Geometric analysis of the Bergman kernel and metric:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York [u.a.]
Springer
2013
|
| Series: | Graduate texts in mathematics
268 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis Klappentext |
| Item Description: | Literaturverz. S. 277 - 285 |
| Physical Description: | XIII, 292 S. graph. Darst. |
| ISBN: | 9781461479239 |
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MARC
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| 245 | 1 | 0 | |a Geometric analysis of the Bergman kernel and metric |c Steven G. Krantz |
| 264 | 1 | |a New York [u.a.] |b Springer |c 2013 | |
| 300 | |a XIII, 292 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Graduate texts in mathematics |v 268 | |
| 500 | |a Literaturverz. S. 277 - 285 | ||
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Record in the Search Index
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|---|---|
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Contents
Introductory Ideas
. 1
]. 1
The Bergman Kernel
. 1
1.1.1
Calculating the Bergman Kernel
. 14
1.1.2
The
Poincaré-Bergman
Distance on the Disc
. 19
1.1.3
Construction of the Bergman Kernel by Way
of
Di fferential
Equations
. 20
1.1.4
Construction of the Bergman Kernel by Way
of
Conformai Invariance.
23
1.2
The
Szegő
and
Poisson—
Szegő
Kernels
. 25
1.3
Formal Ideas of Aronszajn
. 35
1.4
A New Bergman Basis
. 36
1.5
Further Examples
. 39
1.6
A Real Bergman Space
. 40
1.7
The Behavior of the Singularity in a General Setting
. 41
1.8
The Annulus
. 43
1.9
A Direct Connection Between the Bergman and
Szegő
Kernels
_ 45
1.9.1
Introduction
. 45
J
.9.2
The Case of the Disc
. 45
1.9.3
The Unit Ball in
С
. 49
1.9.4
Strongly Pseudoconvex Domains
. 52
1.9.5
Concluding Remarks
. 53
1.10
Multiply Connected Domains
. 53
1.
Π
The Bergman Kernel for a Sobolev Space
. 54
1.12
Ramadanov's Theorem
. 56
1.13
Coda on the
Szegő
Kernel
. 58
1.14
Boundary Localization
. 59
1.14.1
Definitions and Notation
. 60
1.14.2
A Representative Result
. 60
Xl
Contents
1.14.3
The More
General
Result in the Plane
. 62
1.14.4
Domains in Higher-Dimensional Complex Space
. 62
Exercises
. 65
The Bergman Metric
. 71
2.1
Smoothness to the Boundary
of Biholomorphic Mappings
. 71
2.2
Boundary Behavior of the Bergman Metric
. 81
2.3
The Biholomorphic Inequivalence of the Ball and the Polydisc
---- 83
Exercises
. 84
Further Geometric and Analytic Theory
. 87
3.1
Bergman Representative Coordinates
. 87
3.2
The Berezin Transform
. 90
3.2.1
Preliminary Remarks
. 90
3.2.2
Introduction to the
Poisson-Bergman
Kernel
. 91
3.2.3
Boundary Behavior
. 94
3.3
Ideas of Fefferman
. 98
3.4
Results on the Invariant Laplacian
. 100
3.5
The Dirichlet Problem for the Invariant Laplacian on the Ball
. 109
3.6
Concluding Remarks
. 115
Exercises
. 115
Partial Differential Equations
. 117
4.1
The Idea of Spherical
Harmonies
. 117
4.2
Advanced Topics in the Theory of Spherical Harmonics:
The Zonal Harmonics
. 117
4.3
Spherical Harmonics in the Complex Domain and Applications.
_ 130
4.4
An Application to the Bergman Projection
. 141
Exercises
. 145
Further Geometric Explorations
. 147
5.1
Introductory Remarks
.
1
47
5.2
Semicontinuity of Automorphism Groups
. 151
5.3
Convergence of Holomorphic Mappings
. 156
5.3.1
Finite Type in Dimension Two
.
1
56
5.4
The Semicontinuity Theorem
. 166
5.5
Some Examples
. 168
5.6
Further Remarks
. 168
5.7
The
Lu
Qi-Keng Conjecture
. 169
5.8
The
Lu
Qi-Keng Theorem
. 171
5.9
The Dimension of the Bergman Space
. 174
5.10
The Bergman Theory on a Manifold
. 178
5.10.1
Kernel Forms
. 1 78
5.10.2
The Invariant Metric
. 182
5.11
Boundary Behavior of the Bergman Metric
. 184
Exercises
. 1 85
Contents xiii
6
Additional Analytic Topics
. 187
6.1
The Diederich—Fornaess Worm Domain
. 187
6.2
More on the Worm
. 192
6.3
Non-Smooth Versions of the Worm Domain
. 199
6.4
Irregularity of the Bergman Projection
. 200
6.5
Irregularity Properties of the Bergman Kernel
. 205
6.6
The Kohn Projection Formula
. 207
6.7
Boundary Behavior of the Bergman Kernel
. 208
6.7.1 Hörmander's
Result on Boundary Behavior
. 209
6.7.2
The Fefferman's Asymptotic Expansion
. 215
6.8
The Bergman Kernel for a Sobolev Space
. 221
6.9
Regularity of the Dirichlet Problem on a Smoothly
Bounded Domain and
Conformai
Mapping
. 224
6.10
Existence of Certain Smooth Plurisubharmonic Defining
Functions for Strictly Pseudoconvex Domains and Applications
. 228
6.10.1
Introduction
. 228
6.11
Proof of Theorem
6.10.1 . 229
6.12
Application of the Complex
Monge-Ampère
Equation
. 233
6.13
An Example of David Barrett
. 235
6.14
The Bergman Kernel as a Hubert Integral
. 245
Exercises
. 249
7
Curvature of the Bergman Metric
. 251
7.1
What is the Scaling Method?
