Characterization of non-constant lower bound of Ricci curvature via entropy inequality on Wasserstein space

TL;DR

This paper extends the characterization of Ricci curvature bounds to cases where the lower bound varies continuously, using entropy convexity on Wasserstein space, generalizing previous constant-bound results.

Contribution

It introduces a new entropy-based characterization for non-constant Ricci curvature lower bounds, broadening the scope of curvature analysis on Riemannian manifolds.

Findings

Generalizes previous results to variable lower bounds

Provides a new entropy convexity criterion

Connects curvature bounds with probability space convexity

Abstract

When the Ricci curvature of a Riemannian manifold is not lower bounded by a constant, but lower bounded by a continuous function, we give a new characterization of this lower bound through the convexity of relative entropy on the probability space over the Riemannian manifold. Hence, we generalize K.T. Sturm and von Renesse's result (Comm. Pure Appl. Math. 2005) to the case with non-constant lower bound of Ricci curvature.