A class of hypersurfaces in $ \mathbb{E}^{n+1}_{s} $ satisfying $ \Delta \vec{H} = \lambda\vec{H} $
A nondegenerate hypersurface in a pseudo-Euclidean space $ \mathbb{E}^{n+1}_{s} $ is called to have proper mean curvature vector if its mean curvature $ \vec{H} $ satisfies $ \Delta \vec{H} = \lambda \vec{H} $ for a constant $ \lambda $. In 2013, Arvanitoyeorgos and Kaimakamis conjectured <sup>...
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| Language: | English |
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AIMS Press
2022-01-01
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| Series: | AIMS Mathematics |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/math.2022003?viewType=HTML |
| _version_ | 1798035321553682432 |
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| author | Dan Yang Jinchao Yu Jingjing Zhang Xiaoying Zhu |
| author_facet | Dan Yang Jinchao Yu Jingjing Zhang Xiaoying Zhu |
| author_sort | Dan Yang |
| collection | DOAJ |
| description | A nondegenerate hypersurface in a pseudo-Euclidean space $ \mathbb{E}^{n+1}_{s} $ is called to have proper mean curvature vector if its mean curvature $ \vec{H} $ satisfies $ \Delta \vec{H} = \lambda \vec{H} $ for a constant $ \lambda $. In 2013, Arvanitoyeorgos and Kaimakamis conjectured <sup>[<xref ref-type="bibr" rid="b1">1</xref>]</sup>: any hypersurface satisfying $ \Delta \vec{H} = \lambda \vec{H} $ in pseudo-Euclidean space $ \mathbb E_{s}^{n+1} $ has constant mean curvature. This paper will give further support evidences to this conjecture by proving that a linear Weingarten hypersurface $ M^{n}_{r} $ in $ \mathbb E^{n+1}_{s} $ satisfying $ \Delta \vec{H} = \lambda \vec{H} $ has constant mean curvature if $ M^{n}_{r} $ has diagonalizable shape operator with less than seven distinct principal curvatures. |
| first_indexed | 2024-04-11T20:56:37Z |
| format | Article |
| id | doaj.art-8258531450cf4a748a7d7a1a6b9ca8f8 |
| institution | Directory Open Access Journal |
| issn | 2473-6988 |
| language | English |
| last_indexed | 2024-04-11T20:56:37Z |
| publishDate | 2022-01-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | AIMS Mathematics |
| spelling | doaj.art-8258531450cf4a748a7d7a1a6b9ca8f82022-12-22T04:03:39ZengAIMS PressAIMS Mathematics2473-69882022-01-0171395310.3934/math.2022003A class of hypersurfaces in $ \mathbb{E}^{n+1}_{s} $ satisfying $ \Delta \vec{H} = \lambda\vec{H} $Dan Yang0Jinchao Yu1Jingjing Zhang2Xiaoying Zhu31. School of Mathematics, Liaoning University, Shenyang 110044, China1. School of Mathematics, Liaoning University, Shenyang 110044, China2. School of Mechanical Engineering and Automation, Shanghai University, Shanghai 200072, China1. School of Mathematics, Liaoning University, Shenyang 110044, ChinaA nondegenerate hypersurface in a pseudo-Euclidean space $ \mathbb{E}^{n+1}_{s} $ is called to have proper mean curvature vector if its mean curvature $ \vec{H} $ satisfies $ \Delta \vec{H} = \lambda \vec{H} $ for a constant $ \lambda $. In 2013, Arvanitoyeorgos and Kaimakamis conjectured <sup>[<xref ref-type="bibr" rid="b1">1</xref>]</sup>: any hypersurface satisfying $ \Delta \vec{H} = \lambda \vec{H} $ in pseudo-Euclidean space $ \mathbb E_{s}^{n+1} $ has constant mean curvature. This paper will give further support evidences to this conjecture by proving that a linear Weingarten hypersurface $ M^{n}_{r} $ in $ \mathbb E^{n+1}_{s} $ satisfying $ \Delta \vec{H} = \lambda \vec{H} $ has constant mean curvature if $ M^{n}_{r} $ has diagonalizable shape operator with less than seven distinct principal curvatures.https://www.aimspress.com/article/doi/10.3934/math.2022003?viewType=HTMLproper mean curvature vectorlinear weingarten hypersurfacesprincipal curvatures |
| spellingShingle | Dan Yang Jinchao Yu Jingjing Zhang Xiaoying Zhu A class of hypersurfaces in $ \mathbb{E}^{n+1}_{s} $ satisfying $ \Delta \vec{H} = \lambda\vec{H} $ AIMS Mathematics proper mean curvature vector linear weingarten hypersurfaces principal curvatures |
| title | A class of hypersurfaces in $ \mathbb{E}^{n+1}_{s} $ satisfying $ \Delta \vec{H} = \lambda\vec{H} $ |
| title_full | A class of hypersurfaces in $ \mathbb{E}^{n+1}_{s} $ satisfying $ \Delta \vec{H} = \lambda\vec{H} $ |
| title_fullStr | A class of hypersurfaces in $ \mathbb{E}^{n+1}_{s} $ satisfying $ \Delta \vec{H} = \lambda\vec{H} $ |
| title_full_unstemmed | A class of hypersurfaces in $ \mathbb{E}^{n+1}_{s} $ satisfying $ \Delta \vec{H} = \lambda\vec{H} $ |
| title_short | A class of hypersurfaces in $ \mathbb{E}^{n+1}_{s} $ satisfying $ \Delta \vec{H} = \lambda\vec{H} $ |
| title_sort | class of hypersurfaces in mathbb e n 1 s satisfying delta vec h lambda vec h |
| topic | proper mean curvature vector linear weingarten hypersurfaces principal curvatures |
| url | https://www.aimspress.com/article/doi/10.3934/math.2022003?viewType=HTML |
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