Boundary Behaviour of Conformal Maps / Edition 1

by Christian Pommerenke

Overview

We study the boundary behaviour of a conformal map of the unit disk onto an arbitrary simply connected plane domain. A principal aim of the theory is to obtain a one-to-one correspondence between analytic properties of the function and geometrie properties of the domain. In the classical applications of conformal mapping, the domain is bounded by a piecewise smooth curve. In many recent applications however, the domain has a very bad boundary. It may have nowhere a tangent as is the case for Julia sets. Then the conformal map has many unexpected properties, for instance almost all the boundary is mapped onto almost nothing and vice versa. The book is meant for two groups of users. (1) Graduate students and others who, at various levels, want to learn about conformal mapping. Most sections contain exercises to test the understanding. They tend to be fairly simple and only a few contain new material. Pre­ requisites are general real and complex analyis including the basic facts about conformal mapping (e.g. AhI66a). (2) Non-experts who want to get an idea of a particular aspect of confor­ mal mapping in order to find something useful for their work. Most chapters therefore begin with an overview that states some key results avoiding technicalities. The book is not meant as an exhaustive survey of conformal mapping. Several important aspects had to be omitted, e.g. numerical methods (see e.g.

Product Details

ISBN-13:	9783642081293
Publisher:	Springer Berlin Heidelberg
Publication date:	12/07/2010
Series:	Grundlehren der mathematischen Wissenschaften , #299
Edition description:	Softcover reprint of hardcover 1st ed. 1992
Pages:	300
Product dimensions:	6.10(w) x 9.25(h) x 0.24(d)

Table of Contents

1. Some Basic Facts.- 2. Continuity and Prime Ends.- 3. Smoothness and Corners.- 4. Distortion.- 5. Quasidisks.- 6. Linear Measure.- 7. Smirnov and Lavrentiev Domains.- 8. Integral Means.- 9. Curve Families and Capacity.- 10. Hausdorff Measure.- 11. Local Boundary Behaviour.- References.- Author Index.

Editorial Reviews

Studies the boundary behavior of a conformal map of the unit disk onto an arbitrary simply-connected plane domain. Topics include the Koebe distortion theorem, crosscuts and prime ends, starlike domains, Bloch functions, the hyperbolic metric, Plessner's theorem and twisting, Jordan curves, Lavrentiev domains, Hausdorff and Zygmund measures, and the Visser- Ostrowski quotient. Chapters begin with an overview of key results and contain many exercises. Annotation c. Book News, Inc., Portland, OR (booknews.com)

Booknews