عنوان مجله
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Tamkang Journal of Mathematics
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چکیده
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A well-known conjecture of Bang-Yen Chen says that the only
biharmonic submanifolds in the Euclidean spaces 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 with 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 proved for
Lorentzian hypersurfaces with constant ordinary mean curvature in the
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 $\E_1^4$.
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