L2 curvature bounds on manifolds with bounded Ricci curvature

Wenshuai Jiang*, Aaron Naber

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

15 Scopus citations


Consider a Riemannian manifold with bounded Ricci curvature [Ric] ≤ n - 1 and the noncollapsing lower volume bound Vol(B1(p)) > v > 0. The first main result of this paper is to prove that we have the L2 curvature bound R{B1(p)jRmj2(x) dx < C(n; v), which proves the L2 conjecture. In order to prove this, we will need to first show the following structural result for limits. Namely, (Formula Presented) is a GH-limit of noncollapsed manifolds with bounded Ricci curvature, then the singular set S(X) is n-4 rectifiable with the uniform Hausdorff measure estimates (Formula Presented) which, in particular, proves the n-4-finiteness conjecture of Cheeger-Colding. We will see as a consequence of the proof that for (Formula Presented) the tangent cone of X at x is unique and isometric to (Formula Presented) that acts freely away from the origin.

Original languageEnglish (US)
Pages (from-to)107-222
Number of pages116
JournalAnnals of Mathematics
Issue number1
StatePublished - Jan 2021


  • Curvature
  • Ricci
  • Singularity
  • Stratification

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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