{
  "abstract": "We derive an interior planarity estimate for space-time proper mean curvature flows of $n$-dimensional submanifolds of $\\mathbb R^{n+p}$, $n\\geq5$, that are uniformly quadratically pinched up to the Andrews-Baker threshold $\\frac{4}{3n}$. The estimate is parabolically scale-invariant and it requires neither curvature nor entropy bounds. The proof combines Naff's gradient estimates with a localization argument in the spirit of Ecker and Huisken. In dimensions five and six, Naff's estimates alone do not give the gradient terms a sign. We bound these terms from below with an argument we developed in a previous work. As a consequence, for $n\\geq5$, and by known results for $2\\leq n\\leq4$, every space-time proper ancient solution that is uniformly $\\frac{4}{3n}$-pinched is either a static affine $n$-plane or lies in a fixed affine $(n+1)$-dimensional subspace. This replaces the finite entropy and polynomial curvature growth assumptions of Lee, Naff and Zhu by space-time properness, and improves their pinching constant in dimensions two, five and six. For closed flows, we extend Lynch's planarity estimate to a larger range of exponents.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.39877",
  "author_total": 1,
  "authors": [
    {
      "name": "Francesco Ruatta"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/21d5kqazjvav7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.39877",
  "fulltext_url": null,
  "id": "21d5kqazjvav7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.39877",
  "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.DG"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.39877",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/21d5kqazjvav7.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.39877",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.39877"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Planarity estimates and codimension reduction for ancient flows at the Andrews-Baker threshold",
  "title_text": "Planarity estimates and codimension reduction for ancient flows at the Andrews-Baker threshold",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/21d5kqazjvav7",
  "venue": "arXiv",
  "versions": []
}