Related papers: On the Characterization of Polyhedra in Hyperbolic…

In a recent paper Hodgson and Kerckhoff prove a local rigidity theorem for finite volume, three dimensional hyperbolic cone-manifolds. In this paper we extend this result to geometrically finite cone-manifolds. Our methods also give a new…

We define a relative version of the Turaev-Viro invariants for an ideally triangulated compact 3-manifold with non-empty boundary and a coloring on the edges, generalizing the Turaev-Viro invariants [35] of the manifold. We also propose the…

We prove that, both in the hyperbolic and spherical 3-spaces, there exist nonconvex compact boundary-free polyhedral surfaces without selfintersections which admit nontrivial continuous deformations preserving all dihedral angles and study…

We prove that, among all convex hyperbolic polygons with given angles, the perimeter is minimized by the unique polygon with an inscribed circle. The proof relies on work of J.-M.\ Schlenker.

In this paper, we prove the Shafarevich conjecture for proper hyperbolic polycurves, which is a higher dimensional analogue of that for proper hyperbolic curves. First, we study theories of proper hyperbolic polycurves over regular schemes.…

In this paper we derive an extended Circle Pattern Theorem that allows obtuse overlap angles. As a consequence, we characterize a subclass of compact convex hyperbolic polyhedra with possibly obtuse dihedral angles and thus generalize…

A divide is the image of a proper and generic immersion of a compact $1$-manifold into the $2$-disk. Due to A'Campo's theory, each divide is associated with a link in the 3-sphere. In this paper, we reveal a hidden hyperbolic structure in…

In this paper, we study the rigidity of hyperbolic polyhedral 3-manifolds and the volume optimization program of angle structures. We first study the rigidity of decorated 1-3 type hyperbolic polyhedral metrics on 3-manifolds which are…

We give a method for computing upper and lower bounds for the volume of a non-obtuse hyperbolic polyhedron in terms of the combinatorics of the 1-skeleton. We introduce an algorithm that detects the geometric decomposition of good…

Let $P \subset \R^3$ 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…

In 3-dimensional hyperbolic geometry, the classical Schlafli formula expresses the variation of the volume of a hyperbolic polyhedron in terms of the length of its edges and of the variation of its dihedral angles. We prove a similar…

In this paper we present a necessary conditions, that simple close geodesics on regular tetrahedra in the 3-dimensional hyperbolic space must satisfy. Furthermore, we explicitly describe three classes of simple closed geodesics on regular…

We establish a connection between two previously unrelated topics: a particular discrete version of conformal geometry for triangulated surfaces, and the geometry of ideal polyhedra in hyperbolic three-space. Two triangulated surfaces are…

This work provides two sufficient conditions in terms of sections or projections for a convex body to be a polytope. These conditions are necessary as well.

Let $M$ be a non-compact hyperbolic $3$-manifold with finite volume and totally geodesic boundary components. By subdividing mixed ideal polyhedral decompositions of $M$, under some certain topological conditions, we prove that $M$ has an…

We observe that a large part of the volume of a hyperbolic polyhedron is taken by a tubular neighbourhood of its boundary, and use this to give a new proof for the finiteness of arithmetic maximal reflection groups following a recent work…

Although the hyperbolic metric possesses many remarkable properties, it is not defined on arbitrary subdomains of $\mathbb{R}^n$ with $n \geq 2$. This article introduces a new hyperbolic-type metric that provides an alternative approach to…

The symmedian point of a triangle enjoys several geometric and optimality properties, which also serve to define it. We develop a new dynamical coordinatization of the symmedian, which naturally generalizes to other ideal hyperbolic…

We study the supremum of the volume of hyperbolic polyhedra with some fixed combinatorics and with vertices of any kind (real, ideal or hyperideal). We find that the supremum is always equal to the volume of the rectification of the…

We prove that the complement ${\mathcal S}:={\mathbb R}^3\setminus{\mathcal K}$ of a 3-dimensional convex polyhedron ${\mathcal K}\subset{\mathbb R}^3$ and its closure $\overline{{\mathcal S}}$ are polynomial images of ${\mathbb R}^3$. The…