{
  "abstract": "Let $n\\geq 2$. We establish isodiametric-type upper bounds for Steklov eigenvalues of bounded Lipschitz domains $Ω$ in Euclidean space $\\mathbb{R}^n$ and hyperbolic space $\\mathbb{H}^n$. In Euclidean space, we prove \\[ D(Ω)σ_k(Ω)\\leq C(n)k, \\] with optimal dependence on $k$. For domains in either $\\mathbb{R}^n$ or $\\mathbb{H}^n$, we establish \\[ D(Ω)σ_k(Ω)\\leq C(n)k^2\\left(\\frac{|Ω|^{\\frac{1}{n}}}{D(Ω)}\\right)^{\\frac{n}{n-1}}. \\] The exponent of the volume-diameter ratio is optimal in both settings, and with this exponent fixed, the quadratic dependence on $k$ is also optimal. Interpolating the preceding volume-diameter estimate with a volume-normalized Steklov bound, we further obtain \\[ D(Ω)σ_k(Ω)\\leq C(n)k^{1+\\frac{1}{2n-1}} \\] in hyperbolic space. These results answer Open Question 4.37 affirmatively and resolve the hyperbolic part of Open Question 4.35, both posed in [5].",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.30807",
  "author_total": 2,
  "authors": [
    {
      "name": "Pingxin Gu"
    },
    {
      "name": "Yao Wan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8jrz3t7ptt0sp.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.30807",
  "fulltext_url": null,
  "id": "8jrz3t7ptt0sp",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.30807",
  "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.DG"
    ],
    [
      "arxiv",
      "math.SP"
    ],
    [
      "openalex",
      "2610|Spectral Theory in Mathematical Physics"
    ]
  ],
  "openalex": "W7204934096",
  "pdf_url": "https://arxiv.org/pdf/2608.30807",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8jrz3t7ptt0sp.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2608.30807",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.30807"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.SP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Isodiametric-type upper bounds for Steklov eigenvalues",
  "title_text": "Isodiametric-type upper bounds for Steklov eigenvalues",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/8jrz3t7ptt0sp",
  "venue": "arXiv",
  "versions": []
}