## Geometry & Topology

### Hyperbolic cusps with convex polyhedral boundary

#### 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.

#### Article information

Source
Geom. Topol., Volume 13, Number 1 (2009), 457-492.

Dates
Revised: 7 October 2008
Accepted: 17 September 2008
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.gt/1513800186

Digital Object Identifier
doi:10.2140/gt.2009.13.457

Mathematical Reviews number (MathSciNet)
MR2469522

Zentralblatt MATH identifier
1179.57026

Subjects
Primary: 57M50: Geometric structures on low-dimensional manifolds
Secondary: 53C24: Rigidity results

#### Citation

Fillastre, François; Izmestiev, Ivan. Hyperbolic cusps with convex polyhedral boundary. Geom. Topol. 13 (2009), no. 1, 457--492. doi:10.2140/gt.2009.13.457. https://projecteuclid.org/euclid.gt/1513800186

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