{
  "abstract": "We establish two rigidity theorems for positive rupture solutions of the conformally invariant equation $Δ^2 u=u^{-7}$ in $\\mathbb R^3\\setminus\\{0\\}$, which extend continuously to the origin with $u(0)=0$. First, we prove that if the associated conformal metric $g=u^{-4}|dx|^2$ has nonnegative scalar curvature, then every such solution is radially symmetric and has the sharp rupture profile $u(x)\\sim(\\frac{4}{3})^{\\frac{1}{4}}|x|^{\\frac{1}{2}}$ as $x\\to 0$; in particular, the metric is complete at the origin. The principal novelty is a global rigidity theorem requiring neither curvature nor symmetry: the single global condition $u(x)=o(|x|)$ at infinity forces $u(x)\\equiv(\\frac{4}{3})^{\\frac{1}{4}}|x|^{\\frac{1}{2}}$. This result provides a sharp answer to the uniqueness question posed by McKenna and Reichel in a class with no a priori symmetry assumption. Finally, we construct complementary examples demonstrating the essential roles of the curvature and growth hypotheses.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.13820",
  "author_total": 3,
  "authors": [
    {
      "name": "Xia Huang"
    },
    {
      "name": "Yahui Jiang"
    },
    {
      "name": "Xianmei Zhou"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ce77k8sjtfwnk.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.13820",
  "fulltext_url": null,
  "id": "ce77k8sjtfwnk",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.13820",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "openalex",
      "2604|Nonlinear Partial Differential Equations"
    ]
  ],
  "openalex": "W7213261287",
  "pdf_url": "https://arxiv.org/pdf/2609.13820",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ce77k8sjtfwnk.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.13820",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.13820"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Rigidity of positive rupture solutions to a biharmonic equation with critical negative exponent",
  "title_text": "Rigidity of positive rupture solutions to a biharmonic equation with critical negative exponent",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/ce77k8sjtfwnk",
  "venue": "arXiv",
  "versions": []
}