Volume and rigidity of hyperbolic polyhedral 3‐manifolds

Abstract

We investigate the rigidity of hyperbolic cone metrics on 3‐manifolds which are isometric gluing of ideal and hyper‐ideal tetrahedra in hyperbolic spaces. These metrics will be called ideal and hyper‐ideal hyperbolic polyhedral metrics. It is shown that a hyper‐ideal 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.

On the deformation of discrete conformal factors on surfaces

“…a ω is locally convex on A and each a i can be extended continuously to X by constant functions to a function a i on X, then F (x) = x a a i (x)dx i is a C 1 -smooth convex function on X extending F . Lemma 2.6 plays an essential role in [9] and [10]. We use this lemma to extend the definition of F (u).…”

Asymptotic expansion of relative quantum invariants

“…, e |E| } be the set of edges of T . As defined in [16,17], a hyperbolic polyhedral metric on (N, T ) is obtained by replacing each tetrahedron in T by a truncated hyperideal tetrahedron and replacing the gluing homeomorphisms between pairs of the faces by isometries. The cone angle at an edge is the sum of the dihedral angles of the truncated hyperideal tetrahedra around the edge.…”

Twisted Reidemeister torsion and Gram matrices

“…Similar to the construction of a fundamental shadow link complement is the construction of the double of a hyperbolic polyhedral 3-manifold. As defined in [12,13], a hyperbolic polyhedral 3-manifold N is obtained from d truncated hyperideal tetrahedra ∆ 1 , . .…”

“…, ∆ d . It is proved in [13,Theorem 1.2 (b)] that hyperbolic polyhedral 3-manifolds are rigid in the sense that they are up to isometry determined by their cone angles.…”