### A Characterization of Complex Manifolds Biholomorphic to a Circular Domain.

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We consider a class of maximal plurisubharmonic functions and prove several properties of it. We also give a condition of maximality for unbounded plurisubharmonic functions in terms of the Monge-Ampère operator $\left(d{d}^{c}{e}^{u}\right)\u207f$.

We compute the constant sup $(1/degP)\left(ma{x}_{S}log\right|P|-{\int}_{S}log|P\left|d\sigma \right)$ : P a polynomial in ${\u2102}^{n}$, where S denotes the euclidean unit sphere in ${\u2102}^{n}$ and σ its unitary surface measure.

Let $G={K}^{\u2102}$ be a complex reductive group. We give a description both of domains $\Omega \subset G$ and plurisubharmonic functions, which are invariant by the compact group, $K$, acting on $G$ by (right) translation. This is done in terms of curvature of the associated Riemannian symmetric space $M:=G/K$. Such an invariant domain $\Omega $ with a smooth boundary is Stein if and only if the corresponding domain ${\Omega}_{M}\subset M$ is geodesically convex and the sectional curvature of its boundary $S:=\partial {\Omega}_{M}$ fulfills the condition ${K}^{S}\left(E\right)\ge {K}^{M}\left(E\right)+k(E,n)$. The term $k(E,n)$ is explicitly computable...

We study a general Dirichlet problem for the complex Monge-Ampère operator, with maximal plurisubharmonic functions as boundary data.

For a regular, compact, polynomially convex circled set $K$ in ${\mathbf{C}}^{2}$, we construct a sequence of pairs $\{{P}_{n},{Q}_{n}\}$ of homogeneous polynomials in two variables with $\mathrm{deg}\phantom{\rule{0.166667em}{0ex}}{P}_{n}=$$\mathrm{deg}\phantom{\rule{0.166667em}{0ex}}{Q}_{n}...$

We prove that an analytic surface $V$ in a neighborhood of the origin in ${\u2102}^{3}$ satisfies the local Phragmén-Lindelöf condition ${\mathrm{PL}}_{\mathrm{loc}}$ at the origin if and only if $V$ satisfies the following two conditions: (1) $V$ is nearly hyperbolic; (2) for each real simple curve $\gamma $ in ${\mathbb{R}}^{3}$ and each $d\ge 1$, the (algebraic) limit variety ${T}_{\gamma ,d}V$ satisfies the strong Phragmén-Lindelöf condition. These conditions are also necessary for any pure $k$-dimensional analytic variety $V$ to satisify ${\mathrm{PL}}_{\mathrm{loc}}$.

We give a simplified proof of J. P. Rosay's result on plurisubharmonicity of the envelope of the Poisson functional [10].

Let $X$ be a compact Kähler manifold and $\omega $ be a smooth closed form of bidegree $(1,1)$ which is nonnegative and big. We study the classes ${\mathcal{E}}_{\chi}(X,\omega )$ of $\omega $-plurisubharmonic functions of finite weighted Monge-Ampère energy. When the weight $\chi $ has fast growth at infinity, the corresponding functions are close to be bounded. We show that if a positive Radon measure is suitably dominated by the Monge-Ampère capacity, then it belongs to the range of the Monge-Ampère operator on some class ${\mathcal{E}}_{\chi}(X,\omega )$. This is done by establishing...

For complex algebraic varieties V, the strong radial Phragmén-Lindelöf condition (SRPL) is defined. It means that a radial analogue of the classical Phragmén-Lindelöf Theorem holds on V. Here we derive a sufficient condition for V to satisfy (SRPL), which is formulated in terms of local hyperbolicity at infinite points of V. The latter condition as well as the extension of local hyperbolicity to varieties of arbitrary codimension are introduced here for the first time. The proof of the main result...

${C}^{1,1}$ regularity of the solutions of the complex Monge-Ampère equation in ℂPⁿ with the n-root of the right hand side in ${C}^{1,1}$ is proved.

We prove a comparison principle for the log canonical threshold of plurisubharmonic functions under an assumption on complex Monge-Ampère measures.

We give sufficient conditions for unicity of plurisubharmonic functions in Cegrell classes.