Structure Theory of Metric-measure Spaces with Lower Ricci Curvature Bounds I
نویسنده
چکیده
We prove that a metric measure space (X, d,m) satisfying finite dimensional lower Ricci curvature bounds and whose Sobolev space W1,2 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. CONTENTS
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