Reifenberg Parameterizations for Sets with Holes

TL;DR

This paper extends Reifenberg's Topological Disk Theorem to sets with holes, providing conditions under which such sets can be bi-Lipschitzly parameterized by smooth manifolds, especially in the context of Ahlfors-regular sets.

Contribution

It introduces a new extension of Reifenberg's theorem accommodating sets with holes, using square summability of Jones numbers for bi-Lipschitz parameterizations.

Findings

Provides sufficient conditions for bi-Lipschitz parameterization of sets with holes.

Applies to locally Ahlfors-regular sets, yielding large bi-Lipschitz images.

Extends classical Reifenberg theorem to more complex sets.

Abstract

We extend the proof of Reifenberg's Topological Disk Theorem to allow the case of sets with holes, and give sufficient conditions on a set $E$ for the existence of a bi-Lipschitz parameterization of $E$ by a $d$-dimensional plane or smooth manifold. Such a condition is expressed in terms of square summability for the P. Jones numbers $\beta_1(x,r)$. In particular, it applies in the locally Ahlfors-regular case to provide very big pieces of bi-Lipschitz images of $\R^d$.