{
  "abstract": "We prove that every bounded stable entire solution $v$ of the Allen--Cahn equation in $\\R^3$ is one-dimensional. As a consequence, the full De Giorgi conjecture in $\\mathbb{R}^4$ is true. We also obtain local curvature estimates for stable solutions. The proof strategy is inspired by the recent breakthrough work of Chan, Fernández-Real, Figalli and Serra [J. Amer. Math. Soc. 2026], by reducing the stabilty condition for the Allen-Cahn equation to a weak stability condition on a surface (the zero set) and then utilizing Gauss-Bonnet formula. For this purpose, we first use the stability condition to get a sublinear bound for a weighted integral that controls the zeros where the solution is far from planar. If such zeros exist, we isolate a bounded set of them and join $1-v^2$ near this set to derivatives of one-dimensional transitions farther away. By controlling the interaction between these transitions, we derive the weak stability condition on the zero set, which is then used to bound a weighted integral of the squared curvature on the regular part of the zero set by a cutoff gradient integral and a controlled error. We use this inequality to bound the intrinsic area and construct logarithmic cutoffs. The resulting compactly supported test function has a negative contribution near this set that exceeds all joining and cutoff errors, contradicting stability.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.21680",
  "author_total": 6,
  "authors": [
    {
      "name": "Yong Liu"
    },
    {
      "name": "Tianci Luo"
    },
    {
      "name": "Kelei Wang"
    },
    {
      "name": "Juncheng Wei"
    },
    {
      "name": "Yong Wei"
    },
    {
      "name": "Ke Wu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/70gj5whksb8ss.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.21680",
  "fulltext_url": null,
  "id": "70gj5whksb8ss",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.21680",
  "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": "W7213898499",
  "pdf_url": "https://arxiv.org/pdf/2609.21680",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/70gj5whksb8ss.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:07Z",
      "native_id": "2609.21680",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.21680"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Stable De Giorgi conjecture of the Allen--Cahn equation in $\\mathbb{R}^3$",
  "title_text": "Stable De Giorgi conjecture of the Allen--Cahn equation in ℝ^3",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:07Z",
  "url": "https://research.draconitic.com/paper/70gj5whksb8ss",
  "venue": "arXiv",
  "versions": []
}