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This text provides a masterful and systematic treatment of all the basic analytic and geometric aspects of Bergman's classic theory of the kernel and its invariance properties. These include calculation, invariance properties, boundary asymptotics, and asymptotic expansion of the Bergman kernel and metric. Moreover, it presents a unique compendium of results with applications to function theory, geometry, partial differential equations, and interpretations in the language of functional analysis, with emphasis on the several complex variables context. Several of these topics appear here for the first time in book form. Each chapter includes illustrative examples and a collection of exercises which will be of interest to both graduate students and experienced mathematicians.
Graduate students who have taken courses in complex variables
and have a basic background in real and functional analysis will find this textbook appealing. Applicable courses for either main or supplementary usage include those in complex variables, several complex variables, complex differential geometry, and partial differential equations. Researchers in complex analysis, harmonic analysis, PDEs, and complex differential geometry will also benefit from the thorough treatment of the many exciting aspects of Bergman's theory.
Graduate students who have taken courses in complex variables
and have a basic background in real and functional analysis will find this textbook appealing. Applicable courses for either main or supplementary usage include those in complex variables, several complex variables, complex differential geometry, and partial differential equations. Researchers in complex analysis, harmonic analysis, PDEs, and complex differential geometry will also benefit from the thorough treatment of the many exciting aspects of Bergman's theory.
From the book reviews:
"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." (Steven Deckelman, MAA Reviews, December, 2014)
"The monograph under review gives the reader a look at the ever-evolving Bergman theory. ... The book covers a great deal of interesting topics on the Bergman kernel and metric. In particular, it abounds with useful and instructive calculation, much of which cannot be found elsewhere. It will certainly be an invaluable resource for students as well as seasoned researchers in geometric analysis of the Bergman kernel and metric." (Jianbing Su, Mathematical Reviews, October, 2014)
"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." (Steven Deckelman, MAA Reviews, December, 2014)
"The monograph under review gives the reader a look at the ever-evolving Bergman theory. ... The book covers a great deal of interesting topics on the Bergman kernel and metric. In particular, it abounds with useful and instructive calculation, much of which cannot be found elsewhere. It will certainly be an invaluable resource for students as well as seasoned researchers in geometric analysis of the Bergman kernel and metric." (Jianbing Su, Mathematical Reviews, October, 2014)