{
  "abstract": "We resolve an open problem posed by Frank--Loss. Let $n\\ge 3$ and let $φ\\in L^p(\\mathbb{S}^n)$, with $\\frac{n}{n-1}<p<\\infty$, be a nontrivial Dirac zero mode on $\\mathbb{S}^n$, i.e. a nonzero spinor satisfying \\begin{equation*} Dφ= iA\\cdotφ, \\end{equation*} where $D$ is the Dirac operator and $A$ is a vector field with $\\mathrm{d} A^\\flat\\in L^{n/2}$. We prove the sharp lower bound \\begin{equation*} \\|\\mathrm{d} A^\\flat\\|_{\\frac{n}{2}} \\ge 2\\left[\\frac{n}{2}\\right]^{-\\frac12}\\frac{n-1}{n-2}S_n = \\left[\\frac{n}{2}\\right]^{-\\frac12}\\frac{n(n-1)}{2}ω_n^{\\frac{2}{n}}. \\end{equation*} Equality is attainable if and only if $n$ is odd; in that case, modulo conformal and gauge transformations, $φ$ is a Killing spinor and $A$ is a real multiple of the Reeb field associated with $φ$. The case $n=3$ was proved in our recent paper using a different method. In the paper we divide the remaining cases into 3 cases: i) $n\\ge 5$ is odd, ii) $n\\ge 5$ is even and iii) $n=4$. All these cases need to use different methods. For $n\\ge 5$, the argument crucially reduces to an improved Sobolev inequality on $\\mathbb{S}^n$ under a barycenter constraint. Specifically, for $u\\in W^{1,2}(\\mathbb{S}^n)$ we consider \\begin{equation*} \\mathfrak{a}_n := \\inf\\Bigg\\{ \\frac{ \\int \\Big(|\\nabla u|^2 + \\frac{n(n-2)}{4}u^2\\Big) - \\frac{n(n-2)}{4}ω_n^{\\frac{2}{n}}\\|u\\|_{\\frac{2n}{n-2}}^2 }{ ω_n^{\\frac{2}{n}}\\|u\\|_{\\frac{2n}{n-2}}^2 - \\int u^2 } \\,\\Bigg|\\, \\int x|u|^{\\frac{2n}{n-2}}=0,\\ ω_n^{\\frac{2}{n}}\\|u\\|_{\\frac{2n}{n-2}}^2 - \\int u^2>0 \\Bigg\\}, \\end{equation*} and we obtain the following universal estimate \\begin{equation*} \\mathfrak{a}_n > \\frac{n(n-2)}{4(n^2-3n+1)}, \\end{equation*} which is enough for our aim. Determining the exact value of $\\mathfrak{a}_n$ remains open.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.34958",
  "author_total": 2,
  "authors": [
    {
      "name": "Guofang Wang"
    },
    {
      "name": "Mingwei Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0wtjt4d8b118n.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.34958",
  "fulltext_url": null,
  "id": "0wtjt4d8b118n",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.34958",
  "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-ph"
    ],
    [
      "arxiv",
      "math.AP"
    ],
    [
      "arxiv",
      "math.DG"
    ],
    [
      "arxiv",
      "math.MP"
    ],
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      "openalex",
      "2610|Spectral Theory in Mathematical Physics"
    ]
  ],
  "openalex": "W7214744017",
  "pdf_url": "https://arxiv.org/pdf/2609.34958",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0wtjt4d8b118n.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:12Z",
      "native_id": "2609.34958",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.34958"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    }
  ],
  "title": "Another sharp criterion for Dirac zero modes",
  "title_text": "Another sharp criterion for Dirac zero modes",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/0wtjt4d8b118n",
  "venue": "arXiv",
  "versions": []
}