Heat flow and calculus on metric measure spaces with Ricci curvature bounded below - the compact case

TL;DR

This paper reviews calculus tools and main results related to heat flow on compact metric measure spaces with Ricci curvature bounds, introducing new approaches and definitions that extend classical geometric analysis to metric spaces.

Contribution

It introduces a new, stable Ricci curvature lower bound concept for metric measure spaces and connects it with heat flow linearity, expanding the geometric analysis framework.

Findings

Equivalence of L^2-gradient flow and Wasserstein gradient flow

A metric version of Brenier's Theorem

A new definition of Ricci curvature lower bounds

Abstract

We provide a quick overview of various calculus tools and of the main results concerning the heat flow on compact metric measure spaces, with applications to spaces with lower Ricci curvature bounds. Topics include the Hopf-Lax semigroup and the Hamilton-Jacobi equation in metric spaces, a new approach to differentiation and to the theory of Sobolev spaces over metric measure spaces, the equivalence of the L^2-gradient flow of a suitably defined "Dirichlet energy" and the Wasserstein gradient flow of the relative entropy functional, a metric version of Brenier's Theorem, and a new (stronger) definition of Ricci curvature bound from below for metric measure spaces. This new notion is stable w.r.t. measured Gromov-Hausdorff convergence and it is strictly connected with the linearity of the heat flow.