The Geometry of Complex Domains

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The Geometry of Complex Domains by Robert E. Greene
English | PDF | 2011 | 310 Pages | ISBN : 0817641394 | 2.3 MB
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.

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