Year: 2019
Type: article
Abstract: We prove that a metric measure space (X, \sf d, \mathfrak m) satisfying finite dimensional lower Ricci curvature bounds and whose Sobolev space W^{1,2} is Hilbert is rectifiable . That is, a \sf {RCD}^*(K,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.
DOI: 10.4171/jems/874
Authors: Andrea Mondino, Aaron Naber
Institutions: University of Warwick, Northwestern University
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2518878460