Structure theory of metric measure spaces with lower Ricci curvature bounds

Andrea Mondino, Aaron Naber

Research output: Contribution to journalArticlepeer-review

76 Scopus citations


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, an 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. We also show a sharp integral Abresch–Gromoll type inequality for the excess function and an Abresch–Gromoll-type inequality for the gradient of the excess. The argument is new even in the smooth setting.

Original languageEnglish (US)
Pages (from-to)1809-1854
Number of pages46
JournalJournal of the European Mathematical Society
Issue number6
StatePublished - 2019


  • Optimal transport
  • Rectifiable space
  • Ricci curvature
  • Unique tangent space

ASJC Scopus subject areas

  • General Mathematics
  • Applied Mathematics


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