timelike hypersurfaces in einstein space with constant 2nd mean curvature

In this paper, we study timelike hypersurfaces of the minkowski 4-space l4, whose2nd mean curvature vector field is an eigenvector of the cheng-yau operator □, which isdefined as the linear part of the first variation of the 2nd mean curvature of a hypersurfacearising from its normal variations. we show that any timelike hypersurface of l4 satisfying thecondition □h2 = λh2 (where 0 ≤ k ≤ n − 1) belongs to the class of □-biharmonic, □-1-type or□-null-2-type hypersurface. furthermore, we study the weakly convex hypersurfaces (i.e. onwhich all of principle curvatures are nonnegative). we prove that, on any weakly convexlorentz hypersurface satisfying the above condition, the scalar curvature will be constant. asan interesting result, any weakly convex riemannian or lorentzian hypersurfaces, havingassumed to be □-biharmonic, has to be 1-maximal.