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 $\kappa =0$.
Categories: math.DG