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Gauss images of hyperbolic cusps with convex polyhedral boundary

Authors: François Fillastre and Ivan Izmestiev
Journal: Trans. Amer. Math. Soc. 363 (2011), 5481-5536
MSC (2010): Primary 57M50; Secondary 52A55, 52C26, 52C25
Published electronically: April 25, 2011
MathSciNet review: 2813423
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Abstract: We prove that a 3-dimensional hyperbolic cusp with convex polyhedral boundary is uniquely determined by its Gauss image. Furthermore, any spherical metric on the torus with cone singularities of negative curvature and all closed contractible geodesics of length greater than $ 2\pi$ is the metric of the Gauss image of some convex polyhedral cusp. This result is an analog of the Rivin-Hodgson theorem characterizing compact convex hyperbolic polyhedra in terms of their Gauss images.

The proof uses a variational method. Namely, a cusp with a given Gauss image is identified with a critical point of a functional on the space of cusps with cone-type singularities along a family of half-lines. The functional is shown to be concave and to attain its maximum at an interior point of its domain. As a byproduct, we prove rigidity statements with respect to the Gauss image for cusps with or without cone-type singularities.

In a special case, our theorem is equivalent to the existence of a circle pattern on the torus, with prescribed combinatorics and intersection angles. This is the genus one case of a theorem by Thurston. In fact, our theorem extends Thurston's theorem in the same way as Rivin-Hodgson's theorem extends Andreev's theorem on compact convex polyhedra with non-obtuse dihedral angles.

The functional used in the proof is the sum of a volume term and curvature term. We show that, in the situation of Thurston's theorem, it is the potential for the combinatorial Ricci flow considered by Chow and Luo.

Our theorem represents the last special case of a general statement about isometric immersions of compact surfaces.

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Additional Information

François Fillastre
Affiliation: Departement of Mathematics, University of Cergy-Pontoise, UMR CNRS 8088, F-95000 Cergy-Pontoise, France

Ivan Izmestiev
Affiliation: Institut für Mathematik, MA 8-3, Technische Universität Berlin, Straße des 17. Juni 136, D-10623 Berlin, Germany

Keywords: Hyperbolic cusp, convex polyhedral boundary, Gauss image, Rivin-Hodgson theorem, circle pattern.
Received by editor(s): September 9, 2009
Received by editor(s) in revised form: February 4, 2010
Published electronically: April 25, 2011
Additional Notes: The first author was partially supported by Schweizerischer Nationalfonds 200020-113199/1
The second author was supported by the DFG Research Unit 565 “Polyhedral Surfaces”
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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