A gap theorem for complete space-like hypersurface with constant scalar curvature in locally symmetric Lorentz spaces

Let M^n be a complete space-like hypersurface with constant scalar curvature in locally symmetric Lorentz space N_n+1 ^1 , S be the squared norm of the second fundamental form of Mn in N_n+1 ^1 . In this paper, we obtain a gap property of S: if nP ≤ sup S ≤ D(n, P) for some constants P and D(n, P), then either sup S = nP and M^n is totally umbilical, or sup S = D(n, P) and M^n has two distinct principal curvatures.