$L_k$-biharmonic spacelike hypersurfaces in Minkowski $4$-space $mathbb{E}_1^4$

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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 $x : M^2rightarrowmathbb{E}^{3}$ 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 of $M^3$ are distinct. A hypersurface $x: M^3rightarrowmathbb{E}^{4}$ 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.