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

TL;DR

This paper establishes a Sobolev-based criterion for when metric manifolds can be bi-Lipschitzly mapped to Euclidean spaces and characterizes smoothability of Lipschitz manifolds through Sobolev regularity of frames.

Contribution

It introduces a Sobolev condition for measurable coframes that ensures bi-Lipschitz equivalence to Euclidean open sets and provides an analytic criterion for smoothability of Lipschitz manifolds.

Findings

A sufficient Sobolev condition for bi-Lipschitz parametrizations.

An analytic characterization of smoothability in terms of Sobolev regularity.

Extension of the measurable Riemann mapping theorem to metric measure spaces.

Abstract

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.