Hyperbolic cusps with convex polyhedral boundary
Given a hyperbolic metric with cone singularities on the 2-torus \(T^2\), such that all cone angles are less than \(2\pi\), the authors prove that this metric extends from \(T^2\times 0\) to a unique hyperbolic three dimensional cusp \(T^2\times [0,+\infty)\) with convex polyhedral boundary. Infinitesimal rigidity is also discussed. The authors consider not only smooth metrics on \(T^2\times [0,\infty)\), but also metrics having cone singularities along lines \(x\times [0,\infty)\), for \(x\) a cone point of \(T^2\). The main tool is a discrete scalar curvature functional in a polyhedral deformation space. This functional is concave and its critical points correspond to the metric. The paper gives an excellent overview of the subject, and the exposition is clear and well written
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