Calculus of Variations and Geometric Measure Theory

A. Mondino - A. Naber

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

created by mondino on 09 May 2014
modified on 04 May 2017


Accepted Paper

Inserted: 9 may 2014
Last Updated: 4 may 2017

Journal: Journ. European Math Soc. (JEMS)
Year: 2014


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.