On $L_1$-biharmonic timelike hypersurfaces in pseudo-Euclidean space $E_1^4$

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Firooz Pashaie


A well-known conjecture of Bang Yen-Chen says that the only biharmonic Euclidean submanifolds 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 $x: M_1^3\rightarrow\E_1^4$ is called $L_1$-biharmonic if it satisfies the condition $L_1^2x=0$, where $L_1$ is the linearized operator associated to the first variation of 2-th mean curvature vector field on $M_1^3$. According to the multiplicities of principal curvatures, the $L_1$-extension of Chen's conjecture is affirmed for Lorentzian hypersurfaces with constant ordinary mean curvature in 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 $E_1^4$.

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How to Cite
Pashaie, F. (2020). On $L_1$-biharmonic timelike hypersurfaces in pseudo-Euclidean space $E_1^4$. Tamkang Journal of Mathematics, 51(4), 313–332. https://doi.org/10.5556/j.tkjm.51.2020.3188
Author Biography

Firooz Pashaie, Academic member of Universiity of Maragheh, Maragheh, Iran.

Dept of Mathematics, Faculty of Basic Sciences


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