$L_k$-Biharmonic hypersurfaces in the 3-or 4-dimensional Lorentz-Minkowski spaces

Abstract

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^2\psi =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.

Keywords

• linearized operator $l_k$
• $l_k$-biharmonic hypersurface
• $k$-maximal hypersurface
• $k$-th mean curvature