Area estimates for capillary cmc hypersurfaces with nonpositive Yamabe invariant

We prove area estimates for stable capillary $cmc$ (minimal) hypersurfaces $\Sigma$ with nonpositive Yamabe invariant that are properly immersed in a Riemannian $n$-dimensional manifold $M$ with scalar curvature $R^M$ and mean curvature of the boundary $H^{\partial M}$ bounded from below. We also prove a local rigidity result in the case $\Sigma$ is embedded and $J$-energy-minimizing. In this case, we show that $M$ locally splits along $\Sigma$ and is isometric to $(-\epsilon ,\epsilon )\times \Sigma ,d{t}^2+e^{-2Ht}g)$, where $g$ is Einstein or Ricci flat, $H⩾0$ and $\partial \Sigma$ is totally geodesic.