Progress in Mathematics

The Geometry of Complex Domains

Authors: Greene, Robert E., Kim, Kang-Tae, Krantz, Steven G

  • 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
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About this book

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.

About the authors

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.

Reviews

From the reviews:

“The book under review gives an excellent presentation of modern problems related to various characterizations of the holomorphic geometry of domains in Cn and complex manifolds. … The book may be strongly recommended for researchers and Ph.D. students working in complex analysis.” (Marek Jarnicki, Mathematical Reviews, Issue 2012 c)

Table of contents (11 chapters)

  • Preliminaries

    Greene, Robert E. (et al.)

    Pages 1-28

  • Riemann Surfaces and Covering Spaces

    Greene, Robert E. (et al.)

    Pages 29-63

  • The Bergman Kernel and Metric

    Greene, Robert E. (et al.)

    Pages 65-98

  • Applications of Bergman Geometry

    Greene, Robert E. (et al.)

    Pages 99-133

  • Lie Groups Realized as Automorphism Groups

    Greene, Robert E. (et al.)

    Pages 135-159

Buy this book

eBook $129.00
price for USA
  • ISBN 978-0-8176-4622-6
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Download immediately after purchase
Hardcover $169.00
price for USA
  • ISBN 978-0-8176-4139-9
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
Rent the ebook  
  • Rental duration: 1 or 6 month
  • low-cost access
  • online reader with highlighting and note-making option
  • can be used across all devices
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Bibliographic Information

Bibliographic Information
Book Title
The Geometry of Complex Domains
Authors
Series Title
Progress in Mathematics
Series Volume
291
Copyright
2011
Publisher
Birkhäuser Basel
Copyright Holder
Springer Science+Business Media, LLC
eBook ISBN
978-0-8176-4622-6
DOI
10.1007/978-0-8176-4622-6
Hardcover ISBN
978-0-8176-4139-9
Series ISSN
0743-1643
Edition Number
1
Number of Pages
XIV, 303
Number of Illustrations and Tables
14 b/w illustrations
Topics