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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
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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

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 $$\mathbb R^d$$.

• 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