Hyperbolization of cusps with convex boundary

TL;DR

This paper proves that any metric with curvature bounded below by -1 on a torus can be realized as the boundary of a hyperbolic cusp with convex boundary, completing a general theorem about such metrics.

Contribution

It establishes the existence of hyperbolic cusps with convex boundary for all metrics with curvature bounded below, using polyhedral approximation methods.

Findings

Existence of hyperbolic cusps for all such metrics

Completion of the theorem for compact surfaces

Use of polyhedral approximation in the proof

Abstract

We prove that for every metric on the torus with curvature bounded from below by -1 in the sense of Alexandrov there exists a hyperbolic cusp with convex boundary such that the induced metric on the boundary is the given metric. The proof is by polyhedral approximation. This was the last open case of a general theorem: every metric with curvature bounded from below on a compact surface is isometric to a convex surface in a 3-dimensional space form.