Biharmonic Hypersurfaces With Three Distinct Principal Curvatures In Lorentzian Space

In this paper,I study the generalized Chen conjecture of the spacelike biharmonic hyper-surfaces in the Lorentzian space form N₁ⁿ⁺¹(c).For some special spacelike hypersurfaces,I prove the generalized Chen conjecture.Let N₁ⁿ⁺¹(c)be a Lorentzian space form,when c=-1,N₁ⁿ⁺¹(-1)=H₁ⁿ⁺¹is the Anti de Sitter space whose curvature is-1;when c=0,N₁ⁿ⁺¹(0)=R₁ⁿ⁺¹is the Minkowski space;When c=1,N₁ⁿ⁺¹(1)=S₁ⁿ⁺¹is the De Sitter space whose curvature is 1.Let f:Mⁿ?N₁ⁿ⁺¹(c)be a spacelike hypersurface and H the mean curvature of the hypersurface f.If H=0,the spacelike hypersurface is called the maximal hypersurface.Let us denote?the Laplace operator with respect to the first fundamental form I on the hypersurface f.If?H=0,the hypersurface is called a biharmonic hypersurface.Obviously maximal hypersurfaces are biharmonic hypersurfaces.The generalized Chen conjecture states that the biharmonic hypersurface is a maximal hypersurface.This is a local geometry problem.The main conclusion of this paper is that the generalized Chen conjecture holds for the spacelike hypersurfaces in De Sitter space and Minkowski space if the number of its principal curvatures is less than or equal to 3,that is,it is a maximal hypersurface.On the other hand,for spacelike hypersurfaces in Anti-de Sitter space,if its number of principal curvatures is less than or equal to 3,then its mean curvature is constant.These conclusions generalize the existed conclusions.The method used in this paper is the reduction to absurdity.The differential equations of mean curvature are derived by using integrable conditions and biharmonic equations of spacelike hypersurfaces.The conclusion is obtained by analyzing the differential equations.Our method is also applicable to the Riemannian space form.