Let $(M, \dr M)$ be a 3-manifold with incompressible boundary that admits a convex co-compact hyperbolic metric. We consider the hyperbolic metrics on $M$ such that $\dr M$ looks locally like a hyperideal polyhedron, and we characterize the possible dihedral angles.
We find as special cases the results of Bao and Bonahon on hyperideal polyhedra, and those of Rousset on fuchsian hyperideal polyhedra. Our results can also be stated in terms of circle configurations on $\dr M$, they provide an extension of the Koebe theorem on circle packings.
The proof uses some elementary properties of the hyperbolic volume, in particular the Schläfli formula and the fact that the volume of (truncated) hyperideal simplices is a concave function of the dihedral angles.
We study hyperideal polyhedra in the 3-dimensional anti-de Sitter space $AdS^3$, which are defined as the intersection of the projective model of $AdS^3$ with a convex polyhedron in $RP^3$ whose vertices are all outside of $AdS^3$ and whose edges all meet $AdS^3$. We show that hyperideal polyhedra in $AdS^3$ are uniquely determined by their combinatorics and dihedral angles, as well as by the induced metric on their boundary together with an additional combinatorial data, and describe the possible dihedral angles and the possible induced metrics on the boundary.
A hyperbolic semi-ideal polyedron is a polyedron whose vertices lie inside the hyperbolic space $\mathbf{H}^{3}$ or at infinity. A hyperideal polyedron is, in the projective model, the intersection of $\mathbf{H}^{3}$ with a projective polyhedron whose vertices all lie outside of $\mathbf{H}^{3}$, and whose edges all meet $\mathbf{H}^{3}$. We classify semi-ideal polyhedra in terms of their dual metric, using the results of Rivin and Hodgson in \cite{comp} et \cite{idea}. This result is used to obtain the classification of hyperideal polyhedra in terms of their combinatorial type and their dihe…
We consider ``hyperideal'' circle patterns, i.e. patterns of disks appearing in the definition of the Delaunay decomposition associated to a set of disjoint disks, possibly with cone singularities at the center of those disks. Hyperideal circle patterns are associated to hyperideal hyperbolic polyhedra. We describe the possible intersection angles and singular curvatures of those circle patterns, on Euclidean or hyperbolic surfaces with conical singularities. This is related to results on the dihedral angles of ideal or hyperideal hyperbolic polyhedra.
We use a variational principle to prove an existence and uniqueness theorem for planar weighted Delaunay triangulations (with non-intersecting site-circles) with prescribed combinatorial type and circle intersection angles. Such weighted Delaunay triangulations may be interpreted as images of hyperbolic polyhedra with one vertex on and the remaining vertices beyond the infinite boundary of hyperbolic space. Thus the main theorem states necessary and sufficient conditions for the existence and uniqueness of such polyhedra with prescribed combinatorial type and dihedral angles. More generally, w…
A ``hyperideal circle pattern'' in $S^2$ is a finite family of oriented circles, similar to the ``usual'' circle patterns but such that the closed disks bounded by the circles do not cover the whole sphere. Hyperideal circle patterns are directly related to hyperideal hyperbolic polyhedra, and also to circle packings. To each hyperideal circle pattern, one can associate an incidence graph and a set of intersection angles. We characterize the possible incidence graphs and intersection angles of hyperideal circle patterns in the sphere, the torus, and in higher genus surfaces. It is a consequenc…
Hyperideal tetrahedra are the fundamental building blocks of hyperbolic 3-manifolds with geodesic boundary. The study of their geometric properties (in particular, of their volume) has applications also in other areas of low-dimensional topology, like the computation of quantum invariants of 3-manifolds and the use of variational methods in the study of circle packings on surfaces. The Schläfli formula neatly describes the behaviour of the volume of hyperideal tetrahedra with respect to dihedral angles, while the dependence of volume on edge lengths is worse understood. In this paper we prove …
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 1-skeleton. The theorem is proved by applying a sort of volume-increasing flow to any hyperbolic polyhedron. Singularities may arise in the flow because some strata of the polyhedron may degenerate to lower-dimensional objects; when this occurs, we need to study carefully the combinatorics of the resulting polyhedron and continue with the flow, until eventually we …
In [6], Kellerhals and Perren conjectured that the growth rates of the reflection groups given by hyperbolic Coxeter polyhedra are always Perron numbers. We prove that this conjecture is always true for the case of ideal Coxeter polyhedra in $\mathbb{H}^3$. We also find out the ideal Coxeter polyhedron in $\mathbb{H}^3$ with the smallest growth rate. Finally, we show that there are correlations between the volumes and the growth rates of ideal Coxeter polyhedra in $\mathbb{H}^3$ in many cases.
