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

Authors: JIANCHENG LIU, LIN WEI

Abstract: 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 M^n in N^{n+1}_1. In this paper, we obtain a gap property of S: if nP\leq \sup S\leq 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.

Keywords: Constant scalar curvature, space-like hypersurface, second fundamental form, locally symmetric Lorentz space.

Full Text: PDF