{
  "abstract": "Already in $\\mathbb{R}^4$, there are many minimal hypersurfaces but few structural results. The half-space theorem fails there: higher-dimensional catenoids are among the many examples contained in a slab. Nevertheless, we show that in any dimension and codimension, the height of a minimal submanifold controls its volume growth: confinement to a narrow cone forces polynomial growth, and sublinear height forces Euclidean growth, the slowest possible. Thus volume bounds, which regularity theory assumes, follow from height alone. The key is a volume doubling theorem that converts geometric confinement into quantitative rigidity: if an $n$-dimensional stationary integral varifold lies in a thin slab at a given scale, then its volume doubling ratio at that scale is bounded by a universal constant. Iterating this gives the polynomial growth bounds above. Sublinear height is exactly the borderline for Euclidean growth: examples grow at least like $r^{n+4ε^2/5}$ in cones of aperture $ε$. From Euclidean volume growth, we obtain an optimal Bernstein theorem, in all dimensions, for stable minimal hypersurfaces with sublinear height growth, generalizing results of Moser, Bombieri--De Giorgi--Miranda, Trudinger, Caffarelli--Nirenberg--Spruck, and Ecker--Huisken.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.15031",
  "author_total": 2,
  "authors": [
    {
      "name": "Tobias Holck Colding"
    },
    {
      "name": "William P. Minicozzi II"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cpe8y8r62n0b8.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.15031",
  "fulltext_url": null,
  "id": "cpe8y8r62n0b8",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2605.15031",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "arxiv",
      "math.DG"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7161245071",
  "pdf_url": "https://arxiv.org/pdf/2605.15031",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cpe8y8r62n0b8.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:53Z",
      "native_id": "2605.15031",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.15031"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Minimal submanifolds confined in space",
  "title_text": "Minimal submanifolds confined in space",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:53Z",
  "url": "https://research.draconitic.com/paper/cpe8y8r62n0b8",
  "venue": "arXiv",
  "versions": []
}