The principal aim of this text is to address the paucity of conformal maps in higher dimensions. The exposition includes both an analytic proof, due to Nevanlinna, 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 as an independent study text. 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.

David E Blair: Michigan State University, East Lansing, MI

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?