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
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.
Graduate students and researchers interested in mapping theory.
The “measurable Riemann Mapping
Theorem” (or the existence theorem for quasiconformal mappings)
has found a central rôle in a diverse variety of areas such as
holomorphic dynamics, Teichmüller theory, low dimensional
topology and geometry, and the planar theory of PDEs. Anticipating
the needs of future researchers, the authors give an account of the
“state of the art” as it pertains to this theorem, that is,
to the existence and uniqueness theory of the planar Beltrami
equation, and various properties of the solutions to this equation.
The classical theory concerns itself with the uniformly elliptic case
(quasiconformal mappings). Here the authors develop the theory in the
more general framework of mappings of finite distortion and the
associated degenerate elliptic equations.
The authors recount aspects of this classical theory for the uninitiated, and then develop the more general theory. Much of this is either new at the time of writing, or provides a new approach and new insights into the theory. Indeed, it is the substantial recent advances in non-linear harmonic analysis, Sobolev theory and geometric function theory that motivated their approach here. The concept of a principal solution and its fundamental role in understanding the natural domain of definition of a given Beltrami operator is emphasized in their investigations. The authors believe their results shed considerable new light on the theory of planar quasiconformal mappings and have the potential for wide applications, some of which they discuss.