{
  "abstract": "In this paper, we study critical points of second Neumann eigenfunctions on convex domains in two-dimensional space forms from three complementary perspectives: spectral and geometric criteria for the absence of interior critical points, quantitative localization of possible critical points, and explicit bounds for the hot spots constant. Precisely, we first establish monotonicity-radius localization principles in $\\mathbb S^2$ and $\\mathbb H^2$. As a consequence, we obtain a unified diameter criterion for convex domains in these two space forms: if $μ_2(Ω)D^2\\le j_{1,1}^2,$ then every second Neumann eigenfunction on $Ω$ has no interior critical points. Moreover, when interior critical points may exist, we derive explicit quantitative location restrictions in terms of the domain diameters in $\\mathbb{S}^{2}$ and $\\mathbb{H}^{2}$. Finally, we develop an analytic approach to study the \\emph{hot spots constant} $\\mathfrak{C}(Ω)$ on convex domains. For planar convex domains, we improve the Euclidean upper bound to $\\mathfrak{C}(Ω)<1.48$. We further obtain the corresponding hot spots constants for convex domains in non-Euclidean space forms, that is, $\\mathfrak{C}(Ω) < 4$ for $Ω\\subset\\mathbb{S}^{2}$ contained in a hemisphere; $\\mathfrak{C}(Ω) < 11.2$ for $Ω\\subset\\mathbb{H}^{2}$. Our proofs combine the properties of Bessel and Legendre functions, eigenvalue estimates, and Green's identity. Our results quantitatively measure ``how wrong'' the \\emph{hot spots conjecture} can be.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.17882",
  "author_total": 3,
  "authors": [
    {
      "name": "Haiyun Deng"
    },
    {
      "name": "Xuyong Jiang"
    },
    {
      "name": "Xiaoping Yang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5yd9g2vybhz85.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.17882",
  "fulltext_url": null,
  "id": "5yd9g2vybhz85",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.17882",
  "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",
      "2610|Spectral Theory in Mathematical Physics"
    ]
  ],
  "openalex": "W7169830530",
  "pdf_url": "https://arxiv.org/pdf/2607.17882",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-31",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5yd9g2vybhz85.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2607.17882",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.17882"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Quantitative characterization of deviations from the hot spots conjecture on convex domains in two-dimensional space forms",
  "title_text": "Quantitative characterization of deviations from the hot spots conjecture on convex domains in two-dimensional space forms",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/5yd9g2vybhz85",
  "venue": "arXiv",
  "versions": []
}