{
  "id": "2406.09449",
  "version": "v1",
  "published": "2024-06-12T12:21:48.000Z",
  "updated": "2024-06-12T12:21:48.000Z",
  "title": "Smooth solutions to the Christoffel problem in $\\mathbb{H}^{n+1}$",
  "authors": [
    "Li Chen"
  ],
  "comment": "22 pages. arXiv admin note: substantial text overlap with arXiv:2302.01604",
  "categories": [
    "math.DG",
    "math.AP"
  ],
  "abstract": "The famous Christoffel problem is possibly the oldest problem of prescribed curvatures for convex hypersurfaces in Euclidean space. Recently, this problem has been naturally formulated in the context of uniformly $h$-convex hypersurfaces in hyperbolic space by Espinar-G\\'alvez-Mira. Surprisingly, Espinar-G\\'alvez-Mira find that the Christoffel problem in hyperbolic space is essentially equivalent to the Nirenberg-Kazdan-Warner problem on prescribing scalar curvature on $\\mathbb{S}^n$. This equivalence opens a new door to study the Nirenberg-Kazdan-Warner problem. In this paper, we establish a existence of solutions to the Christoffel problem in hyperbolic space by proving a full rank theorem. As a corollary, a existence of solutions to the Nirenberg-Kazdan-Warner problem follows.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-06-12T12:21:48.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "smooth solutions",
      "hyperbolic space",
      "nirenberg-kazdan-warner problem",
      "convex hypersurfaces",
      "full rank theorem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}