{
  "abstract": "Let $M$ be a closed star-shaped hypersurface in $\\mathbb{R}^{n+1}$ and let $η=Hg-h$ be the first Newton transformation of its second fundamental form. For a fixed constant $α>0$ and $2\\le k\\le n-1$, we study the prescribed sum Hessian curvature equation \\[ σ_k(λ(η))+ασ_{k-1}(λ(η))=f(X,ν). \\] Under the natural admissibility condition \\[ λ(η)\\inΓ_{k-1}\\cap \\{λ:σ_k(λ)+ασ_{k-1}(λ)>0\\}, \\] we establish a global curvature estimate for closed star-shaped solutions with a general positive right-hand side depending on the position vector and the unit normal. A key point is that the sum Hessian operator is nonhomogeneous; the resulting lower-order term in the maximum-principle calculation is controlled by a partial curvature estimate. We also use the augmented-variable identity \\[ σ_k(λ)+ασ_{k-1}(λ) =σ_k(λ_1,\\ldots,λ_n,α) \\] to recover the relevant ellipticity and concavity properties. As an application of the a priori estimates, we prove existence and uniqueness for the corresponding prescribed curvature problem under suitable spherical barrier and radial monotonicity assumptions.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.05032",
  "author_total": 3,
  "authors": [
    {
      "name": "Dongjing Ma"
    },
    {
      "name": "Peihe Wang"
    },
    {
      "name": "Hua Zhu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6759nmzxehdp7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.05032",
  "fulltext_url": null,
  "id": "6759nmzxehdp7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.05032",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "arxiv",
      "math.DG"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.05032",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6759nmzxehdp7.ris",
  "sources": [
    {
      "first_seen": "2026-11-04T15:21:29Z",
      "native_id": "2611.05032",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.05032"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Curvature Estimates and Prescribed Curvature Problems for Sum Hessian Equations on Star-Shaped Hypersurfaces",
  "title_text": "Curvature Estimates and Prescribed Curvature Problems for Sum Hessian Equations on Star-Shaped Hypersurfaces",
  "type": "preprint",
  "updated_at": "2026-11-06T18:21:01Z",
  "url": "https://research.draconitic.com/paper/6759nmzxehdp7",
  "venue": "arXiv",
  "versions": []
}