$L_k$-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


[1] M. Aminian, S. M. B. Kashani, Lk-Biharmonic Hypersurfaces in Euclidean Space, Taiwan J Math., 19., no 3 (2015) 861-874.
[2] M. Aminian, S. M. B. Kashani, Lk-Biharmonic Hypersurfaces in Space Form, Acta Math. Vietnam., 42., no 3 (2017) 471-490.
[3] A. Arvanitoyeorgos, F. Defever, G. Kaimakamis and V. J. Papantoniou, Biharmonic Lorentz Hypersurfaces in E4
1 , Paci c J. Math., 229., no 2 (2007) 293-305.
[4] B. Y. Chen, Some Open Problems and Conjetures on Submanifolds of Finite Type, Soochow J. Math., 17., (1991) 169-188.
[5] B. Y. Chen, S. Ishikawa, Biharmonic Surfaces in Pseudo-Euclidean Spaces, Mem. Fac. Sci. Kyushu Univ. ser A., 45., (1991) 323-347.
[6] Deepika., R. S. Gupta, On Biharmonic Lorentz Hypersurfaces with non-Diagonal Shape Operator, Int. Electron. J. Geom., 10., no 1, (2017) 96-111.
[7] Deepika., R. S. Gupta and A. Sharfuddin, Biharmonic Hypersurfaces with Constant Scalar Curvature in E5
s , Kyungpook Math. J., 56., no 1 (2016) 273-293.
[8] F. Defever, G. Kaimakamis, and V. Papantoniou, Biharmonic Hypersurfaces of the 4-Dimensional Semi-Euclidean Space E4 s , J. Math. Anal. Appl., 315., 1 (2006) 276-286.
[9] R. S. Gupta, Biharmonic Hypersurfaces in E5 s , An. St. Univ. Al. I. Cuza., 2., no 2 (2016) 585-593.
[10] J. Hahn, Isoparametric Hypersurfaces in the Pseudo-Riemannian Space Forms, Math. Z., 187., (1984) 195-208.
[11] G. Y. Jiang, 2-Harmonic Maps and Their First and Second Variational Formulas, Chin. Ann. Math. Ser A., 7., (1986) 389-402.
[12] J. Liu, L. Du, Classi cation of Proper Biharmonic Hypersurfaces in Pseudo-Riemannian Space Forms, Di er. Geom. Appl., 41,. (2015) 110-122.
[13] P. Lucas, H.F. Ramirez-Ospina, Hypersurfaces in Lorentz-Minkowski Space Satisfying Lk  = A  + b, Geom. Dedicata., 153., no 1 (2011) 151-175.
[14] P. Lucas, H.F. Ramirez-Ospina, Hypersurfaces in Pseudo-Euclidean Spaces Satisfying a Linear Condition on the Linearized Operator of a Higher Order Mean Curvature, Di er. Geom. Appl., 31.,no 2 (2013) 175-1.89.
[15] M. A. Magid, Isometric Immersions of Lorentz Space with Parallel Second Fundamental Forms, Tsukuba J. Math., 8., no 1 (1984) 31-54.
[16] A. Mohammadpouri, Hypersurfaces with Lr-Pointwise 1-Type Gauss Map, Math. Phys. Anal. Geom., 14., no 1 (2018) 67-77.
[17] A. Mohammadpouri, F. Pashaiei, On the Classi cation of Hypersurfaces in Euclidean Spaces Satisfying Lr 􀀀!
H r+1 = 􀀀! H r+1, Proyecciones J. Math., 35., no 1 (2016) 1-10.
[18] A. Mohammadpouri, F. Pashaie, Lr-Biharmonic Hypersurfaces in E4, Bol. Soc. Paran. Mat., 38., no 5 (2020) 9-18.
[19] A. Mohammadpouri, F. Pashaie, and S. Tajbakhs, L1-Biharmonic Hypersurfaces in Euclidean Spaces with Three Distinct Principal Curvatures, Iran. J. Math. Sci. Inform., 13., no 2 (2018) 49-70.
[20] V. J. Papantoniou, K. Petoumenos, Biharmonic Hypersurfaces of Type M3 2 in E4 2 , Houston J. Math., 38., no 1 (2012) 93-114.
[21] F. Pashaie, A. Mohammadpouri, Lk-Biharmonic Spacelike Hypersurfaces in Minkowski 4-Space E4 1 , Sahand commun. math. anal., 5., no 1 (2017) 21-30.
[22] R. C. Reilly, Variational Properties of Functions of the Mean Curvatures for Hypersurfaces in Space Forms, J. Di erential Geom., 8., no 3 (1973) 465-477.
[23] M. Shams. Solary, Eigenvalues for Tridiagonal 3-Toeplitz Matrices, J. Mahani math. res. cent., 10.,no 2 (2021) 63-72.
[24] H. Urakawa, Geometry of Biharmonic Mappings: Di erential Geometry of Variational Methods, World Scienti c pub Co Inc., Singapore, 2018.