Stability of metric measure spaces with integral Ricci curvature bounds

In this article we study stability and compactness w.r.t. measured Gromov-Hausdorff convergence of smooth metric measure spaces with integral Ricci curvature bounds. More precisely, we prove that a sequence of n-dimensional Riemannian manifolds subconverges to a metric measure space that satisfies the curvature-dimension condition CD (K, n) in the sense of Lott-Sturm-Villani provided the L-p-norm for p > n/2 of the part of the Ricci curvature that lies below K converges to 0. The results also hold for sequences of general smooth metric measure spaces (M, g(M), e(-f) vol(M)) where Bakry-Emery curvature replaces Ricci curvature. Corollaries are a Brunn-Minkowski-type inequality, a Bonnet-Myers estimate and a statement on finiteness of the fundamental group. Together with a uniform noncollapsing condition the limit even satisfies the Riemannian curvature-dimension condition RCD (K, N). This implies volume and diameter almost rigidity theorems. (C) 2021 Elsevier Inc. All rights reserved.

Automated summary

In this article, stability and compactness with respect to measured Gromov-Hausdorff convergence of smooth metric measure spaces with integral Ricci curvature bounds are studied. The results show convergence conditions for subconvergence to spaces satisfying the curvature-dimension condition CD (K, n) and implications for various geometric inequalities and estimates. The study also extends to general smooth metric measure spaces with Bakry-Emery curvature, leading to important corollaries and rigidity theorems.