Decomposition of extremal length and a proof of Shen's conjecture on QED constant and boundary dilatation
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 be improved to be \(1+H(\Omega)\), which was conjectured by the reviewer [loc. cit.].
scientific article; zbMATH DE number 6384022