#### Volume 13, issue 1 (2009)

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Hyperbolic cusps with convex polyhedral boundary

### François Fillastre and Ivan Izmestiev

Geometry & Topology 13 (2009) 457–492
##### Abstract

We prove that a $3$–dimensional hyperbolic cusp with convex polyhedral boundary is uniquely determined by the metric induced on its boundary. Furthermore, any hyperbolic metric on the torus with cone singularities of positive curvature can be realized as the induced metric on the boundary of a convex polyhedral cusp.

The proof uses the discrete total curvature functional on the space of “cusps with particles”, which are hyperbolic cone-manifolds with the singular locus a union of half-lines. We prove, in addition, that convex polyhedral cusps with particles are rigid with respect to the induced metric on the boundary and the curvatures of the singular locus.

Our main theorem is equivalent to a part of a general statement about isometric immersions of compact surfaces.

##### Keywords
Alexandrov's theorem, convex polyhedral boundary, hyperbolic cone-manifold, discrete total curvature
Primary: 57M50
Secondary: 53C24