Univalent Functions and Teichmüller Spaces

Inhaltsverzeichnis

• 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.