As was pointed out by Nikulin [8] and Vinberg [10], a right-angled polyhedron of finite volume in hyperbolic n-space $\mathbb{H}^n$ has at least one cusp for $n\geq 5$. We obtain non-trivial lower bounds on the number of cusps of such polyhedra. For example, right-angled polyhedra of finite volume must have at least three cusps for $n=6$. Our theorem also says that the higher the dimension of a right-angled polyhedron becomes, the more cusps it must have.
Cannon, Swenson, and others have proved numerous theorems about subdivision rules associated to hyperbolic groups with a 2-sphere at infinity. However, few explicit examples are known. We construct an explicit subdivision rule for many 3-manifolds from polyhedral gluings. The manifolds that satisfy the conditions include all closed manifolds created from right-angled hyperbolic polyhedra, as well as many 3-manifolds with toral or hyperbolic boundary.
A discrete conformality for hyperbolic polyhedral surfaces is introduced in this paper. This discrete conformality is shown to be computable. It is proved that each hyperbolic polyhedral metric on a closed surface is discrete conformal to a unique hyperbolic polyhedral metric with a given discrete curvature satisfying Gauss-Bonnet formula. Furthermore, the hyperbolic polyhedral metric with given curvature can be obtained using a discrete Yamabe flow with surgery. In particular, each hyperbolic polyhedral metric on a closed surface with negative Euler characteristic is discrete conformal to a u…
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 considered discretely conformally equivalent if the edge lengths are related by scale factors associated with the vertices. This simple definition leads to a surprisingly rich theory featuring Möbius invariance, the definition of discrete conformal maps as circumcircle preserving piecewise projective maps, and two variational principles. We show how literally…
In 1970, E. M. Andreev published a classification of all three-dimensional compact hyperbolic polyhedra having non-obtuse dihedral angles. Given a combinatorial description of a polyhedron, $C$, Andreev's Theorem provides five classes of linear inequalities, depending on $C$, for the dihedral angles, which are necessary and sufficient conditions for the existence of a hyperbolic polyhedron realizing $C$ with the assigned dihedral angles. Andreev's Theorem also shows that the resulting polyhedron is unique, up to hyperbolic isometry. Andreev's Theorem is both an interesting statement about the …
We determine the lowest volume hyperbolic Coxeter polyhedron whose corresponding hyperbolic polyhedral 3-orbifold contains an essential 2-suborbifold, up to a canonical decomposition along essential hyperbolic triangle 2-suborbifolds.
In this paper we obtain new upper bounds on volumes of right-angled polyhedra in hyperbolic space $\mathbb{H}^3$ in three different cases: for ideal polyhedra with all vertices on the ideal hyperbolic boundary, for compact polytopes with only finite (or usual) vertices, and for finite volume polyhedra with vertices of both types.
The deformation theory of hyperbolic and Euclidean cone-manifolds with all cone angles less then 2π plays an important role in many problems in low dimensional topology and in the geometrization of 3-manifolds. Furthermore, various old conjectures dating back to Stoker about the moduli of convex hyperbolic and Euclidean polyhedra can be reduced to the study of deformations of cone-manifolds by doubling a polyhedron across its faces. This deformation theory has been understood by Hodgson and Kerckhoff when the singular set has no vertices, and by Weiß when the cone angles are less than π. We pr…
The vertex scaling for piecewise linear metrics on polyhedral surfaces was introduced by Luo, who proved the local rigidity by establishing a variational principle and conjectured the global rigidity. Luo's conjecture was solved by Bobenko-Pinkall-Springborn, who also introduced the vertex scaling for piecewise hyperbolic metrics and proved its global rigidity. Bobenko-Pinkall-Spingborn's proof is based on their observation of the connection of vertex scaling and the geometry of polyhedra in $3$-dimensional hyperbolic space and the concavity of the volume of ideal and hyper-ideal tetrahedra. I…
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.
