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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 $Σ$ 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 $Σ$ 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 $Σ$. In the case that the scalar curvature of $M$ is at least $-n(n-1)$ and $Σ$ locally minimizes a certain functional inspired by [30], a neighborhood of $Σ$ in $M$ is isometric to $((-\varepsilon,\varepsilon)\timesΣ,dt^2+e^{2t}g)$, where $g$ is Ricci flat.