On biharmonic hypersurfaces of three curvatures in Minkowski 5-space

Abstract

In this paper, we study the L_k-biharmonic Lorentzian hypersurfaces of the Minkowski 5-space M^5, whose second fundamental form has three distinct eigenvalues. An isometrically immersed Lorentzian hypersurface, x: M^4_1----> M^5, is said to be L_k-biharmonic if it satisfies the condition (L_k)^2 x = 0, where L_k is the linearized operator associated to the 1st variationof the mean curvature vector field of order (k + 1) on M^4_1 . In the special case k = 0, we have L_0 is the well-known Laplace operator \Delta and by a famous conjecture due to Bang-Yen Chen each \Delta-biharmonic submanifold of every Euclidean space is minimal. The conjecture has been affirmed in many Riemannian cases. We obtain similar results confirming the L_k-conjecture on Lorentzian hypersurfaces in M^5 with at least three principal curvatures.