A class of hypersurfaces in $ \mathbb{E}^{n+1}_{s} $ satisfying $ \Delta \vec{H} = \lambda\vec{H} $

Dan Yang; Jinchao Yu; Jingjing Zhang; Xiaoying Zhu; Dan Yang; Jinchao Yu; Jingjing Zhang; Xiaoying Zhu

doi:10.3934/math.2022003

AIMS Mathematics

2022, Volume 7, Issue 1: 39-53. doi: 10.3934/math.2022003

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Research article

A class of hypersurfaces in $ \mathbb{E}^{n+1}_{s} $ satisfying $ \Delta \vec{H} = \lambda\vec{H} $

1.
School of Mathematics, Liaoning University, Shenyang 110044, China
2.
School of Mechanical Engineering and Automation, Shanghai University, Shanghai 200072, China

Received: 03 July 2021 Accepted: 08 September 2021 Published: 29 September 2021
MSC : 53C40, 53C42

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 ^[1]: 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.
- proper mean curvature vector,
- linear Weingarten hypersurfaces,
- principal curvatures
Citation: 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} $[J]. AIMS Mathematics, 2022, 7(1): 39-53. doi: 10.3934/math.2022003

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Abstract

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 ^[1]: 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.

References

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