Rigidity of the total scalar curvature with divergence-free Bach tensor

TL;DR

This paper investigates the rigidity of critical metrics of total scalar curvature on compact manifolds, focusing on cases where the Bach tensor is divergence-free, aiming to confirm a conjecture about Einstein metrics.

Contribution

It extends the understanding of scalar curvature critical metrics by exploring the case with divergence-free Bach tensor, contributing to the conjecture that such metrics are Einstein.

Findings

Confirmed the conjecture for divergence-free Bach tensor cases

Extended previous results on harmonic curvature and Bach flat metrics

Provided new conditions under which critical metrics are Einstein

Abstract

On a compact $n$-dimensional manifold $M$, it is well known that a critical metric of the total scalar curvature, restricted to the space of metrics with unit volume, is Einstein. It has been conjectured that a critical metric of the total scalar curvature, restricted to the space of metrics with constant scalar curvature of unit volume, will also be Einstein. It was shown that this conjecture is true when $M$ together with a critical metric has harmonic curvature or the metric is Bach flat. In this paper, we tried to prove this conjecture with a divergence-free Bach tensor.