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  • © 2011

The Geometry of Complex Domains

Birkhäuser

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  • 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

Part of the book series: Progress in Mathematics (PM, volume 291)

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  • ISBN: 978-0-8176-4622-6
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Table of contents (11 chapters)

  1. Front Matter

    Pages 1-12
  2. Preliminaries

    • Robert E. Greene, Kang-Tae Kim, Steven G. Krantz
    Pages 1-28
  3. Riemann Surfaces and Covering Spaces

    • Robert E. Greene, Kang-Tae Kim, Steven G. Krantz
    Pages 29-63
  4. The Bergman Kernel and Metric

    • Robert E. Greene, Kang-Tae Kim, Steven G. Krantz
    Pages 65-98
  5. Applications of Bergman Geometry

    • Robert E. Greene, Kang-Tae Kim, Steven G. Krantz
    Pages 99-133
  6. Lie Groups Realized as Automorphism Groups

    • Robert E. Greene, Kang-Tae Kim, Steven G. Krantz
    Pages 135-159
  7. The Significance of Large Isotropy Groups

    • Robert E. Greene, Kang-Tae Kim, Steven G. Krantz
    Pages 161-178
  8. Some Other Invariant Metrics

    • Robert E. Greene, Kang-Tae Kim, Steven G. Krantz
    Pages 179-208
  9. Automorphism Groups and Classification of Reinhardt Domains

    • Robert E. Greene, Kang-Tae Kim, Steven G. Krantz
    Pages 209-217
  10. The Scaling Method, I

    • Robert E. Greene, Kang-Tae Kim, Steven G. Krantz
    Pages 219-254
  11. The Scaling Method, II

    • Robert E. Greene, Kang-Tae Kim, Steven G. Krantz
    Pages 255-273
  12. Afterword

    • Robert E. Greene, Kang-Tae Kim, Steven G. Krantz
    Pages 275-282
  13. Back Matter

    Pages 290-309

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.

Keywords

  • Cauchy-Riemann equations
  • Greene-Krantz conjecture
  • automorphism
  • complex geometry
  • complex invariants
  • curvature
  • equivalent embeddings
  • isometry
  • semicontinuity

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)

Authors and Affiliations

  • , Department of Mathematics, University of California, Los Angeles, Los Angeles, USA

    Robert E. Greene

  • , Department of Mathematics, Pohang University Science & Technology, Pohang, Korea, Republic of (South Korea)

    Kang-Tae Kim

  • , Department of Mathematics, Washington University in St. Louis, St. Louis, USA

    Steven G. Krantz

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.

Bibliographic Information

Buying options

eBook
USD 129.00
Price excludes VAT (USA)
  • ISBN: 978-0-8176-4622-6
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Hardcover Book
USD 169.99
Price excludes VAT (USA)