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Teichmüller Theory and Applications to Geometry, Topology, and Dynamics by John Hubbard

Volume I: Teichmüller Theory

Foreword by Clifford Earle:


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.


ISBN-13: 978-0-9715766-2-9
ISBN-10: 0-9715766-2-9
459 pages, Hardcover, $85

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March 1, 2022