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Geometric Analysis of the Bergman Kernel and Metric Softcover reprint of the original 1st ed. 2013 [Minkštas viršelis]

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  • Formatas: Paperback / softback, 292 pages, aukštis x plotis: 235x155 mm, weight: 4686 g, 7 Illustrations, black and white; XIII, 292 p. 7 illus., 1 Paperback / softback
  • Serija: Graduate Texts in Mathematics 268
  • Išleidimo metai: 23-Aug-2016
  • Leidėjas: Springer-Verlag New York Inc.
  • ISBN-10: 1493944290
  • ISBN-13: 9781493944293
Kitos knygos pagal šią temą:
Geometric Analysis of the Bergman Kernel and Metric Softcover reprint of the original 1st ed. 2013Didesnis paveikslėlis
  • Formatas: Paperback / softback, 292 pages, aukštis x plotis: 235x155 mm, weight: 4686 g, 7 Illustrations, black and white; XIII, 292 p. 7 illus., 1 Paperback / softback
  • Serija: Graduate Texts in Mathematics 268
  • Išleidimo metai: 23-Aug-2016
  • Leidėjas: Springer-Verlag New York Inc.
  • ISBN-10: 1493944290
  • ISBN-13: 9781493944293
Kitos knygos pagal šią temą:
Pastovi nuoroda: https://www.kriso.lt/db/9781493944293.html
Raktažodžiai:
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.

This text provides a systematic treatment of all the basic analytic and geometric aspects of Bergman's classic theory of the kernel and its invariance properties. Each chapter includes illustrative examples and a collection of exercises.

Recenzijos

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)

