Hypersurfaces satisfying $△\vec{H}=\hat{I}»\vec{H}$ in $\mathbb{E}_{\lowercase{s}}^{5}$

Abstract

In this paper, we study hypersurfaces $M_r^4$ $(r=0,1,2,3,4)$ satisfying $△\vec{H}=\hat{I}»\vec{H}$ ($\hat{I}»$ 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.

19 pages