Hypersurfaces satisfying $\triangle \vec {H}=\lambda \vec {H}$ in $\mathbb{E}_{\lowercase{s}}^{5}$

Abstract

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.