Volume 13, issue 1 (2009)

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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Hyperbolic cusps with convex polyhedral boundary

François Fillastre and Ivan Izmestiev

Geometry & Topology 13 (2009) 457–492

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.

Alexandrov's theorem, convex polyhedral boundary, hyperbolic cone-manifold, discrete total curvature
Mathematical Subject Classification 2000
Primary: 57M50
Secondary: 53C24
Received: 14 December 2007
Revised: 7 October 2008
Accepted: 17 September 2008
Preview posted: 11 November 2008
Published: 1 January 2009
Proposed: Jean-Pierre Otal
Seconded: Walter Neumann, Benson Farb
François Fillastre
Département de Mathématiques
Université de Cergy-Pontoise / Saint-Martin
2, avenue Adolphe Chauvin
95 302 Cergy-Pontoise Cedex
Ivan Izmestiev
Institut für Mathematik, MA 8-3
Technische Universität Berlin
Str. des 17. Juni 136
D-10623 Berlin