Besse conjecture with vanishing conditions on the Weyl tensor

TL;DR

This paper proves the Besse conjecture for compact manifolds under weaker conditions than harmonic curvature, advancing understanding of critical metrics related to scalar curvature.

Contribution

It establishes the Besse conjecture with vanishing conditions on the Weyl tensor, a weaker assumption than previously required, for dimensions n ≥ 3.

Findings

Proves the Besse conjecture under weaker curvature conditions

Shows critical metrics with vanishing Weyl tensor are Einstein

Extends previous results to broader class of metrics

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 was proposed in 1987 by Besse, but has yet to be proved. In this paper, we prove the Besse conjecture with a weaker condition than harmonic curvature for $n\geq 3$.