Lars Ahlfors' Lectures on Quasiconformal Mappings, based on a course he gave at Harvard University in the spring term of 1964, was first published in 1966 and was soon recognized as the classic it was shortly destined to become. These lectures develop the theory of quasiconformal mappings from scratch, give a self-contained treatment of the Beltrami equation, and cover the basic properties of Teichmuller spaces, including the Bers embedding and the Teichmuller curve. It is remarkable how Ahlfors goes straight to the heart of the matter, presenting major results with a minimum set of prerequisites. Many graduate students and other mathematicians have learned the foundations of the theories of quasiconformal mappings and Teichmuller spaces from these lecture notes. This edition includes three new chapters.The first, written by Earle and Kra, describes further developments in the theory of Teichmuller spaces and provides many references to the vast literature on Teichmuller spaces and quasiconformal mappings. The second, by Shishikura, describes how quasiconformal mappings have revitalized the subject of complex dynamics. The third, by Hubbard, illustrates the role of these mappings in Thurston's theory of hyperbolic structures on 3-manifolds. Together, these three new chapters exhibit the continuing vitality and importance of the theory of quasiconformal mappings.
Readership: Graduate students and research mathematicians interested in complex analysis.
Author(s): Lars V. Ahlfors
Series: University Lecture Series 38
Edition: 2
Publisher: American Mathematical Society
Year: 2006
Language: English
Pages: 162
Cover
Lectures on Quasiconformal Mappings, Second Edition
About this Edition
Copyright
2006 by American Mathematical Society
ISBN 0182836447
Contents
Preface
The Ahlfors Lectures
Acknowledgments
Chapter I: Differentiable Quasiconformal Mappings
Introduction
A. The Problem and Defintion of Grötzach
B. Solution of Grötzach's Problem
C. Composed Mappings
D. Extremal Length
E. A Symmetry Principle
Chapter II: The General Definition
A. The Geometric Approach
B. The Analytic Definition
Chapter III: Extremal Geometric Properties
A. Three Extremal Problems
B. Elliptic and Modular Functions
C. Mori's Theorem
D. Quadruplets
Chapter IV: Boundary Correspondence
A. The M-condition
B. The Sufficiency of the M-condition
C. Quasi-isometry
D. Quasiconformal Reflection
E. The Reverse Inequality
Chapter V: The Mapping Theorem
A. Two Integral Operators
B. Solution of the Mapping Problem
C. Dependence on Parameters
D. The Caldero'n-Zygmund Inequality
Chapter VI: Teichmüller Spaces
A. Preliminaries
B. Beltrami Differentials
C. \Delta Open
D. The Infinitesimal Approach
Editors' Notes
The Additional Chapters
Supplement to Ahlfors's Lectures
1. Quasiconformal mappings and their boundary values
1.1 The metric defintion
1.2 Quasiconformal mappngs with given boundary values
1.3 Holomorphic motions and the \lambda-lemma
1.4 More about the Beltramiequation
1.5 Quasiconformal and Quasiregular maapngs in R^n
2. Definitions and basic properties of Teichmüller space
2.1 The Tiechmüller Space T(So)
2.2 T(So) as an orbit space
2.3 The Tiechmüller modular groups
2.4 The compact case: a fiber bandle approach
2.5 The compact case: hyperbolic metrics
2.6 Thurston's compactification of T(So) and classification of diffeomorphisms
2.7 A funtorial approach to T(So)
3. Extremal quasiconformal mappings and Teichmüller's metric
3.1. Extremal quasiconformal mappings and quadratic differentials.
3.2 Tiechmüller's extremal problem
3.3. Geometry of Teichmüller's metric
3.4. Dynamics of Teichmüller's metric
3.5. Bers's extremal problem
3.6. Extremal mappings In the general case.
4. Royden's theorems
4.1. The teichmüller and Kobayashi metrics
4.2. The infinitesmal metric
4.3. Isometries between the spaces Q(So)
5. Weil-Peterson geometry
5.1. The weil-Peterson metric
5.2. Completion of the WP metric and compactification of the moduli space
5.3. Curvature of the WP metric and the Nielson realization problem
5.4. The WP isometry group
6. Finitely generated Klienian groups
6.1. Definitions
6.2. The Ahlfors finiteness theorem
6.3. Eichler cohomology and the Bers area theorem
6.4. The Teichmüller space of a finitely generated Kleinian group
6.5. Quasi-Fuchsian groups and simultaneous uniforrnization.
6.6. Kleinian groups and hyperbolic 3-rnanifolds
References
Complex Dynamics and Quasiconformal Mappings
1. Preliminanes from the classical theory or complex dynamics
2. No-wandering-domain theorem and qc-deformation
3. Classification of periodic Fatou components and Teichmüller spaces
4. Structural stability and \lambda -lemma
5. Quasiconformal surgery
6. Thurston's theorem
7. Quadratic polynomials
8. Renormalization
Acknowledgements
References
Hyperbolic Structures on Three-Manifolds that Fiber over the Circle
1. The Hyperbolization Theorem
2. Limits of Kleinian Groups
3. Outline of the Proof
4. Sketch of Proof of the Double Limit Theorem
5. Sketch of McMullen Rigidity
References
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