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Ahmed Zeriahi

Publications and source records attributed to Ahmed Zeriahi.

At least 19 recordsLinked to original sources
arXiv2026

Complex Monge-Amp{è}re equations with boundary values: a variational approach

Let $Ω$ ___ C n be a bounded hyperconvex domain and $Φ$\,: $Ω$ $\rightarrow$ R a maximal plurisubharmonic function in $Ω$ (possibly unbounded). We introduce a new class E 1 $Φ$ ($Ω$) of plurisubharmonic functions with boundary values equal to $Φ$ in the sense of Cegrell and finite $Φ$-energy on $Ω$. Then given a Borel measure $μ$ on $Ω$, we apply a variational approach to prove that $μ$ is the Monge-Amp{è}re measure of a function u $\in$ E 1 $Φ$ ($Ω$) if and only if it satisfies the integrability condition We also establish a quantitative version of this condition extending previous results of Cegrell and others. Using a logarithmic transformation, we obtain analogous results for the real Monge-Amp{è}re equations with (unbounded) boundary data.

math.AP↗
arXiv2026

Complex Monge-Amp{è}re equations on Hermitian Manifolds: From bounded to smooth solutions

We study regularity of bounded weak solutions to complex Monge-Amp{è}re equations on compact Hermitian manifolds with right-hand side possibly decreasing in the unknown. Our proof relies on the domination principle and a priori estimates, partially motivated by the Monge-Amp{è}re eigenvalue problem. Our main result roughly says that any bounded solution is smooth in the regular locus of the data. When the right-hand side is strictly positive and smooth, we recover known results by Nie, Ko lodziej-Nguyen for the Hermitian case and Sz{é}kelyhidi-Tosatti for the K{ä}hler case, which rely on the regularizing property of the K{ä}hler-Ricci flow.

math.CV↗
arXiv2026

The Eigenvalue Problem for the complex Monge-Ampère operator

We prove the existence of the first eigenvalue and an associated eigenfunction with Dirichlet condition for the complex Monge-Ampère operator on a bounded strongly pseudoconvex domain in $\C^n$. We show that the eigenfunction is plurisubharmonic, smooth with bounded Laplacian in $Ω$ and boundary values $0$. Moreover it is unique up to a positive multiplicative constant. To this end, we follow the strategy used by P.L. Lions in the real case. However, we have to prove a new theorem on the existence of solutions for some special complex degenerate Monge-Ampère equations. This requires establishing new a priori estimates of the gradient and Laplacian of such solutions using methods and results of L. Caffarelli, J.J. Kohn, L. Nirenberg and J. Spruck \cite{CKNS85} and B. Guan \cite{GuanB98}. Finally we provide a Pluripotential variational approach to the problem and using our new existence theorem, we prove a Rayleigh quotient type formula for the first eigenvalue of the complex Monge-Ampère operator.

math.CV↗
arXiv2025

Kähler-Ricci flows coming out of metric spaces

Given a compact Kähler manifold $X$ and a closed, positive $(1,1)$-current $T$ on $X$, we find sufficient conditions for $T$ to induce a metric structure $(X,d_T)$ which is the Gromov-Hausdorff limit of compact Kähler manifolds either in a "static" way or at time zero of smooth Kähler-Ricci flows. In dimension $1$ we extend works of T. Richard and M. Simon, showing that any oriented compact Alexandrov surface with bounded integral curvature and without cusp is the initial datum of a Kähler-Ricci flow.

math.DG↗
arXiv2025

High Energy plurisubharmonic classes

Let $Ω\Subset \C^n$ be a bounded strongly pseudoconvex domain. For any concave increasing weight $χ: \R^- \longrightarrow \R^-$ such that $χ(0) = 0$, we introduce and study finite energy classes $\mathcal E_χ(Ω)$ of plurisubharmonic functions, using the Orlicz space formalism. We investigate the range of the Monge-Ampère operator on these classes, and conjecture that this should lead to an integral characterization of the image of bounded plurisubharmonic functions, an open problem since the birth of Pluripotential Theory more than forty years ago.

