Univalent Functions and Teichmüller Spaces

ISBN-13: 9781461386544 / Angielski / Miękka / 2011 / 260 str.

O. Lehto

Opis

This monograph grew out of the notes relating to the lecture courses that I gave at the University of Helsinki from 1977 to 1979, at the Eidgenossische Technische Hochschule Zurich in 1980, and at the University of Minnesota in 1982. The book presumably would never have been written without Fred Gehring's continuous encouragement. Thanks to the arrangements made by Edgar Reich and David Storvick, I was able to spend the fall term of 1982 in Minneapolis and do a good part of the writing there. Back in Finland, other commitments delayed the completion of the text. At the final stages of preparing the manuscript, I was assisted first by Mika Seppala and then by Jouni Luukkainen, who both had a grant from the Academy of Finland. I am greatly indebted to them for the improvements they made in the text. I also received valuable advice and criticism from Kari Astala, Richard Fehlmann, Barbara Flinn, Fred Gehring, Pentti Jarvi, Irwin Kra, Matti Lehtinen, I1ppo Louhivaara, Bruce Palka, Kurt Strebel, Kalevi Suominen, Pekka Tukia and Kalle Virtanen. To all of them I would like to express my gratitude. Raili Pauninsalo deserves special thanks for her patience and great care in typing the manuscript. Finally, I thank the editors for accepting my text in Springer-Verlag's well known series. Helsinki, Finland June 1986 Olli Lehto Contents Preface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v Introduction ............................................... .

