Poletsky has introduced a notion of plurisubharmonicity for functions defined on compact sets in ℂⁿ. We show that these functions can be completely characterized in terms of monotone convergence of plurisubharmonic functions defined on neighborhoods of the compact.

To a plurisubharmonic function $u$ on ${\mathbf{C}}^n$ with logarithmic growth at infinity, we may associate the Robin function$${\displaystyle {\rho }_u\left(z\right)=\underset{\lambda \to \infty }{lim\; sup}u\left(\lambda z\right)-log\left(\lambda z\right)}$$defined on ${\mathbf{P}}^{n-1}$, the hyperplane at infinity. We study the classes $L_{+}$, and (respectively) $L_p$ of plurisubharmonic functions which have the form $u=log(1+|z\left|\right)+O\left(1\right)$ and (respectively) for which the function ${\rho }_u$ is not identically $-\infty$. We obtain an integral formula which connects the Monge-Ampère measure on the space ${\mathbf{C}}^n$ with the Robin function on ${\mathbf{P}}^{n-1}$. As an application we obtain a criterion on the convergence of the Monge-Ampère...

A certain linear growth of the pluricomplex Green function of a bounded convex domain of ${ℂ}^N$ at a given boundary point is related to the existence of a certain plurisubharmonic function called a “plurisubharmonic saddle”. In view of classical results on the existence of angular derivatives of conformal mappings, for the case of a single complex variable, this allows us to deduce a criterion for the existence of subharmonic saddles.

Let $S$ be a Riemann surface. Let ${ℍ}^3$ be the $3$-dimensional hyperbolic space and let ${\partial }_{\infty }{ℍ}^3$ be its ideal boundary. In our context, a Plateau problem is a locally holomorphic mapping $\varphi :S\to {\partial }_{\infty }{ℍ}^3=\widehat{ℂ}$. If $i:S\to {ℍ}^3$ is a convex immersion, and if $N$ is its exterior normal vector field, we define the Gauss lifting, $\widehat{ı}$, of $i$ by $\widehat{ı}=N$. Let $\overrightarrow{n}:U{ℍ}^3\to {\partial }_{\infty }{ℍ}^3$ be the Gauss-Minkowski mapping. A solution to the Plateau problem $(S,\varphi )$ is a convex immersion $i$ of constant Gaussian curvature equal to $k\in (0,1)$ such that the Gauss lifting $(S,\widehat{ı})$ is complete and $\overrightarrow{n}\circ \widehat{ı}=\varphi$. In this paper, we show...

Nous donnons une condition suffisante pour l’existence de points périodiques pour une application birationnelle de $ℂ{\mathbb{P}}^2$. Sous cette hypothèse, une estimation précise du nombre de points périodiques de période fixée est obtenue. Nous donnons une application de ce résultat à l’étude dynamique de ces applications, en calculant explicitement l’auto-intersection de leur courant invariant naturellement associé. Nos résultats reposent essentiellement sur le théorème de Bézout donnant le cardinal de l’intersection...

On donne une autre démonstration (sans désingularisation de Hironaka) du théorème de Tamm, qui dit que la partie régulière d’un sous-analytique est sous-analytique. En plus, on montre que pour chaque fonction $f:U\to R$ de classe SUBB (“sous-analytique à l’infini”), où $U$ est un sous-ensemble ouvert et borné dans $\mathbf{R}(n$, il existe un entier $k\in \mathbf{N}$ tel que $f$ est analytique dans $x\in U$ si et seulement si $f$ est de classe $G^k$ ($k$-fois différentiable au sens de Gateaux) dans un voisinage de $x$.

Weighted $L^2$ estimates are obtained for the canonical solution to the $\bar{\partial }$ equation in $L^2({ℂ}^n,e^{-\phi }d\lambda )$, where $\Omega$ is a pseudoconvex domain, and $\phi$ is a strictly plurisubharmonic function. These estimates are then used to prove pointwise estimates for the Bergman projection kernel in $L^2({ℂ}^n,e^{-\phi }d\lambda )$. The weight is used to obtain a factor $e^{-ϵ\rho (z,\zeta )}$ in the estimate of the kernel, where $\rho$ is the distance function in the Kähler metric given by the metric form $i\partial \bar{\partial }\phi$.

In this paper we obtain several characterizations of the pointwise multipliers of the space $BMOA$ in the unit ball $B$ of ${ℂ}^n$. Moreover, if $g_1,...,g_m$ are holomorphic functions on $B$, we prove that $M_g\left(f\right)\left(z\right)=\sum _{j=1}^m{g}_j\left(z\right)f_j\left(z\right)$ maps $BMOA\times ...\times BMOA$ onto $BMOA$ if and only if the functions $g_j$ are multipliers of the space $BMOA$ and satisfy $\sum _{j=1}^m\left|g_j\left(z\right)\right|\ge \delta \>0.$

Let $X$ be a complex manifold with strongly pseudoconvex boundary $M$. If $\psi$ is a defining function for $M$, then $-log\psi$ is plurisubharmonic on a neighborhood of $M$ in $X$, and the (real) 2-form $\sigma =i\partial \overline{\partial }(-log\psi )$ is a symplectic structure on the complement of $M$ in a neighborhood of $M$ in $X$; it blows up along $M$. The Poisson structure obtained by inverting $\sigma$ extends smoothly across $M$ and determines a contact structure on $M$ which is the same as the one induced by the complex structure. When $M$ is compact, the Poisson structure near...