2016/09/30 by Vyron Vellis
Mathematics · #Analytic and geometric function theory #Bounded function #Cantor function #Cantor set #Combinatorics #Computer science #Discrete mathematics #Geometric function theory #Geometry #Holomorphic and Operator Theory #Mathematical analysis #Mathematics #Metric space #Nonlinear Partial Differential Equations #Pure mathematics #Riemann surface #Set (abstract data type) #Uniform boundedness #Uniform continuity #Uniformization (probability theory) #Uniformization theorem #math.MG #msc:30C65 #msc:30L05
paper · pdf · doi:10.1090/ecgd/360
published in Conformal Geometry and Dynamics of the American Mathematical Society 25(5), 88-103 (Serbian Mathematical Society) · 13 pages. This is a small part of an older manuscript called "Extension properties of planar uniform domains"
arxiv created 2021/01/14 · openalex publication_date 2021/08/10 · arxiv updated 2022/02/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
In this note we provide a quasisymmetric taming of uniformly perfect and uniformly disconnected sets that generalizes a result of MacManus [Rev. Mat. Iberoamericana 15 (1999), pp. 267–277] from 2 to higher dimensions. In particular, we show that a compact subset of<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R Superscript n"><mml:semantics><mml:msup><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="double-struck">R</mml:mi></mml:mrow><mml:mi>n</mml:mi></mml:msup><mml:annotation encoding="application/x-tex">\mathbb Rn</mml:annotation></mml:semantics></mml:math></inline-formula>is uniformly perfect and uniformly disconnected if and only if it is ambiently quasiconformal to the standard Cantor set<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper C"><mml:semantics><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi class="MJX-tex-caligraphic" mathvariant="script">C</mml:mi></mml:mrow><mml:annotation encoding="application/x-tex">\mathcal C</mml:annotation></mml:semantics></mml:math></inline-formula>in<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R Superscript n plus 1"><mml:semantics><mml:msup><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="double-struck">R</mml:mi></mml:mrow><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:annotation encoding="application/x-tex">\mathbb Rn+1</mml:annotation></mml:semantics></mml:math></inline-formula>.