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Some years
ago I had the pleasure of attending a Cornell graduate course,
given by my friend John Hubbard, about three theorems of Bill
Thurston. As a life-long fan of Teichmüller spaces, I was
delighted to see how each of these theorems could be formulated
as a nontrivial statement about certain geometrically defined
mappings of a Teichmüller space into itself.
At that
time, Hubbard was already planning a book about Thurston's construction
of these mappings and the analysis of their properties. However,
the ideal reader of that book would need to understand a great
deal about the geometry of Riemann surfaces and their Teichmüller
spaces. Hubbard soon realized that the required background material
could fill a book all by
itself, and this is it.
As his preface
indicates, this book is remarkably self-contained, with thorough
treatments of the uniformization theorem, the geometry of hyperbolic
surfaces, and the properties of quasiconformal mappings that
are needed for its development of Teichmüller theory. These
features will be particular helpful to the topologically inclined
reader mentioned in Hubbard's preface.
The book
also has much to offer to readers who are already familiar with
advanced complex analysis and Teichmüller spaces. There
is novelty even in the discussions of the uniformization theorem
and the geometry of quasiconformal mappings. No other book proves
both Royden's theorem about automorphisms of Teichmüller
spaces and Slodkowski's theorem about holomorphic motions. But
the most important novelty is provided by the author's taste
for hands-on geometric constructions and the enthusiasm with
which he presents them.
This book
will whet the appetite for further reading about quasiconformal
mappings and Teichmüller spaces and their applications.
The second volume, and the books and papers cited in the bibliographies
of both volumes of this work, provide more information about
these very active areas. Even more references can be found in
the supplementary chapters to the 2006 AMS edition of Ahlfors's
classic Lectures on Quasiconformal Mappings, which we
highly recommend.
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