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

Abstract

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 K\"ahler 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 K\"ahler--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.