Vacuum Static Spaces with Positive Isotropic Curvature.

In this paper, we give a complete classification of critical metrics of the volume functional on a compact manifold $M$ with boundary $\partial M$ having positive isotropic curvature. We prove that for a pair $(f, \kappa)$ of a nontrivial smooth function $f: M \to {\Bbb R}$ and a nonnegative real number $\kappa$, if $(M, g)$ having positive isotropic curvature satisfies $$ Ddf - (\Delta f)g - f{\rm Ric} = \kappa g, $$ then $(M, g)$ is isometric to a geodesic ball in ${\Bbb S}^n$ when $\kappa >0$, and either $M$ isometric to ${\Bbb S}^n_+$, or the product $I \times {\Bbb S}^{n-1}$, up to finite cover when $\kappa =0$.