on timelike hypersurfaces of the minkowski 4-space with 1-proper second mean curvature vector

The mean curvature vector field of a submanifold in the euclidean $n$-space is said to be $proper$ if it is an eigenvector of the laplace operator $delta$. it is proven that every hypersurface with proper mean curvature vector field in the euclidean 4-space ${bbb e}^4$ has constant mean curvature. in this paper, we study an extended version of the mentioned subject on timelike (i.e., lorentz) hypersurfaces of minkowski 4-space ${bbb e}^4_1$. let ${textbf x}:m_1^3rightarrow{bbb e}_1^4$ be the isometric immersion of a timelike hypersurface $m^3_1$ in ${bbb e}_1^4$. the second mean curvature vector field ${textbf h}_2$ of $m_1^3$ is called {it 1-proper} if it is an eigenvector of the cheng-yau operator $mathcal{c}$ (which is the natural extension of $delta$). we show that each $m^3_1$ with 1-proper ${textbf h}_2$ has constant scalar curvature. by a classification theorem, we show that such a hypersurface is $mathcal{c}$-biharmonic, $mathcal{c}$-1-type or  null-$mathcal{c}$-2-type. since the shape operator of $m^3_1$ has four possible matrix forms, the results will be considered in four different cases.