Locally convex hypersurfaces immersed in $$\mathbb {H}^n\times \mathbb {R}$$ H n × R

Abstract

We prove a theorem of Hadamard–Stoker type: a connected locally convex complete hypersurface immersed in \(\mathbb {H}^n\times \mathbb {R}\) (\(n\ge 2\)), where \(\mathbb {H}^n\) is n-dimensional hyperbolic space, is embedded and homeomorphic either to the n-sphere or to \(\mathbb {R}^n\). In the latter case it is either a vertical graph over a convex domain in \(\mathbb {H}^n\) or has what we call a simple end.