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.
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Fillastre, F., Izmestiev, I. & Veronelli, G. Hyperbolization of cusps with convex boundary. manuscripta math. 150, 475–492 (2016). https://doi.org/10.1007/s00229-015-0814-y
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DOI: https://doi.org/10.1007/s00229-015-0814-y