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## Proyecciones (Antofagasta)

*Print version* ISSN 0716-0917

#### Abstract

MOHAMMADPOURI, Akram and PASHAIE, Firooz. **On the classification of hypersurfaces in Euclidean spaces satisfying L _{r}H_{r+1} = AH_{r+1}**.

*Proyecciones (Antofagasta)*[online]. 2016, vol.35, n.1, pp.1-10. ISSN 0716-0917. http://dx.doi.org/10.4067/S0716-09172016000100001.

In this paper, we study isometrically immersed hypersurfaces of the Euclidean space E^{n+1} satisfying the condition L_{r}H _{r+i} = λH_{r+1} for an integer r ( 0 ≤ r ≤ n - 1), where H_{r+I }is the (r + 1)th mean curvature vector field on the hypersurface, L_{r} 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 : M^{n} → E^{n+1}, the vector field H_{r+i} be an eigenvector of the operator L_{r} with a constant real eigenvalue λ, we show that, M^{n} has to be an L_{r}-biharmonic, L_{r}-1-type, or L_{r}-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 L_{r} H_{r+i} = λ H_{r}+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 L_{r}-biharmonicity condition on the weakly convex Euclidean hypersurfaces implies the r-minimality.

**Keywords
:
**Linearized operators L_{r}; L_{r}-biharmonic; r-minimal; (r + 1)-th mean curvature; weakly convex..