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Lectures on quasiconformal mappings. With additional chapters by C. J. Earle and I. Kra, M. Shishikura and J. H. Hubbard. 2nd enlarged and revised ed. (English) Zbl 1103.30001

University Lecture Series 38. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3644-7/pbk). viii, 162 p. (2006).
The first edition of this book [L. V. Ahlfors, Lectures on quasiconformal mappings. Princeton, N.J.-Toronto-New York-London: D. Van Nostrand Company (1966; Zbl 0138.06002)] was published in 1966 and was based on a set of lecture notes of a course of the author at Harvard University in 1964. The book has become a classic. In this second edition the original text has been kept unchanged except for editorial notes and for new typography. The most valuable new feature of this edition is that three additional chapters written by C.J. Earle, I. Kra, M. Shishikura and J.H. Hubbard have been added. These three new chapters outline the development since 1964.
The author begins with the definition of a differentiable quasiconformal mapping, a related geometric problem of Grötzsch, and an introduction to the notion of extremal length. In Chapter II the author gives the general geometric and analytic definitions of quasiconformal mappings. Chapter III covers the extremal problems of Grötzsch, Teichmüller and Mori. The boundary correspondence problems and quasicircles are considered in Chapter IV. In Chapter V the author studies quasiconformal mappings as the solutions of the Beltrami equation, and finally in Chapter VI he deals with the Teichmüller spaces. Another important reference on plane quasiconformal mappings is [O. Lehto and K.I. Virtanen, Quasiconformal mappings in the plane. Translated from the German by K.W. Lucas. 2nd ed. Die Grundlehren der mathematischen Wissenschaften. Band 126. Berlin-Heidelberg-New York: Springer-Verlag (1973; Zbl 0267.30016)].
The first of the three additional chapters is written by C.J. Earle and I. Kra. The authors briefly summarize further developments in some areas related to Ahlfors’ book, including quasiconformal mappings with given boundary values, holomorphic motions, quasiconformal and quasiregular mappings in higher dimensions, with an exhaustive list of references. The emphasis is in Teichmüller spaces and relations to Kleinian groups. A comprehensive survey on related development is the two-volume handbook edited by R. Kühnau [R. Kühnau (ed.), Handbook of complex analysis: geometric function theory. Volume 1. Amsterdam: North Holland (2002; Zbl 1057.30001); Volume 2. Amsterdam: Elsevier/North Holland (2005; Zbl 1056.30002)]. The second new chapter by M. Shishikura is devoted to the relation between complex dynamics and quasiconformal mappings. The main topics in this chapter include quasiconformal deformation techniques, classification of periodic Fatou components and characterization of rational maps. The final chapter, by J.H. Hubbard, deals with applications of the techniques developed by Ahlfors and Bers in Thurston’s theory of hyperbolic structures on 3-manifolds.

MSC:

30-02 Research exposition (monographs, survey articles) pertaining to functions of a complex variable
30C62 Quasiconformal mappings in the complex plane
30Cxx Geometric function theory
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