Locally convex hypersurfaces immersed in $H^n \times R$

TL;DR

This paper proves a Hadamard-Stoker type theorem for locally convex hypersurfaces in hyperbolic space times a line, classifying their topology and embedding properties.

Contribution

It establishes that such hypersurfaces are embedded, homeomorphic to either a sphere or Euclidean space, and characterizes their geometric structure.

Findings

Hypersurfaces are either spheres or Euclidean spaces.

They are embedded and homeomorphic to standard topological spaces.

Characterization of hypersurfaces with simple ends or as vertical graphs.

Abstract

We prove a theorem of Hadamard-Stoker type: a connected locally convex complete hypersurface immersed in $H^n \times R$ (n>1), where $H^n$ is n-dimensional hyperbolic space, is embedded and homeomorphic either to the n-sphere or to $R^n$. In the latter case it is either a vertical graph over a convex domain in $H^n$ or has what we call a simple end.