Area-minimizing capillary cones
Abstract
We construct non-flat minimal capillary cones with bi-orthogonal symmetry groups for any dimension and contact angle. These cones interpolate between rescalings of a singular solution to the one-phase problem and the free-boundary cone obtained by halving a Lawson cone along a hyperplane of symmetry. The existence and uniqueness of such cones is proved by solving a nonlinear free boundary equation parametrized by the contact angle and obtaining monotonicity properties for the solutions. The constructed cones are minimizing in ambient dimension or higher, for appropriate contact angles, demonstrating that the regularity theory for minimizing capillary hypersurfaces can have singularities in codimension and completing the capillary regularity theory for contact angles near . We further develop the connection between capillary hypersurfaces and solutions of the one-phase problem, consequently producing new examples of singular minimizing free boundaries for the Alt-Caffarelli functional.
Cite
@article{arxiv.2601.18794,
title = {Area-minimizing capillary cones},
author = {Benjy Firester and Raphael Tsiamis and Yipeng Wang},
journal= {arXiv preprint arXiv:2601.18794},
year = {2026}
}