{
  "abstract": "Yau asked whether every complete Riemannian manifold with bounded Ricci curvature and positive injectivity radius admits an isometric embedding into some Euclidean space with bounded mean curvature. We construct a counterexample: a smooth complete metric on $\\mathbb{R}^4$ with bounded Ricci curvature, positive injectivity radius, and unbounded full curvature. The Gauss equation then forces every $C^2$ isometric immersion of this metric into a finite-dimensional Euclidean space to have unbounded mean curvature. The metric is assembled from disjoint curvature peaks whose Ricci contractions stay uniformly bounded. A short-geodesic curvature estimate and the Jacobi equation give the uniform injectivity-radius bound. The construction works in every dimension at least four, and in dimension four the Ricci curvature may be chosen to tend to zero at infinity.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.00153",
  "author_total": 1,
  "authors": [
    {
      "name": "Haoxuan Cheng"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/13wkn9b6ywj4v.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.00153",
  "fulltext_url": null,
  "id": "13wkn9b6ywj4v",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.00153",
  "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": "W7215725445",
  "pdf_url": "https://arxiv.org/pdf/2610.00153",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/13wkn9b6ywj4v.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:07Z",
      "native_id": "2610.00153",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.00153"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "A counterexample to Yau's bounded-mean-curvature embedding question",
  "title_text": "A counterexample to Yau's bounded-mean-curvature embedding question",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:07Z",
  "url": "https://research.draconitic.com/paper/13wkn9b6ywj4v",
  "venue": "arXiv",
  "versions": []
}