eBook ISBN:  9780821885178 
Product Code:  MEMO/215/1012.E 
List Price:  $71.00 
MAA Member Price:  $63.90 
AMS Member Price:  $42.60 
eBook ISBN:  9780821885178 
Product Code:  MEMO/215/1012.E 
List Price:  $71.00 
MAA Member Price:  $63.90 
AMS Member Price:  $42.60 

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 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

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. Reifenbergflatness 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 lowerAhlfors regularity and a better sufficient biLipschitz condition

14. Big pieces of biLipschitz images and approximation by biLipschitz domains

15. Uniform rectifiability and Ahlforsregular Reifenbergflat 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 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 \(\mathbb R^d\).

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. Reifenbergflatness 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 lowerAhlfors regularity and a better sufficient biLipschitz condition

14. Big pieces of biLipschitz images and approximation by biLipschitz domains

15. Uniform rectifiability and Ahlforsregular Reifenbergflat sets