Lk -biharmonic spacelike hypersurfaces in Minkowski 4 -space E41

Biharmonic surfaces in Euclidean space E3 are firstly studied from a differential 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 E41 with an additional condition that the principal curvatures are distinct. A hypersurface x:M3→E4 is called Lk-biharmonic if L2kx=0 (for k=0,1,2), where Lk is the linearized operator associated to the first 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 E41 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 E41 with constant Hk are k-maximal.