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Fuchsian polyhedra in Lorentzian spaceforms
, 2009
"... Let S be a compact surface of genus> 1, and g be a metric on S of constant curvature K ∈ {−1,0, 1} with conical singularities of negative singular curvature. When K = 1 we add the condition that the lengths of the contractible geodesics are> 2π. We prove that there exists a convex polyhedral ..."
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Cited by 7 (5 self)
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Let S be a compact surface of genus> 1, and g be a metric on S of constant curvature K ∈ {−1,0, 1} with conical singularities of negative singular curvature. When K = 1 we add the condition that the lengths of the contractible geodesics are> 2π. We prove that there exists a convex polyhedral surface P in the Lorentzian spaceform of curvature K and a group G of isometries of this space such that the induced metric on the quotient P/G is isometric to (S, g). Moreover, the pair (P, G) is unique (up to global isometries) among a particular class of convex polyhedra, namely Fuchsian polyhedra. This extends theorems of A.D. Alexandrov and Rivin–Hodgson [Ale42, RH93] concerning the sphere to the higher genus cases, and it is also the polyhedral version of a theorem of Labourie–Schlenker [LS00].
A variational proof of Alexandrov’s convex cap theorem; arXiv: math/0703169. [333
"... We give a variational proof of the existence and uniqueness of a convex cap with the given upper boundary. The proof uses the concavity of the total scalar curvature functional on the space of generalized convex caps. As a byproduct, we prove that generalized convex caps with the fixed boundary are ..."
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Cited by 6 (3 self)
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We give a variational proof of the existence and uniqueness of a convex cap with the given upper boundary. The proof uses the concavity of the total scalar curvature functional on the space of generalized convex caps. As a byproduct, we prove that generalized convex caps with the fixed boundary are globally rigid, that is uniquely determined by their curvatures. 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.
Volume and rigidity of hyperbolic polyhedral 3manifolds
"... We investigate the rigidity of hyperbolic cone metrics on 3manifolds which are isometric gluing of ideal and hyperideal tetrahedra in hyperbolic spaces. These metrics will be called ideal and hyperideal hyperbolic polyhedral metrics. It is shown that a hyperideal hyperbolic polyhedral metric is ..."
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We investigate the rigidity of hyperbolic cone metrics on 3manifolds which are isometric gluing of ideal and hyperideal tetrahedra in hyperbolic spaces. These metrics will be called ideal and hyperideal hyperbolic polyhedral metrics. It is shown that a hyperideal hyperbolic polyhedral metric is determined up to isometry by its curvature and a decorated ideal hyperbolic polyhedral metric is determined up to isometry and change of decorations by its curvature. The main tool used in the proof is the Fenchel dual of the volume function. 1
SHAPES OF POLYHEDRA, MIXED VOLUMES, AND HYPERBOLIC GEOMETRY
"... Abstract. We are generalizing to higher dimensions the BavardGhys construction of the hyperbolic metric on the space of polygons with fixed directions of edges. The space of convex ddimensional polyhedra with fixed directions of facet normals has a decomposition into type cones that correspond to ..."
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Abstract. We are generalizing to higher dimensions the BavardGhys construction of the hyperbolic metric on the space of polygons with fixed directions of edges. The space of convex ddimensional polyhedra with fixed directions of facet normals has a decomposition into type cones that correspond to different combinatorial types of polyhedra. This decomposition is a subfan of the secondary fan of a vector configuration and can be analyzed with the help of Gale diagrams. We construct a family of quadratic forms on each of the type cones using the theory of mixed volumes. The AlexandrovFenchel inequalities ensure that these forms have exactly one positive eigenvalue. This introduces a piecewise hyperbolic structure on the space of similarity classes of polyhedra with fixed directions of facet normals. We show that some of the dihedral angles on the boundary of the resulting conemanifold are equal to pi