Stability of metric measure spaces with integral Ricci curvature bounds

Abstract

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 Lp-norm for p>n2 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,gM,e−fvolM) 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.