### Abstract

Let Y^{n} denote the Gromov-Hausdorff limit of v-noncollapsed Riemannian manifolds with Ric_{Mni} ≥ - (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 ℝ^{k+1} isometrically. Here, we define for all η> 0, 0 < r ≤ 1, the k-th effective singular stratum X_{η,r}^{k} satisfying ∪_{η} ∩_{r} S_{n,r}^{k} = S^{k}. Sharpening the known Hausdorff dimension bound dim S^{k} ≤k, we prove that for all y, the volume of the r-tubular neighborhood of S_{η,r}^{k} satisfies Vol(T_{r}(S_{η,r}^{k}) ∩ B1/2(y)) ≤C(n,v,η)r^{n-k-η}. The proof involves a quantitative differentiation argument. This result has applications to Einstein manifolds. Let B_{r} denote the set of points at which the C^{2}-harmonic radius is ≤r. If also the M_{i}^{n} are Kähler-Einstein with L_{2} curvature bound, {double pipe}Rm{double pipe}L_{2} ≤ C, then Vol(B_{r} ∩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 M_{i}^{n}, we obtain a slightly weaker volume bound on B_{r} which yields an a priori L_{p} 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 language | English (US) |
---|---|

Pages (from-to) | 321-339 |

Number of pages | 19 |

Journal | Inventiones Mathematicae |

Volume | 191 |

Issue number | 2 |

DOIs | |

State | Published - 2013 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Inventiones Mathematicae*,

*191*(2), 321-339. https://doi.org/10.1007/s00222-012-0394-3

**Lower bounds on Ricci curvature and quantitative behavior of singular sets.** / Cheeger, Jeff; Naber, Aaron.

Research output: Contribution to journal › Article

*Inventiones Mathematicae*, vol. 191, no. 2, pp. 321-339. https://doi.org/10.1007/s00222-012-0394-3

}

TY - JOUR

T1 - Lower bounds on Ricci curvature and quantitative behavior of singular sets

AU - Cheeger, Jeff

AU - Naber, Aaron

PY - 2013

Y1 - 2013

N2 - 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 ∪η ∩r 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.

AB - 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 ∪η ∩r 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.

UR - http://www.scopus.com/inward/record.url?scp=84872778967&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84872778967&partnerID=8YFLogxK

U2 - 10.1007/s00222-012-0394-3

DO - 10.1007/s00222-012-0394-3

M3 - Article

VL - 191

SP - 321

EP - 339

JO - Inventiones Mathematicae

JF - Inventiones Mathematicae

SN - 0020-9910

IS - 2

ER -