Structure theory of metric measure spaces with lower Ricci curvature bounds

Abstract

We prove that a metric measure space $(X,\mathsf{d},\mathfrak{m})$ satisfying finite dimensional lower Ricci curvature bounds and whose Sobolev space $W^{1,2}$ is Hilbert is rectifiable. That is, a ${\mathsf{RCD}}^{\ast }(\mathsf{K},\mathsf{N})$-space is rectifiable, and in particular for $\mathfrak{m}$-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. 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.