Contents

Preface vii

The Ahlfors Lectures

Acknowledgments 3

Chapter I. Differentiable Quasiconformal Mappings 5

A. The Problem and Definition of Gr¨ otzsch 5

B. Solution of Gr¨ otzsch’s Problem 8

C. Composed Mappings 8

D. Extremal Length 10

E. A Symmetry Principle 13

F. Dirichlet Integrals 13

Chapter II. The General Definition 15

A. The Geometric Approach 15

B. The Analytic Definition 16

Chapter III. Extremal Geometric Properties 23

A. Three Extremal Problems 23

B. Elliptic and Modular Functions 25

C. Mori’s Theorem 30

D. Quadruplets 34

Chapter IV. Boundary Correspondence 39

A. The M-condition 39

B. The Suﬃciency of the M-condition 42

C. Quasi-isometry 45

D. Quasiconformal Reflection 45

E. The Reverse Inequality 49

Chapter V. The Mapping Theorem 51

A. Two Integral Operators 51

B. Solution of the Mapping Problem 54

C. Dependence on Parameters 58

D. The Calder´ on-Zygmund Inequality 62

Chapter VI. Teichm¨ uller Spaces 67

A. Preliminaries 67

B. Beltrami Differentials 69

C. Δ Is Open 73

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