Quasiconformal Jordan Domains
Copyright policy )Quasiconformal Jordan Domains
Abstract We extend the classical Carathéodory extension theorem to quasiconformal Jordan domains ( Y , d Y ). We say that a metric space ( Y , d Y ) is a quasiconformal Jordan domain if the completion ̄ Y of ( Y , d Y ) has finite Hausdorff 2-measure, the boundary ∂ Y = ̄ Y \ Y is homeomorphic to 𝕊 1 , and there exists a homeomorphism ϕ : 𝔻 →( Y , d Y ) that is quasiconformal in the geometric sense. We show that ϕ has a continuous, monotone, and surjective extension Φ: 𝔻 ̄ → Y ̄. This result is best possible in this generality. In addition, we find a necessary and sufficient condition for Φ to be a quasiconformal homeomorphism. We provide sufficient conditions for the restriction of Φ to 𝕊 1 being a quasisymmetry and to ∂Y being bi-Lipschitz equivalent to a quasicircle in the plane.
- University of Jyväskylä Finland
primary 30l10, metric spaces, carathéodory, metric surface, funktioteoria, quasiconformal Jordan domains, Mathematics - Metric Geometry, Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations, FOS: Mathematics, Matematiikka, Complex Variables (math.CV), Length, area, volume, other geometric measure theory, QA299.6-433, Carathéodory extension theorem, Mathematics - Complex Variables, ta111, Primary 30L10, Secondary 30C65, 28A75, 51F99, 52A38, beurling–ahlfors, Metric Geometry (math.MG), quasiconformal, secondary 30c65, 28a75, 51f99, Carathéodory, metriset avaruudet, Beurling–Ahlfors, Quasiconformal mappings in metric spaces, mittateoria, Mathematics, Analysis
primary 30l10, metric spaces, carathéodory, metric surface, funktioteoria, quasiconformal Jordan domains, Mathematics - Metric Geometry, Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations, FOS: Mathematics, Matematiikka, Complex Variables (math.CV), Length, area, volume, other geometric measure theory, QA299.6-433, Carathéodory extension theorem, Mathematics - Complex Variables, ta111, Primary 30L10, Secondary 30C65, 28A75, 51F99, 52A38, beurling–ahlfors, Metric Geometry (math.MG), quasiconformal, secondary 30c65, 28a75, 51f99, Carathéodory, metriset avaruudet, Beurling–Ahlfors, Quasiconformal mappings in metric spaces, mittateoria, Mathematics, Analysis
selected citations These citations are derived from selected sources.This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).1 popularity This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.Average influence This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).Average impulse This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.Average
Citations provided by BIP!Popularity provided by BIP!