{
  "id": "2407.07062",
  "version": "v1",
  "published": "2024-07-09T17:34:27.000Z",
  "updated": "2024-07-09T17:34:27.000Z",
  "title": "On the Morse index of free-boundary CMC hypersurfaces in the upper hemisphere",
  "authors": [
    "Crísia Ramos de Oliveira"
  ],
  "categories": [
    "math.DG"
  ],
  "abstract": "We prove results for free-boundary hypersurfaces in the upper unit hemisphere $\\mathbb{S}^{n+1}_{+}$ of $\\mathbb{R}^{n+2}$. First we show that if the norm squared of the second fundamental form is constant, the Morse index of a free-boundary minimal hypersurface $\\Sigma\\subset \\mathbb{S}^{n+1}_{+}$ equals: $1$ if $\\Sigma$ is a totally geodesic equator, $n+1$ if $\\Sigma$ is half of the Clifford torus, or it is at least $2(n+1)$ when $\\Sigma$ is not totally geodesic. Next we prove an estimate for the first eigenvalue $\\lambda_1$ of the second variation's Jacobi operator, and show that $\\lambda_1 \\leq -2n$ if $\\Sigma$ is not totally geodesic, with equality iff $\\Sigma$ is half of the minimal Clifford torus. Furthermore, $\\lambda_1 = -n$ iff $\\Sigma$ is totally geodesic. Finally, if $\\Sigma$ is not totally umbilical the Morse index is at least $n+1$, with equality precisely when $\\Sigma$ is the upper $\\mathrm{H}$-torus. For totally umbilical hypersurfaces the Morse index is $1$. We also prove an upper bound for the first eigenvalue of free-boundary $\\mathrm{CMC}$ hypersurfaces, where equality corresponds to totally umbilical hypersurfaces.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-07-09T17:34:27.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "morse index",
      "free-boundary cmc hypersurfaces",
      "upper hemisphere",
      "totally geodesic",
      "totally umbilical hypersurfaces"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}