The free laplacian with attractive boundary conditions
The fundamental group and the spectrum of the Laplacian.
The Fundamental Solution for the Kohn-Laplacian ....b on the Sphere in Cn.
The Gap Phenomena of Yang-Mills Fields over the Complete Manifold.
The gap phenomenon in the dimension study of finite type systems
Several examples of gaps (lacunes) between dimensions of maximal and submaximal symmetric models are considered, which include investigation of number of independent linear and quadratic integrals of metrics and counting the symmetries of geometric structures and differential equations. A general result clarifying this effect in the case when the structure is associated to a vector distribution, is proposed.
The gaps in the spectrum of the Schrödinger operator
We obtain inequalities between the eigenvalues of the Schrödinger operator on a compact domain Ω of a submanifold M in with boundary ∂Ω, which generalize many existing inequalities for the Laplacian on a bounded domain of a Euclidean space. We also establish similar inequalities for a closed minimal submanifold in the unit sphere, which generalize and improve Yang-Yau’s result.
The generalized de Rham-Hodge theory aspects of Delsarte-Darboux type transformations in multidimension
The differential-geometric and topological structure of Delsarte transmutation operators and their associated Gelfand-Levitan-Marchenko type eqautions are studied along with classical Dirac type operator and its multidimensional affine extension, related with selfdual Yang-Mills eqautions. The construction of soliton-like solutions to the related set of nonlinear dynamical system is discussed.
The generalized Lichnerowicz formula and analysis of Dirac operators.
The geodesic hypothesis and non-topological solitons on pseudo-riemannian manifolds
The geometry and spectrum of the one holed torus.
The geometry of Markov diffusion generators
The global Yang-Mills equations depending on an arbitrary metric
The globally homeomorphic solutions to Beltrami system in Cn.
The heat equation and harmonic maps of complete manifolds.
The heat equation in riemannian geometry
The heat equation on manifolds as a gradient flow in the Wasserstein space
We study the gradient flow for the relative entropy functional on probability measures over a riemannian manifold. To this aim we present a notion of a riemannian structure on the Wasserstein space. If the Ricci curvature is bounded below we establish existence and contractivity of the gradient flow using a discrete approximation scheme. Furthermore we show that its trajectories coincide with solutions to the heat equation.
The heat semigroup acting on tensors or differential forms with values in vector bundle
The Hijazi inequalities on complete Riemannian manifolds.
The Hodge laplacian on manifolds with boundary
The inaudible geometry of nilmanifolds.