Critical point equation on four‐dimensional compact manifolds

Abstract

The aim of this article is to study the space of metrics with constant scalar curvature of volume 1 that satisfies the critical point equation for simplicity CPE metrics. It has been conjectured that every CPE metric must be Einstein. Here, we shall focus our attention for 4‐dimensional half conformally flat manifolds M4. In fact, we shall show that for a nontrivial must be isometric to a sphere and f is some height function on