##
**Univalent functions and Teichmüller spaces.**
*(English)*
Zbl 0606.30001

Graduate Texts in Mathematics, 109. New York etc.: Springer-Verlag. XII, 257 p. DM 124.00 (1987).

Teichmüller theory concerns the different conformal structures that may be placed on a surface of a given genus. If we assume that the surface is compact and has genus \(p>1\), then the moduli space of conformal structures (Teichmüller space) is homeomorphic to \(R^{6p-6}\), where R is the real line. One road to this theorem is to employ quasiconformal mappings following Bers. As the author states in his introduction:

”This method is more general, in that it can be applied to non-compact Riemann surfaces....quadratic differentials are (here) Schwarzian derivatives of conformal extensions of quasiconformal mappings considered on the universal covering surface, the extensions being obtained by use of the Beltrami differential equation.”

”The development of the theory of Teichmüller spaces along these lines gives rise to several interesting problems which belong to the classical theory of univalent analytic functions. Consequently, in the early seventies a special branch of the theory of univalent functions, often studied without any connection to Riemann surfaces, began to take shape.”

”The interplay between the theory of univalent functions and the theory of Teichmüller spaces is the main theme of his monograph. We do give a proof of the... classical... theorems of Teichmüller and discuss their consequences. But the emphasis is on the study of the repercussions of Bers’ method, with attention both to univalent functions and to Teichmüller spaces.”

This book painstakingly develops the tools necessary to achieve its stated purpose. The first chapter deals with quasiconformal mappings, the second with univalent functions - specifically those aspects that developed in response to Teichmüller theory, as mentioned in the author’s introduction. These chapters are replete with explicit estimates for geometric and analytic quantities having to do with such crucial items as quasi-disks, Schwarzian derivatives, the distance between simply connected domains and quasiconformal extensions. The development is patient, wonderfully written and dotted with nice examples.

Chapter three is entitled Universal Teichmüller Space, T(1), the simplest example of Bers’ construction and the one that contains the Teichmüller space of any Riemann surface as a subset. Again everything is beautifully explicit; the geometric properties of T(1) are developed - Teichmüller distance, geodesics, completeness. The space is shown contractible and imbeddible in the space of Schwarzian derivatives. So far the book is largely contained in the complex plane, and apart from some results on quasiconformal mappings taken from the author’s well known text with K. I. Virtanen, Quasiconformal mappings in the plane (1973; Zbl 0267.30016). In this chapter, the connection between Teichmüller theory and quasiconformal mappings is exposed.

The fourth chapter deals with Riemann surfaces. It has two nice sections on trajectories of quadratic differentials and geodesics of same. These are very fine sections, not typical in a book on Riemann surfaces. [This chapter relies on some results from the books ”Riemann surfaces” of L. V. Ahlfors and L. Sario (1964; Zbl 0196.338).]

In the last chapter everything comes together in discussion of the Teichmüller spaces of Riemann surfaces. Though the book culminates in the theorem stated in the first paragraph, the complex structure of Teichmüller space is revealed, as are the Bers spaces of holomorphic differentials and their connection to Poincaré theta series. Much of the Teichmüller theory is not dealt with in this chapter, in particular the work of Thurston, Douady-Earle, Masur Wolpert - but there are indications, sign posts, toward other work at several points of this chapter.

The level of workmanship displayed in this book is very rarely seen in these days of camera-ready copy, rush to print mentality. The book is the measured thoughtful product of an elder statesman in the field. Such things are a joy!

”This method is more general, in that it can be applied to non-compact Riemann surfaces....quadratic differentials are (here) Schwarzian derivatives of conformal extensions of quasiconformal mappings considered on the universal covering surface, the extensions being obtained by use of the Beltrami differential equation.”

”The development of the theory of Teichmüller spaces along these lines gives rise to several interesting problems which belong to the classical theory of univalent analytic functions. Consequently, in the early seventies a special branch of the theory of univalent functions, often studied without any connection to Riemann surfaces, began to take shape.”

”The interplay between the theory of univalent functions and the theory of Teichmüller spaces is the main theme of his monograph. We do give a proof of the... classical... theorems of Teichmüller and discuss their consequences. But the emphasis is on the study of the repercussions of Bers’ method, with attention both to univalent functions and to Teichmüller spaces.”

This book painstakingly develops the tools necessary to achieve its stated purpose. The first chapter deals with quasiconformal mappings, the second with univalent functions - specifically those aspects that developed in response to Teichmüller theory, as mentioned in the author’s introduction. These chapters are replete with explicit estimates for geometric and analytic quantities having to do with such crucial items as quasi-disks, Schwarzian derivatives, the distance between simply connected domains and quasiconformal extensions. The development is patient, wonderfully written and dotted with nice examples.

Chapter three is entitled Universal Teichmüller Space, T(1), the simplest example of Bers’ construction and the one that contains the Teichmüller space of any Riemann surface as a subset. Again everything is beautifully explicit; the geometric properties of T(1) are developed - Teichmüller distance, geodesics, completeness. The space is shown contractible and imbeddible in the space of Schwarzian derivatives. So far the book is largely contained in the complex plane, and apart from some results on quasiconformal mappings taken from the author’s well known text with K. I. Virtanen, Quasiconformal mappings in the plane (1973; Zbl 0267.30016). In this chapter, the connection between Teichmüller theory and quasiconformal mappings is exposed.

The fourth chapter deals with Riemann surfaces. It has two nice sections on trajectories of quadratic differentials and geodesics of same. These are very fine sections, not typical in a book on Riemann surfaces. [This chapter relies on some results from the books ”Riemann surfaces” of L. V. Ahlfors and L. Sario (1964; Zbl 0196.338).]

In the last chapter everything comes together in discussion of the Teichmüller spaces of Riemann surfaces. Though the book culminates in the theorem stated in the first paragraph, the complex structure of Teichmüller space is revealed, as are the Bers spaces of holomorphic differentials and their connection to Poincaré theta series. Much of the Teichmüller theory is not dealt with in this chapter, in particular the work of Thurston, Douady-Earle, Masur Wolpert - but there are indications, sign posts, toward other work at several points of this chapter.

The level of workmanship displayed in this book is very rarely seen in these days of camera-ready copy, rush to print mentality. The book is the measured thoughtful product of an elder statesman in the field. Such things are a joy!

Reviewer: M.Sheingorn

### MSC:

30-02 | Research exposition (monographs, survey articles) pertaining to functions of a complex variable |

32G15 | Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) |

30F30 | Differentials on Riemann surfaces |

30F35 | Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization) |