Harmonic Mappings in the Plane [Hardcover]

A comprehensive account, treating both the generalizations of univalent analytic functions and the connections with minimal surfaces.The theory of planar harmonic mappings straddles both complex analysis and differential geometry. Harmonic mappings are natural analogues of analytic univalent functions, but they also provide an essential tool for the study of minimal surfaces. They have been the obj ect of intensive research by complex analysts in the last few years. This book is the first extensive exposition, treating both analytic and geometric aspects of harmonic mappings. This introduction is designed for graduate students and researchers in all areas of mathematics.The theory of planar harmonic mappings straddles both complex analysis and differential geometry. Harmonic mappings are natural analogues of analytic univalent functions, but they also provide an essential tool for the study of minimal surfaces. They have been the obj ect of intensive research by complex analysts in the last few years. This book is the first extensive exposition, treating both analytic and geometric aspects of harmonic mappings. This introduction is designed for graduate students and researchers in all areas of mathematics.Many classical results of geometric function theory extend to harmonic mappings, but basic questions remain unresolved. This book is the first comprehensive account of the theory of planar harmonic mappings, treating both the generalizations of univalent analytic functions and the connections with minimal surfaces. It contains background material in complex analysis and a full development of the classical theory of minimal surfaces, including the Weierstrass-Enneper representation. It introduces non-specialists to a beautiful area of complex analysis and geometry.1. Preliminaries; 2. Local properties of harmonic mappings; 3. Harmonic mappings onto convex regions; 4. Harmonic self-mappings of the disk; 5. Harmonic univalent functions; 6. Extremal problems; 7. l…