{
  "abstract": "We first prove that every smooth boundary metric on a compact manifold extends to a metric with any prescribed positive lower bound for Ricci curvature, whereas extensions under stronger positive \\(k^{\\mathrm{th}}\\)-intermediate Ricci curvature conditions may fail due to local obstructions. For negative sectional curvature, we identify a global obstruction to extension. We then consider the class of compact manifolds defined by the existence of a metric with negative sectional curvature and boundary index at most one. On every manifold in this class, we prove that any smooth boundary metric extends to a metric with negative sectional curvature and strictly convex umbilical boundary. We further show that every such initial metric admits a complete asymptotically hyperbolic isometric extension with the same curvature condition and any prescribed conformal infinity. This class of manifolds is closed under boundary connected sums. As a geometric application of these extension results, every smooth metric on \\(\\mathbb S^n\\) admits a strictly convex isometric embedding into \\(\\mathbb R^{n+1}\\) equipped with a complete metric of negative sectional curvature. The proofs combine neck constructions with corner smoothing for upper curvature bounds.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.21518",
  "author_total": 2,
  "authors": [
    {
      "name": "Wenlong Wang"
    },
    {
      "name": "Jintian Zhu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7b2tj4480k8qz.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.21518",
  "fulltext_url": null,
  "id": "7b2tj4480k8qz",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.21518",
  "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": "W7213881957",
  "pdf_url": "https://arxiv.org/pdf/2609.21518",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7b2tj4480k8qz.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:07Z",
      "native_id": "2609.21518",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.21518"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "The metric extension problem under positive and negative curvature conditions",
  "title_text": "The metric extension problem under positive and negative curvature conditions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:07Z",
  "url": "https://research.draconitic.com/paper/7b2tj4480k8qz",
  "venue": "arXiv",
  "versions": []
}