{
  "abstract": "It is conjectured that a complete, orientable Einstein four-manifold with strictly positive sectional curvature must be isometric to either the round four-sphere $S^4$ or the complex projective plane $\\mathbb{CP}^2$ with the Fubini--Study metric. In this paper, we confirm this conjecture when the manifold is homeomorphic to $S^4$ or $\\mathbb{CP}^2$. More precisely, we show that a complete, orientable Einstein four-manifold $M$ with strictly positive sectional curvature and Euler characteristic $χ(M) \\le 3$ (equivalently, by Freedman's classification theorem, homeomorphic to $S^4$ or $\\mathbb{CP}^2$) is isometric to either the round $S^4$ or the Fubini--Study $\\mathbb{CP}^2$. As an application, we obtain that a complete, orientable Einstein four-manifold with strictly positive sectional curvature admitting a nontrivial Killing field is isometric to either the round $S^4$ or the Fubini--Study $\\mathbb{CP}^2$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.23337",
  "author_total": 1,
  "authors": [
    {
      "name": "Liang Cheng"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8d9yzyjgg3d6a.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.23337",
  "fulltext_url": null,
  "id": "8d9yzyjgg3d6a",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.23337",
  "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": "W7214013310",
  "pdf_url": "https://arxiv.org/pdf/2609.23337",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8d9yzyjgg3d6a.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.23337",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.23337"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "On Positively Curved Einstein Four-Manifolds with Euler Characteristic at Most Three",
  "title_text": "On Positively Curved Einstein Four-Manifolds with Euler Characteristic at Most Three",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/8d9yzyjgg3d6a",
  "venue": "arXiv",
  "versions": []
}