Some Lk-biconservative Lorentzian hypersurfaces in Minkowski 5-space

Abstract

A Lorentzian hypersurface M 41 of Minkowski 5 − space (i.e. E 51 ), defined by an isometric immersion x : M 41 → E 51 , is said to be L k -biconservative if the tangent component of L 2 k x is identically zero, where L k is the k th extension of Laplace operator ∆ = L 0 . The operator L k is the linearized operator arisen from the first variation of ( k + 1)th mean curvature vector field on M 41 . This subject is motivated by a well-known conjecture of Bang-Yen Chen which says that the condition ∆ 2 x = 0 implies the minimality for submanifolds of Euclidean spaces. In this paper, we study L k -biconservative Lorentzian hypersurfaces of E 51 in four different cases based on the matrix representation forms of the shape operator. We show that if such a hypersurface has constant mean curvature and at most two distinct principal curvatures, then its ( k + 1)th mean curvature is constant.