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Quasiconformal Teichmüller Theory
Frederick P. Gardiner, Brooklyn College, CUNY, NY, and Nikola Lakic, Lehman College, CUNY, Bronx, NY
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Mathematical Surveys and Monographs
2000; 372 pp; hardcover
Volume: 76
ISBN-10: 0-8218-1983-6
ISBN-13: 978-0-8218-1983-8
List Price: US$103
Member Price: US$82.40
Order Code: SURV/76
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The Teichmüller space \(T(X)\) is the space of marked conformal structures on a given quasiconformal surface \(X\). This volume uses quasiconformal mapping to give a unified and up-to-date treatment of \(T(X)\). Emphasis is placed on parts of the theory applicable to noncompact surfaces and to surfaces possibly of infinite analytic type.

The book provides a treatment of deformations of complex structures on infinite Riemann surfaces and gives background for further research in many areas. These include applications to fractal geometry, to three-dimensional manifolds through its relationship to Kleinian groups, and to one-dimensional dynamics through its relationship to quasisymmetric mappings. Many research problems in the application of function theory to geometry and dynamics are suggested.

Readership

Graduate students, research and applied mathematicians and physicists interested in functions of a complex variable, several complex variables and analytic spaces, particularly mathematical foundations of deformation theory.

Reviews

"The goal of this book is stated in the excellent preface: `to provide background for applications of Teichmüller theory to dynamical systems' ... The extensive bibliography is instructive ... a very interesting book ... The treatment is clear and methodical ... probably be possible to read this as an introductory text, yet there is much that is relatively new, innovative, and perhaps, percipient."

-- Bulletin of the LMS

"[The authors] have produced a formidable treatise on the modern theories of quasiconformal mappings, Riemann surfaces and Teichmüller spaces. They have gathered, into a unified exposition, results which ... have not previously been found in book form ... Many of the approaches and results are new, others are more detailed than can be found elsewhere ... this monograph is now the standard reference on two-dimensional quasiconformal mappings and Teichmüller theory and is likely to remain so for many years."

-- Mathematical Reviews

"Brings to the literature the current state of the analytic theory of Teichmüuller spaces ... a thorough report on the latest developments ... a solid exposition of most of the classical foundations ... [this book] is a real service to the community ... an important addition to the literature ... topics are discussed very thoroughly; some shed a lot of new light on the material."

-- Bulletin of the AMS

Table of Contents

  • Quasiconformal mapping
  • Riemann surfaces
  • Quadratic differentials, Part I
  • Quadratic differentials, Part II
  • Teichmüller equivalence
  • The Bers embedding
  • Kobayashi's metric on Teichmüller space
  • Isomorphisms and automorphisms
  • Teichmüller uniqueness
  • The mapping class group
  • Jenkins-Strebel differentials
  • Measured foliations
  • Obstacle problems
  • Asymptotic Teichmüller space
  • Asymptotically extremal maps
  • Universal Teichmüller space
  • Substantial boundary points
  • Earthquake mappings
  • Bibliography
  • Index
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