Vacuum Static Spaces with Positive Isotropic Curvature

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