Free Boundary Hypersurfaces with Non-positive Yamabe Invariant in Mean Convex Manifolds

Abstract

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.