Polynomial Hulls of Arcs and Curves

Abstract

It is shown that there exist arcs and simple closed curves in ${\mathbb{C}}^3$ with nontrivial polynomial hulls that contain no analytic discs. It is also shown that in any bounded Runge domain of holomorphy in ${\mathbb{C}}^N$ ($N\ge 2$) there exist polynomially convex arcs and simple closed curves of almost full measure. These results, which strengthen earlier results of the author, are obtained as consequences of a general result about polynomial hulls of arcs and simple closed curves through Cantor sets.