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

This note extends the construction of $D_k$ ALF gravitational instantons in Schroers–Singer to a new case where the nonlinear superposition is given by the $D_1$ Atiyah–Hitchin metric and $k-1$ copies of $A_0$ 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 this gives a large class of $K3$ surfaces containing non-holomorphic minimal spheres.