On the Bergman metric on bounded pseudoconvex domains an approach without the Neumann operator

2014 ◽  
Vol 25 (03) ◽  
pp. 1450025 ◽  
Author(s):  
Gregor Herbort
Keyword(s):  
Distance Function ◽  
Pseudoconvex Domain ◽  
Levi Form ◽  
Pseudoconvex Domains ◽  
Bergman Metric ◽  
Plurisubharmonic Functions ◽  
Euclidean Metric ◽  
Bounded Complex ◽  
Boundary Distance ◽  
Complex Gradient

Let 0 < ε ≤ ½ be fixed. We prove that on a bounded pseudoconvex domain D ⋐ ℂn the Bergman metric grows at least like [Formula: see text] times the euclidean metric, provided that on D there exists a family (φδ)δ of smooth plurisubharmonic functions with a self-bounded complex gradient (uniformly in δ), such that for any δ the Levi form of φδ has eigenvalues ≥ δ-2ε on the set {z ∈ D | δD(z) < δ}. Here, δD denotes the boundary-distance function on D.

Hankel Operators on Pseudoconvex Domains of Finite Type in ℂ2

1998 ◽  
Vol 50 (3) ◽  
pp. 658-672 ◽  
Author(s):  
Frédéric Symesak
Keyword(s):  
Hardy Space ◽  
Finite Type ◽  
Pseudoconvex Domain ◽  
Bergman Spaces ◽  
Hankel Operators ◽  
Weighted Bergman Spaces ◽  
Pseudoconvex Domains ◽  
Sufficient Condition ◽  
Schatten Class ◽  
Strictly Pseudoconvex Domain

AbstractThe aimof this paper is to study small Hankel operators h on the Hardy space or on weighted Bergman spaces,where Ω is a finite type domain in ℂ2 or a strictly pseudoconvex domain in ℂn. We give a sufficient condition on the symbol ƒ so that h belongs to the Schatten class Sp, 1 ≤ p < +∞.

scholarly journals On the existence of Kobayashi and Bergman metrics for Model domains

2020 ◽  
Author(s):  
Nikolay Shcherbina
Keyword(s):  
Entire Function ◽  
Continuous Function ◽  
Pseudoconvex Domain ◽  
Plurisubharmonic Function ◽  
Holomorphic Curves ◽  
List Type ◽  
Bergman Metric ◽  
The Core ◽  
Bounded Smooth ◽  
Bergman Metrics

Abstract We prove that for a pseudoconvex domain of the form $${\mathfrak {A}} = \{(z, w) \in {\mathbb {C}}^2 : v > F(z, u)\}$$ A = { ( z , w ) ∈ C 2 : v > F ( z , u ) } , where $$w = u + iv$$ w = u + i v and F is a continuous function on $${\mathbb {C}}_z \times {\mathbb {R}}_u$$ C z × R u , the following conditions are equivalent: The domain $$\mathfrak {A}$$ A is Kobayashi hyperbolic. The domain $$\mathfrak {A}$$ A is Brody hyperbolic. The domain $$\mathfrak {A}$$ A possesses a Bergman metric. The domain $$\mathfrak {A}$$ A possesses a bounded smooth strictly plurisubharmonic function, i.e. the core $$\mathfrak {c}(\mathfrak {A})$$ c ( A ) of $$\mathfrak {A}$$ A is empty. The graph $$\Gamma (F)$$ Γ ( F ) of F can not be represented as a foliation by holomorphic curves of a very special form, namely, as a foliation by translations of the graph $$\Gamma ({\mathcal H})$$ Γ ( H ) of just one entire function $${\mathcal {H}} : {\mathbb {C}}_z \rightarrow {\mathbb {C}}_w$$ H : C z → C w .

scholarly journals Boundary behavior of the Bergman metric

2002 ◽  
Vol 168 ◽  
pp. 27-40 ◽  
Author(s):  
Bo-Yong Chen
Keyword(s):  
Green Function ◽  
Pseudoconvex Domain ◽  
Boundary Point ◽  
Boundary Behavior ◽  
Sufficient Conditions ◽  
Bergman Metric ◽  
Hölder Continuous ◽  
Pluricomplex Green Function ◽  
Peak Function

AbstractLet Ω be a bounded pseudoconvex domain in Cn. We give sufficient conditions for the Bergman metric to go to infinity uniformly at some boundary point, which is stated by the existence of a Hölder continuous plurisubharmonic peak function at this point or the verification of property (P) (in the sense of Coman) which is characterized by the pluricomplex Green function.

scholarly journals Admissible limits of bloch functions on bounded strongly pseudoconvex domains

1996 ◽  
Vol 54 (1) ◽  
pp. 1-3
Author(s):  
Hu Zhangjian
Keyword(s):  
Pseudoconvex Domain ◽  
Bloch Function ◽  
Pseudoconvex Domains ◽  
Bloch Functions ◽  
Radial Limits ◽  
Strongly Pseudoconvex Domains ◽  
Almost Everywhere ◽  
Strongly Pseudoconvex Domain ◽  
Almost All

Let be a bounded strongly pseudoconvex domain with C2 boundary . In this paper we prove that for a Bloch function in the existance of radial limits at almost all implies the existence of admissible limits almost everywhere on .

A NOTE ON BERGMAN COMPLETENESS

2001 ◽  
Vol 12 (04) ◽  
pp. 383-392 ◽  
Author(s):  
BO-YONG CHEN
Keyword(s):  
Bounded Domain ◽  
Pseudoconvex Domain ◽  
Pseudoconvex Domains

The purpose of this note is to deal with two problems of Kobayashi: 1. Which bounded pseudoconvex domain is Bergman complete? 2. Is a bounded domain Bergman complete if it coincides with its outerhull? We verify Problem 1 for a class of pseudoconvex domains which are not necessary hyperconvex. However, we will show that the answer to Problem 2 is negative in general.

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