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Bi-Lipschitz arcs in metric spaces with controlled geometry

Jacob Honeycutt, Vyron Vellis, Scott Zimmerman

Abstract

We generalize a bi-Lipschitz extension result of David and Semmes from Euclidean spaces to complete metric measure spaces with controlled geometry (Ahlfors regularity and supporting a Poincaré inequality). In particular, we find sharp conditions on metric measure spaces $X$ so that any bi-Lipschitz embedding of a subset of the real line into $X$ extends to a bi-Lipschitz embedding of the whole line. Along the way, we prove that if the complement of an open subset $Y$ of $X$ has small Assouad dimension, then it is a uniform domain. Finally, we prove a quantitative approximation of continua in $X$ by bi-Lipschitz curves.

Bi-Lipschitz arcs in metric spaces with controlled geometry

Abstract

We generalize a bi-Lipschitz extension result of David and Semmes from Euclidean spaces to complete metric measure spaces with controlled geometry (Ahlfors regularity and supporting a Poincaré inequality). In particular, we find sharp conditions on metric measure spaces so that any bi-Lipschitz embedding of a subset of the real line into extends to a bi-Lipschitz embedding of the whole line. Along the way, we prove that if the complement of an open subset of has small Assouad dimension, then it is a uniform domain. Finally, we prove a quantitative approximation of continua in by bi-Lipschitz curves.
Paper Structure (19 sections, 22 theorems, 149 equations, 2 figures)

This paper contains 19 sections, 22 theorems, 149 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Related results
  3. Outline of the proof of Theorem \ref{['thm:1']}
  4. Preliminaries
  5. Porosity and regularity
  6. Poincaré inequality
  7. Modulus of curve families
  8. Uniformity in metric measure spaces
  9. Bi-Lipschitz approximation of curves
  10. Proof of Proposition \ref{['cor:Modi']}
  11. Whitney intervals and a preliminary extension
  12. Reference points
  13. The middle third of each Whitney interval
  14. Proof of Theorem \ref{['thm:1']}
  15. Local modifications around points in E
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $n\geq 3$ be an integer, let $A \subset \mathbb{R}$, and let $f:A \to \mathbb{R}^n$ be a bi-Lipschitz embedding. Then there exists a bi-Lipschitz extension $F:\mathbb{R} \to \mathbb{R}^n$.

Figures (2)

  • Figure 1:
  • Figure 2:

Theorems & Definitions (36)

  • Theorem 1.1: David:1991
  • Theorem 1.2
  • Corollary 1.3
  • Proposition 1.4
  • Proposition 1.5
  • Lemma 2.1: Bonk:2001
  • Lemma 2.2: Kallunki:2001
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 26 more