{
  "abstract": "We study bubble solutions to the critical CR fractional Yamabe equation on the Heisenberg group for the full range \\(0<γ<\\frac{Q}{2}\\), including the admissible integer-order cases. The CR fractional operator is genuinely nonlocal and is described by Gamma-function spectral multipliers. Its structure differs from that of the Euclidean fractional Laplacian, so the linearized analysis cannot be obtained by a direct extension of integer-order arguments. We develop a direct spectral approach based on the Cayley intertwining relation and the complex spherical-harmonic decomposition, which characterizes the kernel of the linearized operator in terms of infinitesimal translation and dilation modes without reducing the equation to an integral equation or requiring a separate decay analysis. As a byproduct, we obtain the exact normalization constant of the standard bubble throughout the admissible range. We then prove three main results: $(i)$ the non-degeneracy of the bubble manifold; $(ii)$ the local rigidity of the bubble manifold, via a uniform invertibility estimate on the orthogonal complement of the tangent space, with an explicit optimal constant; $(iii)$ the Morse index of every positive bubble is exactly one. The results provide a unified spectral and variational framework for the CR Yamabe problem on the Heisenberg group, covering both integer and noninteger orders.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.39607",
  "author_total": 4,
  "authors": [
    {
      "name": "Jianlun Liu"
    },
    {
      "name": "Vicenţiu D. Rădulescu"
    },
    {
      "name": "Hong-Rui Sun"
    },
    {
      "name": "Ziheng Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/64zj0d5tmrg5w.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.39607",
  "fulltext_url": null,
  "id": "64zj0d5tmrg5w",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.39607",
  "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"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.39607",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/64zj0d5tmrg5w.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.39607",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.39607"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Spectral Characterization, Local Rigidity and Morse Index of Bubble Solutions to the CR Fractional Yamabe Problem",
  "title_text": "Spectral Characterization, Local Rigidity and Morse Index of Bubble Solutions to the CR Fractional Yamabe Problem",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/64zj0d5tmrg5w",
  "venue": "arXiv",
  "versions": []
}