Abstract
We study Lorentzian hypersurfaces in the Lorentz-Minkowski 5-space \(\mathbb {E}_1^5\), which are defined by isometric immersions \({{\textbf {x}}}: M_1^4\rightarrow \mathbb {E}_1^5\) satisfying the \(L_1\)-biharmonicity condition \(L_1^2x=0\). The \(L_1\)-biharmonicity condition is an extension of the ordinary biharmonicity condition (i.e. \(L_0^2x=0\)) which has been studied by Bang-Yen Chen on the submanifolds of Euclidean spaces, where \(L_0=\Delta \) is the well-known Laplace operator. The operator \(L_1\) is the linearized map associated to the first variation of the second mean curvature vector field on \(M_1^4\). We discuss on Lorentzian hypersurfaces of \(\mathbb {E}_1^5\) having at most two distinct principal curvatures. After illustrating some examples, we prove that every \(L_1\)-biharmonic Lorentzian hypersurface with at most two distinct principal curvatures is 1-minimal.
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References
Akutagawa, K., Maeta, S.: Biharmonic properly immersed submanifolds in Euclidean spaces. Geom. Dedicata 164, 351–355 (2013)
Alias, L.J., G\(\ddot{u}\)rb\(\ddot{u}\)z, N.: An extension of Takahashi theorem for the linearized operators of the higher order mean curvatures. Geom. Dedicata 121, 113–127 (2006)
Arvanitoyeorgos, A., Defever, F., Kaimakamis, G., Papantoniou, B.J.: Biharmonic Lorentz hypersurfaces in \(\mathbb{E} _1^4\). Pac. J. Math. 229, 293–306 (2007)
Arvanitoyeorgos, A., Defever, F., Kaimakamis, G.: Hypersurfaces in \(\mathbb{E} _s^4\) with proper mean curvature vector. J. Math. Soc. Jpn. 59, 797–809 (2007)
Chen, B.Y.: Total Mean Curvature and Submanifolds of Finite Type, Series in Pure Mathematics. World Scientific Publishing Co, Singapore (2014)
Chen, B.Y.: Some open problems and conjetures on submanifolds of finite type. Soochow J. Math. 17, 169–188 (1991)
Dimitrić, I.: Submanifolds of \(\mathbb{E} ^n\) with harmonic mean curvature vector. Bull. Inst. Math. Acad. Sin. 20, 53–65 (1992)
Eells, J., Wood, J.C.: Restrictions on harmonic maps of surfaces. Topology 15, 263–266 (1976)
Hasanis, T., Vlachos, T.: Hypersurfaces in \(\mathbb{E} ^4\) with harmonic mean curvature vector field. Math. Nachr. 172, 145–169 (1995)
Kashani, S.M.B.: On some \(L_1\)-finite type (hyper)surfaces in \({\mathbb{R} }^{n+1}\). Bull. Korean Math. Soc. 46(1), 35–43 (2009)
Lucas, P., Ramirez-Ospina, H.F.: Hypersurfaces in the Lorentz-Minkowski space satisfying \(L_k\psi = A \psi + b\). Geom. Dedicata 153, 151–175 (2011)
Magid, M.A.: Lorentzian isoparametric hypersurfaces. Pac. J. Math. 118(1), 165–197 (1985)
Pashaie, F., Mohammadpouri, A.: \(L_k\)-biharmonic spacelike hypersurfaces in Minkowski \(4\)-space \(\mathbb{E} _1^4\). Sahand Commun. Math. Anal. 5(1), 21–30 (2017)
O’Neill, B.: Semi-Riemannian Geometry with Applications to Relativity. Academic Press Inc., New York (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), 195–213 (2013)
Pashaie, F., Kashani, S.M.B.: Timelike hypersurfaces in the Lorentzian standard space forms satisfying \( L_kx=Ax+b\). Mediterr. J. Math. 11(2), 755–773 (2014)
Petrov, A.Z.: Einstein Spaces. Pergamon Press, Hungary (1969)
Reilly, R.C.: Variational properties of functions of the mean curvatures for hypersurfaces in space forms. J. Differ. Geom. 8(3), 465–477 (1973)
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The author would like to gratefully thank the anonymous referees for their careful reading of the paper and the suggestions and corrections.
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Pashaie, F. On proper L\(_1\)-biharmonic timelike hypersurfaces with at most two distinct principal curvatures in Lorentz-Minkowski 5-space. Afr. Mat. 34, 51 (2023). https://doi.org/10.1007/s13370-023-01085-1
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DOI: https://doi.org/10.1007/s13370-023-01085-1