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.
This book focuses on gathering the numerous properties and many
different connections with various topics in geometric function theory
that quasidisks possess. A quasidisk is the image of a disk under a
quasiconformal mapping of the Riemann sphere. In 1981 Frederick
W. Gehring gave a short course of six lectures on this topic in
Montreal and his lecture notes “Characteristic Properties of
Quasidisks” were published by the University Press of the University
of Montreal. The notes became quite popular and within the next decade
the number of characterizing properties of quasidisks and their
ramifications increased tremendously. In the late 1990s Gehring and
Hag decided to write an expanded version of the Montreal notes. At
three times the size of the original notes, it turned into much more
than just an extended version. New topics include two-sided
criteria. The text will be a valuable resource for current and future
researchers in various branches of analysis and geometry, and with its
clear and elegant exposition the book can also serve as a text for a
graduate course on selected topics in function theory.
Frederick W. Gehring (1925–2012) was a leading figure in the theory of quasiconformal mappings for over fifty years. He received numerous awards and shared his passion for mathematics generously by mentoring twenty-nine Ph.D. students and more than forty postdoctoral fellows.
Kari Hag received her Ph.D. under Gehring's direction in 1972 and worked with him on the present text for more than a decade.
Graduate students and research mathematicians interested in geometric function theory.
...This text on quasidisks provides a cross section of the plane quasiconformal theory and demonstrates the many ways in which these mappings are related to analysis, topology, geometry and other parts of mathematics. The exposition is very clear and the text is richly illustrated with carefully drawn pictures. This book would be an excellent choice for a first book on plane quasiconformal maps for a graduate student. It is also a valuable source of inspiration for researchers of complex analysis, because the material covers many topics of current interest. It is my guess that this book will be an instant classic in its field.
-- Matti Vuorinen (Turku), Zentralblatt MATH