A gluing construction of 𝐷𝑘 ALF gravitational instantons and existence of non-holomorphic minimal spheres

This note extends the construction of D k Dₖ ALF gravitational instantons in Schroers–Singer [Q. J. Math. 72 (2021), pp. 277–337] to a new case where the non-linear superposition is given by the D 1 D₁ Atiyah–Hitchin metric and k − 1 k-1 copies of A 0 A₀ Taub-NUT metrics. We then give a general class of ALF spaces such that each of them contains a non-holomorphic minimal sphere. Together with Foscolo’s construction [J. Differential Geom. 112 (2019), pp. 79–120] this gives a large class of K 3 K3 surfaces containing non-holomorphic minimal spheres.