{
  "abstract": "We give a convergent explicit representation of the Ricci-flat Kähler metrics supplied by the Calabi--Yau theorem in a one-parameter family of classes on a fixed Kummer K3 surface. An explicit radial background has normalized Monge--Ampère residual \\(O(a^4)\\). Its scalar Green operator \\(G_a\\) is represented by periodic Ewald kernels, separated radial kernels, finite-dimensional Schur complements, and convergent Neumann series. The coefficients are defined by \\(U_{a,1}=-G_af_a\\) and \\(U_{a,n}=G_a\\sum_{j=1}^{n-1}Q_a(U_{a,j},U_{a,n-j})\\), where \\(Q_a\\) is the polarized quadratic Monge--Ampère term. A Green estimate of order \\(a^{-1}\\) gives a first correction of order \\(a^3\\) and a convergence parameter of order \\(a^2\\). A Catalan majorant proves absolute convergence on the entire smooth surface for every sufficiently small fixed \\(a\\). The sum defines a positive form solving the volume equation; comparison identifies the sum with the smooth normalized Calabi--Yau potential. We also record sufficient analytic conditions for the same recursion on collapsing elliptic K3 surfaces.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.11077",
  "author_total": 3,
  "authors": [
    {
      "name": "Jixiang Fu"
    },
    {
      "name": "Yunyang Xiao"
    },
    {
      "name": "Shing-Tung Yau"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6s62mdmm0m942.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.11077",
  "fulltext_url": null,
  "id": "6s62mdmm0m942",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.11077",
  "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.DG"
    ],
    [
      "openalex",
      "2608|Geometry and complex manifolds"
    ]
  ],
  "openalex": "W7221154026",
  "pdf_url": "https://arxiv.org/pdf/2610.11077",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6s62mdmm0m942.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:54Z",
      "native_id": "2610.11077",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.11077"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Explicit Ricci-flat Metrics on Kummer K3 Surfaces",
  "title_text": "Explicit Ricci-flat Metrics on Kummer K3 Surfaces",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:54Z",
  "url": "https://research.draconitic.com/paper/6s62mdmm0m942",
  "venue": "arXiv",
  "versions": []
}