Communications of the
Korean Mathematical Society CKMS

The ordinary mean curvature vector field $\textbf H$ on a submanifold $M$ of a space form is said to be {\it proper} if it satisfies equality $\Delta\textbf H=a\textbf H$ for a constant real number $a$. It is proven that every hypersurface of an Riemannian space form with proper mean curvature vector field has constant mean curvature. In this manuscript, we study the Lorentzian hypersurfaces with proper second mean curvature vector field of four dimensional Lorentzian space forms. We show that the scalar curvature of such a hypersurface has to be constant. In addition, as a classification result, we show that each Lorentzian hypersurface of a Lorentzian 4-space form with proper second mean curvature vector field is $\textrm C$-biharmonic, $\textrm C$-1-type or $\textrm C$-null-2-type. Also, we prove that every $\textbf H_2$-proper Lorentzian hypersurface with constant ordinary mean curvature in a Lorentz 4-space form is 1-minimal.

Keywords: $\C$-finite type, Lorentzian hypersurface, $\C$-biharmonic

MSC numbers: 53A30, 53B30, 53C40, 53C43