Vacuum Static Spaces with Positive Isotropic Curvature

Abstract

In this paper, we study vacuum static spaces with positive isotropic curvature. We prove that if (M n , g, f ), n ≥ 4, is a compact vacuum static space with positive isotropic curvature, then up to finite cover, M is isometric to a sphere S n or the product of a circle S 1 with an (n − 1)-dimensional sphere S n−1 .

Closed generalized Einstein manifolds with positive isotropic curvature

“…, which is exactly a vacuum static space. Thus, applying the main result in [9], we obtain the conclusion.…”

“…First of all, we can easily show that there are no critical points on the set f −1 (0) and each connected component of f −1 (0) is totally geodesic (see Section 2). Motivated by [8] and [9], the strategy of the proof is to introduce a 2-form ω = df ∧ i ∇f z g , consisting of the total differential, df , of the potential function f and the traceless Ricci tensor z g on M defined by z g = Ric g − Scalg n g, and show that ω = 0, under the PIC condition. Here i ∇f z denotes the interior product and Scal g is the scalar curvature of the metric g. Consequently, vanishing of ω = 0 is equivalent that z g is radially flat, i.e., z g (X, ∇f ) = 0 for any vector field X orthogonal to ∇f , and this implies that the potential function f does not have critical points except at minimum and maximum points.…”

Counterexamples to a divergence lower bound for the covariant derivative of skew-symmetric 2-tensor fields

“…The Black Hole Uniqueness Theorem for three-dimensional static solutions with positive scalar curvature and the Besse Conjecture for solutions to the Critical Point Equation are two very famous and related open problems in contemporary geometric analysis. Very recently, some very remarkable advances have been claimed on both of these problems in a series of papers [1,2,3,6,7,8]. In this short note, we point out an issue in the approach proposed in the above mentioned papers, providing counterexamples.…”

Counterexamples to a divergence lower bound for the covariant derivative of skew-symmetric 2-tensor fields

“…The Black Hole Uniqueness Theorem for three-dimensional static solutions with positive scalar curvature and the Besse conjecture for solutions to the critical point equation are two very famous and related open problems in contemporary geometric analysis. Very recently, some very remarkable advances have been claimed on both of these problems in a series of papers [1][2][3][6][7][8]. In this short note, we point out an issue in the approach proposed in the above-mentioned papers, providing counterexamples.…”