Unstable minimal spheres in hyperkähler 4-manifolds with degree one Gauss lift

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.