@article { author = {Pashaie, Firooz and Mohammadpouri, Akram}, title = {$L_k$-biharmonic spacelike hypersurfaces in Minkowski $4$-space $\mathbb{E}_1^4$}, journal = {Sahand Communications in Mathematical Analysis}, volume = {05}, number = {1}, pages = {21-30}, year = {2017}, publisher = {University of Maragheh}, issn = {2322-5807}, eissn = {2423-3900}, doi = {10.22130/scma.2017.20589}, abstract = {Biharmonic surfaces in Euclidean space $\mathbb{E}^3$ are firstly studied from a differential geometric point of view by Bang-Yen Chen, who showed that the only biharmonic surfaces are minimal ones. A surface $x : M^2\rightarrow\mathbb{E}^{3}$ is called biharmonic if $\Delta^2x=0$, where $\Delta$ is the Laplace operator of $M^2$. We study the $L_k$-biharmonic spacelike hypersurfaces in the $4$-dimentional pseudo-Euclidean space $\mathbb{E}_1^4$ with an additional condition that the principal curvatures are distinct. A hypersurface $x: M^3\rightarrow\mathbb{E}^{4}$ is called $L_k$-biharmonic if $L_k^2x=0$ (for $k=0,1,2$), where $L_k$ is the linearized operator associated to the first variation of $(k+1)$-th mean curvature of $M^3$. Since $L_0=\Delta$, the matter of $L_k$-biharmonicity is a natural generalization of biharmonicity. On any $L_k$-biharmonic spacelike hypersurfaces in $\mathbb{E}_1^4$ with distinct principal curvatures, by, assuming $H_k$ to be constant, we get that $H_{k+1}$ is constant. Furthermore, we show that $L_k$-biharmonic spacelike hypersurfaces in $\mathbb{E}_1^4$ with constant $H_k$ are $k$-maximal.}, keywords = {Spacelike hypersurface,Biharmonic,$L_k$-biharmonic,$k$-maximal}, url = {https://scma.maragheh.ac.ir/article_20589.html}, eprint = {https://scma.maragheh.ac.ir/article_20589_41cae243cd77692b496d7ab7a304e79b.pdf} }