On the geometry of complete spacelike hypersurfaces in the anti-de Sitter space

Abstract

Motivated by the nonexistence of closed spacelike hypersurfaces in the anti-de Sitter space $$\mathbb H_1^{n+1}$$ , in this paper we study the geometry of complete spacelike hypersurfaces immersed in $$\mathbb H_1^{n+1}$$ with either constant mean or scalar curvature. In this setting, under suitable restrictions on the norm of the tangential component of a fixed timelike vector, we show that such hypersurfaces must be isometric to certain hyperbolic spaces. Our approach is based on the use of Bochner’s technique jointly with appropriated generalized maximum principles.