In this paper, we study on a Riemannian manifold Mn, isometrically immersed by a map x : M^n → E^{n+1} in the Minkowski space E^{n+1}_1 where the position map x satisfies the condition L^2_1 x = 0. This condition, as an extended version of the biharmonicity (defined by Δ^2 x = 0), is called the L_1-biharmonicity condition, where L_1 stands for the linearized operator of the first variation of 2-th mean curvature of M^n in E^{n+1} A well-known conjecture of Bang-Yen Chen says that any biharmonic Euclidean submanifold has to . be minimal. We discuss an analog of the Chen conjecture, replacing the Laplace operator Δ by L_1. Having assumed that M^n has at least three distinct principal curvatures and constant ordinary mean curvature, we prove that it must be 1-maximal.
