Characterizations of Bounded Ricci Curvature on Smooth and NonSmooth 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.

References in corpus (1)

• Geometric Structures of Collapsing Riemannian Manifolds II: N*-bundles and Almost Ricci Flat Spaces

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• Spectral convergence under bounded Ricci curvature
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• Spectral gap on Riemannian path space over static and evolving manifolds
• Characterization of pinched Ricci curvature by functional inequalities
• Functional inequalities on path space of sub-Riemannian manifolds and applications
• A Bochner Formula on Path Space for the Ricci Flow
• Ricci measure for some singular Riemannian metrics