On $L_1$-biharmonic timelike hypersurfaces in pseudo-Euclidean space $E_1^4$


  • Firooz Pashaie Academic member of Universiity of Maragheh, Maragheh, Iran.




Lorentz hypersurface, Biharmonic, L_1-biharmonic, 1-minimal


A well-known conjecture of Bang Yen-Chen says that the only biharmonic Euclidean submanifolds are minimal ones. In this paper, we consider an extended condition (namely, $L_1$-biharmonicity) on non-degenerate timelike hypersurfaces of the pseudo-Euclidean space $E_1^4$. A Lorentzian hypersurface $x: M_1^3\rightarrow\E_1^4$ is called $L_1$-biharmonic if it satisfies the condition $L_1^2x=0$, where $L_1$ is the linearized operator associated to the first variation of 2-th mean curvature vector field on $M_1^3$. According to the multiplicities of principal curvatures, the $L_1$-extension of Chen's conjecture is affirmed for Lorentzian hypersurfaces with constant ordinary mean curvature in pseudo-Euclidean space $E_1^4$. Additionally, we show that there is no proper $L_1$-biharmonic $L_1$-finite type connected orientable Lorentzian hypersurface in $E_1^4$.

Author Biography

Firooz Pashaie, Academic member of Universiity of Maragheh, Maragheh, Iran.

Dept of Mathematics, Faculty of Basic Sciences


Akutagawa, K., Maeta, S., Biharmonic properly immersed submanifolds in Euclidean spaces, Geom. Dedicata, 164 (2013) 351--355.

Alias, L.J., Gurbuz, N., An extension of Takahashi theorem for the linearized operators of the higher order mean curvatures, Geom. Dedicata 121 (2006), 113--127.

Arvanitoyeorgos, A., Defever, F., Kaimakamis, G., Papantoniou, B. J., Biharmonic Lorentz hypersurfaces in $E_1^4$, Pacific J. Math., 229 (2007), 293--306.

Arvanitoyeorgos, A., Defever, F., Kaimakamis, G., Hypersurfaces in $E_s^4$ with proper mean curvature vector, J. Math. Soc. Japan 59 (2007), 797--809.

Chen, B. Y., Total Mean Curvature and Submanifolds of Finite Type, Series in Pure Mathematics, 2. World Scientific Publishing Co, Singapore, (2014).

Chen, B. Y., Some open problems and conjetures on submanifolds of finite type, Soochow J. Math., 17 (1991), 169--188.

Defever, F., Hypersurfaces of $E^4$ satisfying Delta{H}=lambda H, Michigan. Math. J., 44 (1997), 355--363.

Defever, F., Hypersurfaces of $E^4$ with harmonic mean curvature vector, Math. Nachr., 196 (1998), 61-69.

Dimitric, I., Submanifolds of $E^n$ with harmonic mean curvature vector, Bull. Inst. Math. Acad. Sin., 20 (1992), 53--65.

Eells, J., Wood, J. C., Restrictions on harmonic maps of surfaces, Topology, 15 (1976), 263--266.

Hasanis, T. Vlachos, T., Hypersurfaces in $E^4$ with harmonic mean curvature vector field, Math. Nachr., 172 (1995), 145-169.

Kashani, S. M. B., On some $L_1$-finite type (hyper)surfaces in $R^{n+1}$, Bull. Korean Math. Soc., 46:1 (2009), 35-43.

Lucas, P., Ramirez-Ospina, H. F., Hypersurfaces in the Lorentz-Minkowski space satisfying $L_k psi = A psi + b$, Geom. Dedicata, 153 (2011), 151-175.

Magid, M. A., Lorentzian isoparametric hypersurfaces, Pacific J. of Math., 118 (1), (1985), 165-197.

Pashaie, F., Mohammadpouri, A., $L_k$-biharmonic spacelike hypersurfaces in Minkowski 4-space $E_1^4$, Sahand comm. Math. Anal., 5:1 (2017), 21-30.

O'Neill, B., Semi-Riemannian Geometry with Applicatins to Relativity, Acad. Press Inc., 1983.

Pashaie, F., Kashani, S.M.B., Spacelike hypersurfaces in Riemannian or Lorentzian space forms satisfying $L_kx=Ax+b$, Bull. Iran. Math. Soc. 39 (1), (2013), 195-213.

Pashaie, F., Kashani, S.M.B., Timelike hypersurfaces in the Lorentzian standard space forms satisfying $L_kx=Ax+b$, Mediterr. J. Math. 11 (2), (2014), 755-773.

Petrov, A. Z., Einstein Spaces, Pergamon Press, Hungary, Oxford and New York, 1969.

Reilly, R. C., Variational properties of functions of the mean curvatures for hypersurfaces in space forms, J. Differential Geom., 8, 3 (1973), 465-477.




How to Cite

Pashaie, F. (2020). On $L_1$-biharmonic timelike hypersurfaces in pseudo-Euclidean space $E_1^4$. Tamkang Journal of Mathematics, 51(4), 313-332. https://doi.org/10.5556/j.tkjm.51.2020.3188