Complete pluripolar curves and graphs

Tomas Edlund¹

¹ Department of Mathematics Uppsala University Box 480 SE-751 06 Uppsala, Sweden

Annales Polonici Mathematici, Tome 84 (2004) no. 1, pp. 75-86

Résumé

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.

Détail

DOI : 10.4064/ap84-1-8

Keywords: shown there exist infty functions boundary unit disk whose graphs complete pluripolar moreover natural number functions dense space functions boundary unit disk result implies complete pluripolar closed infty curves dense space closed curves mathbb each closed subset complex plane there continuous function whose graph complete pluripolar