Area-minimizing capillary cones
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914281621028864 |
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| author | Firester, Benjy Tsiamis, Raphael Wang, Yipeng |
| author_facet | Firester, Benjy Tsiamis, Raphael Wang, Yipeng |
| contents | 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 $π/2$. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_18794 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Area-minimizing capillary cones Firester, Benjy Tsiamis, Raphael Wang, Yipeng Differential Geometry Analysis of PDEs 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 $π/2$. 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. |
| title | Area-minimizing capillary cones |
| topic | Differential Geometry Analysis of PDEs |
| url | https://arxiv.org/abs/2601.18794 |