V-static spaces with positive isotropic curvature

Abstract

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 $f:M\to \mathbb{R}$ and a nonnegative real number $\kappa$, if $(M,g)$ having positive isotropic curvature satisfies $$Ddf-(\Delta f)g-f\mathrm{Ric}=\kappa g,$$ then $(M,g)$ is isometric to a geodesic ball in ${\mathbb{S}}^n$ when $\kappa >0$, and either $M$ isometric to ${\mathbb{S}}_{+}^n$, or the product $I\times {\mathbb{S}}^{n-1}$, up to finite cover when $\kappa =0$.