Complete pluripolar graphs in ${\mathbb C}^N$

Nguyen Quang Dieu¹ ; Phung Van Manh¹

¹ Hanoi National University of Education 136 Xuan Thuy Street Cau Giay, Hanoi, Vietnam

Annales Polonici Mathematici, Tome 112 (2014) no. 1, pp. 85-100.

Résumé

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].

Détail

DOI : 10.4064/ap112-1-7

Keywords: cartesian product closed sets mathbb prove there exists function which continuous holomorphic interior vargamma complete pluripolar mathbb using result analytic polyhedron there exists bounded holomorphic function vargamma complete pluripolar mathbb these results high dimensional analogs previous due edlund complete pluripolar curves graphs ann polon math levenberg martin poletsky analytic disks pluripolar sets indiana univ math