### Abstract

In this paper, we are concerned with the regularity of noncollapsed Riemannian manifolds (M^{n},g) with bounded Ricci curvature, as well as their Gromov-Hausdorfflimit spaces (M^{n}_{j}; d_{j}) →dGH(X, d), where d_{j} denotes the Riemannian distance. Our main result is a solution to the codimension 4 conjecture, namely that X is smooth away from a closed subset of codimension 4. We combine this result with the ideas of quantitative stratication to prove a priori L^{q} estimates on the full curvature jRmj for all q<2. In the case of Einstein manifolds, we improve this to estimates on the regularity scale. We apply this to prove a conjecture of Anderson that the collection of 4-manifolds (M^{4}, g) with RicM_{4}≤ 3, Vol(M)> v>0, and diam(M) ≤ D contains at most a nite number of diffeomorphism classes. A local version is used to show that noncollapsed 4-manifolds with bounded Ricci curvature have a priori L^{2} Riemannian curvature estimates.

Original language | English (US) |
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Pages (from-to) | 1093-1165 |

Number of pages | 73 |

Journal | Annals of Mathematics |

Volume | 182 |

Issue number | 3 |

DOIs | |

State | Published - Jan 1 2015 |

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### ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty

### Cite this

*Annals of Mathematics*,

*182*(3), 1093-1165. https://doi.org/10.4007/annals.2015.182.3.5