On critical point equation of compact manifolds with zero radial Weyl curvature

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.