Conformal invariants of QED domains
type: Item
instance of: scholarly article
title: Conformal invariants of QED domains
zbMATH Open document ID: 1063.30017
author: Yuliang Shen
published in: Tôhoku Mathematical Journal. Second Series
publication date: 2005-03-21T00:00:00Z
review text: Let \(\Omega\) be a domain in the extended complex plane \(\overline{{\mathbb C}}\). The aim of this paper is to give inequalities between the boundary quasiextremal distance constant \(M_b(\Omega)\), the quasiextremal distance constant \(M(\Omega)\) and the quasiextremal reflection constant \(R(\Omega)\). These constants were introduced and studied by S. Yang in a series of papers. They are defined as folows. Given two disjoint nondegenerate continua \(A\) and \(B\) in the closure \(\overline{\Omega}\), we let \(\text{mod}(A,B;\Omega)\) be the modulus of the family \(\Gamma(A,B;\Omega)\) of curves that join \(A\) and \(B\) in \(\Omega\). Then, \[ M(\Omega)=\sup{\text{mod}(A,B;{\mathbb C})\over \text{mod}(A,B;\Omega)} \] where the supremum is taken over all pairs \(A\) and \(B\) in \(\overline{\Omega}\). The domain \(\Omega\) is called a QED domain if \(M(\Omega)\) is finite. The notion of QED domain is due to Gehring and Martio. For such a domain, one can define the boundary quasiextremal reflection constant \(R(\Omega)\) by \[ R(\Omega)=\inf K[f], \] the infimum being taken over all quasiconformal reflections \(f\) in \(\partial \Omega\). Finally, one defines \[ M_b(\Omega)=\sup{\text{mod}(A,B;{\mathbb C})\over \text{mod}(A,B;\Omega)} \] where the supremum is taken over all pairs \(A\) and \(B\) in \(\partial \Omega\). The inequality \(M_b(\Omega)\leq M(\Omega)\) is straightforward. \textit{S. Yang} [J. Anal. Math. 75, 173--183 (1998; Zbl 0911.30017)] proved that \(M(\Omega)\leq R(\Omega) +1\) for all smooth domains other than disks and he asked whether \(M(\Omega)\leq R(\Omega) +1\) for all asymptotically conformal extension domains other than disks. Wu and Yang proved that \(M(\Omega)< R(\Omega) +1\) for all asymptotically conformal extension domains other than disks. Here, a Jordan domain \(\Omega\) is said to be an asymptotically conformal extension domain if the Riemann mapping theorem from the unit disk \(D\) to \(\Omega\) has a quasiconformal extension to a…
reviewed by: Athanase Papadopoulos
Mathematics Subject Classification ID: 30C62
Mathematics Subject Classification ID: 30C70
zbMATH DE Number: 2148497
zbMATH Keywords: boundary quasiextremal distance constant
zbMATH Keywords: quasiextremal distance constant
zbMATH Keywords: quasiconformal reflection constant
zbMATH Keywords: quasisymmetric homeomorphism
zbMATH Keywords: QED domain
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1113246678
OpenAlex ID: W2068314118
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DOI: 10.2748/TMJ/1113246678
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description: scientific article; zbMATH DE number 2148497