Complete pluripolar graphs in \({\mathbb C}^N\) (Q2875410)

The authors prove that if \(F\) is the Cartesian product of \(N\) closed sets in \(\mathbb{C}\), then there exists a function \(g\) which is continuous on \(F\) and holomorphic on the interior of \(F\) such that \(\Gamma_g(F):= \big\{(x,g(x)): x\in F\big\}\) is complete pluripolar in \(\mathbb{C}^{N+1}\). This result is then applied to analytic polyhedrons. Their results generalize to higher dimensions those of \textit{T. Edlund} [Ann. Pol. Math. 84, No. 1, 75--86 (2004; Zbl 1098.32015)] and \textit{N. Levenberg} et al. [Indiana Univ. Math. J. 41, No. 2, 515--532 (1992; Zbl 0763.32010)]. The paper also contains the conjecture that if \(D\) is a bounded open pseudoconvex subset of \(\mathbb{C}^N\), then there exists a continuous function \(g\) on \(\overline{D}\) which is holomorphic on \(D\) and such that \(\Gamma_g(\overline{D})\) is complete pluripolar in \(\mathbb{C}^{N+1}\).