A gap theorem for complete constant scaler curvature hypersurfaces in the de Sitter space

Abstract

To each immersed complete space-like hypersurface M with constant normalized scalar curvature R in the de Sitter space S-1(n+1), we associate sup H-2, where H is the mean curvature of M. It is proved that the condition sup H-2 less than or equal to C-n((R) over bar), where (R) over bar = (R - 1) > 0 and C-n((K) over bar) is a constant depending only on R and n, implies that either M is totally umbilical or M is a hyperbolic cylinder. It is also proved the sharpness of this result by showing the existence of a class of new rotation constant scalar curvature hypersurfaces in S-1(n+1) such that sup H-2 > C-n((R) over bar). (C) 2001 Elsevier Science B.V. All rights reserved.