Rigidity of Critical Point Metrics under some Ricci curvature constraints

TL;DR

This paper investigates the conditions under which critical point metrics of the total scalar curvature functional are necessarily Einstein, proving the conjecture under specific Ricci curvature constraints.

Contribution

It proves the Einstein conjecture for critical point metrics when the traceless Ricci operator has constant norm or satisfies certain inequalities in three dimensions.

Findings

Critical point metrics are Einstein if $| ilde{Ric}|$ is constant.

In 3D, the conjecture holds if $tr(( ilde{Ric})^3) \,\geq \, -\frac{R}{12} |\tilde{Ric}|^2$.

The results confirm the conjecture under specific Ricci curvature conditions.

Abstract

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.