Lk-BIHARMONIC SPACELIKE HYPERSURFACES IN MINKOWSKI 4-SPACE E^4_1

Biharmonic surfaces in Euclidean space E³ are rstly studied from a di erential geometric point of view by Bang-Yen Chen, who showed that the only biharmonic surfaces are minimal ones. A surface x : M² →E³ is called biharmonic if Δ²x = 0, where Δ is the Laplace operator of M². We study the Lk- biharmonic spacelike hypersurfaces in the 4-dimentional pseudo- Euclidean space E^4_1 with an additional condition that the principal curvatures of M³ are distinct. A hypersurface x : M³ → E^4 is called Lk-biharmonic if L^2 _kx = 0 (for k = 0,1, 2), where Lk is the linearized operator associated to the rst variation of (k+1)-th mean curvature of M³. Since L0 = Δ, the matter of Lk-biharmonicity is a natural generalization of biharmonicity. On any Lk-biharmonic spacelike hypersurfaces in E^4_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 in E^4_1 with constant Hk are k-maximal.