This paper is concerned with the structure of Gromov-Hausdorff limit spaces $(M^n_i,g_i,p_i)\stackrel{d_{GH}}{\longrightarrow} (X^n,d,p)$ of Riemannian manifolds satisfying a uniform lower Ricci curvature bound $\mathrm{Ric}_{M^n_i}\geq -(n-1)$ as well as the noncollapsing assumption $\mathrm{Vol}(B_1(p_i))>\mathrm{v}>0$. In such cases, there is a filtration of the singular set, $S^0\subset S^1\cdots S^{n-1}:= S$, where $S^k:= \{x\in X:\text{ no tangent cone at $x$ is }(k+1)\text{-symmetric}\}$. Equivalently, $S^k$ is the set of points such that no tangent cone splits off a Euclidean factor $\mathbb{R}^{k+1}$. It is classical from Cheeger-Colding that the Hausdorff dimension of $S^k$ satisfies $\mathrm{dim}\, S^k\leq k$ and $S=S^{n-2}$, i.e., $S^{n-1}\setminus S^{n-2}=\emptyset$. However, little else has been understood about the structure of the singular set $S$.

Our first result for such limit spaces $X^n$ states that $S^k$ is $k$-rectifiable for all $k$. In fact, we will show for $\mathcal H^k$-a.e. $x\in S^k$ that every tangent cone $X_x$ at $x$ is $k$-symmetric, i.e., that $X_x= \mathbb{R}^k\times C(Y)$ where $C(Y)$ might depend on the particular $X_x$. Here $\mathcal{H}^k$ denotes the $k$-dimensional Hausdorff measure. As an application we show for all $0\lt \epsilon\lt \epsilon(n,\mathrm{v})$ there exists an $(n-2)$-rectifiable closed set $S^{n-2}_\epsilon$ with $\mathcal{H}^{n-2}(S_{\epsilon}^{n-2}) < C(n,\mathrm{v},\epsilon)$, such that $X^n\setminus S^{n-2}_\epsilon$ is $\epsilon$-bi-Hölder equivalent to a smooth Riemannian manifold. Moreover, $S=\bigcup_\epsilon S^{n-2}_\epsilon$. As another application, we show that tangent cones are unique $\mathcal H^{n-2}$-a.e.

In the case of limit spaces $X^n$ satisfying a $2$-sided Ricci curvature bound $|\mathrm{Ric}_{M^n_i}|\leq n-1$, we can use these structural results to give a new proof of a conjecture from Cheeger-Colding stating that $S$ is $(n-4)$-rectifiable with uniformly bounded measure. We can also conclude from this structure that tangent cones are unique $\mathcal H^{n-4}$-a.e.

Our analysis builds on the notion of quantitative stratification introduced by Cheeger-Naber, and the neck region analysis developed by Jiang-Naber-Valtorta. Several new ideas and new estimates are required, including a sharp cone-splitting theorem and a geometric transformation theorem, which will allow us to control the degeneration of harmonic functions on these neck regions.

本文涉及满足以下条件的黎曼流形的Gromov-Hausdorff极限空间$（M ^ n_i，g_i，p_i）\ stackrel {d_ {GH}} {\ longrightarrow}（X ^ n，d，p）$的结构统一的下Ricci曲率边界$ \ mathrm {Ric} _ {M ^ n_i} \ geq-（n-1）$以及非崩溃假设$ \ mathrm {Vol}（B_1（p_i））> \ mathrm {v} > 0 $。在这种情况下，将过滤单数集$ S ^ 0 \ subset S ^ 1 \ cdots S ^ {n-1}：= S $，其中$ S ^ k：= \ {x \ in X： \ text {在$ x $处没有切线圆锥为}（k + 1）\ text {-对称} \} $。等价地，$ S ^ k $是这样的点集，使得没有切线圆锥分割出欧几里得因子$ \ mathbb {R} ^ {k + 1} $。根据Cheeger-Colding的经典理论，$ S ^ k $的Hausdorff维满足$ \ mathrm {dim} \，S ^ k \ leq k $和$ S = S ^ {n-2} $，即$ S ^ {n-1} \ setminus S ^ {n-2} = \ emptyset $。然而，

我们对此类限制空间$ X ^ n $的第一个结果表明，对于所有$ k $，$ S ^ k $都是$ k $可纠正的。事实上，我们将显示$ \ mathcal H ^ $ķ$ -ae X \ S中-1K-$即每切线圆锥$ X_x $在$ x $处是$ k $对称的，即$ X_x = \ mathbb {R} ^ k \ times C（Y）$，其中$ C（Y）$可能取决于特定的$ X_x $。在这里，$ \ mathcal {H} ^ k $表示维数为$ k $的Hausdorff测度。作为应用程序，我们针对所有$ 0 \ lt \ epsilon \ lt \ epsilon（n，\ mathrm {v}）$显示一个存在$（n-2）$可校正的封闭集$ S ^ {n-2} _ \带有$ \ mathcal {H} ^ {n-2}（S _ {\ epsilon} ^ {n-2}）<C（n，\ mathrm {v}，\ epsilon）$的epsilon $，使得$ X ^ n \ setminus S ^ {n-2} _ \ epsilon $是$ \ epsilon $-bi-Hölder，等同于光滑的黎曼流形。此外，$ S = \ bigcup_ \ epsilon S ^ {n-2} _ \ epsilon $。作为另一个应用，我们证明切线锥是唯一的$ \ mathcal H ^ {n-2} $-ae

在极限空间$ X ^ n $满足$ 2 $边的Ricci曲率边界$ | \ mathrm {Ric} _ {M ^ n_i} | \ leq n-1 $的情况下，我们可以使用这些结构结果得出来自Cheeger-Colding的一个猜想的新证据，指出$ S $是（n-4）$可校正的，且具有一致的有界测度。我们还可以从该结构得出结论，切线锥是唯一的\\数学H ^ {n-4} $-ae

我们的分析建立在Cheeger-Naber提出的定量分层概念以及Jiang-Naber-Valtorta提出的颈部区域分析的基础上。需要一些新的想法和新的估计，包括尖锐的圆锥分裂定理和几何变换定理，这将使我们能够控制这些颈部区域的谐波函数的退化。