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Proyecciones (Antofagasta)

Print version ISSN 0716-0917

Proyecciones (Antofagasta) vol.35 no.1 Antofagasta Mar. 2016 

Proyecciones Journal of Mathematics Vol. 35, No 1, pp. 1-10, March 2016. Universidad Católica del Norte Antofagasta - Chile

On the classification of hypersurfaces in Euclidean spaces satisfying LrHr+1 = AHr+1

Akram Mohammadpouri

University of Tabriz

Firooz Pashaie

University of Maragheh



In this paper, we study isometrically immersed hypersurfaces of the Euclidean space En+1 satisfying the condition LrH r+i = λHr+1

for an integer r ( 0 ≤ r ≤ n — 1), where Hr+iis the (r + 1)th mean curvature vector field on the hypersurface, Lr is the linearized operator of the first variation of the (r + 1) th mean curvature of hypersurface arising from its normal variations. Having assumed that on a hypersurface x : Mn → En+1, the vector field Hr+i be an eigenvector of the operator Lr with a constant real eigenvalue λ, we show that, Mn has to be an Lr-biharmonic, Lr-1-type, or Lr-null-2-type hypersurface. Furthermore, we study the above condition on a well-known family of hypersurfaces, named the weakly convex hypersurfaces (i.e. on which principal curvatures are nonnegative). We prove that, any weakly convex Euclidean hypersurface satisfying the condition Lr Hr+i = λ Hr+i for an integer r ( 0 ≤ r ≤ n — 1), has constant mean curvature of order (r + 1). As an interesting result, we have that, the Lr-biharmonicity condition on the weakly convex Euclidean hypersurfaces implies the r-minimality.

2010 Mathematics Subject Classification. Primary: 53-02, 53C40, 53C42; Secondary 58G25.

Keywords : Linearized operators Lr, Lr-biharmonic, r-minimal, (r + 1)-th mean curvature, weakly convex.


[1]    Akutagawa, K., Maeta, S., Biharmonic properly immersed submani-folds in Euclidean spaces, Geom. Ded., 164, pp. 351-355, (2013).

[2]    Alias, L. J., Gürb-üz, N., An extension of Takahashi theorem for the linearized operators of the higher order mean curvatures, Geom. Ded., 121, pp. 113-127, (2006).

[3] Aminian, M., Kashani, S. M. B. Lk-biharmonic hypersurfaces in the Euclidean space, Taiwan. J. Math., (2014), DOI: 10.11650/tjm.18.2014.4830.

[4]    Chen, B. Y., Total Mean Curvature and Submanifolds of Finite Type, Ser. Pure Math., World Sci. Pub. Co., Singapore (2014).

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

[6] Defever, F., Hypersurfaces of E4 satisfying AH = AH, Michigan Math. J., 44, pp. 61-69, (1998).

[7]    Defever, F., Hypersurfaces of E4 with harmonio mean curvature vector, Math. Nachr., 196, pp. 61-69, (1998).

[8]    Gupta, R. S., Biharmonic hypersurfaces in space forms with three dis-tinct principal curvatures, arXiv:1412.5479v1[math.DG] 17Dec2014.

[9]    Hardy, G., Littlewood, J., Polya, G., Inequalities, 2nd edit. Cambridge Univ. Press, (1989).

[10]    Hasanis, T., Vlachos, T., Hypersurfaces in E4 with harmonic mean curvature vector field, Math. Nachr., 172, pp. 145-169, (1995).

[11]    Kashani, S. M. B., On some Li-finite type (hyper)surfaces in Rn+1, Bull. Kor. M ath. Soc., 46 (1), pp. 35-43, (2009).

[12]    Luo, Y., Weakly convex biharmonic hypersurfaces in nonpositive curvature space forms are minimal, Results. Math., 65, pp. 49-56, (2014).

[13]    Mohammadpouri, A., Kashani, S. M. B., On some Lk-finite-type Eu-clidean hypersurfaces, ISRN Geom., (2012), 23 p.

[14]    Yang, B. G., Liu, X. M., r-minimal hypersurfaces in space forms, J. Geom. Phys., 59, pp. 685-692, (2009).

Akram Mohammadpouri

Faculty of Mathematical Sciences, University of Tabriz,



e-mail :

Firooz Pashaie

Department of Mathematics, Faculty of Basic Sciences, University of Maragheh,

P. O. Box 55181-83111



e-mail :

Received : March 2015. Accepted : February 2016

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