Regularity of stable capillary minimal hypersurfaces

We develop a regularity and compactness theory for stable capillary minimal hypersurfaces in the half-space ${\mathbb{H}}^{n+1}$ with contact angle $\theta \in (0,\pi )$ and dimension $n\ge 2$. As a consequence, we obtain the generalized Bernstein theorem for embedded complete stable capillary minimal hypersurfaces in ${\mathbb{H}}^{n+1}$ with Euclidean area growth. The key innovation is an integral curvature estimate: by carefully selecting an appropriate tilt excess function, we are able to eliminate the boundary terms arising in the stability inequality. Building on this, we establish a boundary sheeting theorem by refining the arguments in [SS81]. These results, combined with a refined classification of stable capillary minimal cones, lead to the main regularity and compactness theorems.