Free Boundary Hypersurfaces with Non-positive 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 non-positive 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 from below by a non-positive 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 works of Yau [35] and Andersson-Galloway [4], a neighbourhood of $$\Sigma $$ in M is isometric to $$((-\varepsilon , \varepsilon ) \times \Sigma ,\mathrm{{d}}t^{2}+e^{2t}g)$$ , where g is Ricci flat with totally geodesic boundary.