{
  "abstract": "We study entire graphical translating solutions of the mean curvature flow, \\[ \\operatorname{div}\\!\\left(\\frac{\\nabla G}{\\sqrt{1+|\\nabla G|^2}}\\right) =\\frac{1}{\\sqrt{1+|\\nabla G|^2}} \\qquad\\text{in }\\mathbb R^N. \\] Every such graph is mean-convex, since its mean curvature is the vertical component of its unit normal. In dimension two, mean-convex translating solitons are convex, and an entire graphical translator is therefore the rotationally symmetric bowl soliton. In higher dimensions Wang constructed non-rotational entire convex translating graphs. We prove that a further loss of rigidity occurs at the Bernstein dimension: for every $N\\ge 8$ there exists a one-parameter family of entire graphical translators that are mean-convex but not convex. The construction starts from the Bombieri--De Giorgi--Giusti (BDG) entire minimal graph in $\\mathbb R^8$ and develops a singular perturbation theory for the translator equation around it. The main new feature is a transition layer near Simons' cone: the translating term breaks the odd symmetry of the minimal graph, and after a suitable recentering the matching problem is governed by a parabolic inner equation. A detailed analysis of this layer, together with weighted Jacobi theory on the BDG graph and global barriers, yields the desired entire solutions.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.04675",
  "author_total": 3,
  "authors": [
    {
      "name": "Juan Davila"
    },
    {
      "name": "Manuel del Pino"
    },
    {
      "name": "Juncheng Wei"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/35qm9wrywny70.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.04675",
  "fulltext_url": null,
  "id": "35qm9wrywny70",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.04675",
  "language": null,
  "licence": "CC-BY-SA-4.0",
  "licence_label": "CC BY-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "arxiv",
      "math.DG"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7211915918",
  "pdf_url": "https://arxiv.org/pdf/2609.04675",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/35qm9wrywny70.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:18Z",
      "native_id": "2609.04675",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.04675"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "A Bernstein problem for translating solutions to the mean curvature flow",
  "title_text": "A Bernstein problem for translating solutions to the mean curvature flow",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:18Z",
  "url": "https://research.draconitic.com/paper/35qm9wrywny70",
  "venue": "arXiv",
  "versions": []
}