The Geometry of Complex Domains: 291 (Progress in Mathematics) - Hardcover

Über die Autorin bzw. den Autor

Steven G. Krantz received the B.A. degree from the University of California at Santa Cruz and the Ph.D. from Princeton University.  He has taught at UCLA, Princeton, Penn State, and Washington University, where he has most recently served as Chair of the Mathematics Department.

Krantz has directed 18 Ph.D. Students and 9 Masters students, and is winner of the Chauvenet Prize and the Beckenbach Book Award. He edits six journals and is Editor-in-Chief of three.

A prolific scholar, Krantz has published more than 55 books and more than 160 academic papers.

Von der hinteren Coverseite

The geometry of complex domains is a subject with roots extending back more than a century, to the uniformization theorem of Poincaré and Koebe and the resulting proof of existence of canonical metrics for hyperbolic Riemann surfaces. In modern times, developments in several complex variables by Bergman, Hörmander, Andreotti-Vesentini, Kohn, Fefferman, and others have opened up new possibilities for the unification of complex function theory and complex geometry. In particular, geometry can be used to study biholomorphic mappings in remarkable ways. This book presents a complete picture of these developments.

Beginning with the one-variable case―background information which cannot be found elsewhere in one place―the book presents a complete picture of the symmetries of domains from the point of view of holomorphic mappings. It describes all the relevant techniques, from differential geometry to Lie groups to partial differential equations to harmonic analysis. Specific concepts addressed include:

• covering spaces and uniformization;
• Bergman geometry;
• automorphism groups;
• invariant metrics;
• the scaling method.

All modern results are accompanied by detailed proofs, and many illustrative examples and figures appear throughout.

Written by three leading experts in the field, The Geometry of Complex Domains is the first book to provide systematic treatment of recent developments in the subject of the geometry of complex domains and automorphism groups of domains. A unique and definitive work in this subject area, it will be a valuable resource for graduate students and a useful reference for researchers in the field.