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ON THE INFINITESIMAL RIGIDITY OF POLYHEDRA WITH VERTICES IN CONVEX POSITION
"... Abstract. Let P ⊂ R 3 be a polyhedron. It was conjectured that if P is weakly convex (i. e. its vertices lie on the boundary of a strictly convex domain) and decomposable (i. e. P can be triangulated without adding new vertices), then it is infinitesimally rigid. We prove this conjecture under a wea ..."
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Cited by 7 (2 self)
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Abstract. Let P ⊂ R 3 be a polyhedron. It was conjectured that if P is weakly convex (i. e. its vertices lie on the boundary of a strictly convex domain) and decomposable (i. e. P can be triangulated without adding new vertices), then it is infinitesimally rigid. We prove this conjecture under a weak additionnal assumption of codecomposability. The proof relies on a result of independent interest concerning the HilbertEinstein function of a triangulated convex polyhedron. We determine the signature of the Hessian of that function with respect to deformations of the interior edges. In particular, if there are no interior vertices, then the Hessian is negative definite. 1.
Hyperbolic cusps with convex polyhedral boundary
 Geom. Topol
"... 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 ..."
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Cited by 6 (5 self)
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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 conemanifolds with the singular locus a union of halflines. 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. 57M50; 53C24 1
Polyhedral hyperbolic metrics on surfaces
 Geom. Dedicata
"... Abstract. 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 3 and a group G of ..."
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Cited by 5 (3 self)
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Abstract. 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 3 and a group G of isometries of 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.
GAUSS IMAGES OF HYPERBOLIC CUSPS WITH CONVEX POLYHEDRAL BOUNDARY
, 2009
"... 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 ..."
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Cited by 2 (1 self)
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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 RivinHodgson 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 conetype singularities along a family of halflines. The functional is shown to be concave and to attain 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 conetype singularities. In a special case, our theorem is equivalent to 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 RivinHodgson’s theorem extends Andreev’s theorem on compact convex polyhedra with nonobtuse 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.
On weakly convex starshaped polyhedra
, 2007
"... Abstract. Weakly convex polyhedra which are starshaped with respect to one of their vertices are infinitesimally rigid. This is partial answer to the question whether every decomposable weakly convex polyhedron is infinitesimally rigid. The proof uses a recent result of Izmestiev on the geometry of ..."
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Abstract. Weakly convex polyhedra which are starshaped with respect to one of their vertices are infinitesimally rigid. This is partial answer to the question whether every decomposable weakly convex polyhedron is infinitesimally rigid. The proof uses a recent result of Izmestiev on the geometry of convex caps.