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 $\int_{B_1(p)}|\mathrm{Rm}|^2(x)\, dx < C(n,\mathrm{v})$,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.

考虑一个黎曼曲率为Ricci曲率为$ | \ mathrm {Ric} | \ leq n-1 $的黎曼流形，其非塌陷下体积为$ \ mathrm {Vol}（B_1（p））> \ mathrm {v}> 0 $。本文的第一个主要结果是证明我们具有$ L ^ 2 $曲率边界$ \ int_ {B_1（p）} | \ mathrm {Rm} | ^ 2（x）\，dx <C（n， \ mathrm {v}）$，这证明了$ L ^ 2 $的猜想。为了证明这一点，我们将需要首先显示以下极限结构结果。也就是说，如果$（M ^ n_j，d_j，p_j）\ longrightarrow（X，d，p）$是具有Ricci曲率的无塌陷流形的$ \ mathrm {GH} $极限，则奇异集$ \ mathcal { S}（X）$可通过统一的Hausdorff测度估计值$ H ^ {n-4}（\ mathcal {S}（X）\ cap B_1）<C（n，\ mathrm {v} ）$，尤其证明了Cheeger-Colding的$ n-4 $-有限性猜想。