ON L_1-BIHARMONIC TIMELIKE HYPERSURFACES IN PSEUDO-EUCLIDEAN SPACE E_1^4

FIROOZ PASHAIE

《Tamkang Journal of Mathematics》 51卷4期 (2020/12) Pp. 313-332

https://doi.org/10.5556/j.tkjm.51.2020.3188

摘要

A well-known conjecture of Bang-Yen Chen says that the only biharmonic submanifolds in the Euclidean spaces are minimal ones. In this paper, we consider an extended condition (namely, L_1-biharmonicity) on non-degenerate timelike hypersurfaces of the pseudo-Euclidean space E_1^4. A Lorentzian hypersurface x:M_1^3→E_1^4 is called L_1-biharmonic if it satisfies the condition L_1^2x = 0, where L_1 is the linearized operator associated with the first variation of 2th mean curvature vector field on M_1^3. According to the multiplicities of principal curvatures, the L_1-extension of Chen's conjecture is proved for Lorentzian hypersurfaces with constant ordinary mean curvature in the pseudo-Euclidean space E_1^4. Additionally, we show that there is no proper L_1-biharmonic L_1-finite type connected orientable Lorentzian hypersurface in E_1^4.