# Hyperbolization of cusps with convex boundary

@article{Fillastre2015HyperbolizationOC,
title={Hyperbolization of cusps with convex boundary},
author={Franccois Fillastre and Ivan Izmestiev and Giona Veronelli},
journal={Manuscripta Mathematica},
year={2015},
volume={150},
pages={475-492}
}
• Published 2015
• Mathematics
• Manuscripta Mathematica
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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