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 $L^p$-norm for $p>\frac{n}{2}$ 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 $(M,g_M, e^{-f}\mbox{vol}_M)$ 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.