Heat flow and calculus on metric measure spaces with ricci curvature bounded below—The compact case

15Citations
Citations of this article
12Readers
Mendeley users who have this article in their library.
Get full text

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.

Cite

CITATION STYLE

APA

Ambrosio, L., Gigli, N., & Savaré, G. (2013). Heat flow and calculus on metric measure spaces with ricci curvature bounded below—The compact case. In Springer INdAM Series (Vol. 4, pp. 63–115). Springer International Publishing. https://doi.org/10.1007/978-88-470-2592-9_8

Register to see more suggestions

Mendeley helps you to discover research relevant for your work.

Already have an account?

Save time finding and organizing research with Mendeley

Sign up for free