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Abstract

It is shown that there exist $C^{\infty }$ functions on the boundary of the unit disk whose graphs are complete pluripolar. Moreover, for any natural number $k$, such functions are dense in the space of $C^k$ functions on the boundary of the unit disk. We show that this result implies that the complete pluripolar closed $C^\infty $ curves are dense in the space of closed $C^k$ curves in ${\mathbb C}^n$. We also show that on each closed subset of the complex plane there is a continuous function whose graph is complete pluripolar.