{
  "abstract": "We compare the maximum points \\(P_Ω\\) and \\(Q_Ω\\) of the torsion function and the positive first Dirichlet eigenfunction. We begin with a systematic analytic description of both hot spots over the full range of opening angles of circular sectors: uniqueness, strict monotonicity with respect to the angle, distinct small-angle asymptotics, and explicit limiting equations in the slit-disk limit. We also prove the existence of a coincidence angle and establish arithmetic properties of the limiting coordinates. We then consider bounded convex planar domains with a line of symmetry, and we prove the sharp diameter-normalized identity \\[ \\sup_Ω \\frac{|P_Ω-Q_Ω|}{\\operatorname{diam}(Ω)}=\\frac12. \\] Its upper-bound argument uses standard localization in the heart and an elementary geometric enclosure. Sharpness follows from thin convex domains with a vanishing relative width perturbation peaking near one endpoint: the torsion hot spot approaches that endpoint, whereas the ground-state hot spot approaches the midpoint.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.41205",
  "author_total": 4,
  "authors": [
    {
      "name": "Jiaxin He"
    },
    {
      "name": "Qinfeng Li"
    },
    {
      "name": "Shangfeng Wu"
    },
    {
      "name": "Shuangquan Xie"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3gx3p055m0sp3.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.41205",
  "fulltext_url": null,
  "id": "3gx3p055m0sp3",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.41205",
  "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/2610.41205",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3gx3p055m0sp3.ris",
  "sources": [
    {
      "first_seen": "2026-11-02T12:21:20Z",
      "native_id": "2610.41205",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.41205"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "On hot spots of the torsion function and the ground state of the Laplacian",
  "title_text": "On hot spots of the torsion function and the ground state of the Laplacian",
  "type": "preprint",
  "updated_at": "2026-11-04T15:20:52Z",
  "url": "https://research.draconitic.com/paper/3gx3p055m0sp3",
  "venue": "arXiv",
  "versions": []
}