{
  "abstract": "Let $Ω\\subset\\mathbb C$ be a bounded simply connected Lipschitz domain with conformal metric $g=ω|dz|^2$, where $ω\\in C^2(Ω)\\cap L^\\infty(Ω)$ is positive in the interior and may tend to zero at the boundary. If the Gaussian curvature satisfies $K_g\\leq K$, $K>0$, and $KM<4π$, where $M=\\int_Ωω\\,dA$, then the Neumann eigenvalues, numbered by $0=μ_1(Ω;ω)<μ_2(Ω;ω)\\leqμ_3(Ω;ω)\\leq\\cdots$, satisfy $\\frac{1}{μ_2(Ω;ω)}+\\frac{1}{μ_3(Ω;ω)}\\geq\\frac{2}{μ_2(D_K(M))}$. Here $D_K(M)$ is the geodesic disk of area $M$ in the surface of constant curvature $K$. We prove that equality holds exactly when the interiors are isometric. This completes Conjecture 1.2 of Langford--Laugesen; in particular, the model disk maximizes the first positive Neumann eigenvalue in this class.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.35360",
  "author_total": 4,
  "authors": [
    {
      "name": "Hongtao Hu"
    },
    {
      "name": "Meiqi Liu"
    },
    {
      "name": "Qiaoran Wu"
    },
    {
      "name": "Wenming Zou"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ecbqypztn7vrh.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.35360",
  "fulltext_url": null,
  "id": "ecbqypztn7vrh",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.35360",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "openalex",
      "2610|Spectral Theory in Mathematical Physics"
    ]
  ],
  "openalex": "W7214787875",
  "pdf_url": "https://arxiv.org/pdf/2609.35360",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ecbqypztn7vrh.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.35360",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.35360"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Sharp Harmonic-Mean Bounds for Neumann Eigenvalues on Riemannian Surfaces and Counterexamples to Geodesic Ball Optimality on Higher-Dimensional Spheres",
  "title_text": "Sharp Harmonic-Mean Bounds for Neumann Eigenvalues on Riemannian Surfaces and Counterexamples to Geodesic Ball Optimality on Higher-Dimensional Spheres",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/ecbqypztn7vrh",
  "venue": "arXiv",
  "versions": []
}