An Introduction To The Theory Of Higher Dimensional Quasiconformal Mappings
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An Introduction to the Theory of Higher-Dimensional Quasiconformal Mappings
Author: Frederick W. Gehring
language: en
Publisher: American Mathematical Soc.
Release Date: 2017-05-03
This book offers a modern, up-to-date introduction to quasiconformal mappings from an explicitly geometric perspective, emphasizing both the extensive developments in mapping theory during the past few decades and the remarkable applications of geometric function theory to other fields, including dynamical systems, Kleinian groups, geometric topology, differential geometry, and geometric group theory. It is a careful and detailed introduction to the higher-dimensional theory of quasiconformal mappings from the geometric viewpoint, based primarily on the technique of the conformal modulus of a curve family. Notably, the final chapter describes the application of quasiconformal mapping theory to Mostow's celebrated rigidity theorem in its original context with all the necessary background. This book will be suitable as a textbook for graduate students and researchers interested in beginning to work on mapping theory problems or learning the basics of the geometric approach to quasiconformal mappings. Only a basic background in multidimensional real analysis is assumed.
Conformally Invariant Metrics and Quasiconformal Mappings
This book is an introduction to the theory of quasiconformal and quasiregular mappings in the euclidean n-dimensional space, (where n is greater than 2). There are many ways to develop this theory as the literature shows. The authors' approach is based on the use of metrics, in particular conformally invariant metrics, which will have a key role throughout the whole book. The intended readership consists of mathematicians from beginning graduate students to researchers. The prerequisite requirements are modest: only some familiarity with basic ideas of real and complex analysis is expected.
Quasiconformal Mappings in the Plane and Complex Dynamics
This book comprehensively explores the foundations of quasiconformal mappings in the complex plane, especially in view of applications to complex dynamics. Besides playing a crucial role in dynamical systems these mappings have important applications in complex analysis, geometry, topology, potential theory and partial differential equations, functional analysis and calculus of variations, electrostatics and nonlinear elasticity . The work covers standard material suitable for a one-year graduate-level course and extends to more advanced topics, in an accessible way even for students in an initial phase of university studies who have learned the basics of complex analysis at the usual level of a rigorous first one-semester course on the subject. At the frontier of complex analysis with real analysis, quasiconformal mappings appeared in 1859-60 in the cartography work of A. Tissot, well before the term “quasiconformal” was coined by L. Ahlfors in 1935. The detailed study of these mappings began in 1928 by H. Grötzsch, and L. Ahlfors’ seminal work published in 1935 significantly contributed to their development and was considered for awarding him the Fields Medal in 1936. The theory further evolved in 1937 and 1939 with O. Teichmüller’s contributions, and subsequent advancements are partially covered in this book. Organized into ten chapters with eight appendices, this work aims to provide an accessible, self-contained approach to the subject and includes examples at various levels and extensive applications to holomorphic dynamics. Throughout the text, historical notes contextualize advancements over time. A sequel to the author’s previous book, ‘Complex Analysis and Dynamics in One Variable with Applications,’ also published by Springer, this volume might be suitable for students in mathematics, physics, or engineering. A solid background in basic mathematical analysis is recommended to fully benefit from its content.