Liouville theorem for immersed minimal surfaces in any codimension

TL;DR

This paper proves a Liouville theorem for harmonic functions on minimal disks in higher codimension, leading to a Bernstein-type result and establishing regularity of harmonic functions.

Contribution

It extends Liouville and Bernstein theorems to higher codimension minimal surfaces with quadratic area growth, and proves regularity results for harmonic functions.

Findings

Harmonic functions with slowly growing negative parts are constant on minimal disks.

Higher codimensional Bernstein theorem for minimal disks in sub-linearly growing cones.

Uniform Hölder regularity of harmonic functions on these surfaces.

Abstract

For a proper immersed minimal disk in $\bf{R}^N$ with quadratic area growth, we show that any harmonic function whose negative part grows at a slow sub-linear rate is constant. This leads to a higher codimensional Bernstein theorem for minimal disks contained in a sub-linearly growing cone. The catenoid, helicoid and Enneper's family of surfaces together show that this result is optimal. We also show uniform H\"older regularity of harmonic functions.