Abstract : It has been conjectured that, on a compact $3$-dimensional orientable manifold, a critical point of the total scalar curvature restricted to the space of constant scalar curvature metrics of unit volume is Einstein. In this paper we prove this conjecture under a condition that $\ker s_g'^*\neq 0$, which generalizes the previous partial results.
Keywords : total scalar curvature, critical point metric, Einstein metric
