eBook ISBN: | 978-0-8218-8517-8 |
Product Code: | MEMO/215/1012.E |
List Price: | $71.00 |
MAA Member Price: | $63.90 |
AMS Member Price: | $42.60 |
eBook ISBN: | 978-0-8218-8517-8 |
Product Code: | MEMO/215/1012.E |
List Price: | $71.00 |
MAA Member Price: | $63.90 |
AMS Member Price: | $42.60 |
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 215; 2011; 102 ppMSC: Primary 28; 49
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 \(\mathbb R^d\).
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Table of Contents
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Chapters
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1. Introduction
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2. Coherent families of balls and planes
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3. A partition of unity
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4. Definition of a mapping $f$ on $\Sigma _0$
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5. Local Lipschitz graph descriptions of the $\Sigma _k$
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6. Reifenberg-flatness of the image
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7. Distortion estimates for $D\sigma _k$
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8. Hölder and Lipschitz properties of $f$ on $\Sigma _0$
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9. $C^2$-regularity of the $\Sigma _k$ and fields of linear isometries defined on $\Sigma _0$
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10. The definition of $g$ on the whole $\mathbb R^n$
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11. Hölder and Lipschitz properties of $g$ on $\mathbb R^n$
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12. Variants of the Reifenberg theorem
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13. Local lower-Ahlfors regularity and a better sufficient bi-Lipschitz condition
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14. Big pieces of bi-Lipschitz images and approximation by bi-Lipschitz domains
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15. Uniform rectifiability and Ahlfors-regular Reifenberg-flat sets
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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 \(\mathbb R^d\).
-
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