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

TL;DR

This paper classifies hypersurfaces in pseudo-Euclidean 5-space satisfying a specific Laplacian condition on the mean curvature vector, showing they have constant geometric quantities and that biharmonic ones are minimal.

Contribution

It provides a classification of hypersurfaces satisfying b3 c  b3 c  = bb c  in  b5_s^5, demonstrating their geometric properties and minimality of biharmonic hypersurfaces.

Findings

Hypersurfaces with diagonal shape operator have constant mean curvature.

Such hypersurfaces have constant second fundamental form norm and scalar curvature.

Biharmonic hypersurfaces with diagonal shape operator are minimal.

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.