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© 2011

The Geometry of Complex Domains

  • 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

Book

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

Table of contents

  1. Front Matter
    Pages 1-12
  2. Robert E. Greene, Kang-Tae Kim, Steven G. Krantz
    Pages 1-28
  3. Robert E. Greene, Kang-Tae Kim, Steven G. Krantz
    Pages 29-63
  4. Robert E. Greene, Kang-Tae Kim, Steven G. Krantz
    Pages 65-98
  5. Robert E. Greene, Kang-Tae Kim, Steven G. Krantz
    Pages 99-133
  6. Robert E. Greene, Kang-Tae Kim, Steven G. Krantz
    Pages 135-159
  7. Robert E. Greene, Kang-Tae Kim, Steven G. Krantz
    Pages 161-178
  8. Robert E. Greene, Kang-Tae Kim, Steven G. Krantz
    Pages 179-208
  9. Robert E. Greene, Kang-Tae Kim, Steven G. Krantz
    Pages 209-217
  10. Robert E. Greene, Kang-Tae Kim, Steven G. Krantz
    Pages 219-254
  11. Robert E. Greene, Kang-Tae Kim, Steven G. Krantz
    Pages 255-273
  12. Robert E. Greene, Kang-Tae Kim, Steven G. Krantz
    Pages 275-282
  13. Back Matter
    Pages 290-309

About this book

Introduction

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

Authors and affiliations

  1. 1., Department of MathematicsUniversity of California, Los AngelesLos AngelesUSA
  2. 2., Department of MathematicsPohang University Science & TechnologyPohangKorea, Republic of (South Korea)
  3. 3., Department of MathematicsWashington University in St. LouisSt. LouisUSA

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

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)