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Conformal geometry and quasiregular mappings. (English) Zbl 0646.30025
Lecture Notes in Mathematics. 1319. Berlin etc.: Springer-Verlag. xix, 209 p. (1988).
Apart from Yu. G. Reshetnyak’s books, available only in Russian, this is the first book on quasiregular mappings in \(R^ n.\) These mappings form a generalization of plane analytic functions to higher dimensional Euclidean spaces: A continuous mapping f of a domain G into \(R^ n\) is K-quasiregular (QR) if f is ACL\(^ n\) and \(| f'(x)|^ n\leq KJ(x,f)\) a.e. in G.
The book stresses geometric properties of QR-maps. After a good survey on the current state of affairs Chapter I deals with conformal geometry: Möbius transformations, hyperbolic and quasihyperbolic geometry. There are several useful formulas hard to find in the literature. Quasihyperbolic geometry has also been dealt in F. W. Gehring’s lecture notes [F. W. Gehring, Characteristic properties of quasidisks (1982; Zbl 0495.30018)]. Chapter II is devoted to the modulus and capacity. Here the author has mostly collected the known properties of the n-modulus, the main source of information being J. Väisälä’s book on n-dimensional quasiconformal \((=\) homeomorphic QR-) mappings [J. Väisälä: Lectures on n-dimensional quasiconformal mappings (1971; Zbl 0221.30031)]. However, the moduli of Teichmüller and Grötzsch condensers get a careful analysis. At the end of the chapter two conformal invariants, the modulus metric and its dual, are introduced. The author has used these invariants in several occasions to study quasiconformal and QR-mappings, especially their distortion properties.
QR-mappings are introduced in Chapter III. Their topological behavior and the change of the moduli of condensers under QR-maps are stated without proof and the main emphasis is put to the change of the aforementioned conformal invariants. Using these the author develops a rather precise distortion theory: several Hölder-continuity type results are obtained. The last chapter is devoted to the boundary behavior: the author mostly considers quasiconformal mappings since very little is known on the boundary properties of QR-maps. Lindelöf type problems get most attention: When does a quasiconformal mapping of the unit ball of \(R^ n\) into \(R^ n\) have an angular limit at a boundary point? The final, short section is devoted to Dirichlet finite mappings f, i.e. \(\int_{B^ n}| f'(x)|^ n dm(x)<\infty.\)
QR-mappings of B n do not satisfy this condition, but since this is true locally for bounded QR-mappings, it can be used to study uniform continuity of these mappings in the hyperbolic metric.
The book contains plenty of examples and exercises. The bibliography list is extensive. Those interested in the semilocal distortion theory of QR and quasiconformal mappings will find the book most valuable.
Reviewer: O.Martio

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