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Proyecciones (Antofagasta)

Print version ISSN 0716-0917

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

http://dx.doi.org/10.4067/S0716-09172016000100001 

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

Iran 


ABSTRACT

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.


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Akram Mohammadpouri

Faculty of Mathematical Sciences, University of Tabriz,

Tabriz,

Iran

e-mail : pouri@tabrizu.ac.ir

Firooz Pashaie

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

P. O. Box 55181-83111

Maragheh

Iran

e-mail : f-pashaie@maragheh.ac.ir

Received : March 2015. Accepted : February 2016

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