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This book provides the reader with a broad introduction to the geometric methodology in complex analysis. It covers both single and several complex variables, creating a dialogue between the two viewpoints. Regarded as one of the 'grand old ladies' of modern mathematics, complex analysis traces its roots back 500 years. The subject began to flourish with Carl Friedrich Gauss's thesis around 1800. The geometric aspects of the theory can be traced back to the Riemann mapping theorem around 1850, with a significant milestone achieved in 1938 with Lars Ahlfors's geometrization of complex analysis. These ideas inspired many other mathematicians to adopt this perspective, leading to the proliferation of geometric theory of complex variables in various directions, including Riemann surfaces, Teichmüller theory, complex manifolds, extremal problems, and many others. This book explores all these areas, with classical geometric function theory as its main focus. Its accessible and gentle approach makes it suitable for advanced undergraduate and graduate students seeking to understand the connections among topics usually scattered across numerous textbooks, as well as experienced mathematicians with an interest in this rich field. Preface. Introduction. The Riemann Mapping Theorem. The Riemann Mapping Theorem. The Proof of the Riemann Mapping Theorem. A New Proof of the Riemann Mapping Theorem. Some Definitions. Discrete Analytic Function Theory. A Sketch of Thurston's Idea. Details of the Proof. Convergence of Circle Packings to the Riemann Mapping. The Main Result. The Riemann Map. The Schwarz-Christoffel Formula. Carathéodory Convergence Theorem. Construction of the Riemann Map. The Ahlfors Map. A Riemann Mapping Theorem for Two-Connected Domains in the Plane. Proofs of the Lemmas. How to Compute Phi and the Median of Omega. Riemann Multiply Connected Domains. The Mappings on the Canonical Regions. Quasiconformal Mappings. Historical Development of Quasiconformal Mappings. The Planar Theory. The Problem and Definition of Grötzsch. Solution of Grötzsch's Problem. Composed Mappings. Extremal Length. Extremal Length on Riemann Surfaces. Beurling's Criterion for an Extremal Metric. Extremal Length and Quasiconformal Mappings. Manifolds. Principal Spaces of Functions. Definition of a Fréchet Space. Distributions and Weak Derivatives. Definitions of Sobolev Spaces. Definition of Manifolds. Vector Bundles. Almost Complex Manifolds and the overline partial-Operator. Tensor Fields. Riemann Surfaces. The Evolution of the Concept of a Riemann Surface. Riemann Surfaces: The Classical Approach. Examples of Riemann Surfaces. Mappings of Riemann Surfaces. Construction of Riemann Surfaces. Branch Points. Coverings of Riemann Surfaces. Möbius Transformations. Fuchsian Groups. Covering Spaces and Automorphisms in General. Compact Quotients of D and Their Automorphisms. The Automorphism Group of a Riemann Surface of Genus at Least. The Uniformization Theorem. Double Periodicity of Elliptic Functions. Elliptic Curves. How to Look at the Problem. Covering Surfaces and Classical Plane Geometries. The Uniformization Theorem for Riemann Surfaces. Proof of the Uniformization Theorem. Green's Functions. Bipolar Green's Functions. Proof of the Uniformization Theorem. Another Proof of the Uniformization Theorem. Automorphism Groups. Introduction. Automorphism Groups of Planar Domains. Holomorphic Functions and Mappings of Several Variables. Failure of the Riemann Mapping Theorem in Several Complex Variables. Convexity in Complex Analysis. The Theorem of Bun Wong and Jean-Pierre Rosay. Some Startling Examples. Boundary Flatness and an Influential Conjecture. More on the Preceding Examples. The Contact Type of D'Angelo. The Greene/Krantz Conjecture. Closing Remarks. Rigidity of Holomorphic Mappings and a New Schwarz Lemma at the Boundary. Introduction. A Result on the Disc About Boundary Derivatives. A Result for the Ball. Generalization of the Result to Strongly Convex and to Strongly Pseudoconvex Domains. Another Direct Extension. Geometric Interpretations and Return to the Question of Order of Contact. Discussion of the Weakly Pseudoconvex Case and Closing Remarks. The Schwarz Lemma and Its Generalizations. The Schwarz-Pick Lemma. Curvature and the Schwarz Lemma Revisited. A Generalization by Ahlfors. Compact Riemann Surfaces of Genus geqq. Holomorphic Mappings from an Annulus into an Annulus. Invariant Distances on Complex Manifolds. Poincaré Metric. Bergman Metric. Bergman Representative Coordinates. The Lu Qi-Keng Conjecture. The Lu Qi-Keng Theorem. The Kobayashi and Carathéodory Metrics. Kobayashi Distance. Completeness with Respect to the Carathéodory Distance. Carathéodory Distance. Sibony Metric. Hyperbolic Manifolds. Hyperbolicity. On Completeness of Kobayashi Distance. Kobayashi-Hurwitz Theorem. Extension Theorems. Geometric Methods. Invariant Volume Forms. Measure Hyperbolicity. Existence of Bounded psh Functions and Hyperbolicity. Hyperbolicity of a Complex Manifold and Other Equivalent Properties. The Fatou Theory and Related Matters. Review of the Classical Theory of H^p Spaces on the Disc. The Szego and Poisson-Szego Kernels. Three Propositions About the Poisson Kernel. Subharmonicity, Harmonic Majorants, and Boundary Values. Pointwise Convergence for Harmonic Functions on Domains in R^N. Boundary Values of Holomorphic Functions in C^n. Admissible Convergence. The Lindelöf Principle. Additional Tangential Phenomena: Lipschitz Spaces. The Theorem of Bun Wong and Rosay. Smoothness to the Boundary of Biholomorphic Mappings. Solution of the overline partial Problem. The Pompeiu Formula for Solutions of. The overline partial-Problem in C^n. The Leray (Cauchy-Fantappiè) IntegralRepresentation. The Khenkin-Ramirez Integral Representation. Formula for the Solution of the overline partial-Problem. Harmonic Measure. The Idea of Harmonic Measure. Some Examples. Hadamard's Three-Circles Theorem. A Discussion of Interpolation of Linear Operators. The F and M Riesz Theorem. Quadrature. Introduction. Classical Quadrature Domains. The Schwarz Function. Area Quadrature Domains. Riemann Surfaces (the Schottky Double). Teichmüller Theory. What Is Teichmüller Space? Inner Automorphisms. The Fundamental Group. The Space A of Almost Complex Structures. The Space of Riemannian Metrics M. The Manifold M_- of Metrics of Negative Constant Curvature Minus One. Conformal Coordinates and the D-equivariant Equivalence Between M_- and C, and C and A. The L_-Decomposition of the Tangent Space to M_-. Teichmüler's Moduli Space Is a C^infty Manifold of Dimension (genus M)-. Teichmüller's Space Is Diffeomorphic to R^ genus M-. Proof of Poincaré Theorem. The Classical Schwarzian Derivative. The Schwarzian Derivative. Appendix on the Structure Equations and Curvature. Introduction. Expressing Curvature Intrinsically. Coordinates on a Surface. Tangent Planes. Curvature Calculations on Planar Domains. Concluding Remarks. References. Index