Bi-Lipschitz arcs in metric spaces with controlled geometry

TL;DR

This paper extends bi-Lipschitz embedding results from Euclidean spaces to metric measure spaces with controlled geometry, establishing conditions for extension, analyzing domain properties, and approximating continua by bi-Lipschitz curves.

Contribution

It generalizes bi-Lipschitz extension theorems to metric spaces with Ahlfors regularity and Poincaré inequalities, providing sharp conditions and new geometric insights.

Findings

Bi-Lipschitz extension criteria in metric measure spaces

Complement of certain subsets are uniform domains

Approximation of continua by bi-Lipschitz curves

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 Poincar\'e 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.