If one considers an integral varifold $I^m\subseteq M$ with bounded mean curvature, and if $S^k(I)\equiv\{x\in M: \text{ no tangent cone at $x$ is }k+1\text{-symmetric}\}$ is the standard stratification of the singular set, then it is well known that $\dim S^k\leq k$. In complete generality nothing else is known about the singular sets $S^k(I)$. In this paper we prove for a general integral varifold with bounded mean curvature, in particular a stationary varifold, that every stratum $S^k(I)$ is $k$-rectifiable. In fact, we prove for $k$-a.e. point $x\in S^k$ that there exists a unique $k$-plane $V^k$ such that every tangent cone at $x$ is of the form $V\times C$ for some cone $C$. In the case of minimizing hypersurfaces $I^{n-1}\subseteq M^n$ we can go further. Indeed, we can show that the singular set $S(I)$, which is known to satisfy $\dim S(I)\leq n-8$, is in fact $n-8$ rectifiable with uniformly finite $n-8$ measure. An effective version of this allows us to prove that the second fundamental form $A$ has apriori estimates in $L^7_{weak}$ on $I$, an estimate which is sharp as $|A|$ is not in $L^7$ for the Simons cone. In fact, we prove the much stronger estimate that the regularity scale $r_I$ has $L^7_{weak}$-estimates. The above results are in fact just applications of a new class of estimates we prove on the quantitative stratifications $S^k_{\epsilon,r}$ and $S^k_{\epsilon}\equiv S^k_{\epsilon,0}$. Roughly, $x\in S^k_{\epsilon}\subseteq I$ if no ball $B_r(x)$ is $\epsilon$-close to being $k+1$-symmetric. We show that $S^k_\epsilon$ is $k$-rectifiable and satisfies the Minkowski estimate $Vol(B_r\,S_\epsilon^k)\leq C_\epsilon r^{n-k}$. The proof requires a new $L^2$-subspace approximation theorem for integral varifolds with bounded mean curvature, and a $W^{1,p}$-Reifenberg type theorem proved by the authors in \cite{NaVa+}.

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平稳变节的奇异结构和规律

如果考虑具有有界平均曲率的积分可变折线 $I^m\subseteq M$，并且如果 $S^k(I)\equiv\{x\in M: \text{ 在 $x$ 处没有切锥是 }k +1\text{-对称}\}$是奇异集的标准分层，那么众所周知$\dim S^k\leq k$。总的来说，关于奇异集 $S^k(I)$ 一无所知。在本文中，我们证明了具有有界平均曲率的一般积分变数，特别是平稳变数，每个层 $S^k(I)$ 都是 $k$-可修正的。事实上，我们证明 $k$-ae 点 $x\in S^k$ 存在唯一的 $k$-平面 $V^k$，使得 $x$ 处的每个切锥都具有 $V 形式\times C$ 为一些锥 $C$。在最小化超曲面 $I^{n-1}\subseteq M^n$ 的情况下，我们可以更进一步。事实上，我们可以证明奇异集 $S(I)$，已知满足 $\dim S(I)\leq n-8$，实际上 $n-8$ 可以用统一有限的 $n-8$ 度量来修正。一个有效的版本使我们能够证明第二个基本形式 $A$ 在 $L^7_{weak}$ 中对 $I$ 具有先验估计，该估计在 $|A|$ 不在 $L 中时是尖锐的^7$ 用于西蒙斯锥。事实上，我们证明了更强的估计，即正则性尺度 $r_I$ 具有 $L^7_{weak}$-estimates。上述结果实际上只是我们在定量分层 $S^k_{\epsilon,r}$ 和 $S^k_{\epsilon}\equiv S^k_{\epsilon,0 上证明的一类新估计的应用}$。粗略地说，$x\in S^k_{\epsilon}\subseteq I$ 如果没有球 $B_r(x)$ 是 $\epsilon$-接近于 $k+1$-对称。我们证明 $S^k_\epsilon$ 是 $k$-rectifiable 并且满足 Minkowski 估计 $Vol(B_r\,S_\epsilon^k)\leq C_\epsilon r^{nk}$。