Regularity of minimal surfaces with capillary boundary conditions

We prove $\epsilon$-regularity theorems for varifolds with capillary boundary condition in a Riemannian manifold. These varifolds were first introduced by Kagaya–Tonegawa. We establish a uniform first variation control for all such varifolds (and free-boundary varifolds generally) satisfying a sharp density bound and prove that if a capillary varifold has bounded mean curvature and is close to a capillary half-plane with angle not equal to $\frac{\pi }{2}$, then it coincides with a $C^{1,\alpha }$ properly embedded hypersurface. We apply our theorem to deduce regularity at a generic point along the boundary in the region where the density is strictly less than 1.