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 $\mathcal{J}$-energy-minimizing. In this case, we show that $M$ locally splits along $\Sigma$ and is isometric to $(-\varepsilon,\varepsilon)\times \Sigma, dt^2 + e^{-2Ht}g)$, where $g$ is Einstein, or Ricci flat, $H\geq 0$ and $\partial\Sigma$ is totally geodesic.