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

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.