Weakly convex hypersurfaces of pseudo-Euclidean spaces satisfying the condition LkHk+1 = λHk+1
Keywords:Spacelike hypersurfaces, Lk-biharmonic, k-maximal, Weakly convex
In this paper, we try to give a classification of spacelike hypersurfaces of the Lorentz-Minkowski space-time E1n+1, whose mean curvature vector field of order (k+ 1) is an eigenvector of the kth linearized operator Lk, for a non-negative integer k less than n. The operator Lk is defined as the linear part of the first variation of the (k + 1)th mean curvature of a hypersurface arising from its normal variations. We show that any spacelike hypersurface of E1n+1 satisfying the condition LkHk+1 = λHk+1 (where 0 ≤ k ≤ n − 1) belongs to the class of Lk-biharmonic, Lk-1-type or Lk-null-2-type hypersurface. Furthermore, we study the above condition on a well-known family of spacelike hypersurfaces of Lorentz-Minkowski spaces, named the weakly convex hypersurfaces (i.e. on which all of principle curvatures are nonnegative). We prove that, on any weakly convex spacelike hypersurface satisfying the above condition for an integer k (where, 0 ≤ r ≤ n−1), the (k + 1)th mean curvature will be constant. As an interesting result, any weakly convex spacelike hypersurfaces, having assumed to be Lk-biharmonic, has to be k-maximal.
K. Akutagawa and S. Maeta, ”Biharmonic properly immersed submanifolds in Euclidean spaces”, Geometriae dedicata, vol. 164, pp. 351-335, 2013, doi: 10.1007/s10711-012-9778-1
L. J. Alias and N. Gürbüz, ”An extension of Takahashi theorem for the linearized operators of the higher order mean curvatures”, Geometriae dedicata, vol. 121, pp. 113-127, 2006, doi: 10.1007/s10711-006-9093-9
M. Aminian and S. M. B. Kashani, ”Lk-biharmonic hypersurfaces in the Euclidean space”, Taiwanese journal of mathematics, vol. 19, no. 3, pp. 861-874, 2015, doi: 10.11650/tjm.19.2015.4830
A. Caminha, ”On spacelike hypersurfaces of constant sectional curvature lorentz manifolds”, Journal of geometry and physics, vol. 56, no. 7, pp. 1144-1174, 2006, doi: 10.1016/j.geomphys.2005.06.007
B. Y. Chen, ”Some open problems and conjetures on submanifolds of finite type: recent development”, Tamkang journal of mathematics, vol. 45, no. 1, pp. 87-108, 2014, doi: 10.5556/j.tkjm.45.2014.1564
B. Y. Chen, Total mean curvature and submanifolds of finite type, 2nd ed. Singapore: World Scientific, 2015, doi: 10.1142/9237
F. Defever, ”Hypersurfaces of Ē4 satisfying ∆−H⃗ = λ − H⃗”, Michigan mathematics journal, vol. 44, no. 2, pp. 355-363, 1997, doi: 10.1307/mmj/1029005710
R. S. Gupta, ”Biharmonic hypersurfaces in space forms with three distinct principal curvatures”, 2014, arXiv:1412.5479v1
G. Hardy, J. Littlewood, and G. Polya, Inequalities, 2nd ed. Cambridge: Cambridge University Press, 1989.
T. Hasanis and T. Vlachos, ”Hypersurfaces in E4 with harmonic mean curvature vector field”, Mathematischen nachrichten, vol. 172, no. 1, pp. 145-169, 1995, doi: 10.1002/mana.19951720112
S. M. B. Kashani, ”On some L1-finite type (hyper)surfaces in Rn+1”, Bulletin of the Korean Mathematics Society, vol. 46, no. 1, pp. 35-43, 2009, doi: 10.4134/BKMS.2009.46.1.035
Y. Luo, ”Weakly convex biharmonic hypersurfaces in nonpositive curvature space forms are minimal”, Results in mathematics, vol. 65, pp. 49-56, 2014, doi: 10.1007/s00025-013-0328-4
A. Mohammadpouri and S. M. B. Kashani, ”On some Lk-finitetype Euclidean hypersurfaces”, International scholarly research notices, vol. 2012, Art. ID 591296, 2012, doi: 10.5402/2012/591296
A. Mohammadpouri and F. Pashaie, ”L1-biharmonic hypersurfaces with three distinct principal curvatures in Euclidean 5-space”, Functional analysis, approximation and computations, vol. 7, no. 1, pp. 67-75, 2015. [On line]. Available: https://bit.ly/3eFBvG9
B. O’Neill, Semi-Riemannian geometry with applicatins to relativity, 2nd ed. New York, NY: Academic Press Inc., 1983. [On line]. Available: https://bit.ly/3uDff5c
F. Pashaie and S. M. B. Kashani, ”Spacelike hypersurfaces in Riemannian or Lorentzian space forms satisfying Lkx = Ax + b”, Bulletin of the Irannian Mathematic Society, vol. 39, no. 1, pp. 195-213, 2013. [On line]. Available: https://bit.ly/3tCJNTl
How to Cite
Copyright (c) 2021 Firooz Pashaie
This work is licensed under a Creative Commons Attribution 4.0 International License.