group action on a smooth manifold. We prove that the cyclic cohomology of this Hopf algebroid is equal to the de Rham cohomology of invariant differential forms. When the action is…

In this thesis, we study cohomological properties of non-K\"ahler manifolds. In particular, we are concerned in investigating the cohomology of compact (almost-)complex manifolds, and of manifolds endowed with special structures, e.g.,…

This work deals with relations between a bounded cohomological invariant and the geometry of Hermitian symmetric spaces of noncompact type. The invariant, obtained from the K\"ahler class, is used to define and characterize a special class…

Motivated by the study of high energy Steklov eigenfunctions, we examine the semi-classical Robin Laplacian. In the two dimensional situation, we determine an effective operator describing the asymptotic distribution of the negative…

We develop a new approach, based on quantization methods, to study higher symmetries of invariant differential operators. We focus here on conformally invariant powers of the Laplacian over a conformally flat manifold and recover results of…

Double forms are sections of the vector bundles $\Lambda^{k}T^*\mathcal{M}\otimes \Lambda^{m}T^*\mathcal{M}$, where in this work $(\mathcal{M},\mathfrak{g})$ is a compact Riemannian manifold with boundary. We study graded second-order…

These expository notes are dedicated to the study of the topology of configuration spaces of manifolds. We give detailed computations of many invariants, including the fundamental group of the configuration spaces of $\mathbb{R}^2$, the…

In this paper we present examples of nondivergence form second order elliptic operators with continuous coefficients such that $L$ has an irregular boundary point that is regular for the Laplacian. Also for any eigenvalue spread <1 of the…

A symplectic form is called hyperbolic if its pull-back to the universal cover is a differential of a bounded one-form. The present paper is concerned with the properties and constructions of manifolds admitting hyperbolic symplectic forms.…

We consider the Bochner Laplacian on high tensor powers of a positive line bundle on a closed symplectic manifold (or, equivalently, the semiclassical magnetic Schr\"odinger operator with the non-degenerate magnetic field). We assume that…

This note is devoted to the study of the homology class of a compact Poisson transversal in a Poisson manifold. For specific classes of Poisson structures, such as unimodular Poisson structures and Poisson manifolds with closed leaves, we…