Inversion Theory and Conformal Mapping
About this Title
David E. Blair, Michigan State University, East Lansing, MI
Publication: The Student Mathematical Library
Publication Year 2000: Volume 9
ISBNs: 978-0-8218-2636-2 (print); 978-1-4704-1816-8 (online)
MathSciNet review: MR1779832
MSC: Primary 30C65; Secondary 30C62
It is rarely taught in an undergraduate or even graduate curriculum that the only conformal maps in Euclidean space of dimension greater than two are those generated by similarities and inversions in spheres. This is in stark contrast to the wealth of conformal maps in the plane.
The principal aim of this text is to give a treatment of this paucity of conformal maps in higher dimensions. The exposition includes both an analytic proof in general dimension and a differential-geometric proof in dimension three. For completeness, enough complex analysis is developed to prove the abundance of conformal maps in the plane. In addition, the book develops inversion theory as a subject, along with the auxiliary theme of circle-preserving maps. A particular feature is the inclusion of a paper by Carathéodory with the remarkable result that any circle-preserving transformation is necessarily a Möbius transformation, not even the continuity of the transformation is assumed.
The text is at the level of advanced undergraduates and is suitable for a capstone course, topics course, senior seminar or independent study. Students and readers with university courses in differential geometry or complex analysis bring with them background to build on, but such courses are not essential prerequisites.
Advanced undergraduate students and mathematicians interested in conformal mappings in higher-dimensional spaces.
Table of Contents
- Chapter 1. Classical inversion theory in the plane
- Chapter 2. Linear fractional transformations
- Chapter 3. Advanced calculus and conformal maps
- Chapter 4. Conformal maps in the plane
- Chapter 5. Conformal maps in Euclidean space
- Chapter 6. The classical proof of Liouville’s theorem
- Chapter 7. When does inversion preserve convexity?