Weak convergence和取消
Weak convergence and cancellation
摘要：
本文研究了度量空间中积分流序列的弱极限与支撑序列的可能Hausdorff极限之间的关系。由于相消，弱极限一般在豪斯多夫极限的严格子集中得到支持。我们展示充分条件的拓扑结构的支持，确保没有取消发生和支持的弱极限同意的Hausdorff极限的支持。我们使用我们的结果证明可数$\hm^m$ -可求长的Gromov-Hausdorff极限的Lipschitz流形$M_n$序列，所有这些都是$λ$ -线性局部收缩到一定规模$r_0$。在Raanan Schul和第二作者撰写的附录中，证明了Gromov-Hausdorff极限不需要是可计数的$\hm^m$ -可求积的，如果$M_n$有一个公共的局部几何可收缩函数，该函数只是凹的（而不是线性的）。我们也将我们的结果的工作Cheeger-Colding的限制noncollapsing序列的流形与非负里奇曲率。
英文摘要：
In this article, we study the relationship between the weak limit of a sequence of integral currents in a metric space and the possible Hausdorff limit of the sequence of supports. Due to cancellation, the weak limit is in general supported in a strict subset of the Hausdorff limit. We exhibit sufficient conditions in terms of topology of the supports which ensure that no cancellation occurs and that the support of the weak limit agrees with the Hausdorff limit of the supports. We use our results to prove countable $\hm^m$-rectifiability of the Gromov-Hausdorff limit of sequences of Lipschitz manifolds $M_n$ all of which are $λ$-linearly locally contractible up to some scale $r_0$. In an appendix, written by Raanan Schul and the second author, it is shown that the Gromov-Hausdorff limit need not be countably $\hm^m$-rectifiable if the $M_n$ have a common local geometric contractibility function which is only concave (and not linear). We also relate our results to work of Cheeger-Colding on the limits of noncollapsing sequences of manifolds with nonnegative Ricci curvature.
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