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Über dieses Buch

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.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introductory Ideas

Abstract
In the early days of functional analysis—the early twentieth century—people did not yet know what a Banach space was nor a Hilbert space. They frequently studied a particular complete, infinite-dimensional space from a more abstract point of view. The most common space to be studied in this regard was of course L 2. It was when Stefan Bergman took a course from Erhard Schmidt on L 2 of the unit interval I that he conceived of the idea of the Bergman space of square-integrable holomorphic functions on the unit disc D. And the rest is history.
Steven G. Krantz

Chapter 2. The Bergman Metric

Abstract
Poincaré’s theorem (see [KRA1, KRA9] for discussion) that the ball and polydisc are biholomorphically inequivalent shows that there is no Riemann mapping theorem (at least in the traditional sense) in several complex variables. More recent results of Burns, Shnider, and Wells [BSW] and of Greene and Krantz [GRK1, GRK2] confirm how truly dismal the situation is. First, we need a definition.
Steven G. Krantz

Chapter 3. Further Geometric and Analytic Theory

Abstract
The theory of the Bergman kernel gives rise to many important geometric invariants. Among these are the not-very-well-known Bergman representative coordinates. This is a local coordinate system in which a biholomorphic mapping is realized as a linear mapping. Such a result, while initially quite startling, is in fact completely analogous to the result in the Riemannian geometry regarding geodesic normal coordinates. But geodesic normal coordinates are almost never holomorphic—unless the Kähler metric is flat.
Steven G. Krantz

Chapter 4. Partial Differential Equations

Abstract
Spherical harmonics are for many purposes the natural generalization of the Fourier analysis of the circle to higher dimensions. Spherical harmonics are also intimately connected to the representation theory of the orthogonal group. As a result, analogues of the spherical harmonics play an important role in general representation theory.
Steven G. Krantz

Chapter 5. Further Geometric Explorations

Abstract
A domain Ω in \({\mathbb{C}}^{n}\) is a connected, open set. An automorphism of Ω is a biholomorphic self-map. The collection of automorphisms forms a group under the binary operation of composition of mappings. The standard topology on this group is uniform convergence on compact sets, or the compact-open topology. We denote the automorphism group by \(Aut(\Omega )\). When Ω is a bounded domain, the group \(Aut(\Omega )\) is a real (never a complex) Lie group.
Steven G. Krantz

Chapter 6. Additional Analytic Topics

Abstract
The concept of “domain of holomorphy” is central to the function theory of several complex variables. The celebrated solution of the Levi problem tells us that a connected open set (a domain) is a domain of holomorphy if and only if it is pseudoconvex. For us, in the present book, pseudoconvexity is Levi pseudoconvexity; this is defined in terms of the positive semi-definiteness of the Levi form.
Steven G. Krantz

Chapter 7. Curvature of the Bergman Metric

Abstract
We begin with a little introductory material on the scaling method. Then we use these ideas to discuss Klembeck’s theorem about the boundary asymptotics of the curvature of the Bergman metric on a strictly pseudoconvex domain.
Steven G. Krantz

Chapter 8. Concluding Remarks

Abstract
We have endeavored in this book to give the reader a look at the ever-evolving Bergman theory. Some of the results here are 90 years old, and others were proved quite recently.
Steven G. Krantz

Backmatter

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