Area-minimizing capillary cones

Benjy Firester; Raphael Tsiamis; Yipeng Wang

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arXiv · 2601.18794

Area-minimizing capillary cones

Benjy Firester·Raphael Tsiamis·Yipeng Wang

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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 $8$ or higher, for appropriate contact angles, demonstrating that the regularity theory for minimizing capillary hypersurfaces can have singularities in codimension $7$ and completing the capillary regularity theory for contact angles near $\pi/2$. 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.

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Benjy Firester, Raphael Tsiamis, Yipeng Wang. 2026-01-26. Area-minimizing capillary cones. https://arxiv.org/abs/2601.18794

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