Structure Theory of Metric-Measure Spaces with Lower Ricci Curvature Bounds

TL;DR

This paper proves that metric measure spaces with lower Ricci curvature bounds and Hilbert Sobolev spaces are rectifiable, with unique Euclidean tangent cones, using novel maximal function and Almost Splitting Theorem techniques.

Contribution

It establishes rectifiability of $RCD^*(K,N)$-spaces and introduces new gradient estimates and inequalities that are also applicable in smooth settings.

Findings

$RCD^*(K,N)$-spaces are rectifiable.

Unique Euclidean tangent cones at almost every point.

New gradient and excess function inequalities.

Abstract

We prove that a metric measure space $(X,d,m)$ satisfying finite dimensional lower Ricci curvature bounds and whose Sobolev space $W^{1,2}$ is Hilbert is rectifiable. That is, a $RCD^*(K,N)$-space is rectifiable, and in particular for $m$-a.e. point the tangent cone is unique and euclidean of dimension at most $N$. The proof is based on a maximal function argument combined with an original Almost Splitting Theorem via estimates on the gradient of the excess. To this aim we also show a sharp integral Abresh-Gromoll type inequality on the excess function and an Abresh-Gromoll-type inequality on the gradient of the excess. The argument is new even in the smooth setting.