Decomposition of extremal length and a proof of Shen's conjecture on QED constant and boundary dilatation
type: Item
instance of: scholarly article
title: Decomposition of extremal length and a proof of Shen's conjecture on QED constant and boundary dilatation
zbMATH Open document ID: 1325.30020
author: Tao Cheng
author: Shanshuang Yang
published in: Mathematische Zeitschrift
publication date: 2015-01-07T00:00:00Z
review text: The authors mainly study the relations between various constants of Jordan domains in the extended complex plane. To be precise, let be given a Jordan domain \(\Omega\) in the extended complex plane \(\overline{\mathbb C}\), and also a pair of disjoint nondegenerate continua \(A\) and \(B\) in \(\overline{\Omega}\). Denote by \(\text{mod}(A,B;\Omega)\) the modulus of the family \(\Gamma(A,B;\Omega)\) of curves that join \(A\) and \(B\) in \(\Omega.\) Then the quasi-extremal distance (QED) constant \(M(\Omega)\) is defined as \[ M(\Omega)=\sup\frac{\text{mod}(A,B;\mathbb C)}{\text{mod}(A,B;\Omega)},\tag{1} \] where the supremum is taken over all pairs of disjoint continua \(A\) and \(B\) in \(\overline{\Omega}\). It is known that the QED constant \(M(\Omega)\) reflects the geometric properties of the domain \(\Omega\). For example, a domain \(\Omega\) has finite QED constant and is called a QED domain if and only if \(\Omega\) is a quasi-disk. Thus, for a QED Jordan domain \(\Omega\), it is natural to consider the quasiconformal (qc) reflection constant \[ R(\Omega)=\inf\big\{K(f)\:\, f \text{ is a qc reflection}\big\},\tag{2} \] and the boundary dilatation \[ H(\Omega)=\inf\big\{K(f|_{\Omega-E}): f \text{ is a qc reflection and } E \text{ is a compact subset in } \Omega\big\}.\tag{3} \] Then \textit{S. Yang} [Trans. Am. Math. Soc. 334, No. 1, 97--120 (1992; Zbl 0768.30015)] proved that \[ M(\Omega)\leq R(\Omega)+1,\tag{4} \] and conjectured that the above equality always holds, which turned out to be false however. To further investigate inequality (4), the reviewer [Tohoku Math. J., II. Ser. 56, No. 3, 445--466 (2004; Zbl 1063.30017)] proved that, for a Jordan QED domain \(\Omega\) in \(\overline{\mathbb C}\), either there exists a pair of disjoint continua \(A\) and \(B\) in \(\overline{\Omega}\) for which in (1) the supremum is achieved or \( M(\Omega)\leq 2H(\Omega)\). In this paper the authors prove, among other things, that the upper bound \(2H(\Omega)\) can b…
reviewed by: Yuliang Shen
Mathematics Subject Classification ID: 30C62
Mathematics Subject Classification ID: 30C70
zbMATH DE Number: 6384022
zbMATH Keywords: QED constant
zbMATH Keywords: quasiconformal reflection constant
zbMATH Keywords: boundary dilatation
MaRDI profile type: Publication
full work available at URL: s00209-014-1353-z
OpenAlex ID: W2089610440
Wikidata QID: Q122981724
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DOI: 10.1007/S00209-014-1353-Z
description: scientific article; zbMATH DE number 6384022