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Mathematical Reviews (American Mathematical Society)

Teichmüller theory and applications to geometry, topology, and dynamics, Vol. 1,

by John H Hubbard



The following  review appeared in Mathematical Reviews, Issue 2008k; it is reprinted with permission of the American Mathematical Society. Some references and mathematical statements have been omitted. The complete review is available from MathSciNet.

This volume is the first part of a book whose final purpose is to give complete and self-contained proofs of the following four theorems shown by W. Thurston between 1970-1980:

(1) The classification of homeomorphisms of a surface.

(2) The topological classification of rational maps.

(3) A hyperbolization theorem for 3-manifolds that fiber over the circle.

(4) A hyperbolization theorem for Haken 3-manifolds.

    Although these theorems seem to be unrelated, the proofs of the theorems have a common stream, that is, Teichmüller theory.  In the volume  under review, the author presents Teichmüller theory in order to understand the proofs of Thurston's theorems, but as a result, the book becomes an excellent textbook on Teichmüller theory.

    This volume begins with a foreword by Thurston in which the importance and the beauty of Teichmüller theory are discussed from the point of view of his mathematical experiences and his philosophy on mathematics.  The foreword itself is worth reading.

    In Chapter 1, the uniformization theorem of Riemann surfaces is the central object.  It contains fundamental and classical topics, potential theory, Riemann surfaces, etc. But the author explains them from various perspectives. For example, in stating Rado's theorem, which asserts that every Riemann surface is second countable, the author presents a two-dimensional complex manifold that is not second countable and this example should allow the reader to get a feeling for how special is the case of one-dimensional complex manifolds, i.e., of Riemann surfaces.

    Hyperbolic geometry on planar domains is discussed in Chapter 2.  After discussing fundamental results on hyperbolic geometry and introducing the (1/d)-metric traditionally known as the quasi-hyperbolic metric, the author defines the geodesic curvature for a curve, which measures the derivation of the curve from a geodesic.  The geodesic curvature is defined at each point of the curve and controls the curve on the hyperbolic plane.  For example, it is shown that a horocycle is a curve with constant geodesic curvature 1.  This chapter ends with a discussion of hyperbolic trigonometry.

    The uniformization theorem in Chapter 1 implies that almost Riemann surfaces admit the hyperbolic metric.  In Chapter 3, the hyperbolic geometry on Riemann surfaces is discussed, in particular fundamental notions and results (e.g. the Poincaré polygon theorem) on Fuchsian groups.  The author carefully treats noncompact Riemann surfaces, possible, topologically infinite.

    The discussion on quasiconformal mappings in Chapter 4 is very important but might be one of the most difficult parts of this volume.  The reader is required to have the patience to learn several theorems in the field of real analysis.

    Two definitions—an analytic definition and a geometric one—are introduced.  The geometric definition here is a little bit different from the standard one because it is defined by using "skew" for later use, while it is normally defined by the moduli of quadrilaterals.  Skew is defined for a triple (a1, a2, a3) of points in C by

skew(a1, a2, a3)=supi,j,k  |ai-aj|/ |ai-ak|

where the supremum is taken over all triples of distinct {i,j,k}.  It is shown that these two definitions are the same in Theorem 4.5.4, whose proof is difficult.

    The mapping theorem shown in this chapter is one of the most important results in the theory of quasiconformal mappings. This theorem, which is a great achievement of L.. Ahlfors and L. Bers ... , guarantees us that the Beltrami equation ... has a solution  on C ...  It is  essential to understanding the complex structure of Teichmüller spaces.

    In the last part of this chapter, an extension theorem is shown.  The theorem says that the quasi-symmetry of a function on the real  line is a necessary and sufficient condition for the function to have a quasiconformal extension to the upper half plane and it is known as the Beurling-Ahlfors theorem.... In the proof here, the author uses the geometric definition of quasiconformal mappings by "skew".  So, the proof is geometric and different from the original one.

  Chapter 5 is set up as preliminary to the subsequent treatment of Teichmüller theory.  It presents a number of important results.  They are important for Teichmüller theory but are also interesting in their own right. Especially, the Douady-Earle extension theorem... and Slodkowski's theorem... are recent developments and highly interesting.

    The Douady-Earle extension theorem says that for a quasisymmetric homeomorphism of the unit circle, there exists a quasiconformal extension Φ(f) called the Douady-Earle extension to the unit disk such that

Φ(γ1 circ f circ γ2) = γ1 circ Φ(f) circ γ2

holds for every holomorphic automorphism  γ,γ2 of the  unit disk.  The main idea of the proof is the conformal barycenter which is the barycenter of f in the sense of harmonic measure.  The origin of the conformal barycenter comes from a result of P. Tukia...

Slodkowski's theorem is another striking result.  It is an extension of the λ lemma (Theorem 5.2.3) shown by R. Mañé, P. Sad and D. P. Sullivan... The λ lemma itself is also surprising because it shows that quasiconformality on a set of the plane is equivalent to two elementary properties, that is,  injectivity and analyticity on the set. Briefly, Slodkowski's theorem states that the properties of injectivity and analyticity on a subset of the plane extend to the whole plane.  ...

    In Chapter 6, Teichmüller spaces are introduced.  As in Chapter 3, the author explains the theory in full generality, that is, he deals with Riemann surfaces, possibly topologically infinite.  Thus, finite-dimensional Teichmüller spaces, which are the Teichmüller spaces of Riemann surfaces of finite type, are treated as a special case of infinite-dimensional Teichmüller spaces.

    The main topic treated in this chapter is the so-called Ahlfors-Bers theory of Teichmüller spaces.  The analytic structure of Teichmüller spaces is introduced and some basic properties are presented.  The reader may find a number of important subjectsmapping class group, moduli space, tangent and cotangent spaces of Teichmüller space, the universal property of Teichmüller space, etc.,emerging from the mathematical essense.  Hence, this chapter is the heart of the volume.

    Here the reader can see that the results shown in Chapter 5 play important roles.  For example, the contractibility of Teichmüller spaces is proved by the Douady-Earle extension and the equality of the Kobayashi metric and the Teichmüller metric is shown by Slodkowski's theorem.

    The last chapter, Chapter 7, is devoted to the geometry of finite-dimensional Teichmüller spaces.  Thus, this chapter is written with a different geometric flavor.  Two theorems, Royden's theorem on automorphisms of Teichmüller spaces and Wolpert's formula for the Weil-Petersson metric, are proved.

    At the end of the volume, 10 appendices and one glossary may be found.  Each appendix is a brief survey of a theory used in the book and the glossary gives explanations of  terms that are not defined in the book. These are very helpful for the reader.

    In conclusion, the reviewer is convinced that this well-written book will be very useful not only for people who are not familiar with Teichmüller theory, but also for the experts.  This is because the reader is offered everywhere in the volume  the deep insights of the author, who looks at the topics developed from a new vantage point.  Throughout this volume, the author explains in his own words and tries to expose new ideas in each subject treated, using brief comments or historical notes.  This attitude of the author will certain stimulate the reader's mind in mathematics.Hiroshige Shiga (J-TOKYTE; Meguro)




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