@article{Pashaie_2020, title={On $L_1$-biharmonic timelike hypersurfaces in pseudo-Euclidean space $E_1^4$}, volume={51}, url={https://journals.math.tku.edu.tw/index.php/TKJM/article/view/3188}, DOI={10.5556/j.tkjm.51.2020.3188}, abstractNote={<p>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 Chenâ€™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 $E_1^4$.</p>}, number={4}, journal={Tamkang Journal of Mathematics}, author={Pashaie, Firooz}, year={2020}, month={Nov.}, pages={313-332} }