Critical point equation on four‐dimensional compact manifolds

Abstract

The aim of this article is to study the space of metrics with constant scalar curvature of volume 1 that satisfies the critical point equation frakturLg*(f)=Ric˚, for simplicity CPE metrics. It has been conjectured that every CPE metric must be Einstein. Here, we shall focus our attention for 4‐dimensional half conformally flat manifolds M4. In fact, we shall show that for a nontrivial f,M4 must be isometric to a sphere double-struckS4 and f is some height function on S4.

Brown–York mass and positive scalar curvature II: Besse’s conjecture and related problems

“…(In fact, this result can be improved for three dimensional manifolds. For more details, please see [21]; also see [1] for a four dimensional result).…”

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

“…Barros and Ribeiro Jr [3] showed that Conjecture 1 is also true for 4-dimensional half conformally flat manifolds. We highlight that CP 2 endowed with Fubini-Study metric shows that the half-conformally flat condition is weaker than locally conformally flat condition in dimension 4.…”

“…But, since |∇ f | 2 is constant in Σ c follows that H is constant on Σ, so is µ (see also [3]). Then, we may use Codazzi's equation to infer Similarly, we can use (3.8) jointly with (3.2) to conclude that C i j1 = 0 when {i, j} = {1, 2, 3, 4}.…”

“…Therefore, it is natural to ask which geometric implications has the assumption of the harmonicity of the tensor W + on a CPE metric. Inspired by the historical development on the study of CPE conjecture, in this paper, we prove that, in dimension 4, the assumption that (M 4 , g) is locally conformally flat, considered in [13] and [6], as well as half conformally flat considered in [3], can be replaced by the weaker condition that M 4 has harmonic tensor W + . More precisely, we have the following result.…”

The critical point equation and contact geometry

“…For example, Lafontaine proved that the CPE conjecture is true under conformally flat assumption with KerL * g (λ) = 0. Recently, Barros and Ribeiro Jr [3] proved that the CPE conjecture is also true for half conformally flat. Another partial proof of the CPE conjecture was presented by Yun, Chang and Hwang [12].…”