{
  "abstract": "We resolve Case's question (Crelle's Journal, 2024) on the sharp lower threshold for Obata-type rigidity of $I_a=Q+aσ_2(A)$ in positive Einstein conformal classes of dimension $n\\ge4$. Using a new reference-curvature identity, we prove that, on a closed connected manifold, every smooth metric in such a class with nonnegative scalar curvature and constant $I_a$ is Einstein whenever $a\\ge-4$. The scalar-curvature sign assumption can be removed for $-4\\le a\\le-2(n-2)/(n-1)$. We also extend the quotient rigidity theorem of Ge--Wang--Wei to $I_a=ΛR^θ$ for $R>0$, $a\\ge-4$, and $θ\\le1$. Sharpness is established in every dimension by a shooting construction. For each sufficiently small $η>0$, we construct a smooth non-Einstein metric in the round conformal class on $\\mathbb S^n$ with positive scalar curvature and the same constant $I_{-4-η}$ as the unit round metric. The construction smoothly matches a perturbed Schwarzschild neck near the equator with perturbed round caps at both poles.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.10527",
  "author_total": 1,
  "authors": [
    {
      "name": "Wangzhe Wu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bsmcwnv1x214s.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.10527",
  "fulltext_url": null,
  "id": "bsmcwnv1x214s",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.10527",
  "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": "W7212117096",
  "pdf_url": "https://arxiv.org/pdf/2609.10527",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bsmcwnv1x214s.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:42Z",
      "native_id": "2609.10527",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.10527"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "A sharp threshold for mixed $Q$-curvature rigidity",
  "title_text": "A sharp threshold for mixed Q-curvature rigidity",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:42Z",
  "url": "https://research.draconitic.com/paper/bsmcwnv1x214s",
  "venue": "arXiv",
  "versions": []
}