{
  "abstract": "We investigate the third Robin eigenvalue on simply connected domains of area $A$ in surfaces with Gaussian curvature bounded above by $K$. For $K>0$, we discover a transition at the critical parameter $α=-4π+KA$: two equal geodesic disks give a sharp upper bound when $α=-4π$ or $-4π+KA\\leqα\\leq4π$, but the two disk bound fails for $-4π<α<-4π+KA$. In the latter range, we establish a new upper bound involving a single geodesic disk with a modified Robin parameter and prove its sharpness in an explicit subrange. The sharp two-disk inequality persists for nonpositive curvature bounds, and Steklov--Helmholtz analogues are also obtained. Both the two-disk and one-disk bounds arise from a unified two-pole Green-function method, with the one-disk regime emerging from a degenerate area split that produces the $2π$ shift in the effective Robin parameter, while sharpness in the stated subrange is established by boundary oscillations that redistribute Robin boundary mass.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.34134",
  "author_total": 2,
  "authors": [
    {
      "name": "Xiaoliang Wang"
    },
    {
      "name": "Zuoqin Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cfrx93ht4tazg.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.34134",
  "fulltext_url": null,
  "id": "cfrx93ht4tazg",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.34134",
  "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"
    ],
    [
      "arxiv",
      "math.SP"
    ],
    [
      "openalex",
      "2610|Spectral Theory in Mathematical Physics"
    ]
  ],
  "openalex": "W7223944611",
  "pdf_url": "https://arxiv.org/pdf/2610.34134",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cfrx93ht4tazg.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.34134",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.34134"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "On the Isoperimetric Inequality for the Third Robin Eigenvalue on Surfaces",
  "title_text": "On the Isoperimetric Inequality for the Third Robin Eigenvalue on Surfaces",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/cfrx93ht4tazg",
  "venue": "arXiv",
  "versions": []
}