Lorentz hypersurfaces in $E_1^4$ satisfying $\Delta\vec H=\alpha\vec H$

Abstract

A hypersurface $M_1^3$ in the four-dimensional pseudo-Euclidean space $E_1^4$ is called a Lorentz hypersurface if its normal vector is space-like. We show that if the mean curvature vector field of $M_1^3$ satisfies the equation $\Delta\vec H=\alpha\vec H$ ($\alpha$ a constant), then $M_1^3$ has constant mean curvature. This equation is a natural generalization of the biharmonic submanifold equation $\Delta\vec H=\vec0$.