### 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) |
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Pages (from-to) | 321-339 |

Number of pages | 19 |

Journal | Inventiones Mathematicae |

Volume | 191 |

Issue number | 2 |

DOIs | |

State | Published - 2013 |

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

*Inventiones Mathematicae*,

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