In this manuscript, we study a condition (namely, L1-biharmonicity condition) on Lorentz hypersurfaces of 5-dimensional Lorentz space forms that gives 1-minimal hypersurfaces. By an isometric immersion ϕ : M4 1 → M51 (c) we denote a timelike hypersurface M41 in a 5-dimensional Lorentz space form M51 (c). By definition, the L1-biharmonicity condition on M4 1 means that ϕ satisfies the differential equation L21ϕ = 0, where, L1 is the well-known operator introduced by Cheng and Yau. We prove that every L1-biharmonic hypersurface with at most two distinct principal curvatures and constant ordinary mean curvature is 1-minimal.
