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Review
from Zentralblatt MATH
Teichmüller
Theory and
Applications to
Geometry, Topology, and Dynamics
Volume I:
Teichmüller
Theory




"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 inslude, besides complex analysis and
hyperbolic geometry, the theory of representation
of discrete
groups theory, algebraic geometry, lowdimensional
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 filed of lowdimensional 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 lowdimensional topology which were
discovered by Thurston, namely, the classification
of homeomorphisms of surfaces,
the topological classification of rational maps, the
hyperbolization theorem for 3manifolds that fiber over the circle and
the hyperbolization of Haken 3manifolds.
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
FenchelNielsen parametrization of hyperbolic surfaces, and the
WeilPetersson 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 DouadyEarle barycentric
extension theory, the theory of holomorphic motions, Slodkowski's
theorem, Royden's results on the auto morphisms of
Teichmüller spaces,
the author's work on sections of the universal
Teichmüller curve and Wolpert's formula for the WeilPetersson
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." — Athanase
Papadopoulos (Zentralblatt MATH 1102.30001)