math.CV↗
arXiv2025

A new approach to the Monge-Ampère eigenvalue problem

We study the eigenvalue problem for the complex Monge-Ampère operator in bounded hyperconvex domains in $\C^n$, where the right-hand side is a non-pluripolar positive Borel measure. We establish the uniqueness of eigenfunctions in the finite energy class introduced by Cegrell, up to positive multiplicative constants, and provide a Rayleigh quotient type formula for computing the eigenvalue. Under a natural continuity assumption on the measure, we further show that both the eigenvalue and eigenfunctions can be obtained via an iterative procedure starting from any negative finite energy function. Our approach relies on the fine properties of plurisubharmonic envelopes, which allow a partial sublinearization of the nonlinear problem. As far as we know, this method is new, even in the linear case, and not only yields new results but also significantly simplifies existing arguments in the literature. Moreover, it extends naturally to the setting of complex Hessian operators. Finally, by translating our results from the complex Monge-Ampère setting via a logarithmic transformation, we also obtain several interesting analogues for the real Monge-Ampère operator.

math.CV↗
arXiv2025

An Iterative Approach to the Complex Monge-Ampère Eigenvalue Problem

We present an iterative approach to approximate the solution to the Dirichlet complex Monge-Ampère eigenvalue problem on a bounded strictly pseudoconvex domain in $\C^n$. This approach is inspired by a similar approach initiated by F. Abedin, J. Kitagawa who considered the real Monge-Ampère operator on a strictly convex domain in $\R^N$.

math.CV↗
arXiv2023

A variational approach to the eigenvalue problem for complex Hessian operators

Let $1 \leq m \leq n$ be two integers and $Ω\Subset \C^n$ a bounded $m$-hyperconvex domain in $\C^n$. Using a variational approach, we prove the existence of the first eigenvalue and an associated eigenfunction which is $m$-subharmonic with finite energy for general twisted complex Hessian operators of order $m$. Under some extra assumption on the twist measure we prove Hölder continuity of the corresponding eigenfunction. Moreover we give applications to the solvability of more general degenerate complex Hessian equations with the right hand side depending on the unknown function.

math.CV↗
arXiv2023

Diameter of Kähler currents

We establish upper bounds on the diameter of compact Kähler manifolds endowed with Kähler metrics whose volume form satisfies an Orlicz integrability condition. Our results extend previous estimates due to Fu-Guo-Song, Y.Li, and Guo-Phong-Song-Sturm. In particular, they do not involve any constraint on the vanishing of the volume form. Moreover, we show that singular Kähler-Einstein currents have finite diameter, provided that their local potentials are Hölder continuous.

math.DG↗
arXiv2023

Strict positivity of Kähler-Einstein currents

Kähler-Einstein currents, also known as singular Kähler-Einstein metrics, have been introduced and constructed a little over a decade ago. These currents live on mildly singular compact Kähler spaces $X$ and their two defining properties are the following: they are genuine Kähler-Einstein metrics on $X_{\rm reg}$ and they admit local bounded potentials near the singularities of $X$. In this note we show that these currents dominate a Kähler form near the singular locus, when either $X$ admits a global smoothing, or when $X$ has isolated smoothable singularities. Our results apply to klt pairs and allow us to show that if $X$ is any compact Kähler space of dimension $3$ with log terminal singularities, then any singular Kähler-Einstein metric of non-positive curvature dominates a Kähler form.

math.CV↗
arXiv2023

The Continuous Subsolution Problem for Complex Hessian Equations

Let $Ω\subset \mathbb C^n$ be a bounded strictly $m$-pseudoconvex domain ($1\leq m\leq n$) and $μ$ a positive Borel measure on $Ω$. We study the Dirichlet problem for the complex Hessian equation $(dd^c u)^m \wedge β^{n - m} = μ$ on $Ω$. First we give a sufficient condition on the "modulus of diffusion" of the measure $μ$ with respect to the $m$-Hessian capacity which guarantees the existence of a continuous solution to the associated Dirichlet problem with a continuous boundary datum. As an application, we prove that if the equation has a continuous $m$-subharmonic subsolution whose modulus of continuity satisfies a Dini type condition, then the equation has a continuous solution with an arbitrary continuous boundary datum. Moreover when the measure has a finite mass on $Ω$, we give a precise quantitative estimate on the modulus of continuity of the solution. One of the main steps in our proof is to establish a new capacity estimate providing a precise estimate of the modulus of diffusion of the $m$-Hessian measure of a continuous $m$-subharmonic function $φ$ in $Ω$ with zero boundary with respect to the $m$-Hessian capacity in terms of the modulus of continuity of $φ$. Another important ingredient is a new weak stability estimate for the $m$-Hessian measure of a continuous $m$-subharmonic function in $Ω$.

