2005/09/29 by David Hamilton, David H Hamilton, Hamilton, David H
Mathematics · #30C65 #Analytic and geometric function theory #Complex Variables (math.CV) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Holomorphic and Operator Theory #math.CV #msc:30C65
paper · pdf · doi:10.48550/arxiv.math/0509703
8 pages
arxiv created 2005/09/29 · openalex publication_date 2005/09/29 · arxiv updated 2009/12/01 · openalex created_date 2024/04/10 · openalex updated_date 2026/07/28
Ahlfors and Gehring asked for the Riemann Mapping Theorem for quasiconformal mappings (QC) of R3. We summarise our solution: (a) QC reflections are tame (b) T is the fixed set of a QC reflection iff T is a uniform sphere (i.e. the limits of its blowups are topologically flat spheres) (c) T is a quasiphere iff is a uniform & quasisymmetric sphere (d) A domain D is the QC image of the unit ball iff it is a regular ball: uniformly simply connected (i.e. the limits of blowups are topological balls) with boundary mapping from the unit sphere which is regular (i.e. the limits of blowups are boundary maps from the sphere). The latter is equivalent to being locally linearly connected and Lowner.