Flat forms, bi-Lipschitz parametrizations, and smoothability of manifolds
Publications Mathématiques de l'IHÉS, Tome 113 (2011), pp. 1-37.

We give a sufficient condition for a metric (homology) manifold to be locally bi-Lipschitz equivalent to an open subset in R n . 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 R n , our result gives a kind of asymptotic and Lipschitz version of the measurable Riemann mapping theorem as suggested by Sullivan.

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     title = {Flat forms, {bi-Lipschitz} parametrizations, and smoothability of manifolds},
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Heinonen, Juha; Keith, Stephen. Flat forms, bi-Lipschitz parametrizations, and smoothability of manifolds. Publications Mathématiques de l'IHÉS, Tome 113 (2011), pp. 1-37. doi : 10.1007/s10240-011-0032-4. http://archive.numdam.org/articles/10.1007/s10240-011-0032-4/

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