مشخصات پژوهش

A well-known conjecture of Bang Yen-Chen says that the only biharmonic submanifolds in the Euclidean space are minimal ones. The conjecture was proved by Chen in the case of surfaces in $3$-dimensional Euclidean space. In this paper, we consider an extended version of Chen's conjecture (namely, $L_k$-conjecture) on Lorentzian hypersurfaces of the pseudo-Euclidean space E_1^4 (i.e. the Einstein space). A hypersurface x: M_1^3\rightarrow E_1^4 is called L_k-biharmonic (for k=0,1,2) if it satisfies the condition L_k^2x=0, where L_k is the linearized operator associated to the first variation of (k+1)th mean curvature vector field on M_1^3. Since L_0=\Delta the matter of L_k-biharmonicity is a natural generalization of biharmonicity. On any L_k-biharmonic Lorentzian hypersurface in E_1^4 with distinct principal curvatures, assuming $H_k$ to be constant we get that H_{k+1} is constant. Furthermore, we show that L_k-biharmonic spacelike hypersurfaces in E_1^4 with constant H_k are k-minimal.