Stability of metric measure spaces with integral Ricci curvature bounds

摘要

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.