Minimal surfaces and CPE metric

The critical points of the total scalar curvature functional, restricted to closed $n$-dimensional manifolds with constant scalar curvature metrics and unit volume, are termed CPE metrics. In 1987, Arthur L. Besse conjectured that CPE metrics are always Einstein. Using the theory of minimal surfaces, we prove the conjecture for three-dimensional manifolds with $C^\infty$-generic Riemannian metric.