Negative Plurisubharmonic Functions Research Articles

Rooftop envelopes and residual plurisubharmonic functions

Given a negative plurisubharmonic function $\phi $ in a bounded pseudoconvex domain of $\mathbb C^n$, we introduce and study its <em>residual function</em> $g_\phi $ determined by the asymptotic behavior of $\phi $ near its singularity points, both inside

(Pluri)Potential Compactifications

Using pluricomplex Green functions we introduce a compactification of a complex manifold M invariant with respect to biholomorphisms similar to the Martin compactification in the potential theory. For this we show the existence of a norming volume form V on M such that all negative plurisubharmonic functions on M are in L1(M, V ). Moreover, the set of such functions with the norm not exceeding 1 is compact. Identifying a point w ∈ M with the normalized pluricomplex Green function with pole at w we get an imbedding of M into a compact set and the closure of M in this set is the pluripotential compactification.

An ordering of measures induced by plurisubharmonic functions

We study an ordering of measures induced by plurisubharmonic functions. This ordering arises naturally in connection with problems related to negative plurisubharmonic functions. We study maximalit ...

Vector spaces of delta-plurisubharmonic functions and extensions of the complex Monge–Ampère operator

In this paper we shall consider two types of vector ordering on the vector space of differences of negative plurisubharmonic functions, and the problem whether it is possible to construct supremum and infimum. Then we consider two different approaches to define the complex Monge–Ampère operator on these vector spaces, and we solve some Dirichlet problems. We end this paper by stating and discussing some open problems.

Monge–Ampère measures of maximal subextensions of plurisubharmonic functions with given boundary values

The aim of the paper is to investigate the Monge–Ampère measures of maximal subextensions of plurisubharmonic functions with given boundary values. As an application, we study the approximation of negative plurisubharmonic function with given boundary values by an increasing sequence of plurisubharmonic functions defined in larger domains.

An essay on Bergman completeness

We give first of all a new criterion for Bergman completeness in terms of the pluricomplex Green function. Among several applications, we prove in particular that every Stein subvariety in a complex manifold admits a Bergman complete Stein neighborhood basis, which improves a theorem of Siu. Secondly, we give for hyperbolic Riemann surfaces a sufficient condition for when the Bergman and Poincaré metrics are quasi-isometric. A consequence is an equivalent characterization of uniformly perfect planar domains in terms of growth rates of the Bergman kernel and metric. Finally, we provide a noncompact Bergman complete pseudoconvex manifold without nonconstant negative plurisubharmonic functions.

APPROXIMATION OF NEGATIVE PLURISUBHARMONIC FUNCTIONS WITH GIVEN BOUNDARY VALUES

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.

On a Monge–Ampère type equation in the Cegrell class Eχ

Let Omega be a bounded hyperconvex domain in C-n and let mu be a positive and finite measure which vanishes on all pluripolar subsets of Omega. We prove that for every continuous and strictly increasing function chi : (-infinity, 0) -> (-infinity, 0) there exists a negative plurisubharmonic function u which solves the Monge-Ampere type equation -chi(u)(dd(c)u)(n) = d mu. Under some additional assumption the solution u is uniquely determined.

Modulability and duality of certain cones in pluripotential theory

Let p > 0 , and let E p denote the cone of negative plurisubharmonic functions with finite pluricomplex p-energy. We prove that the vector space δ E p = E p − E p , with the vector ordering induced by the cone E p is σ-Dedekind complete, and equipped with a suitable quasi-norm it is a non-separable quasi-Banach space with a decomposition property with control of the quasi-norm. Furthermore, we explicitly characterize its topological dual. The cone E p in the quasi-normed space δ E p is closed, generating, and has empty interior.

Subextension and approximation of negative plurisubharmonic functions

In this thesis we study approximation of negative plurisubharmonic functions by functions defined on strictly larger domains. We show that, under certain conditions, every function u that is defined on a bounded hyperconvex domain Ω in Cn and has essentially boundary values zero and bounded Monge-Ampere mass, can be approximated by an increasing sequence of functions {uj} that are defined on strictly larger domains, has boundary values zero and bounded Monge-Ampere mass. We also generalize this and show that, under the same conditions, the approximation property is true if the function u has essentially boundary values G, where G is a plurisubharmonic functions with certain properties. To show these approximation theorems we use subextension. We show that if Ω_1 and Ω_2 are hyperconvex domains in Cn and if u is a plurisubharmonic function on Ω_1 with given boundary values and with bounded Monge-Ampere mass, then we can find a plurisubharmonic function u defined on Ω_2, with given boundary values, such that u <= u on Ω and with control over the Monge-Ampere mass of u.

