{
  "abstract": "We survey a circle of recent results showing that a minimal submanifold obeys a dichotomy: either it fills up space, spreading out like a space-filling curve, or it is confined, and confinement forces quantitative restrictions. On the confined side these restrictions are both geometric and function-theoretic: Euclidean volume growth, an optimal rate of convergence of the density, complex-curve rigidity for stable surfaces in $\\bf{R}^4$, and a Liouville theorem forcing slowly growing harmonic functions on minimal disks to be constant. A prototypical restriction is Euclidean volume growth, which in these results is forced by the geometry rather than assumed. The mechanism is a volume doubling theorem that converts geometric confinement into quantitative rigidity: a stationary integral varifold trapped in a thin slab at a given scale cannot double its volume by more than a universal factor. We explain how this principle, and the height-excess bounds behind it, produce Euclidean volume growth for submanifolds of sublinearly growing height in every dimension and codimension, the optimal density rate in a slab, the complex-curve and Liouville rigidity above, a higher-codimension Bernstein theorem for disks, one-sided volume bounds, and an optimal stable Bernstein theorem in all dimensions generalizing Moser, Bombieri-De Giorgi-Miranda, Caffarelli-Nirenberg-Spruck and Ecker-Huisken. We describe three regimes with sharp borderlines. In narrow cones the volume grows polynomially with exponent $n+O(\\varepsilon^2)$, and the order $\\varepsilon^2$ is attained; in wide cones the growth can be arbitrarily fast. For holomorphic curves in $\\mathbb{C}^2$, there are also three regimes and the borderline is determined by the Kähler angle.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.29502",
  "author_total": 2,
  "authors": [
    {
      "name": "Tobias Holck Colding"
    },
    {
      "name": "William P. Minicozzi II"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/19bgpfzntbgv9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.29502",
  "fulltext_url": null,
  "id": "19bgpfzntbgv9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.29502",
  "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": "W7172300148",
  "pdf_url": "https://arxiv.org/pdf/2607.29502",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/19bgpfzntbgv9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:53Z",
      "native_id": "2607.29502",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.29502"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "A dichotomy for minimal submanifolds",
  "title_text": "A dichotomy for minimal submanifolds",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:53Z",
  "url": "https://research.draconitic.com/paper/19bgpfzntbgv9",
  "venue": "arXiv",
  "versions": []
}