{
  "abstract": "Motivated by the systematic study of spiral minimal products, we consider harmonic maps obtained by coupling two spherical eigenmaps through a profile curve in $\\mathbb S^3$, with a doubly warped product metric on the source. The source warps are independent of the profile magnitudes. This gives two complementary problems: reconstructing a compatible source metric from prescribed profile or source data, and finding closed profiles when the source metric is fixed. For arbitrary prescribed warps, homogenization of the reduced Lagrangian gives a strongly convex conic Finsler metric whose oriented graph geodesics are exactly the harmonic profiles. In the unit-speed inverse problem, the phase momenta reduce reconstruction to a scalar magnitude equation. We obtain global reconstructions from a prescribed magnitude, one source warp, or a monotone volume factor, together with periodic families in which profiles closing on finite covers are dense. In the fixed-metric problem, a change of time gives an autonomous Neumann system. One fixed source metric then carries continuous families of closed profiles and infinitely many closed graph geodesics. For fixed even $p,q\\geq 2$, every fixed metric in this Neumann class on $\\mathbb S^1\\times\\mathbb S^p\\times\\mathbb S^q$ supports harmonic maps into $\\mathbb S^{p+q+1}$ of degree $4m$ for every $m\\geq 1$. In the matched constant-warp case, these maps reduce to explicit eigenmaps.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.32191",
  "author_total": 2,
  "authors": [
    {
      "name": "Hai Zhang"
    },
    {
      "name": "Yongsheng Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/c1rnd74n3tgx5.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.32191",
  "fulltext_url": null,
  "id": "c1rnd74n3tgx5",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.32191",
  "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": "W7214739986",
  "pdf_url": "https://arxiv.org/pdf/2609.32191",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/c1rnd74n3tgx5.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.32191",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.32191"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Spiral Harmonic Products",
  "title_text": "Spiral Harmonic Products",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/c1rnd74n3tgx5",
  "venue": "arXiv",
  "versions": []
}