{
  "abstract": "We study the critical biharmonic equation \\[ Δ^2 u=u^{\\frac{n+4}{n-4}}, \\] on a complete, connected, and non-compact Riemannian manifold \\((M^n,g)\\) of dimension \\(n\\geq 5\\) with non-negative Ricci curvature. We establish an optimal pointwise second-order derivative estimate by Bernstein's method and the continuity method. Using invariant tensors, we derive a differential inequality that yields a rigidity result. More precisely, if there exists a positive finite energy solution, then the manifold is isometric to the Euclidean space \\(\\mathbb{R}^n\\), and the solution is given by \\[ u(x)= \\frac{ \\left[λ^2 n(n-4)(n^2-4)\\right]^{\\frac{n-4}{8}} } {\\left(1+λ|x-x_0|^2\\right)^{\\frac{n-4}{2}}}, \\qquad λ>0,\\quad x_0\\in\\mathbb{R}^n . \\]",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.07657",
  "author_total": 3,
  "authors": [
    {
      "name": "Huabin Li"
    },
    {
      "name": "Tian Wu"
    },
    {
      "name": "Xiao Zhou"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4pktxmgq59vv2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.07657",
  "fulltext_url": null,
  "id": "4pktxmgq59vv2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.07657",
  "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"
    ],
    [
      "arxiv",
      "math.DG"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7212061940",
  "pdf_url": "https://arxiv.org/pdf/2609.07657",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4pktxmgq59vv2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.07657",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.07657"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Classification of Solutions to a Critical Fourth-Order Equation on Complete Non-compact Manifolds",
  "title_text": "Classification of Solutions to a Critical Fourth-Order Equation on Complete Non-compact Manifolds",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/4pktxmgq59vv2",
  "venue": "arXiv",
  "versions": []
}