Gradient Flows and non-Contraction of Heat Flow on Metric and Minkowski Spaces

Abstract

We show the geometry and heat flow on metric measure spaces. We study the gradient flows ,diffusion semigroups on metric and Wasserstien spaces under lower curvature bounds and over compact Alexandrov spaces. We also study the heat equation on manifolds as a gradient flow in the Wasserstien space. Metric characterizations of Riemannian submersions are considered. We also consider the open map theorem and Trottere product formula for gradient flows in metric Spaces. Acharacterizationof transportation distance between the non-negative measures and applications on gradients flows with Dirichlet boundary conditions are shown. We discuss the measure contraction property and non-contraction of heat flow on metric measure and Minkowski spaces.