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

TL;DR

This paper investigates the scalar curvature of complete maximal spacelike submanifolds in pseudo-hyperbolic spaces, establishing nonpositivity, sharp bounds, and characterizations, with implications for Ricci curvature and Gromov-hyperbolicity.

Contribution

It provides new sharp bounds and characterizations for scalar and Ricci curvatures of maximal spacelike submanifolds in pseudo-hyperbolic spaces, extending previous results.

Findings

Scalar curvature is nonpositive for all such submanifolds.

Achieving the scalar curvature bound at a point implies it holds everywhere.

Explicit descriptions of submanifolds attaining the bounds.

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.