Ahmed Zeriahi
The main goal of this work is to give new and precise generalizations to various classes of plurisubharmonic functions of the classical minimum modulus principle for holomorphic functions of one complex variable, in the spirit of the famous lemma of Cartan-Boutroux. As an application we obtain precise estimates on the size of "plurisubharmonic lemniscates" in terms of appropriate Hausdorff contents.
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We give a new proof of Kiselman's minimum principle for plurisubharmonic functions, based on Ohsawa-Takegoshi extension theorem.
We prove a strong version of the comparison principle for bounded plurisubharmonic function on complex varieties. we then apply our main result to study convergence of Mong-Ampere mesures for bounded plurisubharmonic functions.
We study the minimum sets of plurisubharmonic functions with strictly positive Monge-Ampère densities. We investigate the relationship between their Hausdorff dimension and the regularity of the function. Under suitable assumptions we prove that the minimum set cannot contain analytic subvarieties of large dimension. In the planar case we analyze the influence on the regularity of the right hand side and consider the corresponding free boundary problem with irregular data. We provide sharp examples for the Hausdorff dimension of the minimum set and the related free boundary. We also draw sever…
We study minimum sets of singular plurisubharmonic functions and their relation to upper contact sets. In particular we develop an algorithm checking when a naturally parametrized curve is such a minimum set. The case of Koch curves is studied in detail. We also study the size of the set of upper non-contact points. We show that this set is always of Lebesgue measure zero thus answering an open problem in the viscosity approach to the complex Monge-Ampère equation. Finally, we prove that similarly to the case of convex functions, strictly plurisubharmonic lower tests yield existence of upper t…
Comparison principles for local plurisubharmonic potentials on complex manifolds and applications
本文证明了复流形上Cegrell类局部多重次调和势的加权比较原理，并应用于$\mathbb C^2$中具有特定奇异性的函数，得出极大性等价于Monge-Ampère测度消失且为局部性质。
In this paper, we study a geometric property of a continuous plurisubharmonic function which is a solution of the Monge-Ampère equation and has a convex level set. To prove our main theorem, we show a minimum principle of a maximal plurisubharmonic function. By using our results and Lempert's results, we show a relation between the supports of the Monge-Ampère currents and complex $k$-extreme points of closed balls for the Kobayashi distance in a bounded convex domain in $\mathbb{C}^{n}$.
We prove a disc formula for the largest plurisubharmonic subextension of an upper semicontinuous function on a domain $W$ in a Stein manifold to a larger domain $X$ under suitable conditions on $W$ and $X$. We introduce a related equivalence relation on the space of analytic discs in $X$ with boundary in $W$. The quotient, if it is Hausdorff, is a complex manifold with a local biholomorphism to $X$. We use our disc formula to generalise Kiselman's minimum principle. We show that his infimum function is an example of a plurisubharmonic subextension.
We show that the spaces of $A$-$m$-subharmonic and $B$-$m$-subharmonic functions differ in sufficently high dimensions. We also prove that the Monge-Ampère type operator $\mathcal M_m$ associated to the space of $m$-plurisubharmonic functions does not allow an integral comparison principle except in the classical cases $m=1$ and $m=n$. These answer in the negative two problems posed by A. Sadullaev.
In this paper we study the relation between the weighted energy class $\mathcal{E}_χ$ introduced by S. Benelkouchi, V. Guedj and A. Zeriahi recently with the classes $\mathcal{E}$ and $\mathcal{N}$ studied by Cegrell. Moreover, we establish a generalized comparison principle for the operator $\text{M}_χ$ and, as an application, we prove a slight version of existence of solutions of Monge-Ampère type equation in the class $\mathcal{E}_χ(H,Ø)$.
Many fundamental results of pluripotential theory on the quaternionic space $\mathbb{H}^n$ are extended to the Heisenberg group. We introduce notions of a plurisubharmonic function, the quaternionic Monge-Ampère operator, differential operators $d_0$ and $d_1$ and a closed positive current on the Heisenberg group. The quaternionic Monge-Ampère operator is the coefficient of $ (d_0d_1u)^n$. We establish the Chern-Levine-Nirenberg type estimate, the existence of quaternionic Monge-Ampère measure for a continuous quaternionic plurisubharmonic function and the minimum principle for the quaternioni…
For families of continuous plurisubharmonic functions we show that, in a local sense, separately bounded above implies bounded above.
The aim of this paper is to give a new proof of the complete characterization of measures for which there exist a solution of the Dirichlet problem for the complex Monge-Ampere operator in the set of plurisubharmonic functions with finite pluricomplex energy. The proof uses variational methods.
First, we obtain a new formula for Bremermann type upper envelopes, that arise frequently in convex analysis and pluripotential theory, in terms of the Legendre transform of the convex- or plurisubharmonic-envelope of the boundary data. This yields a new relation between solutions of the Dirichlet problem for the homogeneous real and complex Monge-Ampere equations and Kiselman's minimum principle. More generally, it establishes partial regularity for a Bremermann envelope whether or not it solves the Monge-Ampere equation. Second, we prove the second order regularity of the solution of the fre…
On octonionic Monge-Ampère equation and pluripotential theory associated to octonionic plurisubharmonic functions of two variables
本文研究了二变量八元数 plurisubharmonic 函数的 Monge-Ampère 方程和 pluripotential 理论，解决了八元数非结合性问题，并建立了相关函数的正则性与容量性质。
Comparison principles for Monge-Ampère measures on pluripolar sets
本文基于Bedford-Taylor容量引入多子 harmonic函数的奇异性比较概念，建立了复Monge-Ampère算子在多极集上的比较原理，并通过辅助函数在能量类中的刻画及Monge-Ampère方程的唯一性结果加以应用。
On the Rigidity of Analytic Mappings in Complex Analysis and Geometry
本文研究复几何中解析映射和亚调和函数的刚性性质，通过不同条件证明了映射的拓扑度、纤维结构和极小值的存在性，揭示了李群作用的微分拓扑障碍。
Maximal Plurisubharmonic Functions and Fujii-Seo Determinants in Hilbert spaces
本文提出利用 Fujii-Seo 正则化确定性描述极值全纯亚调和函数，定义 Fujii-Seo 确定性密度并研究其与 Levi 形的谱端点关系，证明极值性与该密度为零等价，建立全局退化条件和比较原理。
We construct a special plurisubharmonic defining function for a smoothly bounded strictly pseudoconvex domain so that the determinant of the complex Hessian vanishes to high order on the boundary. This construction, coupled with regularity of solutions of complex Monge-Ampere equation and the reflection principle, enables us to give a new proof of the Fefferman mapping theorem.
Large deviation principle for determinantal point processes on Stein manifolds
本文研究Stein流形上由典范截面和多重次调和权重函数定义的行列式点过程，给出其满足带Fenchel-Legendre变换速率函数的大偏差原理的条件，还证明相关多重次调和理论结果，是紧Kähler流形对应结果的非紧类比。
Kiselman Minimum Principle and Rooftop Envelopes in Complex Hessian Equations
本文开创了欧氏域中相对于半正 $(1,1)$-形式的 $m$-次调和函数研究，通过 Perron 包络构造证明了解的分解定理与比较原理，并建立屋顶等式与 Kiselman 极小原理类似形式，从而得出 $m$-次调和函数的测地连通性判据。
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