THE DIRICHLET PROBLEM FOR MAXIMAL PLURISUBHARMONIC FUNCTIONS ON ANALYTIC VARIETIES IN ℂn

2009 ◽  
Vol 20 (04) ◽  
pp. 521-528 ◽  
Author(s):  
FRANK WIKSTRÖM
Keyword(s):  
Dirichlet Problem ◽  
Continuous Function ◽  
Plurisubharmonic Function ◽  
Boundary Values ◽  
Regular Domain ◽  
Analytic Variety ◽  
Plurisubharmonic Functions ◽  
Analytic Varieties

Let Ω be a B-regular domain in ℂn and let V be a locally irreducible analytic variety in Ω. Given a continuous function [Formula: see text], we prove that there is a unique maximal plurisubharmonic function u on V with boundary values given by ϕ and furthermore that u is continuous on [Formula: see text].

APPROXIMATION OF NEGATIVE PLURISUBHARMONIC FUNCTIONS WITH GIVEN BOUNDARY VALUES

2010 ◽  
Vol 21 (09) ◽  
pp. 1135-1145 ◽  
Author(s):  
LISA HED
Keyword(s):  
Plurisubharmonic Function ◽  
Boundary Values ◽  
Plurisubharmonic Functions

In this paper, we study the approximation of negative plurisubharmonic functions with given boundary values. We want to approximate a plurisubharmonic function by an increasing sequence of plurisubharmonic functions defined on strictly larger domains.

Green's Forms and Meromorphic Functions on Compact Analytic Varieties

1951 ◽  
Vol 3 ◽  
pp. 108-128 ◽  
Author(s):  
Kunihiko Kodaira
Keyword(s):  
Meromorphic Function ◽  
Meromorphic Functions ◽  
Zero Point ◽  
Positive Definite ◽  
Analytic Surface ◽  
Analytic Variety ◽  
Complex Dimension ◽  
Analytic Varieties ◽  
Complex Analytic Variety ◽  
Zero Points

Let be a compact complex analytic variety of the complex dimension n with a positive definite Kâhlerian metric [4] ; the local analytic coordinates on will be denoted by z = (z 1 z 2, … , zn). Now, suppose a meromorphic function f(z) defined on as given. Then the poles and zero-points of f(z) constitute an analytic surface in consisting of a finite number of irreducible closed analytic surfaces Γ1, Γ2, … , Γk, each of which is a polar or a zero-point variety of f(z).

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 Components of Riemann Surfaces

1954 ◽  
Vol 7 ◽  
pp. 65-83
Author(s):  
Makoto Ohtsuka
Keyword(s):  
Dirichlet Problem ◽  
Riemann Surface ◽  
Conformal Mapping ◽  
Riemann Surfaces ◽  
Boundary Values ◽  
Abstract Riemann Surface

The boundary components of an abstract Riemann surface were defined by B. v. Kérékjértó [7] and utilized in the book [14] written by S. Stoïlow. It is the purpose of the present paper to investigate their images under conformal mapping and to solve the Dirichlet problem with boundary values distributed on them.

scholarly journals Approximation of plurisubharmonic functions on complex varieties

2017 ◽  
Vol 28 (14) ◽  
pp. 1750107
Author(s):  
Nguyen Quang Dieu ◽  
Tang Van Long ◽  
Sanphet Ounheuan
Keyword(s):  
Continuous Function ◽  
Bounded Domain ◽  
Sufficient Conditions ◽  
Continuous Functions ◽  
Plurisubharmonic Functions ◽  
Complex Variety ◽  
Complex Varieties

Let [Formula: see text] be a complex variety in a bounded domain [Formula: see text] in [Formula: see text]. We are interested in finding sufficient conditions on [Formula: see text] so that plurisubharmonic functions which are bounded from above on [Formula: see text] can be approximated from above by continuous functions on [Formula: see text] and plurisubharmonic on [Formula: see text] Next, we discuss the possibility to extend a given real valued continuous function on [Formula: see text] to a maximal plurisubharmonic on [Formula: see text] which is continuous up to the boundary.

scholarly journals On Dirichlet's principle and problem

2012 ◽  
Vol 110 (2) ◽  
pp. 235 ◽  
Author(s):  
Per Åhag ◽  
Urban Cegrell ◽  
Rafal Czyz
Keyword(s):  
Dirichlet Problem ◽  
Variational Methods ◽  
Complete Characterization ◽  
Plurisubharmonic Functions ◽  
Monge Ampere

The aim of this paper is to give a new proof of the complete characterization of measures for which there exists a solution of the Dirichlet problem for the complex Monge-Ampere operator in the set of plurisubharmonic functions with finite pluricomplex energy. The proof uses variational methods.

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