Quasiextremal distance domains and extension of quasiconformal mappings
A domain D in \(\bar R^ 2\) is said to be a K-quasidisc if it is the image of an open disc under a K-quasiconformal self-mapping of \(\bar R^ 2\). The paper under review is concerned with the extension to more general classes of domains D in \(\bar R^ n\) of the following two basic properties of quasidiscs. 1.Extremal distance property. If D is a quasidisc and \(F_ 1\) and \(F_ 2\) are disjoint continua in D, then \(mod(\Gamma)\leq M mod(\Gamma_ D)\) where \(\Gamma\) and \(\Gamma_ D\) are the families of curves which join \(F_ 1\) and \(F_ 2\) in \(\bar R^ 2\) and D, respectively, and where M is a constant which depends only on D and mod(\(\Gamma)\) is the modulus of \(\Gamma\). 2. Extension property. If D is a quasidisc and if f is a quasiconformal mapping of D onto a domain D' in \(\bar R^ 2\), then f has a quasiconformal extension to \(\bar R^ 2\) if and only if D' is a quasidisc. The authors prove that properties 1 and 2 are related in the following sense. If D and D' are as in 2 and if D has the extremal distance property, then D and D' have the extension property iff D' has the extremal distance property. The authors call domains satisfying property 1 QED-domains (quasi-extremal distance domains) and study their geometric properties. They also show that a simply-connected plane domain with a non-degenerated boundary is a QED-domain if it is a quasidisc.
scientific article; zbMATH DE number 3959880
scientific article