This highly original book examines a rich tapestry of themes and concepts, including complex geometry, Finsler metrics, Bergman and Kahler geometry, curvatures, differential invariants, boundary asymptotics of geometries, group actions, and moduli spaces.

This work examines a rich tapestry of themes and concepts that provides a comprehensive treatment of an important area of mathematics, while simultaneously covering a broader area of the geometry of domains in complex space. At once authoritative and accessible, this text touches upon many important parts of modern mathematics: complex geometry, Finsler metrics, Bergman and Kahler geometry, curvatures, differential invariants, boundary asymptotics of geometries, group actions, and moduli spaces.

Unique, authoritative text on a dynamic and active subject area written by three founders of the field Comprehensive treatment of the topic, with abundant examples and references Both accessible to beginners and meaningful for experienced researchers in the field Useful as a textbook in graduate courses on complex analysis Includes supplementary material: sn.pub/extras

Autorentext

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.

Klappentext

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 casebackground information which cannot be found elsewhere in one placethe 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 inthe field.

Inhalt

Preface.- 1 Preliminaries.- 2 Riemann Surfaces and Covering Spaces.- 3 The Bergman Kernel and Metric.- 4 Applications of Bergman Geometry.- 5 Lie Groups Realized as Automorphism Groups.- 6 The Significance of Large Isotropy Groups.- 7 Some Other Invariant Metrics.- 8 Automorphism Groups and Classification of Reinhardt Domains.- 9 The Scaling Method, I.- 10 The Scaling Method, II.- 11 Afterword.- Bibliography.- Index.