Reifenberg Parameterizations For Sets With Holes (memoirs Of The American Mathematical Society)

January 2012, volume 215, number 1012 (third of 5 numbers). Chapter 1. Introduction 8 Chapter 2. Coherent families of balls and planes 16 Chapter 3. A partition of unity 20 Chapter 4. Definition of a mapping f on 0 22 Chapter 5. Local Lipschitz graph descriptions of the k 26 Chapter 6. Reifenberg-flatness of the image 34 Chapter 7. Distortion estimates for Dk 38 Chapter 8. Hölder and Lipschitz properties of f on 0 44 Chapter 9. C2-regularity of the k and fields of linear isometries defined on 0 48 Chapter 10. The definition of g on the whole Rn 56 Chapter 11. Hölder and Lipschitz properties of g on Rn 62 Chapter 12. Variants of the Reifenberg theorem 72 Chapter 13. Local lower-Ahlfors regularity and a better sufficient bi-Lipschitz condition 82 Chapter 14. Big pieces of bi-Lipschitz images and approximation by bi-Lipschitz domains 92 Chapter 15. Uniform rectifiability and Ahlfors-regular Reifenberg-flat sets 98 Bibliography 108 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 $\mathbb R^d$.