Bi-Lipschitz arcs in metric spaces with controlled geometry

• Jacob Honeycutt ^[2] ; Vyron Vellis ^[2] ; Scott Zimmerman ^[1]
  1. [1] Ohio State University

     Ohio State University

     City of Columbus, Estados Unidos

     
  2. [2] The University of Tennessee, Knoxville, USA
• Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 40, Nº 5, 2024, págs. 1887-1916
• Idioma: inglés
• DOI: 10.4171/RMI/1484
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• Resumen

  • 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 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.