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.
Categories: math.DG math-ph math.MP