Memoirs of the American Mathematical Society 2011; 102 pp; softcover Volume: 215 ISBN10: 0821853104 ISBN13: 9780821853108 List Price: US$71 Individual Members: US$42.60 Institutional Members: US$56.80 Order Code: MEMO/215/1012
 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 biLipschitz 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 Ahlforsregular case to provide very big pieces of biLipschitz 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 \(\Sigma_k\)
 Reifenbergflatness 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 lowerAhlfors regularity and a better sufficient biLipschitz condition
 Big pieces of biLipschitz images and approximation by biLipschitz domains
 Uniform rectifiability and Ahlforsregular Reifenbergflat sets
 Bibliography
