Explicit Green construction and estimates for elliptic K3 surfaces

Prove the elliptic Green construction and the corresponding global estimates needed to implement the recursive Calabi–Yau metric construction on collapsing elliptic K3 surfaces.

Background

The paper develops an explicit Green-operator construction and convergent quadratic recursion for Ricci-flat metrics on a fixed Kummer K3 surface. In its elliptic K3 outlook, it explains that the same recursion would apply to a Gross–Wilson-type background obtained by gluing semi-flat and Ooguri–Vafa metrics, provided suitable analytic estimates were available.

An explicit treatment would require constructing local Green kernels near the nodal fibers, assembling them into a global inverse, and proving the analytic bounds controlling the exponentially small residual, the Green operator, the quadratic correction operator, and the resulting metric correction. These ingredients are not established for elliptic K3 surfaces in the paper.

References

The elliptic Green construction and the corresponding global estimates are not proved in this paper.

— Explicit Ricci-flat Metrics on Kummer K3 Surfaces  (2610.11077 - Fu et al., 8 Oct 2026) in Introduction, discussion preceding Section 1; Section 5, “The elliptic K3 setting”

The preceding proofs establish finite analytic constants and a positive convergence threshold. Numerical upper bounds for these constants, a certified numerical value of a_0, and certified component errors remain to be determined; an interval-arithmetic implementation is not supplied here.

— Explicit Ricci-flat Metrics on Kummer K3 Surfaces  (2610.11077 - Fu et al., 8 Oct 2026) in Section 4.6, “Scope of the numerical error estimates”