{
  "abstract": "Using a reference-curvature technique, we prove four main theorems. (i) For $n\\geq4$, we remove the mean-curvature pinching from the Case-Wang rigidity theorem for $σ_2$-flat conformal hemispheres with $A_g\\in\\overline{Γ_2^+}$ and positive constant natural boundary curvature. (ii) On every closed positive Einstein manifold of dimension $n\\geq5$, each conformal metric of constant $σ_2$-curvature is Einstein, and is homothetic to the background unless the background is a round sphere, thereby resolving the $σ_2$-Obata problem posed by Chang, Gursky, and Yang. (iii) We extend the spherical constant-$σ_2/σ_1$ classification and sharp integral comparison to closed positive Einstein manifolds. The comparison holds throughout the positive-scalar-curvature conformal class, with equality only for homotheties on nonround backgrounds. (iv) We establish global quantitative $σ_2$ stability on every fixed nonround positive Einstein background of dimension $n\\geq5$, throughout its positive-scalar-curvature conformal class.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.10523",
  "author_total": 1,
  "authors": [
    {
      "name": "Wangzhe Wu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3dbveqykyv951.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.10523",
  "fulltext_url": null,
  "id": "3dbveqykyv951",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.10523",
  "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.DG"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7212139523",
  "pdf_url": "https://arxiv.org/pdf/2609.10523",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3dbveqykyv951.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2609.10523",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.10523"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Rigidity, sharp inequalities, and stability for $σ_2$-curvature by a reference curvature technique",
  "title_text": "Rigidity, sharp inequalities, and stability for σ_2-curvature by a reference curvature technique",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/3dbveqykyv951",
  "venue": "arXiv",
  "versions": []
}