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Boundary Behaviour of Conformal Maps

  • Book
  • © 1992

Overview

Authors:
  1. Christian Pommerenke

    1. Fachbereich Mathematik, Technische Universität, Berlin 12, Germany

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Part of the book series: Grundlehren der mathematischen Wissenschaften (GL, volume 299)

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Table of contents (11 chapters)

  1. Front Matter

    Pages I-IX
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  2. Some Basic Facts

    • Christian Pommerenke
    Pages 1-17

  3. Continuity and Prime Ends

    • Christian Pommerenke
    Pages 18-40

  4. Smoothness and Corners

    • Christian Pommerenke
    Pages 41-70

  5. Distortion

    • Christian Pommerenke
    Pages 71-93

  6. Quasidisks

    • Christian Pommerenke
    Pages 94-125

  7. Linear Measure

    • Christian Pommerenke
    Pages 126-154

  8. Smirnov and Lavrentiev Domains

    • Christian Pommerenke
    Pages 155-172

  9. Integral Means

    • Christian Pommerenke
    Pages 173-194

  10. Curve Families and Capacity

    • Christian Pommerenke
    Pages 195-221

  11. Hausdorff Measure

    • Christian Pommerenke
    Pages 222-244

  12. Local Boundary Behaviour

    • Christian Pommerenke
    Pages 245-267

  13. Back Matter

    Pages 269-300
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Keywords

About this book

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 understand­ ing. 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 tech­ nicalities. 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.

Authors and Affiliations

  • Fachbereich Mathematik, Technische Universität, Berlin 12, Germany

    Christian Pommerenke

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