{
  "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. 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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.33643",
  "author_total": 2,
  "authors": [
    {
      "name": "Luis J. Alias"
    },
    {
      "name": "Miguel A. Meroño"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2shgtqwf9dexv.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.33643",
  "fulltext_url": null,
  "id": "2shgtqwf9dexv",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.33643",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.DG"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7214791031",
  "pdf_url": "https://arxiv.org/pdf/2609.33643",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2shgtqwf9dexv.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2609.33643",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.33643"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "A short proof of Bernstein's theorem via the Omori--Yau maximum principle",
  "title_text": "A short proof of Bernstein's theorem via the Omori--Yau maximum principle",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/2shgtqwf9dexv",
  "venue": "arXiv",
  "versions": []
}