远离奇异集的光滑收敛
Smooth Convergence Away from Singular Sets
摘要：
我们考虑黎曼流形$M$上的度量序列$g_j$，它在远离奇异集$S\subset M$的紧集上平滑地收敛到$M\setminus S$上的度量$g_\infty$。我们证明定理描述$M_j=(M, g_j)$收敛时的Gromov-Hausdorff意义上的度量完成，$(M_\infty,d_\infty)$，$(M\setminus S, g_\infty)$。为了得到这些定理，我们研究了序列的内在平坦极限。本文利用一种新的方法--半球嵌入方法，得到了关于具有非纯子域的黎曼流形之间的Gromov-Hausdorff距离和内蕴平坦距离的显式估计.这篇论文在CAG上发表七年后，布赖恩·艾伦发现了一个与定理1.3的公开声明相反的例子。注意定理4.6（这是在其他论文中引用的关键定理）仍然是正确的。我们增加了一个假设来纠正定理1.3的陈述及其结果。这个v4包括蓝色的更正，介绍结尾的勘误表，以及附录中的Brian Allen的示例。一份勘误表也被送到了《华尔街日报》。
英文摘要：
We consider sequences of metrics, $g_j$, on a Riemannian manifold, $M$, which converge smoothly on compact sets away from a singular set $S\subset M$, to a metric, $g_\infty$, on $M\setminus S$. We prove theorems which describe when $M_j=(M, g_j)$ converge in the Gromov-Hausdorff sense to the metric completion, $(M_\infty,d_\infty)$, of $(M\setminus S, g_\infty)$. To obtain these theorems, we study the intrinsic flat limits of the sequences. A new method, we call hemispherical embedding, is applied to obtain explicit estimates on the Gromov-Hausdorff and Intrinsic Flat distances between Riemannian manifolds with diffeomorphic subdomains. Seven years after the publication of this paper in CAG, Brian Allen discovered a counter example to the published statement of Theorem 1.3. Note that Theorem 4.6 (which is the key theorem cited in other papers) remains correct. We have added an hypothesis to correct the statement of Theorem 1.3 and its consequences. This v4 includes corrections in blue, an erratum at the end of the introduction, and Brian Allen's example in an appendix. An erratum is also being sent to the journal.
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