Curvature in the balance: the Weyl functional and scalar curvature of $4$-manifolds

Abstract

The infimum of the Weyl functional is shown to be surprisingly small on many compact $4$-manifolds that admit positive-scalar-curvature metrics. Results are also proved that systematically compare the scalar and self-dual Weyl curvatures of certain almost-Kähler $4$-manifolds.