On the scalar curvature of complete maximal spacelike submanifolds in pseudo-hypebolic spaces

Abstract

We study in this article the curvature of complete maximal spacelike submanifolds in pseudo-hyperbolic spaces. We show that the scalar curvature of these submanifolds is nonpositive in every signature. This gives, together with a result of Ishihara, a sharp bound on the scalar curvature of complete maximal spacelike submanifolds in pseudo-hyperbolic spaces of every signature. We show that achieving the bound at a point is equivalent to achieving it identically, and explicitely describe the submanifolds achieving the bound. When the codimension is equal to 1, we deduce a sharp upper bound on the Ricci curvature of complete maximal hypersurfaces in Anti-de Sitter spaces, and characterize the hypersurfaces achieving it. Finally, we discuss the link between scalar curvature and Gromov-hyperbolicity for complete maximal spacelike submanifolds in pseudo-hyperbolic spaces.