Regularity of Einstein manifolds and the codimension 4 conjecture

In this paper, we are concerned with the regularity of noncollapsed Riemannian manifolds (Mⁿ,g) with bounded Ricci curvature, as well as their Gromov-Hausdorfflimit spaces (Mⁿ_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 (M⁴, g) with RicM₄≤ 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 L² Riemannian curvature estimates.