Characterizations of Bounded Ricci Curvature on Smooth and NonSmooth Spaces

TL;DR

This paper characterizes bounded Ricci curvature on smooth and non-smooth spaces, linking geometric bounds to analysis on path spaces and extending these concepts to general metric measure spaces.

Contribution

It introduces new characterizations of bounded Ricci curvature on smooth spaces and develops analytic tools to extend these notions to non-smooth metric measure spaces.

Findings

Bounded Ricci curvature controls path space analysis.

Properties of smooth bounded Ricci spaces extend to non-smooth spaces.

New definitions enable analysis of Ricci bounds on general metric measure spaces.

Abstract

There are two primary goals to this paper. In the first part of the paper we study smooth metric measure spaces (M^n,g,e^{-f}dv_g) and give several ways of characterizing bounds -Kg\leq \Ric+\nabla^2f\leq Kg on the Ricci curvature of the manifold. In particular, we see how bounded Ricci curvature on M controls the analysis of path space P(M) in a manner analogous to how lower Ricci curvature controls the analysis on M. In the second part of the paper we develop the analytic tools needed to in order to use these new characterizations to give a definition of bounded Ricci curvature on general metric measure spaces (X,d,m). We show that on such spaces many of the properties of smooth spaces with bounded Ricci curvature continue to hold on metric-measure spaces with bounded Ricci curvature.