The Geometry of Complex Domains [Hardcover]

This work examines a rich tapestry of themes and concepts and provides a comprehensive treatment of an important area of mathematics, while simultaneously covering a broader area of the geometry of domains in complex space. At once authoritative and accessible, this text touches upon many important parts of modern mathematics: complex geometry, equivalent embeddings, Bergman and Kahler geometry, curvatures, differential invariants, boundary asymptotics of geometries, group actions, and moduli spaces.

The Geometry of Complex Domains can serve as a coming of age book for a graduate student who has completed at least one semester or more of complex analysis, and will be most welcomed by analysts and geometers engaged in current research.

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.

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)

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.

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Krantz has directed 18 Ph.D. Students alSo