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

Abstract

If a smooth compact \(4\)-manifold \(M\) admits a Kahler–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 Kahler.