arXiv subjects

Mohammed Salouf

Publications and source records attributed to Mohammed Salouf.

4 recordsLinked to original sources

arXiv2025

Monge-Ampère equations with prescribed singularities on compact Hermitian manifolds

Given a compact complex manifold $X$, we study the existence and the uniqueness of weak solutions to degenerate Monge-Ampère equations on $X$ with prescribed singularities when the reference form is semipositive and big, while the right hand side is a non-pluripolar positive Radon measure. This generalizes our previous work to more general hermitian manifolds and also to the case of solutions with prescribed singularities.

math.CV↗

arXiv2025

The Range of the Monge-Ampère operator $(ω+ dd^c .)^n$ in bounded domains

Let $Ω$ be a bounded strictly pseudoconvex domain of $\mathbb{C}^n$. We solve degenerate complex Monge-Ampère equations of the form $(ω+ dd^c φ)^n = μ$ in the generalized Cegrell classes $\mathcal{K}(Ω,ω,H)$, where $H \in \mathcal{E}(Ω)$ is maximal, $ω$ is a smooth real $(1,1)$-form defined in a neighborhood of $\barΩ$ and $μ$ is a positive Radon measure. This generalizes the previous work of the last author \cite{Sal25} to the case of non-continuous functions $H$ and also to the case of measures $μ$ which do not vanish on pluripolar sets.

math.CV↗

arXiv2024

Degenerate complex Monge-Ampère equations on some compact Hermitian manifolds

Let $X$ be a compact complex manifold which admits a hermitian metric satisfying a curvature condition introduced by Guan-Li. Given a semipositive form $θ$ with positive volume, we define the Monge-Ampère operator for unbounded $θ$-psh functions and prove that it is continuous with respect to convergence in capacity. We then develop pluripotential tools to study degenerate complex Monge-Ampère equations in this context, extending recent results of Tosatti-Weinkove, Kolodziej-Nguyen, Guedj-Lu and many others who treat bounded solutions.

math.CV↗

arXiv2023

Degenerate complex Monge-Ampère equations with non-Kähler forms in bounded domains

In this paper, we study weak solutions to complex Monge-Ampère equations of the form $(ω+ dd^c φ)^n= F(φ,.)dμ$ on a bounded strictly pseudoconvex domain in $\mathbb{C}^n$, where $ω$ is a smooth $(1,1)$-form, $0\leq F$ is a continuous non-decreasing function, and $μ$ is a positive non-pluripolar measure. Our results extend previous works of Kołodziej and Nguyen \cite{KN15,KN23a,KN23b} who study bounded solutions, as well as Cegrell \cite{Ceg98,Ceg04,Ceg08}, Czyż \cite{Cz09}, Benelkourchi \cite{Ben09,Ben15} and others who treat the case when $ω=0$ and/or $F=1$.

math.CV↗