流形的平坦形式、双Lipschitz参数化和光滑性
Flat forms, bi-Lipschitz parametrizations, and smoothability of manifolds
摘要：
给出了度量（同调）流形局部双Lipschitz等价于$\rn$中开子集的一个充分条件。该条件是一个Sobolev条件的可测余框架的平坦1-形式。结合D. Sullivan，我们的方法也产生了一个解析的光滑性的Lipschitz流形的Sobolev正则性的余切结构的框架方面的表征。在证明中，我们利用了平坦链与平坦形式之间的对偶性，以及最近建立的度量测度空间上的微分分析。当专门用于$\rn$时，我们的结果给出了Sullivan提出的可测黎曼映射定理的一种渐近和Lipschitz形式.
英文摘要：
We give a sufficient condition for a metric (homology) manifold to be locally bi-Lipschitz equivalent to an open subset in $\rn$. The condition is a Sobolev condition for a measurable coframe of flat 1-forms. In combination with an earlier work of D. Sullivan, our methods also yield an analytic characterization for smoothability of a Lipschitz manifold in terms of a Sobolev regularity for frames in a cotangent structure. In the proofs, we exploit the duality between flat chains and flat forms, and recently established differential analysis on metric measure spaces. When specialized to $\rn$, our result gives a kind of asymptotic and Lipschitz version of the measurable Riemann mapping theorem as suggested by Sullivan.
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