7. Stable minimal surfaces [02IA]

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7. Stable minimal surfaces

In this final section we exploit our gluing construction of hyperkähler metrics on the K3 surface to deduce some information about their submanifold geometry.

It is well known that holomorphic submanifolds of a Kähler manifold are volume minimising. It is a classical problem in the theory of minimal submanifolds in Kähler manifolds to understand to what extent volume minimising submanifolds must be holomorphic or anti-holomorphic.

In [32] Micallef showed that every stable minimal surface in a flat $4$ –torus must be holomorphic with respect to a complex structure compatible with the metric. (Note however that this is no longer the case for higher dimensional tori [4].) In view of Micallef’s result it was thought for some time that a similar result could hold for the K3 surface endowed with a hyperkähler metric. Partial results in this direction were established by Micallef–Wolfson [30, Theorem 5.3] and motivation for the conjecture came from the fact that, given an arbitrary hyperkähler metric on the K3 surface, every homology class can be represented by the sum of surfaces each of which is holomorphic with respect to some complex structure compatible with the metric. However, Micallef–Wolfson [31] have eventually shown that no analogue of the result for $4$ –tori holds for the K3 surface. Indeed they found a class $\alpha\in H_{2}(K3,\mathbb{Z})$ and a hyperkähler metric $g$ on the K3 surface such that the volume minimiser in $\alpha$ decomposes into a sum of branched minimal surfaces $\Sigma_{1}\cup\dots\cup\Sigma_{k}$ not all of which can be holomorphic with respect to some complex structure compatible with $g$ .

We can use our gluing construction to construct further (simpler) examples of strictly stable minimal spheres with respect to some hyperkähler metric on the K3 surface which cannot be holomorphic for any complex structure compatible with the metric.

Theorem 7.1.

There exist hyperkähler metrics on the K3 surface that contain a strictly stable minimal sphere which is not holomorphic with respect to any complex structure compatible with the metric.

Proof.

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. ∎