Extremal lengths and quasiconformal maps (Q2702311)

The paper under review has the goal to illustrate developments of the theory of quasiconformal mappings after \textit{L. V. Ahlfors'} paper [J. Anal. Math. 3, 1-58 (1954; Zbl 0057.06506)]. The following topics are considered: quasisymmetry and quasiconformality, extremal length, modulus and capacity, discrete capacity, boundary extension, Möbius invariant extensions, quasiconformal extension of univalent functions, quasiconformal reflections and quasicircles, mapping problems in higher dimensions. The paper starts with the metric definition of quasiconformality and quasisymmetry and points out the relation between these. Few results about quasiconformality and quasisymmetry in Banach spaces are mentioned. It should be remarked that the first person using systematically a concept similar to the extremal length method was H. Grötzsch [\textit{L. V. Ahlfors}, Conformal invariants: Topics in geometric function theory (1973; Zbl 0272.30012), p. 50]. Of course it is not possible to give a complete survey over all results since 1954 in all these fields in such a short paper. The author gives Ahlfors' basic ideas and definitions and some further developments. In some cases the reviewer missed important new results or hints to survey papers. Two examples: qc extensions of univalent functions -- a hint to a self-contained up-to-date presentation would be helpful, for example \textit{Ch. Pommerenke} [Boundary behaviour of conformal maps (1992; Zbl 0762.30001), p. 112], quasiconformal reflections and quasicircles -- the reviewer missed hints to quasiconformal reflection and Fredholm eigenvalue, quasiconformal reflection and Grunsky inequality (a summary is given by \textit{R. Kühnau} [Jahresber. Dtsch. Math.-Ver. 90, No. 2, 90-109 (1988; Zbl 0638.30021)]).NEWLINENEWLINEFor the entire collection see [Zbl 0944.00101].