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Quasiconformal Teichmüller theory. (English) Zbl 0949.30002
Mathematical Surveys and Monographs. 76. Providence, RI: American Mathematical Society (AMS). xix, 372 p. (2000).
Teichmüller space is a universal classification space for complex structures on a surface of given quasiconformal type. The Teichmüller theory originally started with the Riemann module problem. However, it has been applied to many other fields. The purpose of the present book is to provide background for applications of the Teichmüller theory to dynamical systems and in particular to iterations of rational maps and conformal dynamics, to Kleinian groups and three-dimensional manifolds, to Fuchsian groups and Riemann surfaces, and to one-dimensional dynamics. None of these topics is dealt with in this book. But it provides the Teichmüller theory and new results that are connected with these topics. This book gives a good exposition of the main known theorems regardless of dimensionality, emphasizing the theory and techniques that apply generally. This book contains many new developments of the study of the Teichmüller theory in the recent years, for instance, the study of asymptotic Teichmüller spaces. Many of the results concerning asymptotic Teichmüller spaces presented in the text represent the joint works of the authors with Clifford Earle. In Chapter 9, an important progress on the extremal quasiconformal mapping by V. Božin, N. Lakic, V. Marković and M. Mateljević [“Unique extremality”, J. Anal. Math. 75, 299-338 (1998; Zbl 0929.30017)] is presented.
(Remark of the editor: The authors of the book have notified to Zentralblatt that they had intended to include in the bibliography the reference to the paper “Unique extremality” for the central importance to Chapter 9.) In the later chapters, the authors explore a variety of topics including measured foliations, height mappings, a generalization of classical slit mapping theorems, and a construction of earthquakes based in the idea of applying a limiting process to finite earthquakes. At the end of each chapter, exercises and sometimes open problems are provided.
This book contains 18 chapters: 1. Quasiconformal mappings; 2. Riemann surfaces; 3. Quadratic differentials (part I); 4. Quadratic differentials (part II); 5. Teichmüller equivalence; 6. The Bers embedding; 7. Kobayashi’s metric on Teichmüller space; 8. Isomorphisms and automorphisms; 9. Teichmüller uniqueness; 10. The mapping class group; 11. Jenkins-Strebel differentials; 12. Measured foliations; 13. Obstacle problems; 14. Asymptotic Teichmüller spaces; 15. Asymptotically extremal maps; 16. Universal Teichmüller space; 17. Substantial boundary points; 18. Earthquake mappings.
Reviewer: Li Zhong (Kowloon)

MSC:
30-02 Research exposition (monographs, survey articles) pertaining to functions of a complex variable
32-02 Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces
30F60 Teichmüller theory for Riemann surfaces
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
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