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Hyperbolic cusps with
convex polyhedral boundary
François Fillastre and Ivan Izmestiev |
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Geometry & Topology 13 (2009) 457–492
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Abstract |
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We prove that a –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
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Mathematical Subject Classification 2000
Primary: 57M50
Secondary: 53C24
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References |
Publication
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
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Authors |
| François Fillastre | |
| Département de Mathématiques Université de Cergy-Pontoise / Saint-Martin 2, avenue Adolphe Chauvin 95 302 Cergy-Pontoise Cedex France |
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| Ivan Izmestiev | |
| Institut für Mathematik, MA
8-3 Technische Universität Berlin Str. des 17. Juni 136 D-10623 Berlin Germany |
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