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 $n \geq 2$. As a consequence, we obtain the generalized Bernstein theorem for embedded complete stable capillary minimal hypersurfaces in $\mathbb{H}^{4}$ 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.
This work gives a regularity and compactness theory for stable capillary minimal hypersurfaces for every contact angle $\theta\in(0,\pi)$.
Regularity and compactness
The singular set is empty below dimension $n_\theta$, discrete in dimension $n_\theta$, and has dimension at most $n-n_\theta$ above it, where $n_\theta$ equals $7$, $6$, or $5$ according to the distance of $\theta$ from $90^\circ$.
Cone classification
For $n=3,4$ and every contact angle $\theta\in(0,\pi)$, a stable capillary minimal hypercone with an isolated singularity is flat.
Bernstein consequence
For $2\leq n\leq4$, every properly embedded complete two-sided stable capillary minimal hypersurface in $\mathbb H^{n+1}$ with Euclidean area growth is flat.
The main difficulty is designing a capillary tilt function that both satisfies a differential Schoen inequality and is compatible with the capillary boundary condition, so that the boundary terms cancel after a carefully weighted integration. Once this inequality is established, the De Giorgi iteration used by Bellettini yields a boundary sheeting theorem, which then leads to the regularity and compactness theory.
My research interests include Geometric Analysis and Partial Differential Equations.