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Reifenberg parameterizations for sets with holes
About this Title
Guy David, Mathématiques, Bâtiment 425, Université de Paris-Sud 11, 91405 Orsay Cedex, France and Tatiana Toro, University of Washington, Department of Mathematics, Box 354350, Seattle, WA 98195–4350
Publication: Memoirs of the American Mathematical Society
Publication Year:
2012; Volume 215, Number 1012
ISBNs: 978-0-8218-5310-8 (print); 978-0-8218-8517-8 (online)
DOI: https://doi.org/10.1090/S0065-9266-2011-00629-5
Published electronically: April 13, 2011
Keywords: Reifenberg,
topological disk,
bi-Lipschitz parameterizations.
MSC: Primary 28A75, 49Q05; Secondary 49Q20, 49K99
Table of Contents
Chapters
- 1. Introduction
- 2. Coherent families of balls and planes
- 3. A partition of unity
- 4. Definition of a mapping $f$ on $\Sigma _0$
- 5. Local Lipschitz graph descriptions of the $\Sigma _k$
- 6. Reifenberg-flatness of the image
- 7. Distortion estimates for $D\sigma _k$
- 8. Hölder and Lipschitz properties of $f$ on $\Sigma _0$
- 9. $C^2$-regularity of the $\Sigma _k$ and fields of linear isometries defined on $\Sigma _0$
- 10. The definition of $g$ on the whole $\mathbb R^n$
- 11. Hölder and Lipschitz properties of $g$ on $\mathbb R^n$
- 12. Variants of the Reifenberg theorem
- 13. Local lower-Ahlfors regularity and a better sufficient bi-Lipschitz condition
- 14. Big pieces of bi-Lipschitz images and approximation by bi-Lipschitz domains
- 15. Uniform rectifiability and Ahlfors-regular Reifenberg-flat 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 $\mathbb R^d$.
Résumé. On généralise la démonstration du théorème du disque topologique de Reifenberg pour inclure le cas d’ensembles ayant des trous, et on donne des conditions suffisantes sur l’ensemble $E$ pour l’existence de paramétrage de $E$ par un plan affine ou une variété de dimension $d$. L’une de ces conditions porte sur la sommabilité des carrés des nombres de P. Jones $\beta _1(x,r)$, et s’applique en particulier aux ensembles localement Ahlfors-réguliers et à l’existence de très grand morceaux d’images bi-Lipschitziennes de $\mathbb R^d$.
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