The convergence of hulls of curves

TL;DR

The paper investigates the behavior of polynomial hulls of curves in complex spaces, showing convergence properties and approximation techniques for rectifiable simple closed curves and arcs.

Contribution

It establishes that limits of curves with nontrivial polynomial hulls also have nontrivial hulls and provides methods to approximate curves by polynomially convex ones.

Findings

Limit curves inherit nontrivial polynomial hulls.

Rectifiable curves can be approximated by polynomially convex curves.

Every rectifiable arc is contained in a polynomially convex simple closed curve.

Abstract

It is shown that a simple closed curve in $\mathbb C^n$ that is a uniform limit of rectifiable simple closed curves each of which has nontrivial polynomial hull has itself nontrivial polynomial hull. In case the limit curve is rectifiable, the hull of the limit is shown to be the limit of the hulls. It is also shown that every rectifiable simple closed curve in $\mathbb C^n$, $n\geq 2$, can be approximated in total variation norm by a polynomially convex, rectifiable simple closed curve that coincides with the original curve except on an arbitrarily small segment. As a corollary, it is shown that every rectifiable arc in $\mathbb C^n$, $n\geq 2$, is contained in a polynomially convex, rectifiable simple closed curve.