Heat Flow on Alexandrov Spaces



We prove that on compact Alexandrov spaces with curvature bounded below the gradient flow of the Dirichlet energy in the equation image-space produces the same evolution as the gradient flow of the relative entropy in the equation image-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 equation image-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.