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The Theory and Practice of Conformal Geometry

Steven G. Krantz
Dover Publications
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Steven Deckelman
, on

With this new book, in the tradition of Zeev Nehari’s Conformal Mapping, Steven G. Krantz once again displays masterful skill at rendering accessible the often advanced and complex ideas of conformal geometry to non-specialists and advanced undergraduates.

The field of conformal geometry is often ascribed to have begun with Riemann’s 1851 proof of his famous mapping theorem. The book begins with a description of several proof approaches to the Riemann mapping theorem and its generalizations to multiply connected domains. There is a very nice introduction to the discrete analytic function approach that makes use of the circle-packing ideas of Thurston, Rodin and Sullivan. Other topics treated include invariant metrics and geometry, normal families, automorphism groups, the Schwartz lemma as well as harmonic measure, extremal length, anlaytic capacity and a glimpse at several complex variables. Drawing on these ideas, some new results are also given.

Many of the proofs in the book are more along the lines of a sketch of general ideas so that the reader does not get lost in a sea of technical details. Though this is not a textbook, there are many exercises for the reader. This book will be of interest to anyone interested in learning more about conformal geometry, its history and ramifications. There is also a wealth of material that could be used in complex analysis courses.

Steven Deckelman is a professor of mathematics at the University of Wisconsin-Stout, where he has been since 1997. He received his Ph.D from the University of Wisconsin-Madison in 1994 for a thesis in several complex variables written under Patrick Ahern. Some of his interests include complex analysis, mathematical biology and the history of mathematics.

The table of contents is not available.