Complete pluripolar graphs in ${\mathbb C}^N$
Volume 112 / 2014
Annales Polonici Mathematici 112 (2014), 85-100
MSC: Primary 32U15; Secondary 32E20, 32U30.
DOI: 10.4064/ap112-1-7
Abstract
Let $F$ be the Cartesian product of $N$ closed sets in $\mathbb C$. We prove that there exists a function $g$ which is continuous on $F$ and holomorphic on the interior of $F$ such that $\varGamma _g (F):=\{(z, g(z)): z \in F\}$ is complete pluripolar in $\mathbb C^{N+1}$. Using this result, we show that if $D$ is an analytic polyhedron then there exists a bounded holomorphic function $g$ such that $\varGamma _g (D)$ is complete pluripolar in $\mathbb C^{N+1}$. These results are high-dimensional analogs of the previous ones due to Edlund [Complete pluripolar curves and graphs, Ann. Polon. Math. 84 (2004), 75–86] and Levenberg, Martin and Poletsky [Analytic disks and pluripolar sets, Indiana Univ. Math. J. 41 (1992), 515–532].
