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 flows and harmonic maps of complete noncompact Riemannian manifolds.
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 Holomorphic or Anti-Holomorphic Character of Harmonic Maps into Irreducible Compact Quotients of Polydiscs.
The homology of some groups of diffeomorphisms.
The homotopy axiom in semialgebraic cohomology.
The homotopy type of the space of degree 0 immersed plane curves.
The space Bi0 = Imm0 (S1, R2) / Diff (S1) of all immersions of rotation degree 0 in the plane modulo reparameterizations has homotopy groups π1(Bi0) = Z, π2(Bi0) = Z, and πk(Bi0) = 0 for k ≥ 3.
The H-theorem for Boltzmann type equations.
The image in of the set of closed 2-forms with preassigned kernel
The inaudible geometry of nilmanifolds.
The Indes Theorem for k-Fold Connected Minimal Surfaces.
The index of a vector field tangent to a hypersurface and the signature of the relative jacobian determinant
Given a real analytic vector field tangent to a hypersurface with an algebraically isolated singularity we introduce a relative Jacobian determinant in the finite dimensional algebra associated with the singularity of the vector field on . We show that the relative Jacobian generates a 1-dimensional non-zero minimal ideal. With its help we introduce a non-degenerate bilinear pairing, and its signature measures the size of this point with sign. The signature satisfies a law of conservation of...
The index of analytic vector fields and Newton polyhedra
We prove that if f:(ℝⁿ,0) → (ℝⁿ,0) is an analytic map germ such that and f satisfies a certain non-degeneracy condition with respect to a Newton polyhedron Γ₊ ⊆ ℝⁿ, then the index of f only depends on the principal parts of f with respect to the compact faces of Γ₊. In particular, we obtain a known result on the index of semi-weighted-homogeneous map germs. We also discuss non-degenerate vector fields in the sense of Khovanskiĭand special applications of our results to planar analytic vector fields....
The Index of Manifolds with Toral Actions and Geometric Interpretations of the ... (..., (S1, Mn)) Invariant of Atiyah and Singer.
The index of projective families of elliptic operators.
The index of signature operators on Lipschitz manifolds
The index of the second variation of a control system