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

Abstract

Abstract. 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.

Integration by parts on generalized manifolds and applications on quasiregular maps

“…One motivation for our study arises from the difficult question of deciding when a given metric space is locally bi-Lipschitz equivalent to a Euclidean space, or when there exist locally homeomorphic quasiregular maps from the given space to a Euclidean space. For those and related questions, see for instance [4], [5], [7] and [8].…”

Uniformization of two-dimensional metric surfaces

“…We will here concentrate on QC and QS maps. Concerning the existence of bi-Lipschitz parametrizations, we only briefly note that interesting sufficient conditions and counterexamples have been found both in the 2-dimensional ( [13], [22], [38], [44], [52], [53]) and higherdimensional cases ( [4], [28], [30], [32], [50]). …”

Mappings of finite distortion from generalized manifolds

“…Here the first map is induced by f and the second one is the isomorphism provided by assumption (8). The composed map sends the generator of H Using the definition and basic properties of Alexander-Spanier cohomology it is not too difficult to see that topological degree enjoys the properties listed in Lemma 2.33.…”

“…According to [5, Lemma 3.2.25] such orientation above always exists. Because S is an oriented, generalized n-manifold, there is a fixed orientation of U induced by the mapping in (8), provided that U is connected. Fix x ∈ U, such that apT an(x, U ) ∈ T U exists.…”

“…In [13], Heinonen and Sullivan characterized metric spaces that are cohomology n-manifolds and admit local BLD-maps into R n . Moreover, in [8], Heinonen and Keith showed that an Ahlfors n-regular, locally linearly contractible homology n-manifold in R n has local bi-Lipschitz parametrizations if it admits local Cartan-Whitney presentations in a certain Sobolev class. In fact, they gave an analytic characterization for topological manifolds among all the homology n-manifolds.…”