We classify the 3-dimensional hyperbolic polyhedral orbifolds that contain no embedded essential 2-suborbifolds, up to decomposition along embedded hyperbolic triangle orbifolds (turnovers). We give a necessary condition for a 3-dimensional hyperbolic polyhedral orbifold to contain an immersed (singular) hyperbolic turnover, we classify the triangle subgroups of the fundamental groups of orientable 3-dimensional hyperbolic tetrahedral orbifolds in the case when all of the vertices of the tetrahedra are non-finite, and we provide a conjectural classification of all the triangle subgroups of the…
By Andreev theorem acute-angled polyhedra of finite volume in a hyperbolic space $\mathbb H^{3}$ are uniquely determined by combinatorics of their 1-skeletons and dihedral angles. For a class of compact right-angled polyhedra and a class of ideal right-angled polyhedra estimates of volumes in terms of the number of vertices were obtained by Atkinson in 2009. In the present paper upper estimates for both classes are improved.
An equiangular hyperbolic Coxeter polyhedron is a hyperbolic polyhedron where all dihedral angles are equal to π/n for some fixed integer n at least 2. It is a consequence of Andreev's theorem that either n=3 and the polyhedron has all ideal vertices or that n=2. Volume estimates are given for all equiangular hyperbolic Coxeter polyhedra.
This article defines a pair of combinatorial operations on the combinatorial structure of compact right-angled hyperbolic polyhedra in dimension three called decomposition and edge surgery. It is shown that these operations simplify the combinatorics of such a polyhedron, while keeping it within the class of right-angled objects, until it is a disjoint union of Löbell polyhedra, a class of polyhedra which generalizes the dodecahedron. Furthermore, these combinatorial operations are shown to have geometric realizations which are volume decreasing. This allows for an organization of the volumes …
We present a notion of mutation of hyperbolic polyhedra, analogous to mutation in knot theory, and then present a general question about commensurability of mutant pairs of polyhedra. We motivate that question with several concrete examples of mutant pairs for which commensurability is unknown. The polyhedra we consider are compact, so techniques involving cusps that are typically used to distinguishing mutant pairs of knots are not applicable. Indeed, new techniques may need to be developed to study commensurability of mutant pairs of polyhedra.
This paper investigates the combinatorial $α$-curvature for vertex scaling of piecewise hyperbolic metrics on polyhedral surfaces, which is a parameterized generalization of the classical combinatorial curvature. A discrete uniformization theorem for combinatorial $α$-curvature is established, which generalizes Gu-Guo-Luo-Sun-Wu's discrete uniformization theorem for classical combinatorial curvature. We further introduce combinatorial $α$-Yamabe flow and combinatorial $α$-Calabi flow for vertex scaling to find piecewise hyperbolic metrics with prescribed combinatorial $α$-curvatures. To handle…
We review several results related to the characterization of polyhedra in hyperbolic 3-space. In particular we present Rivin's theorem that gives a characterization of compact convex hyperbolic polyhedra, and Hodgson's proof of the Adreev's theorem. We also review the analogous characterization of ideal polyhedra, and give a family of counter-examples that proves that hyperbolic polyhedra are not determined by edge lengths.
We formulate a generalization of the volume conjecture for planar graphs. Denoting by <G, c> the Kauffman bracket of the graph G whose edges are decorated by real "colors" c, the conjecture states that, under suitable conditions, certain evaluations of <G,kc> grow exponentially as k goes to infinity and the growth rate is the volume of a truncated hyperbolic hyperideal polyhedron whose one-skeleton is G (up to a local modification around all the vertices) and with dihedral angles given by c. We provide evidence for it, by deriving a system of recursions for the Kauffman brackets of planar grap…
In this note, we improve Nikulin's inequality in the case of right-angled hyperbolic polyhedra. The new inequality allows to give much shorter proofs of the known dimension bounds. We also improve Nonaka's lower bound on the number of ideal vertices for right-angled hyperbolic polyhedra.
