Fuchsian polyhedra in Lorentzian space-forms

Trans. Amer. Math. Soc.

Izmestiev

2011

Self Cite

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 2π 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.

“…The proof would be close to those in [Sch04,Fil09]. However, one of the steps would be to prove the local rigidity of convex polyhedral cusps with respect to their Gauss images.…”

Section: Theorem B Convex Polyhedral Cusps Are Infinitesimally Rigidmentioning

confidence: 68%

Section: Franç Ois Fillastre and Ivan Izmestievmentioning

confidence: 90%

See 1 more Smart Citation

Gauss images of hyperbolic cusps with convex polyhedral boundary

Trans. Amer. Math. Soc.

Izmestiev

2011

Self Cite

“…Corresponding realization theorems are proved by the first author [12] for hyperbolic space and by Schlenker [26] and the first author [11] for Lorentzian space-forms.…”

Section: Theorem 12 (Rivinmentioning

confidence: 96%

Hyperbolic cusps with convex polyhedral boundary

Izmestiev²

2009

Geom. Topol.

Self Cite

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.

“…It seems that it does not exist yet similar results in the anti-de Sitter space. Space-like convex Fuchsian polyhedra in Minkowski and anti-de Sitter spaces are studied in [11,36,38]. It is possible that there exists convex Fuchsian polyhedra in these spaces for which the induced metric is not everywhere space-like (for example it may contain light-like edges).…”

Section: Some Open Questionsmentioning

confidence: 99%

Polyhedral hyperbolic metrics on surfaces

2008

Geom Dedicata

Self Cite

Let S be a topologically finite surface, and g be a hyperbolic metric on S with a finite number of conical singularities of positive singular curvature, cusps and complete ends of infinite area. We prove that there exists a convex polyhedral surface P in hyperbolic space H 3 and a group G of isometries of H 3 such that the induced metric on the quotient P/G is isometric to g. Moreover, the pair (P, G) is unique among a particular class of convex polyhedra.