Partial generalizations of some conjectures in Lorentzian manifolds

Abstract

In this paper, we mainly investigate complete or compact spacelike hypersurfaces with constant mean curvature or constant scalar curvature in Lorentzian manifolds L₁ⁿ⁺¹. We give a new estimate of the Laplacian ΔS of the squared length S of the second fundamental form of such spacelike hypersurfaces. Finally, we give partial generalizations of some Conjectures in Lorentzian manifolds L₁ⁿ⁺¹.