Flat forms, bi-Lipschitz parametrizations, and smoothability of manifolds

Heinonen, Juha; Keith, Stephen

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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