Lehto, O. 　著

目次

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.