Weakly convex hypersurfaces of pseudo-Euclidean spaces satisfying the condition L k H k+1 = λH k+1 - Pashaie,Firooz (Universidad Católica del Norte, Departamento de Matemáticas, 2021)
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.