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

• Mohammadpouri, Akram ^[1] ; Pashaie, Firooz ^[2]
  1. [1] University of Tabriz

     University of Tabriz

     Irán

     
  2. [2] University of Maragheh

     University of Maragheh

     Irán

     
• Localización: Proyecciones: Journal of Mathematics, ISSN 0716-0917, ISSN-e 0717-6279, Vol. 35, Nº. 1, 2016, págs. 1-10
• Idioma: inglés
• DOI: 10.4067/S0716-09172016000100001
• Enlaces
  • Texto completo
• Resumen

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

• Referencias bibliográficas
  • Citas [1] Akutagawa, K., Maeta, S., Biharmonic properly immersed submanifolds in Euclidean spaces, Geom. Ded., 164, pp. 351-355, (2013).
  • [2] Alias, L. J., Gurbuz, N., An extension of Takahashi theorem for the linearized operators of the higher order mean curvatures, Geom. Ded.,...
  • [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 ∆- H → = λ- H →, Michigan Math. J., 44, pp. 61-69, (1998).
  • [7] Defever, F., Hypersurfaces of E4 with harmonic mean curvature vector, Math. Nachr., 196, pp. 61-69, (1998).
  • [8] Gupta, R. S., Biharmonic hypersurfaces in space forms with three distinct 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 L1-finite type (hyper)surfaces in Rn+1, Bull. Kor. Math. 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 Euclidean 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).‏ ‏ ‏ ‏