Explicit Ricci-flat Metrics on Kummer K3 Surfaces

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.