Conformal Geometry and Quasiregular Mappings
This book is an introduction to the theory of spatial quasiregular mappings intended for the uninitiated reader. At the same time the book also addresses specialists in classical analysis and, in particular, geometric function theory. The text leads the r
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1319 Matti Vuorinen
Conformal Geometry and Quasiregular Mappings
Sprinqer-Verlaj, Berlin Heidelberg New York London Paris Tokyo
Author
Matti Vuorinen Department of Mathematics, University of Helsinki Hallitusk. 15,00100 Helsinki, Finland
Mathematics Subject Classification (19130): 30C60
ISBN 3-540-19342-1 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-19342-1 Springer-Verlag New York Berlin Heidelberg
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9, 1965, in its version of June 24, 1985, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law.
© Springer-Verlag Berlin Heidelberg 1988 Printed in Germany Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr. 2146/3140-543210
Contents Preface
V
Introduction A survey of quasiregular mappings Notation and terminology
Chapter I.
CONFORMAL GEOMETRY
VII IX XVI
1
1.
Mobius transformations in n-space
2.
Hyperbolic geometry
19
3.
Quasihyperbolic geometry
33
4.
Some covering problems
41
Chapter II.
MODULUS AND CAPACITY
1
48
5.
The modulus of a curve family
48
6.
The modulus as a set function
72
7.
The capacity of a condenser
81
8.
Conformal invariants
Chapter III. 9.
QUASIREGULAR MAPPINGS
Topological properties of discrete open mappings
102 120 121
10. Some properties of quasiregular mappings
127
11. Distortion theory
137
12. Uniform continuity properties
152
13. Normal quasiregular mappings
162
Chapter IV.
BOUNDARY BEHAVIOR
173
14. Some properties of quasiconformal mappings
174
15. Lindelof-type theorems
181
16. Dirichlet-finite mappings
187
Some open problems
193
Bibliography
194
Index
208
Preface This book is based on my lectures on quasiregular mappings in the euclidean nspace R n given at the University of Helsinki in 1986. It is assumed that the reader is familiar with basic real analysis or with some basic facts about quasiconformal mappings (an excellent reference is pp. 1-50 in J. Viiisiilii's book [V7]), but otherwise I have tried to make the text as self-contained and easily accessible as possible. For the reader's convenience and for the sake of easy reference I have included without proof most of those results from [V7] which will be exploited here. I have also included a brief review of those properties of Mobius transformations in R n which will be used throughout. In order to make the text more useful for students I have included nearly a hundred exercises, which are scattered throughout the book. They are of varying difficulty, with hints for solution provided for some. For specialists in the field I have included a list of open problems at the end of the book
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