Ahmed Zeriahi

Intrinsic capacities on compact Kähler manifolds

We study fine properties of quasiplurisubharmonic functions on compact Kähler manifolds. We define and study several intrinsic capacities which characterize pluripolar sets and show that locally pluripolar sets are globally “quasi-pluripolar.”

Viscosity Solutions to Degenerate Complex Monge-Ampère Equations

We develop an alternative approach to Degenerate complex Monge-Ampere equations on compact Kahler manifolds based on the concept of viscosity solutions and compare systematically viscosity concepts with pluripotential theoretic ones. We generalize to the Kahler case a theorem due to Dinew and Zhang in the projective case to the effect that their pluripotential solutions constructed previously by the authors are continuous.

Extension of plurisubharmonic functions with growth control

Abstract Suppose that X is an analytic subvariety of a Stein manifold M and that φ is a plurisubharmonic (psh) function on X which is dominated by a continuous psh exhaustion function u of M. Given any number c > 1, we show that φ admits a psh extension to M which is dominated by cu+ on M. We use this result to prove that any ω-psh function on a subvariety of the complex projective space is the restriction of a global ω-psh function, where ω is the Fubini–Study Kähler form.

Subextension of plurisubharmonic functions with bounded Monge-Ampere mass.

Abstract 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–Ampere 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 c ϕ j ) n +∞. It is known that the complex Monge–Ampere 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–Ampere 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 c ϕ) 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–Ampere masses on Ω. To cite this article: U. Cegrell, A. Zeriahi, C. R. Acad. Sci. Paris, Ser. I 336 (2003).

Weak solutions to degenerate complex Monge–Ampère flows I

Studying the (long-term) behavior of the Kähler-Ricci flow on mildly singular varieties, one is naturally lead to study weak solutions of degenerate parabolic complex Monge–Ampère equations. The purpose of this article, the first of a series on this subject, is to develop a viscosity theory for degenerate complex Monge–Ampère flows in domains of

Polya's inequalities, global uniform integrability and the size of plurisubharmonic lemniscates

{\mathbb {C}}^n

Convergence of Weak Kähler–Ricci Flows on Minimal Models of Positive Kodaira Dimension

Cn.

Dirichlet Problem in Domains of ℂ n

The main goal of this paper is to establish new uniform estimates on the size of sublevel sets of plurisubharmonic functions (called plurisubharmonic lemniscates) in terms of Hausdorff?Riesz measures and capacities of certain orders. We first prove a new uniform version of Skodas integrability theorem for a given class of plurisubharmonic functions in terms of Borel measures of Hausdorff?Riesz type of certain orders with a precise estimate of the integrability exponent in terms of Lelong numbers of the class and the order of the measures. Then we present several applications of this result. We first deduce uniform estimates on the size of plurisubharmonic lemniscates associated to functions from some important classes of plurisubharmonic functions in terms of Hausdorff?Riesz measures. We also derive a new comparison inequality between certain Hausdorff?Riesz capacities and the pluricomplex logarithmic capacity for borelean sets of a fixed bounded domain in