$L^2$ curvature bounds on manifolds with bounded Ricci curvature

Abstract

Consider a Riemannian manifold with bounded Ricci curvature $|\mathrm{Ric}|\leq n-1$ and the noncollapsing lower volume bound $\mathrm{Vol}(B_1(p))>\mathrm{v}>0$. The first main result of this paper is to prove that we have the $L^2$ curvature bound $⨏_{B_1(p)}|\mathrm{Rm}|^2(x)\, dx < C(n,\mathrm{v})$,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 $\mathrm{GH}$-limit of noncollapsed manifolds with bounded Ricci curvature, then the singular set $\mathcal{S}(X)$ is $n-4$ rectifiable with the uniform Hausdorff measure estimates $H^{n-4}(\mathcal{S}(X)\cap B_1) < C(n,\mathrm{v})$ 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. $x\in \mathcal{S}(X)$, the tangent cone of $X$ at $x$ is unique and isometric to $\mathbb{R}^{n-4}\times C(S^3/\Gamma_x)$ for some $\Gamma_x\subseteq O(4)$ that acts freely away from the origin.