L_k-Biharmonic hypersurfaces in the ۳-or ۴-dimensional Lorentz-Minkowski spaces

A hypersurface M^n in the Lorentz-Minkowski space \mathbb{L}^{n+۱} is called L_k -biharmonic if the position vector \psi satisfies the condition L_k^۲\psi =۰, where L_k is the linearized operator of the (k+۱)-th mean curvature of M for a fixed k=۰,۱,\ldots,n-۱. This definition is a natural generalization of the concept of a biharmonic hypersurface. We prove that any L_k -biharmonic surface in \mathbb{L}^۳ is k-maximal. We also prove that any L_k -biharmonic hypersurface in \mathbb{L}^۴ 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.

کلیدواژه ها:

Linearized operator L_k ، L_k-biharmonic hypersurface ، k-maximal hypersurface ، k-th mean curvature