A note on static spaces and related problems

Abstract

Abstract. In this paper we study static spaces introduced in [10,12,9,13,7] and Riemannian manifolds possessing solutions to the critical point equation introduced in [1,11,3,4]. In both cases on the manifolds there is a function f satisfying the equation f Ric = ∇ 2 f + Φg. With a similar idea used in [6,5], we have made progress in solving the classifying problem raised in [9] of vacuum static spaces and in proving the conjecture made in [1] about manifolds admitting solutions to the critical point equation in general dimensions. We obtain even stronger results in dimension 3.

On the Mass of Static Metrics with Positive Cosmological Constant: II

“…The question of whether these are the only ones is still open, although there are some partial results. For instance, in [35,39] it is proven that these models are the only locally conformally flat static metrics, in [45] this result has been extended to the Bach-flat case and in [24] the case of cyclic parallel Ricci tensor has been discussed. Some pinching conditions implying the same classification are provided in [5,9].…”

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 point equation on four‐dimensional compact manifolds

“…On the other hand, it is well‐known that the second fundamental form of Σ is given by where Moreover, its mean curvature is Therefore, we combine the fundamental equation with to conclude that near the point q the second fundamental form is given by where μ is equal to the mean curvature. But, mutatis mutandis in Lemma 3.3 of follow that H is constant on Σ, so is μ, this also appeared recently in Corollary 3.2 of . In particular, at q when Proceeding, we can apply the same arguments used by Chen‐Wang in the proof of Lemma to conclude that provided q is not critical point of f .…”

“…where μ is equal to the mean curvature. But, mutatis mutandis in Lemma 3.3 of [4] follow that H is constant on , so is μ, this also appeared recently in Corollary 3.2 of [17]. In particular, R 1 jkl = 0 at q when { j, k, l} = {2, 3, 4}.…”

“…The conjecture proposed in [3] in 1987 may be restated in terms of CPE (see e.g. [11] , [12], [13] and [17]). More exactly, the quoted authors proposed the following conjecture.…”