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Geometric Analysis of the Bergman Kernel and Metric

Steven G. Krantz
Publication Date: 
Number of Pages: 
Graduate Texts in Mathematics 268
[Reviewed by
Steven Deckelman
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For those of us who were weaned on books like Steven G. Krantz’s Function Theory of Several Complex Variables, this book is an exciting and welcome addition to the Krantz bookshelf. The expository writings of Steven G. Krantz are characterized by their lucidity and the author’s knack for deconstructing complex ideas in ways that reveal their underlying unity, intuition, motivation and place in the larger tapestry of mathematics. Krantz’s Geometric Analysis of the Bergman Kernel and Metric is another book in this same tradition. Although a number of recent books treating Bergman theory in one complex variable have appeared recently, this book is unique in that it treats Bergman theory in several complex variables. The book includes a general introduction and overview of the Bergman kernel and metric including numerous examples of methods for their calculation. Related ideas such as the Szegö and Poisson-Szegö kernel are also included. Later chapters explore more specialized topics such as Bergman representative coordinates, the Berezin transform, the Poisson-Bergman or Berezin kernel, partial differential equations, worm domains and curvature of the Bergman metric including the scaling method. Attention is also given to the subject of boundary asymptotic of the Bergman kernel and metric. The author is a leading current researcher in this area.

At the end of each chapter there are exercises. This book is a compendium on the current state of Bergman theory in several complex variables would be an excellent source for a graduate course on Bergman theory in several complex variables. It is a must have for graduate students in this area as well as mathematicians wanting to learn more about this theory. It begins with the basics and ends at the forefront of current research. The prerequisites are minimal and include only some prior exposure to graduate analysis and several complex variables at the introductory level.

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.