Polynomial Hulls of Arcs and Curves

TL;DR

This paper demonstrates the existence of arcs and curves in complex spaces with nontrivial polynomial hulls lacking analytic discs, and shows such structures exist in bounded Runge domains, advancing understanding of polynomial convexity.

Contribution

It introduces new constructions of polynomially convex arcs and curves in complex spaces, extending previous results through a general theorem involving Cantor sets.

Findings

Existence of arcs with nontrivial polynomial hulls without analytic discs in ${f C}^3$

Presence of polynomially convex arcs in bounded Runge domains of holomorphy in ${f C}^N$

Extension of earlier results through a general theorem involving Cantor sets

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 ${\mathbb C}^N$ ($N \geq 2$) 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.