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arXiv · 2605.15038

Liouville theorem for immersed minimal surfaces in any codimension

Tobias Holck Colding·William P. Minicozzi II
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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.

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Tobias Holck Colding, William P. Minicozzi II. 2026-05-14. Liouville theorem for immersed minimal surfaces in any codimension. https://arxiv.org/abs/2605.15038

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