Free boundary hypersurfaces with nonpositive Yamabe invariant in mean convex manifolds

We obtain some estimates on the area of the boundary and on the volume of a certain free boundary hypersurface $\Sigma$ with nonpositive Yamabe invariant in a Riemannian $n$-manifold with bounds for the scalar curvature and the mean curvature of the boundary. Assuming further that $\Sigma$ is locally volume-minimizing in a manifold $M^n$ with scalar curvature bounded below by a nonpositive constant and mean convex boundary, we conclude that locally $M$ splits along $\Sigma$. In the case that the scalar curvature of $M$ is at least $-n(n-1)$ and $\Sigma$ locally minimizes a certain functional inspired by [30], a neighborhood of $\Sigma$ in $M$ is isometric to $((-\varepsilon,\varepsilon)\times\Sigma,dt^2+e^{2t}g)$, where $g$ is Ricci flat.