Hyperbolic cusps with convex polyhedral boundary

Fillastre, François¹ ; Izmestiev, Ivan²

¹ Département de Mathématiques, Université de Cergy-Pontoise / Saint-Martin, 2, avenue Adolphe Chauvin, 95 302 Cergy-Pontoise Cedex, France
² Institut für Mathematik, MA 8-3, Technische Universität Berlin, Str. des 17. Juni 136, D-10623 Berlin, Germany

Geometry & topology, Tome 13 (2009) no. 1, pp. 457-492.

Résumé

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.

Détail

DOI : 10.2140/gt.2009.13.457

Keywords: Alexandrov's theorem, convex polyhedral boundary, hyperbolic cone-manifold, discrete total curvature

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