Monge-Ampère measures on subvarieties
type: Item
instance of: scholarly article
title: Monge-Ampère measures on subvarieties
zbMATH Open document ID: 1302.32040
author: Per Åhag
author: Urban Cegrell
author: Hoàng Hiệp Phạm
published in: Journal of Mathematical Analysis and Applications
publication date: 2014-10-31T00:00:00Z
full work available at URL: https://arxiv.org/abs/1403.7412
Mathematics Subject Classification ID: 32W20
Mathematics Subject Classification ID: 32U05
Mathematics Subject Classification ID: 32U15
zbMATH DE Number: 6362987
zbMATH Keywords: complex Monge-Ampère operator
zbMATH Keywords: Dirichlet problem
zbMATH Keywords: pluripolar set
zbMATH Keywords: plurisubharmonic function
MaRDI profile type: Publication
OpenAlex ID: W2000123285
arXiv ID: 1403.7412
cites work: Monge-Ampère measures on pluripolar sets
cites work: Partial pluricomplex energy and integrability exponents of plurisubharmonic functions
cites work: A product property for the pluricomplex energy
cites work: The Dirichlet problem for a complex Monge-Ampère equation
cites work: A new capacity for plurisubharmonic functions
cites work: A thermodynamical formalism for Monge-Ampère equations, Moser-Trudinger inequalities and Kähler-Einstein metrics
cites work: Equlibrium measure of a product subset of ℂⁿ
cites work: Pluricomplex energy
cites work: A general Dirichlet problem for the complex Monge–Ampère operator
cites work: Subextension and approximation of negative plurisubharmonic functions
cites work: Fundamental solutions of the complex Monge-Ampère equation
cites work: The complex Monge–Ampère operator
cites work: Mesures de Monge-Ampère et mesures pluriharmoniques. (Monge-Ampère measures and pluriharmonic measures)
cites work: Nombres de Lelong généralisés, théorèmes d'intégralité et d'analyticité. (Generalized Lelong numbers, integrability and analyticity theorems)
cites work: A sharp lower bound for the log canonical threshold
cites work: Characterization of Monge-Ampère measures with Hölder continuous potentials
cites work: Exponential estimates for plurisubharmonic functions
cites work: Holomorphic discs and the complex Monge-Ampère equation.
cites work: Notions of convexity
cites work: Q4524284
cites work: Extremal plurisubharmonic functions and invariant pseudodistances
cites work: The complex Monge-Ampère equation
cites work: Monge-Ampere equation on compact Kahler manifolds
cites work: The complex Monge-Ampère equation and pluripotential theory
cites work: Q4538023
cites work: La métrique de Kobayashi et la représentation des domaines sur la boule
cites work: Solving the degenerate complex Monge-Ampère equation with one concentrated singularity
cites work: A comparison principle for the complex Monge-Ampère operator in Cegrell’s classes and applications
cites work: Pluripolar sets and the subextension in Cegrell's classes
cites work: Complex Monge Ampere Equations
cites work: On the singularities of the pluricomplex Green's function
cites work: Analyticity of sets associated to Lelong numbers and the extension of closed positive currents
DOI: 10.1016/J.JMAA.2014.09.061
Recommended article: Monge–Ampère measures of $\mathcal F$-plurisubharmonic functions
Recommended article: Monge-Ampère measures on pluripolar sets
Recommended article: Monge–Ampère measures of maximal subextensions of plurisubharmonic functions with given boundary values
Recommended article: Monge-Ampere Measures Associated to Extremal Plurisubharmonic Functions in C n
Recommended article: Monge–Ampère measures of plurisubharmonic exhaustions associated to the Lie norm of holomorphic maps
Recommended article: Subextension of plurisubharmonic functions without changing the Monge-Ampère measures and applications
Recommended article: Complex Monge-Ampère equation for measures supported on real submanifolds
Recommended article: Monge–Ampère boundary measures
Recommended article: An equality on Monge-Ampère measures
Recommended article: On measures in sub-Riemannian geometry
review text: Let \(\Omega\subset \mathbb C^n\) be a bounded hyperconvex domain, let \(\mathcal E(\Omega)\) be the maximal class of negative plurisubharmonic functions for which the complex Monge-Ampère operator \((dd^c\cdot)^n\) is well defined. For a complex manifold \(X\) let NEWLINE\[NEWLINE \mathcal E(X)=\big\{u\in \text{PSH}(X): \forall \, x\in X,\, \exists \text{ a neighborhood } W \text{ of }z\text{ such that } u\in \mathcal E(W)\big\}. NEWLINE\]NEWLINENEWLINENEWLINEIn this paper the authors study the Dirichlet problem for the complex Monge-Ampère operator \((dd^cu)^n=\mu\), for singular measures \(\mu\) (i.e., for measures supported by pluripolar sets).NEWLINENEWLINEThe main theorem is the following one. Let \(S\) be a subvariety in \(\Omega\) with dimension \(k<n\). If \(v\in \mathcal E(S)\) satisfies \((dd^cv)^k(\{v>-\infty\})=0\) then there exists \(u\in \mathcal E(\Omega)\) such that \((dd^cu)^n=(dd^cv)^k\).NEWLINENEWLINEThe toric case is also considered. Let \(\Delta\) be the unit ball in \(\mathbb C\). Let \(u\in \mathcal E(\Delta^n)\) be such that \newline \(u(z_1,\dots,z_k,z_{k+1},\dots,z_n)=u(z_1,\dots,z_k,|z_{k+1}|,\dots,|z_n|)\). Then there exists a Borel pluripolar set \(E\subset \Delta^k\) such that \((dd^cu)^n((\Delta^k\setminus E)\times \{0\}))=0\). In particular there is no function \(u\in \mathcal E(\Delta^n)\) such that NEWLINE\[NEWLINE u(z_1,\dots,z_k,z_{k+1},\dots,z_n)=u(z_1,\dots,z_k,|z_{k+1}|,\dots,|z_n|) \;\text{ and } \;(dd^cu)^n=\mu\times \delta_0^{n-k}, NEWLINE\]NEWLINE where \(\mu\) is any non-negative Radon measure on \(\Delta^k\) vanishing on all pluripolar sets.
description: scientific article; zbMATH DE number 6362987