1 Introductory Ideas
1(70)
1.1 The Bergman Kernel
1(24)
1.1.1 Calculating the Bergman Kernel
14(5)
1.1.2 The Poincare-Bergman Distance on the Disc
19(1)
1.1.3 Construction of the Bergman Kernel by Way of Differential Equations
20(3)
1.1.4 Construction of the Bergman Kernel by Way of Conformal Invariance
23(2)
1.2 The Szego and Poisson--Szego Kernels
25(10)
1.3 Formal Ideas of Aronszajn
35(1)
1.4 A New Bergman Basis
36(3)
1.5 Further Examples
39(1)
1.6 A Real Bergman Space
40(1)
1.7 The Behavior of the Singularity in a General Setting
41(2)
1.8 The Annulus
43(2)
1.9 A Direct Connection Between the Bergman and Szego Kernels
45(8)
1.9.1 Introduction
45(1)
1.9.2 The Case of the Disc
45(4)
1.9.3 The Unit Ball in Cn
49(3)
1.9.4 Strongly Pseudoconvex Domains
52(1)
1.9.5 Concluding Remarks
53(1)
1.10 Multiply Connected Domains
53(1)
1.11 The Bergman Kernel for a Sobolev Space
54(2)
1.12 Ramadanov's Theorem
56(2)
1.13 Coda on the Szego Kernel
58(1)
1.14 Boundary Localization
59(12)
1.14.1 Definitions and Notation
60(1)
1.14.2 A Representative Result
60(2)
1.14.3 The More General Result in the Plane
62(1)
1.14.4 Domains in Higher-Dimensional Complex Space
62(3)
Exercises
65(6)
2 The Bergman Metric
71(16)
2.1 Smoothness to the Boundary of Biholomorphic Mappings
71(10)
2.2 Boundary Behavior of the Bergman Metric
81(2)
2.3 The Biholomorphic Inequivalence of the Ball and the Polydisc
83(4)
Exercises
84(3)
3 Further Geometric and Analytic Theory
87(30)
3.1 Bergman Representative Coordinates
87(3)
3.2 The Berezin Transform
90(8)
3.2.1 Preliminary Remarks
90(1)
3.2.2 Introduction to the Poisson--Bergman Kernel
91(3)
3.2.3 Boundary Behavior
94(4)
3.3 Ideas of Fefferman
98(2)
3.4 Results on the Invariant Laplacian
100(9)
3.5 The Dirichlet Problem for the Invariant Laplacian on the Ball
109(6)
3.6 Concluding Remarks
115(2)
Exercises
115(2)
4 Partial Differential Equations
117(30)
4.1 The Idea of Spherical Harmonics
117(1)
4.2 Advanced Topics in the Theory of Spherical Harmonics: The Zonal Harmonics
117(13)
4.3 Spherical Harmonics in the Complex Domain and Applications
130(11)
4.4 An Application to the Bergman Projection
141(6)
Exercises
145(2)
5 Further Geometric Explorations
147(40)
5.1 Introductory Remarks
147(4)
5.2 Semicontinuity of Automorphism Groups
151(5)
5.3 Convergence of Holomorphic Mappings
156(10)
5.3.1 Finite Type in Dimension Two
156(10)
5.4 The Semicontinuity Theorem
166(2)
5.5 Some Examples
168(1)
5.6 Further Remarks
168(1)
5.7 The Lu Qi-Keng Conjecture
169(2)
5.8 The Lu Qi-Keng Theorem
171(3)
5.9 The Dimension of the Bergman Space
174(4)
5.10 The Bergman Theory on a Manifold
178(6)
5.10.1 Kernel Forms
178(4)
5.10.2 The Invariant Metric
182(2)
5.11 Boundary Behavior of the Bergman Metric
184(3)
Exercises
185(2)
6 Additional Analytic Topics
187(64)
6.1 The Diederich--Fornæss Worm Domain
187(5)
6.2 More on the Worm
192(7)
6.3 Non-Smooth Versions of the Worm Domain
199(1)
6.4 Irregularity of the Bergman Projection
200(5)
6.5 Irregularity Properties of the Bergman Kernel
205(2)
6.6 The Kohn Projection Formula
207(1)
6.7 Boundary Behavior of the Bergman Kernel
208(13)
6.7.1 Hormander's Result on Boundary Behavior
209(6)
6.7.2 The Fefferman's Asymptotic Expansion
215(6)
6.8 The Bergman Kernel for a Sobolev Space
221(3)
6.9 Regularity of the Dirichlet Problem on a Smoothly Bounded Domain and Conformal Mapping
224(4)
6.10 Existence of Certain Smooth Plurisubharmonic Defining Functions for Strictly Pseudoconvex Domains and Applications
228(1)
6.10.1 Introduction
228(1)
6.11 Proof of Theorem 6.10.1
229(4)
6.12 Application of the Complex Monge--Ampere Equation
233(2)
6.13 An Example of David Barrett
235(10)
6.14 The Bergman Kernel as a Hilbert Integral
245(6)
Exercises
249(2)
7 Curvature of the Bergman Metric
251(22)
7.1 What is the Scaling Method?
251(1)
7.2 Higher Dimensional Scaling
252(9)
7.2.1 Nonisotropic Scaling
252(2)
7.2.2 Normal Convergence of Sets
254(1)
7.2.3 Localization
255(6)
7.3 Klembeck's Theorem with C2-Stability
261(12)
7.3.1 The Main Goal
261(1)
7.3.2 The Bergman Metric near Strictly Pseudoconvex Boundary Points
262(1)
Exercises
263(10)
8 Concluding Remarks
273(2)
Table of Notation 275(2)
Bibliography 277(10)
Index 287
Steven G. Krantz is one of Springers and Birkhäusers most prolific and popular authors in the field of functional analysis, geometric analysis, and partial differential equations. Krantz is the series editor of Birkhäusers Cornerstones graduate text series and the founder and editor-in-chief of The Journal of Geometric Analysis, considered a society journal previously published by the AMS and often acts as an advisor to several senior editors at Springer/ Birkhäuser . He is also editor-in-chief of the Journal of Mathematical Analysis and Applications. Professor Krantz is currently the editor-in-chief of the AMS Notices and also edits for The American Mathematical Monthly, Complex Analysis and Elliptical Equations, and The Bulletin of the American Mathematical Monthly. Krantz is also known for his wide breadth of expertise in several areas of mathematics such as harmonica analysis, differential geometry, and Lie groups, to name a few. Notable awards include Chauvenet Prize (1992), Beckenbach Book Award (1994), Kemper Prize (1994), Outstanding Academic Book Award (1998), Washington University Faculty Mentor Award (2007).