On complete spacelike hypersurfaces with constant scalar curvature in the de Sitter space

Abstract

Let M n be a complete spacelike hypersurface with constant normalized scalar curvature R in the de Sitter Space S n+1 1 . Let H the mean curvature and suppose that R = (R − 1) > 0 and R ≤ sup H 2 ≤ C R , where C R is a constant depending only on R and n. It is proved that either sup H 2 = R and M n is totally umbilical, or sup H 2 = C R and M n is the hyperbolic cylinder H 1 (1 − coth 2 r) × S n−1 (1 − tanh 2 r).

The Riemannian product structures of spacelike hypersurfaces with constant k-th mean curvature in the de Sitter spaces

“…Especially, the interest focuses on characterizing the totally umbilical properties of such hypersurfaces. A classical result due to Q.-M. Cheng and S. Ishikawa [9] states that the totally round spheres are the only compact spacelike hypersurfaces in S n+1 1 (c) with constant normalized scalar curvature R < c. For a more closely study related to the complete spacelike hypersurfaces in S n+1 1 (c) with constant scalar curvature, we refer to [5,6,[16][17][18]23] and the references therein.…”

On the Stability of Spacelike Hypersurfaces With Higher Order Mean Curvature in a De Sitter Space

“…In this sense, Barbosa and Oliker [5] computed the second variation formula and obtained in the de Sitter space S n+1 1 that spheres maximize the area functional for volume-preserving variations, which is consistent with the definition of stability. Later, researches of [6,11], they obtained an extension of the result in [5] for spacelike hypersurfaces with constant scalar curvature, respectively.…”