A classification of Biconservative Hypersurfaces in the Minkowski spaces

In this paper, we study Lorentzian biconservative hypersurfaces for which the gradient of their mean curvature $H$ is lightlike, i.e. $⟨\text{gr}H,\text{gr}H⟩=0$. We establish the non-existence of such hypersurfaces in the Minkowski spaces by conducting a rigorous analysis of both the Codazzi and Gauss equations.