Examples of compact embedded mean convex $λ$-hypersurfaces
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911556632051712 |
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| author | Cheng, Qing-Ming Lai, Junqi Wei, Guoxin |
| author_facet | Cheng, Qing-Ming Lai, Junqi Wei, Guoxin |
| contents | There is a well-known conjecture asserts that the round sphere should be the only compact embedded self-shrinker (i.e. $0$-hypersurface) which is diffeomorphic to a sphere. S. Brendle confirmed the conjecture for 2-dimensional $0$-hypersurfaces. For any dimensional $λ$-hypersurfaces, if $λ<0$, we constructed compact convex embedded $λ$-hypersurface which is diffeomorphic to a sphere and is not a round sphere. In this paper, for $λ>0$, we construct a compact mean convex embedded $λ$-hypersurface which is diffeomorphic to a sphere and is not a round sphere. In fact, for $λ>0$, there are no compact convex embedded $λ$-hypersurfaces which are diffeomorphic to spheres except a round sphere. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_29371 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Examples of compact embedded mean convex $λ$-hypersurfaces Cheng, Qing-Ming Lai, Junqi Wei, Guoxin Differential Geometry There is a well-known conjecture asserts that the round sphere should be the only compact embedded self-shrinker (i.e. $0$-hypersurface) which is diffeomorphic to a sphere. S. Brendle confirmed the conjecture for 2-dimensional $0$-hypersurfaces. For any dimensional $λ$-hypersurfaces, if $λ<0$, we constructed compact convex embedded $λ$-hypersurface which is diffeomorphic to a sphere and is not a round sphere. In this paper, for $λ>0$, we construct a compact mean convex embedded $λ$-hypersurface which is diffeomorphic to a sphere and is not a round sphere. In fact, for $λ>0$, there are no compact convex embedded $λ$-hypersurfaces which are diffeomorphic to spheres except a round sphere. |
| title | Examples of compact embedded mean convex $λ$-hypersurfaces |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2603.29371 |