Quasiconformal non-parametrizability of almost smooth spheres

2015/05/31 by Pekka Pankka, Vyron Vellis
Mathematics · #Analytic and geometric function theory #Contractible space #Euclidean distance #Euclidean geometry #Geometric Analysis and Curvature Flows #Homeomorphism (graph theory) #Intrinsic metric #Metric (unit) #Metric connection #Metric space #Nonlinear Partial Differential Equations #SPHERES #math.MG #msc:30C65 #msc:30L10

paper · pdf · doi:10.1007/s00029-016-0292-4

28 pages, 3 figures

openalex created_date 2016/06/24 · arxiv created 2016/10/10 · openalex publication_date 2016/10/22 · arxiv updated 2017/07/03 · openalex updated_date 2026/08/05

Abstract

We show that, for each n≥ 3, there exists a smooth Riemannian metric g on a punctured sphere \mathbbSⁿ∖ \x₀\ for which the associated length metric extends to a length metric d of \mathbbSⁿ with the following properties: the metric sphere (\mathbbSⁿ,d) is Ahlfors n-regular and linearly locally contractible but there is no quasiconformal homeomorphism between (\mathbbSⁿ,d) and the standard Euclidean sphere \mathbbSⁿ.

Citations

• Lectures on Analysis on Metric Spaces
• Quasisymmetric parametrizations of two-dimensional metric spheres
• Quasiconformal maps in metric spaces with controlled geometry
• A Homeomorphism Between the 3-Sphere and the Sum of Two Solid Horned Spheres
• Strange actions of groups on spheres