TY - JOUR
AU - Pashaie, Firooz
PY - 2020/11/01
Y2 - 2020/11/24
TI - On $L_1$-biharmonic timelike hypersurfaces in pseudo-Euclidean space $E_1^4$
JF - Tamkang Journal of Mathematics
JA - Tamkang J. Math.
VL - 51
IS - 4
SE -
DO - 10.5556/j.tkjm.51.2020.3188
UR - https://journals.math.tku.edu.tw/index.php/TKJM/article/view/3188
SP - 313-332
AB - <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 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 $E_1^4$.</p>
ER -