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

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+I is the (r + 1)th mean curvature vector field on the hypersurface, Lr is the linearized operator...

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Autores principales:	Mohammadpouri,Akram, Pashaie,Firooz
Lenguaje:	English
Publicado:	Universidad Católica del Norte, Departamento de Matemáticas 2016
Materias:	Linearized operators Lr Lr-biharmonic r-minimal (r + 1)-th mean curvature weakly convex.
Acceso en línea:	http://www.scielo.cl/scielo.php?script=sci_arttext&pid=S0716-09172016000100001
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spelling	oai:scielo:S0716-091720160001000012016-06-14On the classification of hypersurfaces in Euclidean spaces satisfying LrHr+1 = AHr+1Mohammadpouri,AkramPashaie,Firooz Linearized operators Lr Lr-biharmonic r-minimal (r + 1)-th mean curvature weakly convex. 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+I is 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.info:eu-repo/semantics/openAccessUniversidad Católica del Norte, Departamento de MatemáticasProyecciones (Antofagasta) v.35 n.1 20162016-03-01text/htmlhttp://www.scielo.cl/scielo.php?script=sci_arttext&pid=S0716-09172016000100001en10.4067/S0716-09172016000100001
institution	Scielo Chile
collection	Scielo Chile
language	English
topic	Linearized operators Lr Lr-biharmonic r-minimal (r + 1)-th mean curvature weakly convex.
spellingShingle	Linearized operators Lr Lr-biharmonic r-minimal (r + 1)-th mean curvature weakly convex. Mohammadpouri,Akram Pashaie,Firooz On the classification of hypersurfaces in Euclidean spaces satisfying LrHr+1 = AHr+1
description	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+I is 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.
author	Mohammadpouri,Akram Pashaie,Firooz
author_facet	Mohammadpouri,Akram Pashaie,Firooz
author_sort	Mohammadpouri,Akram
title	On the classification of hypersurfaces in Euclidean spaces satisfying LrHr+1 = AHr+1
title_short	On the classification of hypersurfaces in Euclidean spaces satisfying LrHr+1 = AHr+1
title_full	On the classification of hypersurfaces in Euclidean spaces satisfying LrHr+1 = AHr+1
title_fullStr	On the classification of hypersurfaces in Euclidean spaces satisfying LrHr+1 = AHr+1
title_full_unstemmed	On the classification of hypersurfaces in Euclidean spaces satisfying LrHr+1 = AHr+1
title_sort	on the classification of hypersurfaces in euclidean spaces satisfying lrhr+1 = ahr+1
publisher	Universidad Católica del Norte, Departamento de Matemáticas
publishDate	2016
url	http://www.scielo.cl/scielo.php?script=sci_arttext&pid=S0716-09172016000100001
work_keys_str_mv	AT mohammadpouriakram ontheclassificationofhypersurfacesineuclideanspacessatisfyinglrhr1ahr1 AT pashaiefirooz ontheclassificationofhypersurfacesineuclideanspacessatisfyinglrhr1ahr1
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On the classification of hypersurfaces in Euclidean spaces satisfying LrHr+1 = AHr+1

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