  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.
