Related papers: Modified Curvatures on Manifolds with Boundary and…

We study a Riemannian manifold equipped with a density which satisfies the Bakry--\'Emery Curvature-Dimension condition (combining a lower bound on its generalized Ricci curvature and an upper bound on its generalized dimension). We first…

It is known that by dualizing the Bochner-Lichnerowicz-Weitzenb\"{o}ck formula, one obtains Poincar\'e-type inequalities on Riemannian manifolds equipped with a density, which satisfy the Bakry-\'Emery Curvature-Dimension condition…

We characterize lower bounds for the Bakry-Emery Ricci tensor of nonsymmetric diffusion operators by convexity of entropy on the $L^2$-Wasserstein space, and define a curvature-dimension condition for general metric measure spaces together…

The boundary double layer potential, or the Neumann-Poincare operator, is studied on the Sobolev space of order 1/2 along the boundary, coinciding with the space of charges giving rise to double layer potentials with finite energy in the…

In this paper we introduce a notion of Poincar\'e exponent for isometric representations of discrete groups on Hilbert spaces. Similarly as growth exponents control the geometry this exponent is shown to control the size of spectral gaps.…

This paper investigates conformal deformations of the scalar curvature and mean curvature on complete Riemannian manifolds with boundary. We establish sufficient conditions for the existence of conformal deformations to complete metrics…

Some equivalent gradient and Harnack inequalities of a diffusion semigroup are presented for the curvature-dimension condition of the associated generator. As applications, the first eigenvalue, the log-Harnack inequality, the heat kernel…

We demonstrate lower bounds for the eigenvalues of compact Bakry-Emery manifolds with and without boundary. The lower bounds for the first eigenvalue rely on a generalised maximum principle which allows gradient estimates in the Riemannian…

As a continuation to \cite{MRW} where the Poincar\'e and log-Sobolev inequalities were studied for the sticky-reflected Brownian motion on Riemannian manifolds with boundary, this paper establishes the super and weak Poincar\'e inequalities…

We study spectral properties and geometric functional inequalities on Riemannian manifolds of dimension $\ge3$ with (finite or countably many) conical singularities $\{z_i\}_{i\in\mathfrak I}$ in the neighborhood of which the largest lower…

Using the curvature-dimension inequality proved in Part~I, we look at consequences of this inequality in terms of the interaction between the sub-Riemannian geometry and the heat semigroup $P_t$ corresponding to the sub-Laplacian. We give…

We prove comparison theorems for the sub-Riemannian distortion coefficients appearing in interpolation inequalities. These results, which are equivalent to a sub-Laplacian comparison theorem for the sub-Riemannian distance, are obtained by…

We introduce diffusion geometry as a new framework for geometric and topological data analysis. Diffusion geometry uses the Bakry-Emery $\Gamma$-calculus of Markov diffusion operators to define objects from Riemannian geometry on a wide…

We study curvature dimension inequalities for the sub-Laplacian on contact Riemannian manifolds. This new curvature dimension condition is then used to obtain: 1) Geometric conditions ensuring the compactness of the underlying manifold…

We study graphs with nonnegative Bakry-\'Emery curvature or Ollivier curvature outside a finite subset. For such a graph, via introducing the discrete Gromov-Hausdorff convergence we prove that the space of bounded harmonic functions is…

We derive an integral inequality between the mean curvature and the scalar curvature of the boundary of any scalar flat conformal compactifications of Poincar{\'e}-Einstein manifolds. As a first consequence , we obtain a sharp lower bound…

Let $L_t:=\Delta_t+Z_t$ for a $C^{1,1}$-vector field $Z$ on a differential manifold $M$ with boundary $\partial M$, where $\Delta_t$ is the Laplacian induced by a time dependent metric $g_t$ differentiable in $t\in [0,T_c)$. We first…

We revisit the problem of obtaining uniform gradient estimates for Dirichlet and Neumann heat semigroups on Riemannian manifolds with boundary. As applications, we obtain isoperimetric inequalities, using Ledoux's argument, and uniform…

In this paper, we obtain the reverse Bakry-\'Emery type estimates for a class of hypoelliptic diffusion operator by coupling method. The (right and reverse) Poincar\'e inequalities and the (right and reverse) logarithmic Sobolev…

Let G = (V, p, $\mu$) be a (finite or infinite) weighted graph with bounded geometry. Assuming that G satisfies the classical curvaturedimension condition of Bakry-Emery CD(K, n) with K $\ge$ 0 (for the usual Laplacian), we prove that the…