# Weakly convex hypersurfaces of pseudo-Euclidean spaces satisfying the condition LkHk+1 = λHk+1

## DOI:

https://doi.org/10.22199/issn.0717-6279-3584## Keywords:

Spacelike hypersurfaces, Lk-biharmonic, k-maximal, Weakly convex## Abstract

*In this paper, we try to give a classification of spacelike hypersurfaces of the Lorentz-Minkowski space-time E _{1}^{n+1}, whose mean curvature vector field of order (k+ 1) is an eigenvector of the kth linearized operator L_{k}, for a non-negative integer k less than n. The operator L_{k} is defined as the linear part of the first variation of the (k + 1)th mean curvature of a hypersurface arising from its normal variations. We show that any spacelike hypersurface of E_{1}^{n+1 }satisfying the condition L_{k}H_{k+1} = λH_{k+1} (where 0 ≤ k ≤ n − 1) belongs to the class of L_{k}-biharmonic, L_{k}-1-type or L_{k}-null-2-type hypersurface. Furthermore, we study the above condition on a well-known family of spacelike hypersurfaces of Lorentz-Minkowski spaces, named the weakly convex hypersurfaces (i.e. on which all of principle curvatures are nonnegative). We prove that, on any weakly convex spacelike hypersurface satisfying the above condition for an integer k (where, 0 ≤ r ≤ n−1), the (k + 1)th mean curvature will be constant. As an interesting result, any weakly convex spacelike hypersurfaces, having assumed to be L_{k}-biharmonic, has to be k-maximal. *

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*Proyecciones (Antofagasta, On line)*, vol. 40, no. 3, pp. 711-719, May 2021.

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Copyright (c) 2021 Firooz Pashaie

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