Weyl Curvature, Del Pezzo Surfaces, and Almost-Kähler Geometry

If a smooth compact 4-manifold $$M$$M admits a Kähler–Einstein metric g of positive scalar curvature, Gursky (Ann Math 148:315–337, 1998) showed that its conformal class [g] is an absolute minimizer of the Weyl functional among all conformal classes with positive Yamabe constant. Here we prove that, with the same hypotheses, [g] also minimizes of the Weyl functional on a different open set of conformal classes, most of which have negative Yamabe constant. An analogous minimization result is then proved for Einstein metrics g which are Hermitian, but not Kähler.