An extension of biconservative timelike hypersurfaces in Einstein space

Abstract

A well-known conjecture of Bang-Yen Chen says that the only biharmonic Euclidean submanifolds are minimal ones, which affirmed by himself for surfaces in 3-dimensional Euclidean space, E³. We consider an extended version of Chen conjecture (namely, Lk-conjecture) on Lorentzian hypersurfaces of the pseudo-Euclidean space E⁴₁ (i.e. the Einstein space). The biconservative submanifolds in the Euclidean spaces are submanifolds with conservative stress-energy with respect to the bienergy functional. In this paper, we consider an extended condition (namely, Lk-biconservativity) on non-degenerate timelike hypersurfaces of the Einstein space E⁴₁ . A Lorentzian hypersurface x : M³₁ → E⁴₁ is called Lk-biconservative if the tangent part of L²k x vanishes identically. We show that Lk-biconservativity of a timelike hypersurface of E⁴₁ (with constant kth mean curvature and some additional conditions) implies that its (k + 1) th mean curvature is constant.