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

Document Type : Research Paper

Authors

Faculty of Mathematics, Statistics and Computer Sciences, University of Tabriz, Tabriz, Iran

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

References

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History

  • Receive Date: 01 June 2022
  • Revise Date: 12 September 2022
  • Accept Date: 22 September 2022

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