Critical metrics of the total scalar curvature functional on 4-manifolds

The purpose of this paper is to investigate the critical points of the total scalar curvature functional restricted to space of metrics with constant scalar curvature of unitary volume, for simplicity CPE metrics. It was conjectured in $1980$'s that every CPE metric must be Einstein. We prove that a $4$-dimensional CPE metric with harmonic tensor $W^+$ must be isometric to a round sphere $\Bbb{S}^4.$