{
  "abstract": "We study the log-concavity property of first Dirichlet eigenfunctions on domains in $\\mathbb{CP}^2$. For every smooth $1$-convex domain $Ω\\subset\\mathbb{CP}^2$, we prove the quantitative estimate \\[ \\nabla^2(-\\log u) > \\max\\left\\{ψ(s),\\frac85\\right\\}g, \\text{ where } ψ(|\\grad f|^2) = \\frac{s}{\\sqrt{1+s}}-\\log(1+s), \\] for its first Dirichlet eigenfunction $u$. In particular, $u$ is strictly log-concave. As consequences, we obtain a uniform convexity estimate for the regular level sets of $u$ and the fundamental gap bound $λ_2-λ_1>46/5$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.22659",
  "author_total": 1,
  "authors": [
    {
      "name": "Yusen Xia"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5597pxft61etd.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.22659",
  "fulltext_url": null,
  "id": "5597pxft61etd",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.22659",
  "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"
    ],
    [
      "arxiv",
      "math.DG"
    ],
    [
      "arxiv",
      "math.SP"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7213999127",
  "pdf_url": "https://arxiv.org/pdf/2609.22659",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5597pxft61etd.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.22659",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.22659"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Log-Concavity of First Dirichlet Eigenfunctions on $\\mathbb{CP}^2$",
  "title_text": "Log-Concavity of First Dirichlet Eigenfunctions on ℂℙ^2",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/5597pxft61etd",
  "venue": "arXiv",
  "versions": []
}