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PreprintCC BY 4.0arXiv

Explicit Ricci-flat Metrics on Kummer K3 Surfaces

  • Jixiang Fu
  • Yunyang Xiao
  • Shing-Tung Yau
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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(a4). Its scalar Green operator Ga is represented by periodic Ewald kernels, separated radial kernels, finite-dimensional Schur complements, and convergent Neumann series. The coefficients are defined by Ua,1=−Gafa and Ua,n=Ga∑j=1n−1Qa(Ua,j,Ua,n−j), where Qa is the polarized quadratic Monge--Ampère term. A Green estimate of order a−1 gives a first correction of order a3 and a convergence parameter of order a2. 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.

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Cite

BibTeXRIS
BibTeX
@misc{fu2026explicit,
  title = {{Explicit Ricci-flat Metrics on Kummer K3 Surfaces}},
  author = {Jixiang Fu and Yunyang Xiao and Shing-Tung Yau},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.11077},
  eprint = {2610.11077},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.11077},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Explicit Ricci-flat Metrics on Kummer K3 Surfaces
AU  - Jixiang Fu
AU  - Yunyang Xiao
AU  - Shing-Tung Yau
PY  - 2026
DA  - 2026/10/15
PB  - arXiv
DO  - 10.48550/arxiv.2610.11077
UR  - https://arxiv.org/abs/2610.11077
AB  - 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∑_j=1^n−1Q_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.
ER  -