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

Abstract

In this paper, we try to give a classification of spacelike hypersurfaces of the Lorentz-Minkowski space-time E₁ⁿ⁺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 E₁ⁿ⁺¹ satisfying the condition L_kH_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.