Bi-Lipschitz Pieces between Manifolds
Bi-Lipschitz Pieces between Manifolds
复制标题
歧管之间的 Bi-Lipschitz 件
DOI：
10.4171/rmi/883
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发表时间：
2013
期刊：
影响因子：
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通讯作者：
Guy C. David
中科院分区：
文献类型：
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作者：
Guy C. David
一个著名的问题问如下：如果$X$和$Y$是度量测度空间和$f：X\rightarrow Y$是一个Lipschitz映射，其图像有积极的措施，那么必须$f$有大块上，它是bi-Lipschitz？方法的大卫（谁不是本作者）和Semmes的基础上，我们回答这个问题的肯定Lipschitz映射之间的某些类型的Ahlfors $s$-经常，拓扑$d$-流形。一般来说，这些流形不需要在任何欧几里得空间中是双Lipschitz可嵌入的。为了证明这个结果，我们使用了一些事实的Gromov-Hausdorff收敛流形和拓扑定理的Bonk和Kleiner。这也给出了某些度量流形的一致求长性的一个新证明。
A well-known class of questions asks the following: If $X$ and $Y$ are metric measure spaces and $f:X\rightarrow Y$ is a Lipschitz mapping whose image has positive measure, then must $f$ have large pieces on which it is bi-Lipschitz? Building on methods of David (who is not the present author) and Semmes, we answer this question in the affirmative for Lipschitz mappings between certain types of Ahlfors $s$-regular, topological $d$-manifolds. In general, these manifolds need not be bi-Lipschitz embeddable in any Euclidean space. To prove the result, we use some facts on the Gromov-Hausdorff convergence of manifolds and a topological theorem of Bonk and Kleiner. This also yields a new proof of the uniform rectifiability of some metric manifolds.