On nonnegatively curved hypersurfaces in hyperbolic space

TL;DR

This paper proves that complete, nonnegatively curved hypersurfaces in hyperbolic space are properly embedded, confirming a conjecture with the exception of certain covering maps of equidistant surfaces.

Contribution

It establishes a key geometric property of hypersurfaces in hyperbolic space, confirming a longstanding conjecture and clarifying the structure of such hypersurfaces.

Findings

Complete, nonnegatively curved hypersurfaces are properly embedded in hyperbolic space.

The only exceptions are covering maps of equidistant surfaces in hyperbolic 3-space.

The conjecture of Alexander and Currier is proven with these conditions.

Abstract

In this paper we prove the conjecture of Alexander and Currier that states, except for covering maps of equidistant surfaces in hyperbolic 3-space, a complete, nonnegatively curved immersed hypersurface in hyperbolic space is necessarily properly embedded.