Title: Rigidity of volume-minimizing hypersurfaces in Riemannian 5-manifolds,
Abstract: 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 $\Sigma$ 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 $\Sigma$. Furthermore, if
$\Sigma$ saturates the respective upper bound and $M$ has nonnegative Ricci
curvature, then $\Sigma$ is isometric to $\mathbb{S}^4$ up to scaling and $M$
splits in a neighborhood of $\Sigma$. Also, we obtain a rigidity result for the
Riemannian cover of $M$ when $\Sigma$ minimizes the volume in its homotopy
class and saturates the upper bound.