$L^2$ Curvature Bounds on Manifolds with Bounded Ricci Curvature

Consider a Riemannian manifold with bounded Ricci curvature $|\Ric|\leq n-1$ and the noncollapsing lower volume bound $\Vol(B_1(p))>\rv>0$. The first main result of this paper is to prove that we have the $L^2$ curvature bound $\fint_{B_1(p)}|\Rm|^2 < C(n,\rv)$, which proves the $L^2$ conjecture. In order to prove this, we will need to first show the following structural result for limits. Namely, if $(M^n_j,d_j,p_j) \longrightarrow (X,d,p)$ is a $GH$-limit of noncollapsed manifolds with bounded Ricci curvature, then the singular set $\cS(X)$ is $n-4$ rectifiable with the uniform Hausdorff measure estimates $H^{n-4}\big(\cS(X)\cap B_1\big)<C(n,\rv)$, which in particular proves the $n-4$-finiteness conjecture of Cheeger-Colding. We will see as a consequence of the proof that for $n-4$ a.e. $x\in \cS(X)$ that the tangent cone of $X$ at $x$ is unique and isometric to $\dR^{n-4}\times C(S^3/\Gamma_x)$ for some $\Gamma_x\subseteq O(4)$ which acts freely away from the origin.