Minimal hulls of compact sets in $\mathbb R^3$

TL;DR

This paper characterizes the minimal surface hull of compact sets in D and the null hull in C^3, using sequences of conformal minimal discs and minimal currents, extending classical theorems in complex analysis.

Contribution

It provides new characterizations of minimal and null hulls via conformal minimal discs and minimal currents, and extends classical theorems to these contexts.

Findings

Characterization of minimal hulls using conformal minimal discs.

Extension of Alexander-Stolzenberg-Wermer theorem to null hulls.

A polynomial hull version of Bochner's tube theorem.

Abstract

The main result of this paper is a characterization of the minimal surface hull of a compact set $K$ in $\mathbb R^3$ by sequences of conformal minimal discs whose boundaries converge to $K$ in the measure theoretic sense, and also by $2$-dimensional minimal currents which are limits of Green currents supported by conformal minimal discs. Analogous results are obtained for the null hull of a compact subset of $\mathbb C^3$. We also prove a null hull analogue of the Alexander-Stolzenberg-Wermer theorem on polynomial hulls of compact sets of finite linear measure, and a polynomial hull version of classical Bochner's tube theorem.