Rectifiability of singular sets of noncollapsed limit spaces with Ricci curvature bounded below

Jeff Cheeger, Wenshuai Jiang, Aaron Naber

Research output: Contribution to journalArticlepeer-review


This paper is concerned with the structure of Gromov-Hausdorff limit spaces (Formula Presented) of Riemannian manifolds satisfying a uniform lower Ricci curvature bound Ric (Formula Presented)) as well as the noncollapsing assumption Vol(B1(pi)) > v > 0. In such cases, there is a filtration of the singular set, (Formula Presented) no tangent cone at x is (k + 1)-symmetricg. Equivalently, Sk is the set of points such that no tangent cone splits off a Euclidean factor Rk+1. It is classical from Cheeger-Colding that the Hausdorff dimension of Sk satisfies dim (Formula Presented) and (Formula Presented). However, little else has been understood about the structure of the singular set S. Our first result for such limit spaces Xn states that Sk is k-rectifiable for all k. In fact, we will show for Hk-a.e. x 2 Sk that every tangent cone Xx at x is k-symmetric, (Formula Presented) where C(Y ) might depend on the particular Xx. Here Hk denotes the k-dimensional Hausdorff measure. As an application we show for all (Formula Presented)there exists an (n-2)-rectiffiable closed set (Formula Presented) with(Formula Presented), such that (Formula Presented) is equivalent to a smooth Riemannian manifold. Moreover, (Formula Presented). As another application, we show that tangent cones are unique (Formula Presented). In the case of limit spaces Xn satisfying a 2-sided Ricci curvature bound (Formula Presented), we can use these structural results to give a new proof of a conjecture from Cheeger-Colding stating that S is (n-4)-rectifiable with uniformly bounded measure. We can also conclude from this structure that tangent cones are unique (Formula Presented). Our analysis builds on the notion of quantitative stratification introduced by Cheeger-Naber, and the neck region analysis developed by Jiang-Naber-Valtorta. Several new ideas and new estimates are required, including a sharp cone-splitting theorem and a geometric transformation theorem, which will allow us to control the degeneration of harmonic functions on these neck regions.

Original languageEnglish (US)
Pages (from-to)407-538
Number of pages132
JournalAnnals of Mathematics
Issue number2
StatePublished - Mar 2021


  • Curvature
  • Rectifiability
  • Ricci
  • Stratification

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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