Hypersurfaces in Pseudo-Euclidean Space with Condition $$\triangle \mathbf{H }=\lambda \mathbf{H }$$

Abstract

We study hypersurfaces in the pseudo-Euclidean space, whose mean curvature vector satisfies the equation: Laplacian of the vector is parallel to the vector (with constant factor), and the second fundamental form has constant norm. We prove that every such hypersurface of diagonalizable shape operator with at most six distinct principal curvatures has constant mean curvature and constant scalar curvature, and if the above factor is zero then the hypersurface is minimal. We classify locally such non-minimal hypersurfaces with extremal value of the norm of the mean curvature vector. Further, we provide some examples of such hypersurfaces.