Four Dimensional Hypersurfaces with Proper Mean Curvature Vector Field in Pseudo-Riemannian Space Forms

Abstract

In this paper, we prove that four-dimensional hypersurface M ⁴_r with proper mean curvature vector field (i.e., \(\Delta{\vec H}\) is proportional to \({\vec H}\)) in pseudo-Riemannian space form N ⁵_s (c) has constant mean curvature, and give the value or range of this constant. As an application, we obtain that biharmonic hypersurfaces in N ⁵_s (c) are minimal in some specific case.