{
  "abstract": "We develop a weighted scalar spectral geometric description of Sacks-Uhlenbeck $α$-harmonic maps into round spheres and identify its convex geometry realization for $α$-harmonic diffeomorphisms onto the round $2$-sphere. More precisely, we establish a one-to-one correspondence, modulo orientation preserving isometries of $\\mathbb{S}^2$, between orientation preserving $α$-harmonic diffeomorphisms onto the round $2$-sphere and positive Codazzi tensors satisfying a determinant bound and a compatibility condition with the prescribed metric on $\\mathbb{S}^2$. These tensors are the radii tensors of auxiliary strictly convex surfaces, and the weighted scalar forms in spectral geometry equal $-α$ times the second variations of the enclosed volumes with respect to the support functions. Using these spectral geometry and convex geometry results, we prove a sharp Morse index characterization of isometric equatorial parametrizations among $α$-harmonic $2$-spheres in the round $n$-sphere. We also prove that every $α$-harmonic diffeomorphism between unit round $2$-spheres is orthogonal for all $α>1$. Equivalently, the only admissible positive Codazzi tensor whose reconstructed metric on $\\mathbb{S}^2$ has Gaussian curvature $1$ is $(κ+2)^{α-1}\\bar{g}$, where $\\bar{g}$ is the unit round metric.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.00642",
  "author_total": 2,
  "authors": [
    {
      "name": "Rui Gao"
    },
    {
      "name": "Miaomiao Zhu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6prccd2xxykd9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.00642",
  "fulltext_url": null,
  "id": "6prccd2xxykd9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.00642",
  "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"
    ],
    [
      "arxiv",
      "math.SP"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.00642",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6prccd2xxykd9.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.00642",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.00642"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Spectral and convex geometry of $α$-harmonic maps and sharp Morse index rigidity",
  "title_text": "Spectral and convex geometry of α-harmonic maps and sharp Morse index rigidity",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/6prccd2xxykd9",
  "venue": "arXiv",
  "versions": []
}