Area estimates for capillary cmc hypersurfaces with nonpositive Yamabe invariant

We prove area estimates for stable capillary $cmc$ (minimal) hypersurfaces $\mathrm{\Sigma }$ 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 $H^{\mathrm{\partial }M}$ bounded from below. We also prove a local rigidity result in the case $\mathrm{\Sigma }$ is embedded and $\mathcal{J}$-energy-minimizing. In this case, we show that $M$ locally splits along $\mathrm{\Sigma }$ and is isometric to $(-\epsilon ,\epsilon )\times \mathrm{\Sigma },d{t}^2+e^{-2Ht}g)$, where $g$ is Einstein, or Ricci flat, $H\ge 0$ and $\mathrm{\partial }\mathrm{\Sigma }$ is totally geodesic.