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arXiv · 2609.33643

A short proof of Bernstein's theorem via the Omori--Yau maximum principle

Luis J. Alias·Miguel A. Meroño
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Abstract

We give a short, self-contained proof of Bernstein's theorem: every entire minimal graph in $R^3$ is a plane. The only global tool is the Omori-Yau maximum principle, which holds on entire minimal graphs with their induced metric. We apply it exactly once to a single explicit function $F$ built from the surface's angle function and principal curvatures. A pointwise differential inequality for $F$, combined with the Omori-Yau principle, forces it to attain its minimum value everywhere, which yields the planarity of the surface and hence Bernstein's theorem.

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BibTeXRIS

Luis J. Alias, Miguel A. Meroño. 2026-09-27. A short proof of Bernstein's theorem via the Omori--Yau maximum principle. https://arxiv.org/abs/2609.33643

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