{
  "abstract": "We study the Jacobi spectrum and low-index rigidity of complex Hopf fibrations. We first give a representation-theoretic description of the Jacobi operator of a homogeneous Riemannian fibration and apply it to \\[ π:(\\mathbb{S}^{2n+1},g_ε)\\longrightarrow(\\mathbb{C}P^n,h), \\] where $g_ε$ is a Berger metric and $h$ is the Fubini--Study metric. For $n\\geq3$, we determine all negative and zero eigenvalues and obtain explicit formulas for the Morse index and nullity for every $ε>0$. In particular, for the unit round metric, $\\mathrm{Ind}(π)=2n+2$ and $\\mathrm{Null}(π)=n(3n+5)$. We then extend Rivière's rigidity argument from $\\mathbb{S}^3\\to\\mathbb{S}^2$ to higher dimensions. If $ϕ:\\mathbb{S}^{2n+1}\\to\\mathbb{C}P^n$ is harmonic and $\\mathrm{Ind}(ϕ)=2n+2$, then $dϕ_x\\circ dϕ_x^\\ast$ commutes with the complex structure of $\\mathbb{C}P^n$ at every point. Consequently, up to an isometry of the domain, $ϕ$ factors through the Hopf fibration as $ϕ=P\\circπ\\circ f$, where $P:\\mathbb{C}P^n\\to\\mathbb{C}P^n$ is holomorphic.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.07815",
  "author_total": 1,
  "authors": [
    {
      "name": "Luiz Lara"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2n72eqhmp2ccr.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.07815",
  "fulltext_url": null,
  "id": "2n72eqhmp2ccr",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.07815",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.DG"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7220903706",
  "pdf_url": "https://arxiv.org/pdf/2610.07815",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2n72eqhmp2ccr.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:26Z",
      "native_id": "2610.07815",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.07815"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Harmonic maps from $\\mathbb{S}^{2n+1}$ to $\\mathbb{C}P^n$ with low Morse index",
  "title_text": "Harmonic maps from 𝕊^2n+1 to ℂP^n with low Morse index",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:26Z",
  "url": "https://research.draconitic.com/paper/2n72eqhmp2ccr",
  "venue": "arXiv",
  "versions": []
}