Complex submanifolds of Kähler manifolds are prototypical examples of stable, minimal submanifolds of higher codimension. In 1990, Yau asked whether it was possible to classify stable minimal spheres in hyperkähler 4-manifolds, proposing that all stable minimal spheres are holomorphic for some element of the S^2-family of Kähler structures.
However, Yau’s proposal can not be true because the only stable minimal sphere in the Atiyah-Hitchin manifold has degree one Gauss lift, i.e., each point is holomorphic with respect to a distinct complex structure and, hence, it satisfies a first-order equation. In this talk, I will discuss joint work with L. Foscolo, where we construct examples of unstable minimal spheres with degree one Gauss lift, which are topologically indistinguishable from the Atiyah-Hitchin sphere. This shows that there is no characterisation of stable minimal surfaces in hyperkähler 4-manifolds in terms of topological data.