An extension of Liebmann's Theorem to hypersurfaces with boundary
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866912552379744256 |
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| author | Cruz, Flávio França Nelli, Barbara |
| author_facet | Cruz, Flávio França Nelli, Barbara |
| contents | Liebmann's Theorem asserts that a compact, connected, convex surface with constant mean curvature (CMC) in the Euclidean space must be a totally umbilical sphere. In this article we extend Liebmann's result to hypersurfaces with boundary. More precisely, we prove that a locally convex, embedded, compact, connected CMC hypersurface bounded by a closed strictly convex $(n-1)$-dimensional submanifold in a hyperplane $Π^n\subset \mathbb{R}^{n+1}$ lies in one of the two halfspace determined by $Π$ and inherits the symmetries of the boundary. Consequently, spherical caps are the only such hypersurfaces with non-zero constant mean curvature bounded by a $(n-1)-$sphere. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_03368 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | An extension of Liebmann's Theorem to hypersurfaces with boundary Cruz, Flávio França Nelli, Barbara Differential Geometry 53C42, 35J60 Liebmann's Theorem asserts that a compact, connected, convex surface with constant mean curvature (CMC) in the Euclidean space must be a totally umbilical sphere. In this article we extend Liebmann's result to hypersurfaces with boundary. More precisely, we prove that a locally convex, embedded, compact, connected CMC hypersurface bounded by a closed strictly convex $(n-1)$-dimensional submanifold in a hyperplane $Π^n\subset \mathbb{R}^{n+1}$ lies in one of the two halfspace determined by $Π$ and inherits the symmetries of the boundary. Consequently, spherical caps are the only such hypersurfaces with non-zero constant mean curvature bounded by a $(n-1)-$sphere. |
| title | An extension of Liebmann's Theorem to hypersurfaces with boundary |
| topic | Differential Geometry 53C42, 35J60 |
| url | https://arxiv.org/abs/2412.03368 |