Spis treści

I Quasiconformal Mappings.- to Chapter I.- 1. Conformal Invariants.- 1.1 Hyperbolic metric.- 1.2 Module of a quadrilateral.- 1.3 Length-area method.- 1.4 Rengel’s inequality.- 1.5 Module of a ring domain.- 1.6 Module of a path family.- 2. Geometric Definition of Quasiconformal Mappings.- 2.1 Definitions of quasiconformality.- 2.2 Normal families of quasiconformal mappings.- 2.3 Compactness of quasiconformal mappings.- 2.4 A distortion function.- 2.5 Circular distortion.- 3. Analytic Definition of Quasiconformal Mappings.- 3.1 Dilatation quotient.- 3.2 Quasiconformal diffeomorphisms.- 3.3 Absolute continuity and differentiability.- 3.4 Generalized derivatives.- 3.5 Analytic characterization of quasiconformality.- 4. Beltrami Differential Equation.- 4.1 Complex dilatation.- 4.2 Quasiconformal mappings and the Beltrami equation.- 4.3 Singular integrals.- 4.4 Representation of quasiconformal mappings.- 4.5 Existence theorem.- 4.6 Convergence of complex dilatations.- 4.7 Decomposition of quasiconformal mappings.- 5. The Boundary Value Problem.- 5.1 Boundary function of a quasiconformal mapping.- 5.2 Quasisymmetric functions.- 5.3 Solution of the boundary value problem.- 5.4 Composition of Beurling-Ahlfors extensions.- 5.5 Quasi-isometry.- 5.6 Smoothness of solutions.- 5.7 Extremal solutions.- 6. Quasidiscs.- 6.1 Quasicircles.- 6.2 Quasiconformal reflections.- 6.3 Uniform domains.- 6.4 Linear local connectivity.- 6.5 Arc condition.- 6.6 Conjugate quadrilaterals.- 6.7 Characterizations of quasidiscs.- II Univalent Functions.- to Chapter II.- 1. Schwarzian Derivative.- 1.1 Definition and transformation rules.- 1.2 Existence and uniqueness.- 1.3 Norm of the Schwarzian derivative.- 1.4 Convergence of Schwarzian derivatives.- 1.5 Area theorem.- 1.6 Conformal mappings of a disc.- 2. Distance between Simply Connected Domains.- 2.1 Distance from a disc.- 2.2 Distance function and coefficient problems.- 2.3 Boundary rotation.- 2.4 Domains of bounded boundary rotation.- 2.5 Upper estimate for the Schwarzian derivative.- 2.6 Outer radius of univalence.- 2.7 Distance between arbitrary domains.- 3. Conformal Mappings with Quasiconformal Extensions.- 3.1 Deviation from Möbius transformations.- 3.2 Dependence of a mapping on its complex dilatation.- 3.3 Schwarzian derivatives and complex dilatations.- 3.4 Asymptotic estimates.- 3.5 Majorant principle.- 3.6 Coefficient estimates.- 4. Univalence and Quasiconformal Extensibility of Meromorphic Functions.- 4.1 Quasiconformal reflections under Möbius transformations.- 4.2 Quasiconformal extension of Conformal mappings.- 4.3 Exhaustion by quasidiscs.- 4.4 Definition of Schwarzian domains.- 4.5 Domains not linearly locally connected.- 4.6 Schwarzian domains and quasidiscs.- 5. Functions Univalent in a Disc.- 5.1 Quasiconformal extension to the complement of a disc.- 5.2 Real analytic solutions of the boundary value problem.- 5.3 Criterion for univalence.- 5.4 Parallel strips.- 5.5 Continuous extension.- 5.6 Image of discs.- 5.7 Homeomorphic extension.- III Universal Teichmüller Space.- to Chapter III.- 1. Models of the Universal Teichmüller Space.- 1.1 Equivalent quasiconformal mappings.- 1.2 Group structures.- 1.3 Normalized Conformal mappings.- 1.4 Sewing problem.- 1.5 Normalized quasidiscs.- 2. Metric of the Universal Teichmüller Space.- 2.1 Definition of the Teichmüller distance.- 2.2 Teichmüller distance and complex dilatation.- 2.3 Geodesics for the Teichmüller metric.- 2.4 Completeness of the universal Teichmüller space.- 3. Space of Quasisymmetric Functions.- 3.1 Distance between quasisymmetric functions.- 3.2 Existence of a section.- 3.3 Contractibility of the universal Teichmüller space.- 3.4 Incompatibility of the group structure with the metric.- 4. Space of Schwarzian Derivatives.- 4.1 Mapping into the space of Schwarzian derivatives.- 4.2 Comparison of distances.- 4.3 Imbedding of the universal Teichmüller space.- 4.4 Schwarzian derivatives of univalent functions.- 4.5 Univalent functions and the universal Teichmüller space.- 4.6 Closure of the universal Teichmüller space.- 5. Inner Radius of Univalence.- 5.1 Definition of the inner radius of univalence.- 5.2 Isomorphic Teichmüller spaces.- 5.3 Inner radius and quasiconformal extensions.- 5.4 Inner radius and quasiconformal reflections.- 5.5 Inner radius of sectors.- 5.6 Inner radius of ellipses and polygons.- 5.7 General estimates for the inner radius.- IV Riemann Surfaces.- to Chapter IV.- 1. Manifolds and Their Structures.- 1.1 Real manifolds.