Minimality of free-boundary axial hyperplanes in high dimensional circular cones via calibration
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866908841010003968 |
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| author | Vianello, Giacomo |
| author_facet | Vianello, Giacomo |
| contents | Consider an $(n+1)$-dimensional circular cone with opening angle $α\in (0,π)$. Using a free-boundary adaptation of the classical calibration method, we prove that, for $n \geq 4$, there exists a threshold $\barα(n) \in (0,π)$ such that if $α\geq \barα(n)$, that is, the cone is wide enough, the intersection of the cone with an axial hyperplane is area-minimizing with respect to free-boundary variations inside the cone. This provides a counterexample to a recent Vertex-skipping Theorem proved by the author in collaboration with G.P. Leonardi, at least for $n\geq4$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_06601 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Minimality of free-boundary axial hyperplanes in high dimensional circular cones via calibration Vianello, Giacomo Analysis of PDEs Differential Geometry Consider an $(n+1)$-dimensional circular cone with opening angle $α\in (0,π)$. Using a free-boundary adaptation of the classical calibration method, we prove that, for $n \geq 4$, there exists a threshold $\barα(n) \in (0,π)$ such that if $α\geq \barα(n)$, that is, the cone is wide enough, the intersection of the cone with an axial hyperplane is area-minimizing with respect to free-boundary variations inside the cone. This provides a counterexample to a recent Vertex-skipping Theorem proved by the author in collaboration with G.P. Leonardi, at least for $n\geq4$. |
| title | Minimality of free-boundary axial hyperplanes in high dimensional circular cones via calibration |
| topic | Analysis of PDEs Differential Geometry |
| url | https://arxiv.org/abs/2601.06601 |