Regularity of Einstein Manifolds and the Codimension 4 Conjecture

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-Hausdorff limit spaces $(M^n_j,d_j)\stackrel{d_{GH}}{\longrightarrow} (X,d)$, 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 stratification to prove a priori $L^q$ estimates on the full curvature $|Rm|$ 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 $|Ric_{M^4}|\leq 3$, $Vol(M)>v>0$, and $diam(M)\leq D$ contains at most a finite number of diffeomorphism classes. A local version of this is used to show that noncollapsed $4$-manifolds with bounded Ricci curvature have a priori $L^2$ Riemannian curvature estimates.