Teichmüller theory and applications to geometry, topology, and dynamics. Volume 1: Teichmüller theory. With contributions by Adrien Douady, William Dunbar, and Roland Roeder, Sylvain Bonnot, David Brown, Allen Hatcher, Chris Hruska, Sudeb Mitra. (English) Zbl 1102.30001

Hubbard, John Hamal

Ithaca, NY: Matrix Editions (ISBN 0-9715766-2-9/hbk). xx, 459 p. (2006).

The subject of Teichmüller theory is the study of moduli of Riemann surfaces. This subject has interconnections and applications in several areas in mathematics which include, besides complex analysis and hyperbolic geometry, the theory of representation of discrete groups theory, algebraic geometry, low-dimensional manifolds, symplectic geometry, dynamical systems, number theory, topological quantum field theory, string theory (and there are others).
Teichmüller theory was founded by Teichmüller in the 1930s, with techniques that come from geometric complex analysis, and it was developed on these grounds, during the three decades that followed Teichmüller’s death, by Ahlfors, Bers and others. In the 1970s, Thurston introduced in that theory beautiful new techniques of hyperbolic geometry. His approach highlighted Teichmüller theory as a central object in the field of low-dimensional topology.
The book under review is the first volume in a series of two. It gives a comprehensive treatment of the foundations of Teichmüller theory with its two aspects, complex analysis and hyperbolic geometry. The second part in the series will cover four applications of Teichmüller theory to low-dimensional topology which were discovered by Thurston, namely, the classification of homeomorphisms of surfaces, the topological classification of rational maps, the hyperbolization theorem for 3-manifolds that fiber over the circle and the hyperbolization of Haken 3-manifolds.
Volume 1 of Hubbard’s book covers all the important classical aspects of Teichmüller’s theory. It includes beautiful treatments of classical theorems such as Poincaré’s uniformization theorem, a development of the theory of quasiconformal maps and of the solution to Beltrami’s partial differential equation, a study of the space of quadratic differentials, of Schwarzian derivatives and of boundary values of extremal maps, and a beautiful treatment of models of the hyperbolic plane, hyperbolic trigonometry, fundamental domains, the Fenchel-Nielsen parametrization of hyperbolic surfaces, and the Weil-Petersson geometry of Teichmüller space.
The book also includes treatments of more specialized topics that one can hardly find (and, for some of them, cannot find at all) in other textbooks on Teichmüller theory. We mention here Mumford’s compactness theorem, the Douady-Earle barycentric extension theory, the theory of holomorphic motions, Slodowski’s theorem, Royden’s results on the automorphisms of Teichmüller spaces, the author’s work on sections of the universal Teichmüller curve and Wolpert’s formula for the Weil-Petersson symplectic form.
This is an invaluable book. It treats a wonderful subject, and it is written by a great mathematician. It is now an essential reference for every student and every researcher in the field.

Reviewer: Athanase Papadopoulos (Strasbourg)

MSC:

30-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to functions of a complex variable
30F10	Compact Riemann surfaces and uniformization
30F30	Differentials on Riemann surfaces
30F60	Teichmüller theory for Riemann surfaces
30C35	General theory of conformal mappings
30C70	Extremal problems for conformal and quasiconformal mappings, variational methods
32G15	Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
57M50	General geometric structures on low-dimensional manifolds

Keywords:

Teichmüller space; hyperbolic plane; quasiconformal map; Weil-Petersson metric; Teichmüller metric; Poincaré metric; modulus; Fenchel-Nielsen coordinates; Beltrami differential; quadratic differential; extremal map; conformal map; Slodowski’s theorem; Fuchsian group; Douady-Earle extension