On biharmonic hypersurfaces of three curvatures in Minkowski 5-space

In this paper, we study the Lk-biharmonic Lorentzian hypersurfaces of the Minkowski 5-space M5 , whose second fundamental form has three distinct eigenvalues. An isometrically immersed Lorentzian hypersurface, x : M4 1 → M5 , is said to be Lk-biharmonic if it satisfies the condition L 2 kx = 0, where Lk is the linearized operator associated to the 1st variation of the mean curvature vector field of order (k + 1) on M4 1 . In the special case k = 0, we have L0 is the well-known Laplace operator ∆ and by a famous conjecture due to Bang-Yen Chen each ∆-biharmonic submanifold of every Euclidean space is minimal. The conjecture has been affirmed in many Riemanian cases. We obtain similar results confirming the Lk-conjecture on Lorentzian hypersurfaces in M5 with at least three principal curvatures.