Critical metrics of the total scalar curvature functional on 4-manifolds

TL;DR

This paper investigates critical points of the total scalar curvature functional on 4-manifolds with constant scalar curvature, proving that under certain conditions, such metrics are isometric to a round sphere.

Contribution

It proves that 4-dimensional CPE metrics with harmonic W+ tensor are necessarily isometric to a round sphere, advancing understanding of the conjecture that all CPE metrics are Einstein.

Findings

4D CPE metrics with harmonic W+ are spherical

Supports the conjecture that CPE metrics are Einstein in specific cases

Provides conditions under which CPE metrics are classified as round spheres

Abstract

The purpose of this paper is to investigate the critical points of the total scalar curvature functional restricted to space of metrics with constant scalar curvature of unitary volume, for simplicity CPE metrics. It was conjectured in $1980$'s that every CPE metric must be Einstein. We prove that a $4$-dimensional CPE metric with harmonic tensor $W^+$ must be isometric to a round sphere $\Bbb{S}^4.$