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Bi-Lipschitz arcs in metric spaces with controlled geometry (2307.06931v2)

Published 13 Jul 2023 in math.MG

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.

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