The authors 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 \(\mathbb R^d\).

Table of Contents

• Introduction
• Coherent families of balls and planes
• A partition of unity
• Definition of a mapping \(f\) on \(\Sigma_0\)
• Local Lipschitz graph descriptions of the \(\Sigma_k\)
• Reifenberg-flatness of the image
• Distortion estimates for \(D\sigma_k\)
• Hölder and Lipschitz properties of \(f\) on \(\Sigma_0\)
• \(C^2\)-regularity of the \(\Sigma_k\) and fields of linear isometries defined on \(\Sigma_0\)
• The definition of \(g\) on the whole \(\mathbb R^n\)
• Hölder and Lipschitz properties of \(g\) on \(\mathbb R^n\)
• Variants of the Reifenberg theorem
• Local lower-Ahlfors regularity and a better sufficient bi-Lipschitz condition
• Big pieces of bi-Lipschitz images and approximation by bi-Lipschitz domains
• Uniform rectifiability and Ahlfors-regular Reifenberg-flat sets
• Bibliography