. 251
7.2
Higher Dimensional Scaling
. 252
7.2.1
Nonisotropic Scaling
. 252
7.2.2
Normal Convergence of Sets
. 254
7.2.3
Localization
. 255
7.3
Klembeck's Theorem with C2-Stability
. 261
7.3.1
The Main Goal
. 261
7.3.2
The Bergman Metric near Strictly
Pseudoconvex Boundary Points
. 262
Exercises
. 263
8
Concluding Remarks
. 273
Table of Notation
. 275
Bibliography
. 277
Index
. 287
Steven
G.
Krantz
Geometric Analysis of the Bergman Kernel and Metric
With Steven Krantz's usual flair for clarity and motivation, this text provides a
masterful and systematic treatment of all the basic analytic and geometric aspects of
Bergman's theory including calculation,
invariance
properties, boundary asymptotics,
and asymptotic expansions of the Bergman kernel and metric. This text includes a
unique compendium of results with applications to function theory, geometry, partial
differential equations, and interpretations in terms of functional analysis. 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 both to
graduate students and to experienced mathematicians.
Graduate students who have taken courses in complex variables and have a basic
background in real and functional analysis will be able to access this textbook.
Applicable courses for either main or supplementary text 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.
Mathematics
ISBN
978-1-4614-7923-9
i!
II
9
І781
461
H
479239
►
springer.com |
| any_adam_object | 1 |
| author | Krantz, Steven G. 1951- |
| author_GND | (DE-588)130535907 |
| author_facet | Krantz, Steven G. 1951- |
| author_role | aut |
| author_sort | Krantz, Steven G. 1951- |
| author_variant | s g k sg sgk |
| building | Verbundindex |
| bvnumber | BV041411199 |
| classification_rvk | SI 990 SK 780 |
| ctrlnum | (OCoLC)862965496 (DE-599)BSZ394130553 |
| dewey-full | 515.98 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515.98 |
| dewey-search | 515.98 |
| dewey-sort | 3515.98 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV041411199 |
| illustrated | Illustrated |
| indexdate | 2024-09-24T00:16:24Z |
| institution | BVB |
| isbn | 9781461479239 |
| language | English |
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| physical | XIII, 292 S. graph. Darst. |
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| publisher | Springer |
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| series | Graduate texts in mathematics |
| series2 | Graduate texts in mathematics |
| spelling | Krantz, Steven G. 1951- Verfasser (DE-588)130535907 aut Geometric analysis of the Bergman kernel and metric Steven G. Krantz New York [u.a.] Springer 2013 XIII, 292 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Graduate texts in mathematics 268 Literaturverz. S. 277 - 285 Bergman-Kernfunktion (DE-588)4236138-2 gnd rswk-swf Bergman-Metrik (DE-588)4144650-1 gnd rswk-swf Bergman-Kernfunktion (DE-588)4236138-2 s DE-604 Bergman-Metrik (DE-588)4144650-1 s Erscheint auch als Online-Ausgabe 978-1-4614-7924-6 Graduate texts in mathematics 268 (DE-604)BV000000067 268 Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026858494&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026858494&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Krantz, Steven G. 1951- Geometric analysis of the Bergman kernel and metric Graduate texts in mathematics Bergman-Kernfunktion (DE-588)4236138-2 gnd Bergman-Metrik (DE-588)4144650-1 gnd |
| subject_GND | (DE-588)4236138-2 (DE-588)4144650-1 |
| title | Geometric analysis of the Bergman kernel and metric |
| title_auth | Geometric analysis of the Bergman kernel and metric |
| title_exact_search | Geometric analysis of the Bergman kernel and metric |
| title_full | Geometric analysis of the Bergman kernel and metric Steven G. Krantz |
| title_fullStr | Geometric analysis of the Bergman kernel and metric Steven G. Krantz |
| title_full_unstemmed | Geometric analysis of the Bergman kernel and metric Steven G. Krantz |
| title_short | Geometric analysis of the Bergman kernel and metric |
| title_sort | geometric analysis of the bergman kernel and metric |
| topic | Bergman-Kernfunktion (DE-588)4236138-2 gnd Bergman-Metrik (DE-588)4144650-1 gnd |
| topic_facet | Bergman-Kernfunktion Bergman-Metrik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026858494&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026858494&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000000067 |
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