On subextension of pluriharmonic and plurisubharmonic functions

The problem of subextension of plurisubharmonic functions is considered. Recently it was shown by Cegrell–Zeriahi, that subextension is always possible for negative plurisubharmonic functions in the energy class ℱ. In this paper we construct, for every hyperconvex domain Ω, a negative plurisubharmonic function in the class ℰ which cannot be subextended. Given any pseudoconvex domain we construct a pluriharmonic function that cannot be subextended.

GREEN FUNCTIONS WITH SINGULARITIES ALONG COMPLEX SPACES

We study properties of a Green function GAwith singularities along a complex subspace A of a complex manifold X. It is defined as the largest negative plurisubharmonic function u satisfying locally u≤ log |ψ|+C, where ψ=(ψ1,…,ψm), ψ1,…,ψmare local generators for the ideal sheaf ℐAof A, and C is a constant depending on the function u and the generators. A motivation for this study is to estimate global bounded functions from the sheaf ℐAand thus proving a "Schwarz lemma" for ℐA.

Green functions with analytic singularities

We study properties of a Green function with singularities determined by a closed complex subspace A of a complex manifold X. It is defined as the largest negative plurisubharmonic function u satisfying locally u ⩽ log | ψ | + O ( 1 ) , where ψ = ( ψ 1 , … , ψ m ) with ψ 1 , … , ψ m local generators for the ideal sheaf I A of A. To cite this article: A. Rashkovskii, R. Sigurdsson, C. R. Acad. Sci. Paris, Ser. I 340 (2005).

Subextension of plurisubharmonic functions with bounded Monge–Ampère mass

Let Ω⋐ C n be a hyperconvex domain. Denote by E 0(Ω) the class of negative plurisubharmonic functions ϕ on Ω with boundary values 0 and finite Monge–Ampère mass on Ω. Then denote by F(Ω) the class of negative plurisubharmonic functions ϕ on Ω for which there exists a decreasing sequence ( ϕ) j of plurisubharmonic functions in E 0(Ω) converging to ϕ such that sup j∫ Ω(dd cϕ j) n+∞. It is known that the complex Monge–Ampère operator is well defined on the class F(Ω) and that for a function ϕ∈ F(Ω) the associated positive Borel measure is of bounded mass on Ω. A function from the class F(Ω) is called a plurisubharmonic function with bounded Monge–Ampère mass on Ω. We prove that if Ω and Ω are hyperconvex domains with Ω⋐ Ω⋐ C n and ϕ∈ F(Ω), there exists a plurisubharmonic function ϕ ̃ ∈ F( Ω) such that ϕ ̃ ⩽ϕ on Ω and ∫ Ω (dd c ϕ ̃ ) n⩽∫ Ω(dd cϕ) n. Such a function is called a subextension of ϕ to Ω. From this result we deduce a global uniform integrability theorem for the classes of plurisubharmonic functions with uniformly bounded Monge–Ampère masses on Ω. To cite this article: U. Cegrell, A. Zeriahi, C. R. Acad. Sci. Paris, Ser. I 336 (2003).

Approximation of plurisubharmonic functions by multipole Green functions

For a strongly hyperconvex domain D C C n we prove that multipole pluricomplex Green functions are dense in the cone in L 1 (D) of negative plurisubharmonic functions with zero boundary values.

Plurisubharmonic extremal functions, Lelong numbers and coherent ideal sheaves

We introduce a new type of pluricomplex Green function which has a logarithmic pole along a complex subspace A of a complex manifold X. It is the largest negative plurisubharmonic function on X whose Lelong number is at least the Lelong number of log max{|f_1|,...,|f_m|}, where f_1,...,f_m are local generators for the ideal sheaf of A. The pluricomplex Green function with a single logarithmic pole or a finite number of weighted poles is a very special case of our construction. We give several equivalent definitions of this function and study its properties, including boundary behaviour, continuity, and uniqueness. This is based on and extends our previous work on disc functionals and their envelopes.

The invariant pseudometric related to negative plurisubharmonic functions

The invariant pseudometric related to negative plurisubharmonic functions