In [30, Proposition 5.5] Micallef–Wolfson show that the double cover of the Atiyah–Hitchin manifold, the rotationally symmetric $D_{1}$ ALF space, contains a strictly stable minimal $2$–sphere $\Sigma$ with $[\Sigma]\cdot[\Sigma]=-4$. Since every holomorphic curve $\Sigma$ of genus $\gamma$ in a hyperkähler $4$–manifold must have $[\Sigma]\cdot[\Sigma]=2\gamma-2$ by the adjunction formula, this minimal $2$–sphere cannot be holomorphic with respect to any complex structure. One can also use the isometric action of $SU(2)$ on the Atiyah–Hitchin metric to prove this fact: the $SU(2)$ action preserves the metric but rotates the complex structures (equivalently, the hyperkähler triple) and the minimal $2$–sphere is an $SU(2)$–orbit. Hence the periods $\int_{\Sigma}{\omega_{i}}$ are forced to vanish.

Now, consider an approximate hyperkähler metric $g_{\epsilon}$ obtained in Section 5 by using the rotationally symmetric $D_{1}$ ALF space as one of the building blocks. Thus $g_{\epsilon}$ contains a strictly stable minimal sphere $\Sigma$ with $[\Sigma]\cdot[\Sigma]=-4$.

Because of strict stability, $\Sigma$ has no Jacobi fields. Then we can invoke White’s Implicit Function Theorem for minimal immersions with respect to variations of the ambient metric [40, Theorem 2.1] to deform $\Sigma$ into a minimal immersion with respect to the hyperkähler metric produced by Theorem 6.15 starting from $g_{\epsilon}$. As before, this minimal $2$–sphere cannot be holomorphic with respect to any complex structure because of its self-intersection number. It is strictly stable by continuity of the spectrum of the Jacobi operator. ∎