An extended bi-conservativity condition on hypersurfaces of the Minkowski spacetime

Abstract

Isoparametric hypersurfaces of Lorentz-Minkowski spaces,classied by M.A. Magid in 1985, is related to the famous family of bi-conservative hypersurfaces. Such a hypersurface has conservative stress-energy with respect to the bienergy functional. A timelike (Lorentzian)hypersurface x : M_1^n ----> E_1^{n+1}, isometrically immersed into the Lorentz-Minkowski space E_1^{n+1} , is said to be biconservative if the tangent com-ponent of vector eld \Delta^2 x on M_1^n is identically zero. In this paper,we study on L_k-extension of biconservativity condition. The map L_k on a hypersurface (as the kth extension of Laplace operator L_0 = \Delta) is the linearized operator arisen from the rst variation of (k + 1)th mean curvature of hypersurface. After illustrating some examples, we prove that an L_k-biconservative timlike hypersurface of E_1^{n+1}, with atmost two distinct principal curvatures and some additional conditions,is isoparametric.