Bi-Lipschitz arcs in metric spaces with controlled geometry

  • Jacob Honeycutt

    The University of Tennessee, Knoxville, USA
  • Vyron Vellis

    The University of Tennessee, Knoxville, USA
  • Scott Zimmerman

    The Ohio State University, Columbus, USA
Bi-Lipschitz arcs in metric spaces with controlled geometry cover
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Abstract

In this paper, 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.

Cite this article

Jacob Honeycutt, Vyron Vellis, Scott Zimmerman, Bi-Lipschitz arcs in metric spaces with controlled geometry. Rev. Mat. Iberoam. 40 (2024), no. 5, pp. 1887–1916

DOI 10.4171/RMI/1484