The mean curvature vector field of a submanifold in the Eu- clidean n-space is said to be proper if it is an eigenvector of the Laplace operator ∆. It is proven that every hypersurface with proper mean cur- vature vector field in the Euclidean 4-space E4 has constant mean cur- vature. In this paper, we study an extended version of the mentioned subject on timelike (i.e., Lorentz) hypersurfaces of Minkowski 4-space E4 1. Let x : M 3 1 → E4 1 be the isometric immersion of a timelike hyper- surface M 3 1 in E4 1. The second mean curvature vector field H2 of M 3 1 is called 1-proper if it is an eigenvector of the Cheng-Yau operator C (which is the natural extension of ∆). We show that each M 3 1 with 1-proper H2 has constant scalar curvature. By a classification theorem, we show that such a hypersurface is C-biharmonic, C-1-type or null-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.
