Pinched self-dual Weyl curvature on Einstein four-manifolds

TL;DR

This paper proves that certain four-dimensional Einstein manifolds with specific curvature conditions are anti-self-dual, revealing a deep link between curvature pinching and geometric structure.

Contribution

It establishes that compact Einstein four-manifolds with harmonic, pinched self-dual Weyl curvature are necessarily anti-self-dual, introducing an almost-Kähler structure in the analysis.

Findings

Manifolds with harmonic, pinched self-dual Weyl curvature are anti-self-dual.

Existence of an almost-Kähler structure outside the zero set of the self-dual Weyl curvature.

Curvature pinching conditions imply strong geometric restrictions.

Abstract

We show that a compact oriented riemannian four-manifold with harmonic and pinched self-dual Weyl curvature is anti-self-dual if the type is nonpositive. The main part is to show that there is an almost-K\"ahler structure outside the zero set of the self-dual Weyl curvature.