Lower bounds on Ricci curvature and quantitative behavior of singular sets

Jeff Cheeger, Aaron Naber

Research output: Contribution to journalArticle

Abstract

Let Yn denote the Gromov-Hausdorff limit of v-noncollapsed Riemannian manifolds with RicMni ≥ - (n - 1). The singular set S ⊂ Y has a stratification S ⊂ S ⊂. ⊂S, where y ∈S if no tangent cone at y splits off a factor ℝk+1 isometrically. Here, we define for all η> 0, 0 < r ≤ 1, the k-th effective singular stratum Xη,rk satisfying ∪ηr Sn,rk = Sk. Sharpening the known Hausdorff dimension bound dim Sk ≤k, we prove that for all y, the volume of the r-tubular neighborhood of Sη,rk satisfies Vol(Tr(Sη,rk) ∩ B1/2(y)) ≤C(n,v,η)rn-k-η. The proof involves a quantitative differentiation argument. This result has applications to Einstein manifolds. Let Br denote the set of points at which the C2-harmonic radius is ≤r. If also the Min are Kähler-Einstein with L2 curvature bound, {double pipe}Rm{double pipe}L2 ≤ C, then Vol(Br ∩B 1/2(y)) ≤c(n,v,C)r for all y. In the Kähler-Einstein case, without assuming any integral curvature bound on the Min, we obtain a slightly weaker volume bound on Br which yields an a priori Lp curvature bound for all p <2. The methodology developed in this paper is new and is applicable in many other contexts. These include harmonic maps, minimal hypersurfaces, mean curvature flow and critical sets of solutions to elliptic equations.

Original languageEnglish (US)
Pages (from-to)321-339
Number of pages19
JournalInventiones Mathematicae
Volume191
Issue number2
DOIs
StatePublished - 2013

ASJC Scopus subject areas

  • Mathematics(all)

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