{
  "abstract": "Let $(M,g)$ be a compact Riemann surface with smooth boundary $\\partial M$. Let $K(x)>0$, $h(x)> 0$ or $h(x)\\equiv 0$ be two smooth functions on $M$ and $\\partial M$ respectively. In this paper, we consider the following double mean field type equation with a nonlinear Neumann boundary condition related to a prescribed curvature problem \\begin{equation}\\label{equation:01} \\begin{cases} \\displaystyle -Δ_g u+\\frac{4π}{|M|} =2ρ\\frac{Ke^u}{\\int_MKe^u\\,dv_g}, & \\text{in }M,\\\\[1.2em] \\displaystyle \\partial_νu =2ρ'\\frac{he^{u/2}}{\\int_{\\partial M}he^{u/2}\\,ds_g}, & \\text{on }\\partial M, \\end{cases} \\end{equation} where $ρ> 0,\\ ρ'\\geq 0$ and $ρ+ρ'=2π$, $ν$ is the outward unit normal vector to $\\partial M$. When $h\\equiv 0$, the second equation changes to $\\partial_νu=0$ on $\\partial M$. By introducing a boundary integration constant, we obtain a sufficient condition under which \\eqref{equation:01} admits a solution. As a geometric application, we solve the prescribed curvature equation for both the Gauss curvature and the geodesic curvature in the case that $M$ is topologically a disk. When $(M,g)$ is the standard hemisphere $\\mathbb S^2_+$, let $p\\in \\partial \\mathbb S^2_+$ be a maximum point of $$ Φ(q)=h(q)+\\sqrt{h(q)^2+K(q)}, \\qquad q\\in\\partial \\mathbb S^2_+. $$ Suppose \\begin{equation*} K_ν(p)+2Φ(p)(-Δ_{\\partial \\mathbb S^2_+})^{1/2}h(p)<0. \\end{equation*} Then the prescribed curvature problem \\begin{align*} -Δ_g u+\\frac{4π}{|M|}=2Ke^u, \\text{in }\\mathbb S^2_+,\\ \\ \\partial_νu=2he^{u/2}, \\text{on }\\partial \\mathbb S^2_+. \\end{align*} admits a smooth solution.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.17571",
  "author_total": 4,
  "authors": [
    {
      "name": "Jiayu Li"
    },
    {
      "name": "Lei Liu"
    },
    {
      "name": "Yanjun Liu"
    },
    {
      "name": "Mingyuan Zhu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/avwm5v24anz5z.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.17571",
  "fulltext_url": null,
  "id": "avwm5v24anz5z",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.17571",
  "language": null,
  "licence": "CC0-1.0",
  "licence_label": "CC0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/publicdomain/zero/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.DG"
    ],
    [
      "openalex",
      "2604|Nonlinear Partial Differential Equations"
    ]
  ],
  "openalex": "W7221835492",
  "pdf_url": "https://arxiv.org/pdf/2610.17571",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/avwm5v24anz5z.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:19Z",
      "native_id": "2610.17571",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.17571"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "On the existence problem for a double mean field equation and its application to prescribed curvature problems",
  "title_text": "On the existence problem for a double mean field equation and its application to prescribed curvature problems",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:19Z",
  "url": "https://research.draconitic.com/paper/avwm5v24anz5z",
  "venue": "arXiv",
  "versions": []
}