The Geometry of Complex Domains
(Sprache: Englisch)
This highly original book examines a rich tapestry of themes and concepts, including complex geometry, Finsler metrics, Bergman and Kahler geometry, curvatures, differential invariants, boundary asymptotics of geometries, group actions, and moduli spaces.
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Produktinformationen zu „The Geometry of Complex Domains “
This highly original book examines a rich tapestry of themes and concepts, including complex geometry, Finsler metrics, Bergman and Kahler geometry, curvatures, differential invariants, boundary asymptotics of geometries, group actions, and moduli spaces.
Klappentext zu „The Geometry of Complex Domains “
This work examines a rich tapestry of themes and concepts and provides a comprehensive treatment of an important area of mathematics, while simultaneously covering a broader area of the geometry of domains in complex space. At once authoritative and accessible, this text touches upon many important parts of modern mathematics: complex geometry, equivalent embeddings, Bergman and Kahler geometry, curvatures, differential invariants, boundary asymptotics of geometries, group actions, and moduli spaces.
The Geometry of Complex Domains can serve as a "coming of age" book for a graduate student who has completed at least one semester or more of complex analysis, and will be most welcomed by analysts and geometers engaged in current research.
Inhaltsverzeichnis zu „The Geometry of Complex Domains “
I. Geometric Background- Complex analysis preliminaries: Riemann Surfaces
- Normal families results: B. Cartan's Theorem (on point fixed, identity differential implies identity)
- The problem of equivalence in one and several complex variables (Generalities)
- The Riemann mapping theorem and the unformization theorem
- Why these theorems fail in higher dimensions (e.g., why domains near the ball are typically not the ball in the circular domain case as an example)
- Moduli spaces for one-dimensional domains of finite connectivity with comments on why the situation is different in several variables
- The existence of local invariants of the boundary from the viewpoint of counting parameters (why Tanaka-Chern-Moser invariants exist)
- Generic inequivalence of domains
- Braun-Kaup-Upmeier's Theorem on Reinhardt domains that are equivalent are linearly equivalent and thus many examples of inequivalent domains
II. The Equivalence problem from the intrinsic viewpoint
- Intrinsic metrics: Why Cartan's theorem implies that they exist in principle
- Kobayashi metric basics
- Caratheodory metric basics
- Kahler metrics in general (notational summary--curvature of etc.)
- The Bergman kernel function
- The Bergman metric
- Specific examples and the general results on how the metric varies with the domain (Ramadanov's theorem)
- The continuous variation with domain (Greene and Krantz)---outline of proof only
- Applications of the continuous dependence of Bergman metric on the domain: closure of equivalence classes. Existence of fixed points. (include Lempert's convexity thing here?) NO semicontinuity result--later for that
- Constant curvature: Lu Qi-Keng's theorem
III. More on the Kobayashi and Caratheodory Metrics
- Invariance properties of the metrics
- Applications to existence and non-existence of holomorphic mappings (Eisenman theory)
- Semi-continuity of automorphism groups
IV. Automorphisms and Mappings
- Generalities and the automorphism
... mehr
group as a Lie group
- Normal families results of the general sort (some of those will be done in effect in the introduction)
- The scaling method and its consequences (this might need to occupy several sections)
- Bedford and Pinchuk on automorphisms of smooth boundary things in C^2
- V. Complex Manifolds
- Stein manifold generalities
- Invariant metrics (and when they exist) on manifolds
- (this will be fairly extended--e.g. Greene and Wu conditions for the Bergman metric and their faster than quadratic negative curvature condition for Kobayashi hyperbolic)
- Conditions for the automorphism group to be a Lie group
- More on Riemann surfaces
- Complex Manifolds with Many Automorphisms Oeljeklaus and Huckleberry theory
- A Summary of classical homogeneous space results
- The Mostow-Siu example
- Normal families results of the general sort (some of those will be done in effect in the introduction)
- The scaling method and its consequences (this might need to occupy several sections)
- Bedford and Pinchuk on automorphisms of smooth boundary things in C^2
- V. Complex Manifolds
- Stein manifold generalities
- Invariant metrics (and when they exist) on manifolds
- (this will be fairly extended--e.g. Greene and Wu conditions for the Bergman metric and their faster than quadratic negative curvature condition for Kobayashi hyperbolic)
- Conditions for the automorphism group to be a Lie group
- More on Riemann surfaces
- Complex Manifolds with Many Automorphisms Oeljeklaus and Huckleberry theory
- A Summary of classical homogeneous space results
- The Mostow-Siu example
... weniger
Autoren-Porträt von Robert E. Greene, Kang-Tae Kim, Steven G. Krantz
Steven G. Krantz received the B.A. degree from the University of California at Santa Cruz and the Ph.D. from Princeton University. He has taught at UCLA, Princeton, Penn State, and Washington University, where he has most recently served as Chair of the Mathematics Department.
Krantz has directed 18 Ph.D. Students and 9 Masters students, and is winner of the Chauvenet Prize and the Beckenbach Book Award. He edits six journals and is Editor-in-Chief of three.
A prolific scholar, Krantz has published more than 55 books and more than 160 academic papers.
Bibliographische Angaben
- Autoren: Robert E. Greene , Kang-Tae Kim , Steven G. Krantz
- 2011, 303 Seiten, Maße: 16,5 x 24,5 cm, Gebunden, Englisch
- Verlag: Springer
- ISBN-10: 0817641394
- ISBN-13: 9780817641399
Sprache:
Englisch
Rezension zu „The Geometry of Complex Domains “
From the reviews:"The book under review gives an excellent presentation of modern problems related to various characterizations of the holomorphic geometry of domains in Cn and complex manifolds. ... The book may be strongly recommended for researchers and Ph.D. students working in complex analysis." (Marek Jarnicki, Mathematical Reviews, Issue 2012 c)
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