Rigidity of Critical Point Metrics under some Ricci curvature constraints

A critical point metric is a critical point of the total scalar curvature functional restricted to the space of constant scalar curvature metrics on a closed manifold with unit volume. It was conjectured in 1980's that every critical point metric must be Einstein. In this paper, we prove that this conjecture is true if the norm of the traceless Ricci operator $|\widetilde{Ric}|$ is constant. For $3$-dimensional case, we prove that the conjecture is true, if the traceless Ricci operator satisfies $tr((\widetilde{Ric})^3)\geq -\frac{R}{12}|\widetilde{Ric}|^2$, where $R$ denotes the scalar curvature. where R denotes the scalar curvature.