{
  "abstract": "Let $X:M^n\\to\\R^{n+1}$ be an $n$-dimensional complete self-shrinker. We obtain a complete classification of complete self-shrinkers with positive constant scalar curvature. More precisely, we prove that the round sphere $S^n(\\sqrt n)$ and the standard generalized cylinder $S^k(\\sqrt k)\\times \\R^{n-k}$ for $2\\leq k\\leq n-1$ are the only complete self-shrinkers with positive constant scalar curvature. The key difficulty is to characterize the residual case in Cheng-Li-Wei \\cite{CLW}: $R>0$, $S<1$, and $\\sup_M S=1$, where $R$ and $S$ denote the scalar curvature and the squared norm of the second fundamental form, respectively. For this case, it seems a hard task that the generalized maximum principle yields a useful information. In order to overcome this substantial difficulty, our key ingredient is to get a uniform positive lower bound for the Bakry-Émery Ricci curvature so that we can make use of the comparison theorem of Wei-Wylie \\cite{WeiWylie} to conclude that the Gaussian volume is finite. Furthermore, the gap theorems on $S$ are given.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.12794",
  "author_total": 3,
  "authors": [
    {
      "name": "Qing-Ming Cheng"
    },
    {
      "name": "Fengjiang Li"
    },
    {
      "name": "Guoxin Wei"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/99bvj4aq73h2w.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.12794",
  "fulltext_url": null,
  "id": "99bvj4aq73h2w",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.12794",
  "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": "W7213183623",
  "pdf_url": "https://arxiv.org/pdf/2609.12794",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/99bvj4aq73h2w.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:11Z",
      "native_id": "2609.12794",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.12794"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "A Classification of Complete Self-shrinkers",
  "title_text": "A Classification of Complete Self-shrinkers",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:11Z",
  "url": "https://research.draconitic.com/paper/99bvj4aq73h2w",
  "venue": "arXiv",
  "versions": []
}