Weiss monotonicity and capillary hypersurfaces

Previous work of the authors established the rigorous limiting behavior of minimizing capillary surfaces to minimizers of the Alt–Caffarelli functional as the capillary angle tends to zero. We prove here that in this limit, the capillary area-density converges to the Weiss energy density. We apply this to obtain angle-independent curvature estimates and regularity results for capillary minimizers.