Hypersurfaces satisfying $\triangle \vec {H}=λ\vec {H}$ in $\mathbb{E}_{\lowercase{s}}^{5}$
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| Main Authors: | , |
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| Format: | Preprint |
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2024
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| _version_ | 1866929499523776512 |
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| author | Gupta, Ram Shankar Arvanitoyeorgos, Andreas |
| author_facet | Gupta, Ram Shankar Arvanitoyeorgos, Andreas |
| contents | In this paper, we study hypersurfaces $M_{r}^{4}$ $(r=0, 1, 2, 3, 4)$ satisfying $\triangle \vec{H}=λ\vec{H}$ ($λ$ a constant) in the pseudo-Euclidean space $\mathbb{E}_{s}^{5}$ $(s=0, 1, 2, 3, 4, 5)$. We obtain that every such hypersurface in $\mathbb{E}_{s}^{5}$ with diagonal shape operator has constant mean curvature, constant norm of second fundamental form and constant scalar curvature.
Also, we prove that every biharmonic hypersurface in $\mathbb{E}_{s}^{5}$ with diagonal shape operator must be minimal. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_08630 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Hypersurfaces satisfying $\triangle \vec {H}=λ\vec {H}$ in $\mathbb{E}_{\lowercase{s}}^{5}$ Gupta, Ram Shankar Arvanitoyeorgos, Andreas Differential Geometry 53D12, 53C40, 53C42 In this paper, we study hypersurfaces $M_{r}^{4}$ $(r=0, 1, 2, 3, 4)$ satisfying $\triangle \vec{H}=λ\vec{H}$ ($λ$ a constant) in the pseudo-Euclidean space $\mathbb{E}_{s}^{5}$ $(s=0, 1, 2, 3, 4, 5)$. We obtain that every such hypersurface in $\mathbb{E}_{s}^{5}$ with diagonal shape operator has constant mean curvature, constant norm of second fundamental form and constant scalar curvature. Also, we prove that every biharmonic hypersurface in $\mathbb{E}_{s}^{5}$ with diagonal shape operator must be minimal. |
| title | Hypersurfaces satisfying $\triangle \vec {H}=λ\vec {H}$ in $\mathbb{E}_{\lowercase{s}}^{5}$ |
| topic | Differential Geometry 53D12, 53C40, 53C42 |
| url | https://arxiv.org/abs/2409.08630 |