In this paper, we study timelike hypersurfaces of the Minkowski 4-space L4, whose 2nd mean curvature vector eld is an eigenvector of the Cheng-Yau operator , which is defi ned as the linear part of the rst variation of the 2nd mean curvature of a hypersurface arising from its normal variations. We show that any timelike hypersurface of L4 satisfying the condition H2 = H2 (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. on which all of principle curvatures are nonnegative). We prove that, on any weakly convex Lorentz hypersurface satisfying the above condition, the scalar curvature will be constant. As an interesting result, any weakly convex Riemannian or Lorentzian hypersurfaces, having assumed to be -biharmonic, has to be 1-maximal.
