# Rigidity of volume-minimizing hypersurfaces in Riemannian 5-manifolds

## TL;DR

This paper establishes volume bounds and rigidity results for hypersurfaces in 5-manifolds with scalar curvature constraints, extending previous results to non-Einstein manifolds and characterizing cases of equality.

## Contribution

It generalizes volume and rigidity results for hypersurfaces in 5-manifolds without the Einstein condition, linking volume bounds to curvature and topological splitting.

## Key findings

- Upper bound for hypersurface volume in terms of traceless Ricci curvature
- Rigidity when the bound is saturated and Ricci curvature is nonnegative
- Splitting of the manifold near the hypersurface under certain conditions

## 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.