@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}
}