The local Phragmén-Lindelöf condition for analytic varieties in complex n-space was introduced by Hörmander and plays an important role in various areas of analysis. Recently, new necessary geometric properties for a variety satisfying this condition were derived by the present authors. These results are now applied to investigate the homogeneous polynomials P with real coefficients that are stable in the following sense: Whenever f is a holomorphic function that is defined in some neighborhood...

We give an overview of the recent developments in plurifine pluripotential theory, i.e. the theory of plurifinely plurisubharmonic functions.

Let $D_j$ be a bounded hyperconvex domain in ${ℂ}^{n_j}$ and set $D=D₁\times \cdots \times D_s$, j=1,...,s, s ≥ 3. Also let ${\Omega }_{\pi }$ be the image of D under the proper holomorphic map π. We characterize those continuous functions $f:\partial {\Omega }_{\pi }\to ℝ$ that can be extended to a real-valued pluriharmonic function in ${\Omega }_{\pi }$.

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 $X$ 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...

We update the state of the subject approximately 20 years after the publication of T. Bloom, L. Bos, C. Christensen, and N. Levenberg, Polynomial interpolation of holomorphic functions in ℂ and ℂⁿ, Rocky Mountain J. Math. 22 (1992), 441-470. This report is mostly a survey, with a sprinkling of assorted new results throughout.

For μ a positive measure, we estimate the pluricomplex potential of μ, $P_{\mu }\left(x\right)={\int }_{\Omega }g(x,y)d\mu \left(y\right)$, where g(x,y) is the pluricomplex Green function (relative to Ω) with pole at y.