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Reifenberg Parameterizations for Sets with Holes
Guy David, Université de Paris Sud, Orsay, France, and Tatiana Toro, University of Washington, Seattle, WA

Memoirs of the American Mathematical Society
2011; 102 pp; softcover
Volume: 215
ISBN-10: 0-8218-5310-4
ISBN-13: 978-0-8218-5310-8
List Price: US$71
Individual Members: US$42.60
Institutional Members: US$56.80
Order Code: MEMO/215/1012
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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
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