On 1-index of Unstable Spacelike Hypersurfaces in Pseudo-Euclidean Spheres

In mathematical physics, the stable hypersurfaces of constant mean curvature in pseudo-Euclidian spheres have been interested by many researchers on general relativity. As an extension, the notion of index of stability has been introduced for unstable ones. The stability index (as a rate of distance from being stable) is defined in terms of the Laplace operator $Delta$ as the trace of Hessian tensor. In this paper, we study an extension of stability index (namely, 1-index) of hypersurfaces with constant scalar curvature in pseudo-Euclidian sphere $S_1^{n+1}$. 1-index is defined based on the Cheng-Yau operator $Box$ as a natural extension of $Delta$.