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Moduli of families of curves for conformal and quasiconformal mappings. (English) Zbl 0999.30001
Lecture Notes in Mathematics. 1788. Berlin: Springer. ix, 211 p. (2002).
The goal of the author is to present a reasonably selfcontained approach to the Ahlfors-Beurling extremal length method and its various applications to Teichmüller spaces and to the study of univalent functions. It is assumed that the reader is familiar with a graduate level course on complex analysis. In particular, conformal mappings are used extensively. In the beginning of the book the author gives a long review of the literature which he has used as source. Some of the important sources listed by the author are the monographs by J. Jenkins [Trans. Am. Math. Soc. 88, 207-213 (1958; Zbl 0082.06201)], G. V. Kuz’mina [”Moduli of families of curves and quadratic differentials”, Proc. Steklov Inst. Math. 139, 1-231 (1982; Zbl 0491.30013)], M. Ohtsuka [Dirichlet problem, extremal length, and prime ends (1970; Zbl 0197.08404)] and K. Strebel [Quadratic differentials (1984; Zbl 0547.30001)].

MSC:
30-02 Research exposition (monographs, survey articles) pertaining to functions of a complex variable
30C75 Extremal problems for conformal and quasiconformal mappings, other methods
30C55 General theory of univalent and multivalent functions of one complex variable
30C62 Quasiconformal mappings in the complex plane
30F10 Compact Riemann surfaces and uniformization
30F60 Teichmüller theory for Riemann surfaces
30C35 General theory of conformal mappings
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