Quasisymmetric parametrizations of two-dimensional metric spheres

2001/07/24 by Mario Bonk, Bruce Kleiner · 183 citations
Mathematics · #Combinatorics #Contractible space #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry and complex manifolds #Mathematical analysis #Mathematics #Metric (unit) #Metric space #Physics #Pure mathematics #SPHERES #math.GR #math.GT #math.MG #msc:51F99 #msc:57M99

paper · pdf · doi:10.1007/s00222-002-0233-z

published in Inventiones mathematicae 150(1), 127-183 (Springer Science+Business Media)

arxiv created 2001/07/24 · openalex publication_date 2002/10/01 · arxiv updated 2015/06/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

We study metric spaces homeomorphic to the 2-sphere, and find conditions under which they are quasisymmetrically homeomorphic to the standard 2-sphere. As an application of our main theorem we show that an Ahlfors 2-regular, linearly locally contractible metric 2-sphere is quasisymmetrically homeomorphic to the standard 2-sphere, answering a question of Heinonen and Semmes.

Citations

• Quasiconformal maps in metric spaces with controlled geometry

Cited by

• Modulus and the Poincar� inequality on metric measure spaces
• Canonical parameterizations of metric disks
• Extension and trace theorems for noncompact doubling spaces
• Hyperbolic groups with planar boundaries
• Monotone Sobolev extensions in metric surfaces and applications to uniformization
• The Heinonen-Semmes problems after thirty years
• Discrete length-volume inequalities and lower volume bounds in metric spaces
• Quasisymmetric Koebe Uniformization of metric surfaces
• Uniformization with infinitesimally metric measures
• Geometry and quasisymmetric parametrization of Semmes spaces
• On quasisymmetry of quasiconformal mappings and its applications
• Regularity of Harmonic Maps from Polyhedra to CAT(1) Spaces
• Harmonic branched coverings and uniformization of CAT(k) spheres
• Weak Capacity and Critical Exponents
• The quasiconformal subinvariance property of John domains in \protect \IRⁿ and its application
• Uniform perfectness for quasi-metric spaces
• Geometric characterizations of inner uniformity through Gromov hyperbolicity
• Flat forms, bi-Lipschitz parametrizations, and smoothability of manifolds
• A metric sphere not a quasisphere but for which every weak tangent is Euclidean
• Quasisymmetric Koebe uniformization with weak metric doubling measures
• On the conformal gauge of a compact metric space
• Uniformization of metric surfaces: A survey
• Geometric function theory: the art of pullback factorization
• Equivalence of quasiregular mappings on subRiemannian manifolds via the Popp extension
• Invertible Carnot Groups
• Non-removability of the Sierpinski Gasket
• Quasisymmetric structures on surfaces
• Nonsmooth calculus
• Conformal dimension and Gromov hyperbolic groups with 2–sphere boundary
• Combinatorial Modulus on Boundary ofRight-Angled Hyperbolic Buildings
• Uniformization of Cantor sets with bounded geometry
• Uniformization of cofat domains on metric two-spheres
• Quasiconformal non-parametrizability of almost smooth spheres
• Oded Schramm: From circle packing to SLE
• Quasisymmetries of finitely ramified Julia sets. [europepmc]