Gauss images of hyperbolic cusps with convex polyhedral boundary
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- by François Fillastre and Ivan Izmestiev PDF
- Trans. Amer. Math. Soc. 363 (2011), 5481-5536 Request permission
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
- Email: francois.fillastre@u-cergy.fr
- Ivan Izmestiev
- Affiliation: Institut für Mathematik, MA 8-3, Technische Universität Berlin, Straße des 17. Juni 136, D-10623 Berlin, Germany
- Email: izmestiev@math.tu-berlin.de
- 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” - © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 363 (2011), 5481-5536
- MSC (2010): Primary 57M50; Secondary 52A55, 52C26, 52C25
- DOI: https://doi.org/10.1090/S0002-9947-2011-05325-0
- MathSciNet review: 2813423