{
  "abstract": "We prove a sheeting theorem for stable solutions to the Allen--Cahn equation in ambient dimensions $2$ through $10$. We show that weak convergence of the measures $\\varepsilon|Du_\\varepsilon|^2\\,dX$ to a hyperplane with integer multiplicity implies that the zero sets are disjoint smooth graphs. As a consequence, stable entire solutions in $\\mathbb R^N$, $N\\le7$, with $O(R^{N-1})$ energy growth are one-dimensional. When $W''(-1)=W''(1)$, every blowdown of an entire solution with finite Morse index and this energy growth has multiplicity one for $4\\le N\\le10$. For even potentials, we also obtain multiplicity one for limits with bounded energy and Morse index on closed manifolds of dimensions $3$ through $7$, under bumpiness or positive Ricci curvature.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.08499",
  "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/1ch3m9e5jsch8.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.08499",
  "fulltext_url": null,
  "id": "1ch3m9e5jsch8",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.08499",
  "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"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7220999542",
  "pdf_url": "https://arxiv.org/pdf/2610.08499",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1ch3m9e5jsch8.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:26Z",
      "native_id": "2610.08499",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.08499"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Sheeting property and rigidity for stable solutions to the Allen--Cahn equation",
  "title_text": "Sheeting property and rigidity for stable solutions to the Allen--Cahn equation",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:26Z",
  "url": "https://research.draconitic.com/paper/1ch3m9e5jsch8",
  "venue": "arXiv",
  "versions": []
}