The Geometry of Complex Domains

ISBN-13: 9780817641399 / Angielski / Twarda / 2011 / 303 str.

Robert Everist Greene

Opis

This highly original work, written by the creators of the multivariable theory of automorphisms, is a rich tapestry of themes and concepts, and a comprehensive treatment of an important area of mathematics. From Poincare's work on biholomorphic inequivalence in 1906, it became clear that the structures of the automorphism groups of domains in multi-dimensional complex space are more complex, and more interesting, than those in the complex plane. The authors build on this theme and trace the evolution of the classical theory to the modern theory, which is today a cornerstone of geometric analysis. The text begins with an introductory chapter on the concept of an automorphism group in which the theory in one complex variable is presented, emphasizing the classical ideas of Schwarz, Jobe, and others. Also examined is the theory of planar domains of multiple but finite connectivity, principally develped by Heins in the 1940s and 1950s. The authors treatment progresses to the theory in several complex variables with the so-called "classical domains" of E. Cartan, the Siegel domains of type I, II, and III, and the more modern theory of automorphism groups of smoothly bounded domains."

Recenzje

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)

Spis treści

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.

O autorze

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.

Blurb

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.