Item Successfully Added to Cart
An error was encountered while trying to add the item to the cart. Please try again.
OK
Please make all selections above before adding to cart
OK
Share this page via the icons above, or by copying the link below:
Copy To Clipboard
Successfully Copied!
Reifenberg Parameterizations for Sets with Holes
 
Guy David and Université de Paris Sud, Orsay, France
Tatiana Toro University of Washington, Seattle, WA
Reifenberg Parameterizations for Sets with Holes
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
Reifenberg Parameterizations for Sets with Holes
Click above image for expanded view
Reifenberg Parameterizations for Sets with Holes
Guy David and Université de Paris Sud, Orsay, France
Tatiana Toro University of Washington, Seattle, WA
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
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 2152011; 102 pp
    MSC: 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 \(\mathbb R^d\).

  • 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
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 2152011; 102 pp
MSC: 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 \(\mathbb R^d\).

  • 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
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Please select which format for which you are requesting permissions.