Stability of metric measure spaces with integral Ricci curvature bounds

TL;DR

This paper investigates the stability and convergence of smooth metric measure spaces with integral Ricci curvature bounds, establishing conditions under which they converge to spaces satisfying curvature-dimension conditions, with implications for geometric inequalities and topological properties.

Contribution

It proves that sequences of manifolds with integral Ricci bounds converge to $CD(K,n)$ spaces under $L^p$-norm conditions, extending results to Bakry-Emery curvature and deriving geometric and topological corollaries.

Findings

Sequences converge to $CD(K,n)$ spaces under $L^p$-norm conditions.

Results extend to Bakry-Emery curvature for smooth metric measure spaces.

Corollaries include Brunn-Minkowski inequality, Bonnet-Myers estimate, and finiteness of fundamental group.

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.