Structure theory of metric measure spaces with lower Ricci curvature bounds

  • Andrea Mondino

    University of Warwick, Coventry, UK
  • Aaron Naber

    Northwestern University, Evanston, USA
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Abstract

We prove that a metric measure space satisfying finite dimensional lower Ricci curvature bounds and whose Sobolev space is Hilbert is rectifiable. That is, a -space is rectifiable, and in particular for -a.e. point the tangent cone is unique and euclidean of dimension at most . 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.

Cite this article

Andrea Mondino, Aaron Naber, Structure theory of metric measure spaces with lower Ricci curvature bounds. J. Eur. Math. Soc. 21 (2019), no. 6, pp. 1809–1854

DOI 10.4171/JEMS/874