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

This note extends the construction of Dₖ ALF gravitational instantons in Singer--Schroers to a new case where the nonlinear superposition is given by the D₁ Atiyah--Hitchin metric and $k-1$ copies of 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 this gives a large class of $K3$ surfaces containing non-holomorphic minimal spheres.