{
  "id": "2405.10843",
  "version": "v1",
  "published": "2024-05-17T15:13:49.000Z",
  "updated": "2024-05-17T15:13:49.000Z",
  "title": "On the index of minimal hypersurfaces in $\\mathbb{S}^{n+1}$ with $λ_1<n$",
  "authors": [
    "Hang Chen",
    "Peng Wang"
  ],
  "comment": "9 pages",
  "categories": [
    "math.DG"
  ],
  "abstract": "In this paper, we prove that a closed minimal hypersurface in $\\SSS$ with $\\lambda_1<n$ has Morse index at least $n+4$, providing a partial answer to a conjecture of Perdomo. As a corollary, we re-obtain a partial proof of the famous Urbano Theorem for minimal tori in $\\mathbb{S}^3$: a minimal torus in $\\mathbb{S}^3$ has Morse index at least $5$, with equality holding if and only if it is congruent to the Clifford torus. The proof is based on a comparison theorem between eigenvalues of two elliptic operators, which also provides us simpler new proofs of some known results on index estimates of both minimal and $r$-minimal hypersurfaces in a sphere.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-05-17T15:13:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53A10",
      "53C42"
    ],
    "keywords": [
      "morse index",
      "minimal torus",
      "partial proof",
      "closed minimal hypersurface",
      "index estimates"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}