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.