{
  "id": "2302.00466",
  "version": "v1",
  "published": "2023-02-01T14:26:44.000Z",
  "updated": "2023-02-01T14:26:44.000Z",
  "title": "Hypersurfaces of $\\mathbb{S}^2\\times\\mathbb{S}^2$ with constant sectional curvature",
  "authors": [
    "Haizhong Li",
    "Luc Vrancken",
    "Xianfeng Wang",
    "Zeke Yao"
  ],
  "comment": "24 pages",
  "categories": [
    "math.DG"
  ],
  "abstract": "In this paper, we classify the hypersurfaces of $\\mathbb{S}^2\\times\\mathbb{S}^2$ with constant sectional curvature. By applying the so-called Tsinghua principle, which was first discovered by the first three authors in 2013 at Tsinghua University, we prove that the constant sectional curvature can only be $\\frac{1}{2}$ and the product angle function $C$ defined by Urbano is identically zero. We show that any such hypersurface is a parallel hypersurface of a minimal hypersurface in $\\mathbb{S}^2\\times\\mathbb{S}^2$ with $C=0$, and we establish a one-to-one correspondence between the involving minimal hypersurface and the famous ``sinh-Gordon equation'' $$ (\\frac{\\partial^2}{\\partial u^2}+\\frac{\\partial^2}{\\partial v^2})h =-\\tfrac{1}{\\sqrt{2}}\\sinh(\\sqrt{2}h). $$ As a byproduct, we give a complete classification of the hypersurfaces of $\\mathbb{S}^2\\times\\mathbb{S}^2$ with constant mean curvature and constant product angle function $C$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-02-01T14:26:44.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53C42",
      "53B25"
    ],
    "keywords": [
      "constant sectional curvature",
      "minimal hypersurface",
      "constant product angle function",
      "constant mean curvature",
      "tsinghua principle"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}