Gap theorems on critical point equation of the total scalar curvature with divergence-free Bach tensor

TL;DR

This paper investigates conditions under which critical metrics of total scalar curvature are Einstein, proving gap theorems for divergence-free Bach tensor metrics in dimensions 4 and 5 or higher.

Contribution

It establishes new gap theorems for critical metrics with divergence-free Bach tensor, advancing understanding of the conjecture for specific curvature conditions.

Findings

Proves gap properties for divergence-free Bach tensor in dimensions ≥5.

Establishes similar results for 4-dimensional manifolds.

Provides partial confirmation of Besse's conjecture under new conditions.

Abstract

On a compact $n$-dimensional manifold, 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 be Einstein. This conjecture, proposed in 1987 by Besse, has not been resolved except when $M$ has harmonic curvature or the metric is Bach flat. In this paper, we prove some gap properties under divergence-free Bach tensor condition for $n\geq 5$, and a similar condition for $n=4$.