lk-biharmonic hypersurfaces in the 3-or 4-dimensional lorentz-minkowski spaces

A hypersurface $ m^n $ in the lorentz-minkowski space $mathbb{l}^{n+1} $ is called $ l_k $-biharmonic if the position vector $ psi $ satisfies the condition $ l_k^2psi =0$, where $ l_k$ is the linearized operator of the $(k+1)$-th mean curvature of $ m $ for a fixed $k=0,1,ldots,n-1$. this definition is a natural generalization of the concept of a biharmonic hypersurface. we prove that any $ l_k $-biharmonic surface in $ mathbb{l}^3 $ is $k$-maximal. we also prove that any $ l_k $-biharmonic hypersurface in $ mathbb{l}^4 $ with constant $ k$-th mean curvature is $ k $-maximal. these results give a partial answer to the chen’s conjecture for $l_k$-operator that $l_k$-biharmonicity implies $l_k$-maximality.