English

Uniformization of Cantor sets with bounded geometry

Metric Geometry 2022-02-23 v5

Abstract

In this note we provide a quasisymmetric taming of uniformly perfect and uniformly disconnected sets that generalizes a result of MacManus from 2 to higher dimensions. In particular, we show that a compact subset of Rn\mathbb{R}^n is uniformly perfect and uniformly disconnected if and only if it is ambiently quasiconformal to the standard Cantor set C\mathcal{C} in Rn+1\mathbb{R}^{n+1}.

Keywords

Cite

@article{arxiv.1609.08763,
  title  = {Uniformization of Cantor sets with bounded geometry},
  author = {Vyron Vellis},
  journal= {arXiv preprint arXiv:1609.08763},
  year   = {2022}
}

Comments

13 pages. This is a small part of an older manuscript called "Extension properties of planar uniform domains"

R2 v1 2026-06-22T16:03:43.619Z