{
  "abstract": "We study blow-up and compactness for the critical nonlinear Dirac equation \\[ D_{g_0}ψ_i = H_i|ψ_i|^{\\frac{2}{n-1}}ψ_i \\qquad \\text{on }\\mathbb S^n,\\quad n\\ge 3, \\] where $g_0$ is the round metric and $H_i\\to H>0$ in $C^0(\\mathbb S^n)$. Under a first-threshold bound on the critical mass, we prove that every blowing-up sequence has, after passing to a subsequence, a unique blow-up point $p$ satisfying $H(p)=\\max_{\\mathbb S^n}H$. The rescaled solutions converge to a ground-state Dirac bubble that carries the entire limiting critical mass. Under local uniform $C^1$ bounds on $H_i$, we use the Kelvin transform and a second rescaling to prove that $p$ is an isolated simple blow-up point. We also derive sharp pointwise and gradient estimates in cylindrical coordinates and identify the normalized Green limit. If the coefficients converge in $C^2$ near $p$, the Pohozaev identity yields $\\operatorname{Hess}_{g_0}H(p)=0$. Consequently, if a positive $H\\in C^2(\\mathbb S^n)$ satisfies $Δ_{g_0}H(p)<0$ at every global maximum, then solutions with critical mass bounded by a level slightly above the first threshold are uniformly bounded in $L^\\infty(\\mathbb S^n)$. This bound remains uniform under small $C^2$ perturbations of $H$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.33507",
  "author_total": 2,
  "authors": [
    {
      "name": "Ting Li"
    },
    {
      "name": "Zhongwei Tang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5cczsb1nc22ag.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.33507",
  "fulltext_url": null,
  "id": "5cczsb1nc22ag",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.33507",
  "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.AP"
    ],
    [
      "arxiv",
      "math.DG"
    ],
    [
      "openalex",
      "2604|Nonlinear Partial Differential Equations"
    ]
  ],
  "openalex": "W7223832489",
  "pdf_url": "https://arxiv.org/pdf/2610.33507",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5cczsb1nc22ag.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.33507",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.33507"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Blow-up analysis and compactness for critical Dirac equations on $\\mathbb S^n$",
  "title_text": "Blow-up analysis and compactness for critical Dirac equations on 𝕊^n",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/5cczsb1nc22ag",
  "venue": "arXiv",
  "versions": []
}