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

Abstract

We give a short, self-contained proof of Bernstein's theorem: every entire minimal graph in R³ 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. As a by-product of our approach, we show that the same method, with a different function and with some extra work, also proves in a simple and self-contained way the generalization of Bernstein's theorem given by Osserman that a complete minimal surface in R³ whose normals omit a neighbourhood of some direction must be a plane.

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BibTeX

@misc{alias2026short,
  title = {{A short proof of Bernstein's theorem via the Omori--Yau maximum principle}},
  author = {Luis J. Alias and Miguel A. Meroño},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.33643},
  eprint = {2609.33643},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.33643},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - A short proof of Bernstein's theorem via the Omori--Yau maximum principle
AU  - Luis J. Alias
AU  - Miguel A. Meroño
PY  - 2026
DA  - 2026/10/29
PB  - arXiv
DO  - 10.48550/arxiv.2609.33643
UR  - https://arxiv.org/abs/2609.33643
AB  - 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. As a by-product of our approach, we show that the same method, with a different function and with some extra work, also proves in a simple and self-contained way the generalization of Bernstein's theorem given by Osserman that a complete minimal surface in R^3 whose normals omit a neighbourhood of some direction must be a plane.
ER  -

Abstract

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