In this paper, we study hypersurfaces $M_{ r}^{ 4}$ $(r=0, 1, 2, 3, 4)$ satisfying $\triangle \vec{ H}=\lambda \vec{ H}$ ($\lambda$ a constant) in the pseudo-Euclidean space $\mathbb{ E}_{ s}^{ 5}$ $(s=0, 1, 2, 3, 4, 5)$. We obtain that every such hypersurface in $\mathbb{ E}_{ s}^{ 5}$ with diagonal shape operator has constant mean curvature, constant norm of second fundamental form and constant scalar curvature. Also, we prove that every biharmonic hypersurface in $\mathbb{ E}_{ s}^{ 5}$ with diagonal shape operator must be minimal.
Categories: math.DG