Lk-BIHARMONIC SPACELIKE HYPERSURFACES INMINKOWSKI 4-SPACE E4

Biharmonic surfaces in Euclidean space E3 are firstlystudied from a differential geometric point of view by Bang-YenChen, who showed that the only biharmonic surfaces are minimalones. A surface x : M 2 ! E3 is called biharmonic if ∆2x =0, where ∆ is the Laplace operator of M 2. We study the Lkbiharmonic spacelike hypersurfaces in the 4-dimentional pseudoEuclidean space E4 1 with an additional condition that the principalcurvatures of M 3 are distinct. A hypersurface x : M 3 ! E4 is calledLk-biharmonic if L2 kx = 0 (for k = 0; 1; 2), where Lk is the linearized operator associated to the first variation of (k + 1)-th meancurvature of M 3. Since L0 = ∆, the matter of Lk-biharmonicity isa natural generalization of biharmonicity. On any Lk-biharmonicspacelike hypersurfaces in E4 1 with distinct principal curvatures, by,assuming Hk to be constant, we get that Hk+1 is constant. Furthermore, we show that Lk-biharmonic spacelike hypersurfaces inE4 1 with constant Hk are k-maximal

Keywords:

Spacelike hypersurface , Biharmonic , Lk-biharmonic , kmaximal