{
  "abstract": "We study mean field equations with multiple positive singularities on flat tori. We establish a uniform uniqueness criterion in terms of the total singular mass and the scale-invariant spectral quantity $λ_1(\\mathbb T)|\\mathbb T|$, where $λ_1(\\mathbb T)$ denotes the first positive eigenvalue of the Laplacian. Combining nodal-set analysis for Jacobi fields with a cylindrical Alexandrov--Bol type inequality and sharp small-capacity asymptotics, we obtain the optimal leading coefficient $8/π^2$ in the total-mass uniqueness bound as $λ_1(\\mathbb T)|\\mathbb T|\\to0$. Sharpness is demonstrated by examples with two symmetric singularities on degenerating rectangular tori that admit at least three distinct solutions. For the case of two singularities with total mass $2ρ$, we further derive uniqueness and multiplicity results near $ρ=4π$ from the critical-point structure of the associated symmetrized Green function. In particular, when its only critical points are the four nondegenerate two-torsion points, the equation admits a unique solution for $ρ$ just below $4π$ and exactly three solutions for $ρ$ just above $4π$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.29991",
  "author_total": 2,
  "authors": [
    {
      "name": "Zhijie Chen"
    },
    {
      "name": "Shihong Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bxpsyr3a11cn5.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.29991",
  "fulltext_url": null,
  "id": "bxpsyr3a11cn5",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.29991",
  "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",
      "2604|Nonlinear Partial Differential Equations"
    ]
  ],
  "openalex": "W7214305790",
  "pdf_url": "https://arxiv.org/pdf/2609.29991",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bxpsyr3a11cn5.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:15Z",
      "native_id": "2609.29991",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.29991"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Uniqueness and nonuniqueness for mean field equations with multiple singularities on flat tori",
  "title_text": "Uniqueness and nonuniqueness for mean field equations with multiple singularities on flat tori",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/bxpsyr3a11cn5",
  "venue": "arXiv",
  "versions": []
}