Isometric immersions in the hyperbolic space with their image contained in a horoball (Q5890233)

Suppose that \(M\) is a complete Riemannian manifold with scalar curvature bounded from below. If \(M\) has an isometric, codimension-1 immersion into a horoball of a real or complex hyperbolic space, then the author proves a sharp lower bound for the supremum of the norm of the mean curvature. The author next characterizes horospheres in real or complex hyperbolic space as those proper isometric, codimension-1 immersions of a complete Riemannian manifold whose image is contained between two parallel horospheres, whose mean curvature is bounded from above by the sharp lower bound already discussed, and whose group of symmetries is sufficiently large.