Let $(M, \partial M)$ be a compact 3-manifold with boundary which admits a complete, convex co-compact hyperbolic metric. For each hyperbolic metric $g$ on $M$ such that $\dr M$ is smooth and strictly convex, the induced metric on $\dr M$ has curvature $K>-1$, and each such metric on $\dr M$ is obtained for a unique choice of $g$. A dual statement is that, for each $g$ as above, the third fundamental form of $\dr M$ has curvature $K<1$, and its closed geodesics which are contractible in $M$ have length $L>2π$. Conversely, any such metric on $\dr M$ is obtained for a unique choice of $g$. We ar…
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 3-orbifolds with planar singular locus and underlying manifold the 3-sphere. The volume bounds follow from techniques related to the proof of Thurston's Orbifold Theorem, Schläfli's formula, and previous results of the author giving volume bounds for right-angled hyperbolic polyhedra.
An algorithm for determining the list of smallest volume right-angled hyperbolic polyhedra in dimension 3 is described. This algorithm has been implemented on computer using the program Orb to compute volumes, and the first 825 polyhedra in the list have been determined.
We define the injectivity radius of a Coxeter polyhedron in H^3 to be half the shortest translation length among hyperbolic/loxodromic elements in the orientation-preserving reflection group. We show that, for finite-volume polyhedra, this number is always less than 2.6339..., and for compact polyhedra it is always less than 2.1225... .
In this paper, we adopt combinatorial Ricci curvature flow methods to study the existence of cusped hyperbolic structure on 3-manifolds with torus boundary. For general pseudo 3-manifolds, we prove the long-time existence and the uniqueness for the extended Ricci flow for decorated hyperbolic polyhedral metrics. We prove that the extended Ricci flow converges to a decorated hyperbolic polyhedral metric if and only if there exists a decorated hyperbolic polyhedral metric of zero Ricci curvature. If it is the case, the flow converges exponentially fast. These results apply for cusped hyperbolic …
We study convex polyhedra in $\mathbb{R}\mathbb{P}^3$ with all their vertices on a sphere. We do not require, in particular, that the polyhedra lie in the interior of the sphere, hence the term "weakly inscribed". Such polyhedra can be interpreted as ideal polyhedra, if we regard $\mathbb{R}\mathbb{P}^3$ as a combination of the hyperbolic space and the de Sitter space, with the sphere as the common ideal boundary. We have three main results: (1) the $1$-skeleta of weakly inscribed polyhedra are characterized in a purely combinatorial way, (2) the exterior dihedral angles are characterized by l…
The main result is that every complete finite area hyperbolic metric on a sphere with punctures can be uniquely realized as the induced metric on the surface of a convex ideal polyhedron in hyperbolic 3-space. A number of other observations are included.
To any prime alternating link, we associate a collection of hyperbolic right-angled ideal polyhedra by relating geometric, topological and combinatorial methods to decompose the link complement. The sum of the hyperbolic volumes of these polyhedra is a new geometric link invariant, which we call the right-angled volume of the alternating link. We give an explicit procedure to compute the right-angled volume from any alternating link diagram, and prove that it is a new lower bound for the hyperbolic volume of the link.
We suggest a method of computing volume for a simple polytope $P$ in three-dimensional hyperbolic space $\mathbb{H}^3$. This method combines the combinatorial reduction of $P$ as a trivalent graph $Γ$ (the $1$-skeleton of $P$) by $I-H$, or Whitehead, moves (together with shrinking of triangular faces) aligned with its geometric splitting into generalised tetrahedra. With each decomposition (under some conditions) we associate a potential function $Φ$ such that the volume of $P$ can be expressed through a critical values of $Φ$. The results of our numeric experiments with this method suggest th…
We are generalizing to higher dimensions the Bavard-Ghys construction of the hyperbolic metric on the space of polygons with fixed directions of edges. The space of convex d-dimensional 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 Alexandrov-Fenchel inequalitie…
