Embedded cylindrical and doughnut-shaped $λ$-hypersurfaces
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866910490603552768 |
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| author | Cheng, Qing-Ming Lai, Junqi Wei, Guoxin |
| author_facet | Cheng, Qing-Ming Lai, Junqi Wei, Guoxin |
| contents | In the paper, we construct, for $λ>0$, complete embedded and non-convex $λ$-hypersurfaces, which are diffeomorphic to a cylinder. Hence, one can not expect that $λ$-hypersurfaces share a common conclusion on the planar domain conjecture even if the planar domain conjecture of T. Ilmanen for self-shrinkers of mean curvature flow are solved by Brendle \cite{B} affirmatively. Furthermore, for a fixed $λ<0$ which may have small $|λ|$, we can construct two compact embedded $λ$-hypersurfaces which are diffeomorphic to $\mathbb{S}^{1}\times \mathbb{S}^{n-1}$, but they are not isometric to each other. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_11123 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Embedded cylindrical and doughnut-shaped $λ$-hypersurfaces Cheng, Qing-Ming Lai, Junqi Wei, Guoxin Differential Geometry In the paper, we construct, for $λ>0$, complete embedded and non-convex $λ$-hypersurfaces, which are diffeomorphic to a cylinder. Hence, one can not expect that $λ$-hypersurfaces share a common conclusion on the planar domain conjecture even if the planar domain conjecture of T. Ilmanen for self-shrinkers of mean curvature flow are solved by Brendle \cite{B} affirmatively. Furthermore, for a fixed $λ<0$ which may have small $|λ|$, we can construct two compact embedded $λ$-hypersurfaces which are diffeomorphic to $\mathbb{S}^{1}\times \mathbb{S}^{n-1}$, but they are not isometric to each other. |
| title | Embedded cylindrical and doughnut-shaped $λ$-hypersurfaces |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2406.11123 |