Conformal K\"{a}hler rigidity of Einstein four-manifolds

For a compact, connected, oriented Einstein four-manifold, we prove that, if the largest eigenvalue of the self-dual Weyl curvature $W^+$ is everywhere simple, then, after at worst passing to a double cover, the metric is conformally Kähler with positive scalar curvature; more generally, this result holds for metrics with harmonic self-dual Weyl curvature.

For Einstein metrics satisfying a uniform simplicity hypothesis on the largest eigenvalue, we further prove that either $W^+\equiv 0$, or $W^+$ nowhere vanishes and the previous conclusion holds.

We also obtain extensions to complete Ricci-flat four-manifolds and an optimal pinching theorem for the holomorphic sectional curvature of compact Kähler--Einstein surfaces.

The proof combines LeBrun's conformal normalization with the resulting weighted divergence equation and new first-order identities.

A zero-capacity argument allows this method to be used across the zero set of $W^+$ and at infinity in the noncompact case.

Finally, K3 surfaces and multicentered Gibbons--Hawking gravitational instantons show that our assumptions are sharp.