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<p>We prove that uniformly disconnected subsets of metric measure spaces with controlled geometry (complete, Ahlfors regular, supporting a Poincaré inequality, and a mild topological condition) are contained in a quasisymmetric arc. This generalizes a result of MacManus [<CitationRef CitationID="CR29">29</CitationRef>] from Euclidean spaces to abstract metric setting. Along the way, we prove a geometric strengthening of the classical Denjoy-Riesz theorem in metric measure spaces. Finally, we prove that the complement of a uniformly disconnected set in such a metric space is uniform, quantitatively.</p>

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Quasisymmetric rectifiability of uniformly disconnected sets

  • 期刊论文

  • 发表日期：

    2025年12月22日

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  • Jacob Honeycutt,
  • Vyron Vellis

摘要

We prove that uniformly disconnected subsets of metric measure spaces with controlled geometry (complete, Ahlfors regular, supporting a Poincaré inequality, and a mild topological condition) are contained in a quasisymmetric arc. This generalizes a result of MacManus [29] from Euclidean spaces to abstract metric setting. Along the way, we prove a geometric strengthening of the classical Denjoy-Riesz theorem in metric measure spaces. Finally, we prove that the complement of a uniformly disconnected set in such a metric space is uniform, quantitatively.