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Uniform domains and the ubiquitous quasidisk. (English) Zbl 0615.30016
A domain $$D\subset {\bar {\mathbb{R}}}^ 2$$ is a K-quasidisk if it is the image of an euclidean disk under a K-quasiconformal selfmapping of $${\bar {\mathbb{R}}}^ 2$$. A simple metric characterization of quasidisks is due to L. Ahlfors. The article summarizes several equivalent characterizations for quasidisks. Many of these have been initiated by the author. Equivalent analytic characterizations for bounded plane simply connected domains D: (i) A BMO extension domain (Jones), (ii) a Sobolev extension domain (Jones) and (iii) a quasiextremal distance domain (Gehring- Martio). Equivalent metric conditions: (i) uniform (Martio-Sarvas), (ii) linearly locally connected (Gehring), (iii) decomposable (Martin) and (iv) $$k_ D\leq cj_ D$$ where $$k_ D$$ is the quasihyperbolic metric and $j_ D(z_ 1,z_ 2)=\log (1+| z_ 1-z_ 2| /d(z_ 1,\partial D))(1+| z_ 1+z_ 2| /d(z_ 2,\partial D))$ (Jones, Gehring-Osgood). There are also a number of function theoretic characterizations based on the quasiconformal extension property, the Schwarzian derivative and its counterpart $$| f''/f'|$$. The article completes in a nice way the lecture notes [F. W. Gehring, Characteristic properties of quasidisks (1982; Zbl 0495.30018)] of the author and contains a comprehensive list of references.
Reviewer: O.Martio

##### MSC:
 30C62 Quasiconformal mappings in the complex plane
##### Keywords:
uniform domains; quasidisks; Schwarzian derivative