math.CV↗
arXiv2023

Weighted Green functions for complex Hessian operators

Let $1\leq m\leq n$ be two fixed integers. Let $Ω\Subset \mathbb C^n$ be a bounded $m$-hyperconvex domain and $\mathcal A \subset Ω\times ]0,+ \infty[$ a finite set of weighted poles. We define and study properties of the $m$-subharmonic Green function of $Ω$ with prescribed behaviour near the weighted set $A$. In particular we prove uniform continuity of the exponential Green function in both variables $(z,\mathcal A)$ in the metric space $\bar Ω\times \mathcal F$, where $\mathcal F$ is a suitable family of sets of weighted poles in $Ω\times ]0,+ \infty[$ endowed with the Hausdorff distance. Moreover we give a precise estimates on its modulus of continuity. Our results generalize and improve previous results concerning the pluricomplex Green function du to P. Lelong.

math.CV↗
arXiv2022

On the extension of quasiplurisubharmonic functions

Let $(V,ω)$ be a compact Kähler manifold such that $V$ admits a cover by Zariski-open Stein sets with the property that $ω$ has a strictly plurisubharmonic exhaustive potential on each element of the cover. If $X\subset V$ is an analytic subvariety, we prove that any $ω|_X$-plurisubharmonic function on $X$ extends to a $ω$-plurisubharmonic function on $V$. This result generalizes a previous result of ours on the extension of singular metrics of ample line bundles. It allows one to show that any transcendental Kähler class in the real Neron-Severi space $NS_{\mathbb R}(V)$ has this extension property.

math.CV↗
arXiv2021

Continuity of singular Kähler-Einstein potentials

In this note, we investigate some regularity aspects for solutions of degenerate complex Monge-Ampère equations (DCMAE) on singular spaces. First, we study the Dirichlet problem for DCMAE on singular Stein spaces, showing a general continuity result. A consequence of our results is that Kähler-Einstein potentials are continuous at isolated singularities. Next, we establish the global continuity of solutions to DCMAE when the reference class belongs to the real Néron-Severi group. This yields in particular the continuity of Kähler-Einstein potentials on any irreducible Calabi-Yau variety.

math.CV↗
arXiv2020

The Hölder continuous subsolution theorem for complex Hessian equations

Let $Ω\Subset \mathbb C^n$ be a bounded strongly $m$-pseudoconvex domain ($1\leq m\leq n$) and $μ$ a positive Borel measure with finite mass on $Ω$. Then we solve the Hölder continuous subsolution problem for the complex Hessian equation $(dd^c u)^m \wedge β^{n - m} = μ$ on $Ω$. Namely, we show that this equation admits a unique Hölder continuous solution on $Ω$ with a given Hölder continuous boundary values if it admits a Hölder continuous subsolution on $Ω$. The main step in solving the problem is to establish a new capacity estimate showing that the $m$-Hessian measure of a Hölder continuous $m$-subharmonic function on $Ω$ with zero boundary values is dominated by the $m$-Hessian capacity with respect to $Ω$ with an (explicit) exponent $τ> 1$.

math.CV↗
arXiv2020

Remarks on the modulus of continuity of subharmonic functions

We introduce different classical characteristics used to regularize a subharmonic function and compare them. As an application we give a complete proof of a useful characterization of the modulus of continuity of such functions in terms of these characteristics under a technical condition. This result is extended to quasi-plurisubharmonic functions on a compact Hermitian manifold.

math.CV↗
arXiv2018

The pluripotential Cauchy-Dirichlet problem for complex Monge-Ampere flows

We develop the first steps of a parabolic pluripotential theory in bounded strongly pseudo-convex domains of Cn. We study certain degenerate parabolic complex Monge-Amp{è}re equations, modelled on the K{ä}hler-Ricci flow evolving on complex algebraic varieties with Kawamata log-terminal singularities. Under natural assumptions on the Cauchy-Dirichlet boundary data, we show that the envelope of pluripotential subsolutions is semi-concave in time and continuous in space, and provides the unique pluripotential solution with such regularity.

math.DG↗