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**Harmonic mappings in the plane.**
*(English)*
Zbl 1055.31001

Cambridge Tracts in Mathematics 156. Cambridge: Cambridge University Press (ISBN 0-521-64121-7/hbk). xii, 212 p. (2004).

Harmonic mappings in the plane are univalent functions of a complex variable. Conformal mappings are a special case where the real and imaginary parts are conjugate harmonic functions. Harmonic mappings were studied classically in differential geometry because they provide isothermal parameters for minimal surfaces. More recently, they have been studied by complex analysts. An extensive theory has resulted from the efforts of many mathematicians. It has turned out that numerous results of conformal mapping theory have their counterparts or analogues in this more general setup. Some of these topics are growth and distortion theorems, coefficient problems, and mapping theorems.

This monograph is the first one dedicated to this topic. The exposition is self-contained, only familiarity with complex analysis is required. The usual methods of complex analysis work here, occasionally reinforced with techniques from the theory of minimal surfaces or quasiconformal mappings. Often the results are sharp and geometrically attractive. Some special classes of univalent functions, such as convex functions, have their counterparts here. In this way, classical results of complex analysis find many new applications. As this beautiful monograph shows, this new area has already reached a mature state, but many challenging problems still remain.

This monograph is the first one dedicated to this topic. The exposition is self-contained, only familiarity with complex analysis is required. The usual methods of complex analysis work here, occasionally reinforced with techniques from the theory of minimal surfaces or quasiconformal mappings. Often the results are sharp and geometrically attractive. Some special classes of univalent functions, such as convex functions, have their counterparts here. In this way, classical results of complex analysis find many new applications. As this beautiful monograph shows, this new area has already reached a mature state, but many challenging problems still remain.

Reviewer: Matti Vuorinen (Turku)

### MSC:

31-02 | Research exposition (monographs, survey articles) pertaining to potential theory |

53C43 | Differential geometric aspects of harmonic maps |

58E20 | Harmonic maps, etc. |

30C35 | General theory of conformal mappings |

31A05 | Harmonic, subharmonic, superharmonic functions in two dimensions |

30C45 | Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.) |