{
  "abstract": "This paper is concerned with the Weyl problem, \\emph{i.e.}, the existence of isometric immersions or embeddings of two-spheres into the three-dimensional Euclidean space or general ambient three-manifolds. We establish two results on the degenerate Weyl problem, namely when the Gaussian curvature $K_g$ is only nonnegative rather than strictly positive. First, for a smooth metric $g$ on the two-sphere, if $K_g$ is strictly positive except at finitely many points where the Hessian of $K_g$ is positive definite, then $g$ admits a global $C^{2,1}$-isometric embedding into $\\mathbb{R}^3$. It appears to be the first result on the degenerate Weyl problem with purely intrinsic conditions on $g$. Second, for a general simply-connected ambient three-manifold $(\\mathcal{M},{\\overline{g}})$, if $K_g \\geq K_0 \\geq {\\rm sec}_{\\overline{g}}$ for some constant $K_0$, $(K_g-K_0)^{-1/2} \\in L^p$ with $p \\geq 2$, and a certain uniform pinching condition holds for approximate nondegenerate isometric immersions, then there exists a $W^{3,p}$-isometric immersion. Alongside we also resolve the nondegenerate Weyl problem (\\emph{i.e.}, when $K_g>0$) into general simply-connected ambient three-manifolds for $g$, ${\\overline{g}} \\in C^{2,1}$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.28905",
  "author_total": 2,
  "authors": [
    {
      "name": "Siran Li"
    },
    {
      "name": "Xiangxiang Su"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bb72rgkfn5npf.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.28905",
  "fulltext_url": null,
  "id": "bb72rgkfn5npf",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.28905",
  "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.AP"
    ],
    [
      "arxiv",
      "math.DG"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7214339773",
  "pdf_url": "https://arxiv.org/pdf/2609.28905",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bb72rgkfn5npf.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.28905",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.28905"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "On the degenerate Weyl problem on isometric immersions",
  "title_text": "On the degenerate Weyl problem on isometric immersions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/bb72rgkfn5npf",
  "venue": "arXiv",
  "versions": []
}