{
  "abstract": "Let $E_τ:=\\mathbb{C}/(\\mathbb{Z}+\\mathbb{Z}τ)$ be a flat torus and $G(z)=G(z;τ)$ be the Green function on $E_τ$ with the singularity at $0$. Consider the multiple Green function $G_{n}$ on $(E_τ)^{n}$: \\[ G_{n}(z_{1},\\cdots,z_{n}):=\\sum_{i<j}G(z_{i}-z_{j})-n\\sum_{i=1}^{n}G(z_{i}). \\] A critical point is called trivial if $\\{z_{1},\\cdots,z_{n}\\}=\\{-z_{1},\\cdots,-z_{n}\\}$ in $E_τ$. Lin and Wang (Ann. Math. 2010) proved that $G_1=-G$ has exactly three trivial critical points and at most one pair of nontrivial critical points (depends on the choice of $τ$). For general $n\\geq 2$, Chai, Lin ang Wang (Camb. J. Math. 2015) proved that $G_n$ has at most $2n+1$ trivial critical points, but how many nontrivial critical points might exist remained completely open there. In this paper, we prove that $G_n$ has at most $n$ pairs of nontrivial critical points, and this upper bound is optimal at least for $n=2,3$. The key idea is to show that nontrivial critical points (if exist) are always non-degenerate and contribute the same degree $(-1)^{n+1}$. Applications to the curvature equation $Δu+e^{u}=8πn δ_{0}$ and the integral Lamé equation $y''=[n(n+1)\\wp(z)+B]y$ will be given. In particular, we show that the kernel of the linearized operator is one-dimensional for any solution of the curvature equation.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.07830",
  "author_total": 2,
  "authors": [
    {
      "name": "Zhijie Chen"
    },
    {
      "name": "Chang-Shou Lin"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8ebxrdj04ksxk.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.07830",
  "fulltext_url": null,
  "id": "8ebxrdj04ksxk",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.07830",
  "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"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7220930095",
  "pdf_url": "https://arxiv.org/pdf/2610.07830",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8ebxrdj04ksxk.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:26Z",
      "native_id": "2610.07830",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.07830"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Structure of critical points of multiple Green functions on flat torus and applications",
  "title_text": "Structure of critical points of multiple Green functions on flat torus and applications",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:26Z",
  "url": "https://research.draconitic.com/paper/8ebxrdj04ksxk",
  "venue": "arXiv",
  "versions": []
}