- 1.2 Complex analytic manifolds.- 1.3 Border of a surface.- 1.4 Differentials on Riemann surfaces.- 1.5 Isothermal coordinates.- 1.6 Riemann surfaces and quasiconformal mappings.- 2. Topology of Covering Surfaces.- 2.1 Lifting of paths.- 2.2 Covering surfaces and the fundamental group.- 2.3 Branched covering surfaces.- 2.4 Covering groups.- 2.5 Properly discontinuous groups.- 3. Uniformization of Riemann Surfaces.- 3.1 Lifted and projected Conformal structures.- 3.2 Riemann mapping theorem.- 3.3 Representation of Riemann surfaces.- 3.4 Lifting of continuous mappings.- 3.5 Homotopic mappings.- 3.6 Lifting of differentials.- 4. Groups of Möbius Transformations.- 4.1 Covering groups acting on the plane.- 4.2 Fuchsian groups.- 4.3 Elementary groups.- 4.4 Kleinian groups.- 4.5 Structure of the limit set.- 4.6 Invariant domains.- 5. Compact Riemann Surfaces.- 5.1 Covering groups over compact surfaces.- 5.2 Genus of a compact surface.- 5.3 Function theory on compact Riemann surfaces.- 5.4 Divisors on compact surfaces.- 5.5 Riemann-Roch theorem.- 6. Trajectories of Quadratic Differentials.- 6.1 Natural parameters.- 6.2 Straight lines and trajectories.- 6.3 Orientation of trajectories.- 6.4 Trajectories in the large.- 6.5 Periodic trajectories.- 6.6 Non-periodic trajectories.- 7. Geodesics of Quadratic Differentials.- 7.1 Definition of the induced metric.- 7.2 Locally shortest curves.- 7.3 Geodesic polygons.- 7.4 Minimum property of geodesics.- 7.5 Existence of geodesies.- 7.6 Deformation of horizontal arcs.- V Teichmüller Spaces.- to Chapter V.- 1. Quasiconformal Mappings of Riemann Surfaces.- 1.1 Complex dilatation on Riemann surfaces.- 1.2 Conformal structures.- 1.3 Group isomorphisms induced by quasiconformal mappings.- 1.4 Homotopy modulo the boundary.- 1.5 Quasiconformal mappings in homotopy classes.- 2. Definitions of Teichmüller Space.- 2.1 Riemann space and Teichmüller space.- 2.2 Teichmüller metric.- 2.3 Teichmüller space and Beltrami differentials.- 2.4 Teichmüller space and Conformal structures.- 2.5 Conformal structures on a compact surface.- 2.6 Isomorphisms of Teichmüller spaces.- 2.7 Modular group.- 3. Teichmüller Space and Lifted Mappings.- 3.1 Equivalent Beltrami differentials.- 3.2 Teichmüller space as a subset of the universal space.- 3.3 Completeness of Teichmüller spaces.- 3.4 Quasi-Fuchsian groups.- 3.5 Quasiconformal reflections compatible with a group.- 3.6 Quasisymmetric functions compatible with a group.- 3.7 Unique extremality and Teichmüller metrics.- 4. Teichmüller Space and Schwarzian Derivatives.- 4.1 Schwarzian derivatives and quadratic differentials.- 4.2 Spaces of quadratic differentials.- 4.3 Schwarzian derivatives of univalent functions.- 4.4 Connection between Teichmüller spaces and the universal space.- 4.5 Distance to the boundary.- 4.6 Equivalence of metrics.- 4.7 Bers imbedding.- 4.8 Quasiconformal extensions compatible with a group.- 5. Complex Structures on Teichmüller Spaces.- 5.1 Holomorphic functions in Banach spaces.- 5.2 Banach manifolds.- 5.3 A holomorphic mapping between Banach spaces.- 5.4 An atlas on the Teichmüller space.- 5.5 Complex analytic structure.- 5.6 Complex structure under quasiconformal mappings.- 6. Teichmüller Space of a Torus.- 6.1 Covering group of a torus.- 6.2 Generation of group isomorphisms.- 6.3 Conformal equivalence of tori.- 6.4 Extremal mappings of tori.- 6.5 Distance of group isomorphisms from the identity.- 6.6 Representation of the Teichmüller space of a torus.- 6.7 Complex structure of the Teichmüller space of torus.- 7. Extremal Mappings of Riemann Surfaces.- 7.1 Dual Banach spaces.- 7.2 Space of integrable holomorphic quadratic differentials.- 7.3 Poincaré theta series.- 7.4 Infinitesimally trivial differentials.- 7.5 Mappings with infinitesimally trivial dilatations.- 7.6 Complex dilatations of extremal mappings.- 7.7 Teichmüller mappings.- 7.8 Extremal mappings of compact surfaces.- 8. Uniqueness of Extremal Mappings of Compact Surfaces.- 8.1 Teichmüller mappings and quadratic differentials.- 8.2 Local representation of Teichmüller mappings.- 8.3 Stretching function and the Jacobian.- 8.4 Average stretching.- 8.5 Teichmüller’s uniqueness theorem.- 9. Teichmüller Spaces of Compact Surfaces.- 9.1 Teichmüller imbedding.- 9.2 Teichmüller space as a ball of the euclidean space.- 9.3 Straight lines in Teichmüller space.- 9.4 Composition of Teichmüller mappings.- 9.5 Teichmüller discs.- 9.6 Complex structure and Teichmüller metric.- 9.7 Surfaces of finite type.