We prove that on compact Alexandrov spaces with curvature bounded below the gradient flow of the Dirichlet energy in the \documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath, amssymb} \pagestyle{empty} \begin{document} \begin{align*}L^2\end{align*} \end{document}‐space produces the same evolution as the gradient flow of the relative entropy in the \documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath, amssymb} \pagestyle{empty} \begin{document} \begin{align*}L^2\end{align*} \end{document}‐Wasserstein space. This means that the heat flow is well‐defined by either one of the two gradient flows. Combining properties of these flows, we are able to deduce the Lipschitz continuity of the heat kernel as well as Bakry‐Émery gradient estimates and the \documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath, amssymb} \pagestyle{empty} \begin{document} \begin{align*}\Gamma_2\end{align*} \end{document}‐condition. Our identification is established by purely metric means, unlike preceding results relying on PDE techniques. Our approach generalizes to the case of heat flow with drift. © 2012 Wiley Periodicals, Inc.

We investigate the m-relative entropy, which stems from the Bregman divergence, on weighted Riemannian and Finsler manifolds. We prove that the displacement K-convexity of the m-relative entropy is equivalent to the combination of the nonnegativity of the weighted Ricci curvature and the K-convexity of the weight function. We use this to show appropriate variants of the Talagrand, HWI and the logarithmic Sobolev inequalities, as well as the concentration of measures. We also prove that the gradient flow of the m-relative entropy produces a solution to the porous medium equation or the fast diffusion equation.

We prove that if a metric measure space satisfies the volume doubling condition and the Caffarelli-Kohn-Nirenberg inequality with the same exponent n ≥ 3, then it has exactly the n-dimensional volume growth. As an application, if an n-dimensional Finsler manifold of non-negative n-Ricci curvature satisfies the Caffarelli-Kohn-Nirenberg inequality with the sharp constant, then its flag curvature is identically zero. In the particular case of Berwald spaces, such a space is necessarily isometric to a Minkowski space.

Abstract. We extend results proved by the second author (Amer. J. Math., 2009) for nonnegatively curved Alexandrov spaces to general compact Alexandrov spaces X with curvature bounded below. The gradient flow of a geodesically convex functional on the quadratic Wasserstein space (P(X), W 2 ) satisfies the evolution variational inequality. Moreover, the gradient flow enjoys uniqueness and contractivity. These results are obtained by proving a first variation formula for the Wasserstein distance.