On Critical Point Equation of Compact Manifolds with Zero radial Weyl Curvature

TL;DR

This paper investigates critical metrics of the total scalar curvature functional on compact manifolds with zero radial Weyl curvature, establishing conditions under which these metrics are isometric to standard spheres, especially in low dimensions.

Contribution

It proves that CPE metrics with nonnegative sectional curvature in 3D and certain pinching conditions in higher dimensions are isometric to standard spheres, extending understanding of geometric structures under curvature constraints.

Findings

3D CPE metrics with nonnegative sectional curvature are standard spheres.

Higher-dimensional CPE metrics satisfying pinching conditions are standard spheres.

Vanishing Weyl tensor conditions imply metrics are isometric to spheres.

Abstract

Let $\mathcal{C}$ be the space of smooth metrics $g$ on a given compact manifold $M^{n}$ ($n\geq3$) with constant scalar curvature and unitary volume. The goal of this paper is to study the critical point of the total scalar curvature functional restricted to the space $\mathcal{C}$ (we shall refer to this critical point as CPE metrics) under assumption that $(M,g)$ has zero radial Weyl curvature. Among the results obtained, we emphasize that in 3-dimension we will be able to prove that a CPE metric with nonnegative sectional curvature must be isometric to a standard $3$-sphere. We will also prove that a $n$-dimensional, $4\leq n\leq10,$ CPE metric satisfying a $L^{n/2}$-pinching condition will be isometric to a standard sphere. In addition, we shall conclude that such critical metrics are isometrics to a standard sphere under fourth-order vanishing condition on the Weyl tensor.