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 selfcontained proofs of the following four theorems shown
by W. Thurston between 19701980:
(1) The
classification of homeomorphisms of a surface.
(2) The
topological classification of rational maps.
(3) A
hyperbolization theorem for 3manifolds that fiber over the circle.
(4) A
hyperbolization theorem for Haken 3manifolds.
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 twodimensional 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 onedimensional 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 quasihyperbolic 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 (a_{1},
a_{2},
a_{3})
of points in C by
skew(a_{1},
a_{2},
a_{3})=sup_{i,j,k }
a_{i}a_{j}/ a_{i}a_{k}
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 f 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 quasisymmetry 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 BeurlingAhlfors 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 DouadyEarle extension theorem...
and
Slodkowski's theorem... are recent developments and highly interesting.
The DouadyEarle extension theorem
says that for a quasisymmetric homeomorphism f of the
unit circle, there exists a quasiconformal extension Φ(f) called the
DouadyEarle extension to the unit disk such that
Φ(γ_{1}
circ f
circ γ_{2})
= γ_{1} circ Φ(f)
circ γ_{2}
holds for every holomorphic
automorphism γ_{1 },γ_{2} 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, finitedimensional Teichmüller spaces, which
are the
Teichmüller spaces of Riemann surfaces of finite type, are
treated
as a special case of infinitedimensional Teichmüller spaces.
The main topic treated in this chapter is the socalled AhlforsBers
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 subjects—mapping 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 DouadyEarle 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
finitedimensional 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 WeilPetersson 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 wellwritten 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
(JTOKYTE; Meguro)
To order
(within the United States)
