Rigidity of volume-minimizing hypersurfaces in Riemannian 5-manifolds

In this paper we generalize the main result of [4] for manifolds that are not necessarily Einstein. In fact, we obtain an upper bound for the volume of a locally volume-minimizing closed hypersurface $\mathrm{\Sigma }$ of a Riemannian 5-manifold $M$ with scalar curvature bounded from below by a positive constant in terms of the total traceless Ricci curvature of $\mathrm{\Sigma }$. Furthermore, if $\mathrm{\Sigma }$ saturates the respective upper bound and $M$ has nonnegative Ricci curvature, then $\mathrm{\Sigma }$ is isometric to ${\mathbb{S}}^4$ up to scaling and $M$ splits in a neighborhood of $\mathrm{\Sigma }$. Also, we obtain a rigidity result for the Riemannian cover of $M$ when $\mathrm{\Sigma }$ minimizes the volume in its homotopy class and saturates the upper bound.