Compact hyperbolic tetrahedra with non-obtuse dihedral angles

Lightlike and ideal tetrahedra

We give a unified description of tetrahedra with lightlike faces in 3d anti-de Sitter, de Sitter and Minkowski spaces and of their duals in 3d anti-de Sitter, hyperbolic and half-pipe spaces. We show that both types of tetrahedra are determined by a generalized cross-ratio with values in a commutative 2d real algebra that generalizes the complex numbers. Equivalently, tetrahedra with lightlike faces are determined by a pair of edge lengths and their duals by a pair of dihedral angles. We prove that the dual tetrahedra are precisely the generalized ideal tetrahedra introduced by Danciger. Final…

使用牛顿法构造双曲多面体

Constructing hyperbolic polyhedra using Newton's Method

本文利用牛顿法和同伦方法实现了Andreev定理的存在性证明，从而构造具有非钝二面角的紧致三维双曲多面体，并应用于构造反射群生成的双曲3-orbifold并研究其体积和闭测地线谱。

Deforming ideal solid tori

We describe the deformation space of a solid torus with boundary modelled on convex ideal hyperbolic polyhedra. This deformation space is given by natural Gauss--Bonnet type inequalities on the dihedral angles. The result extends to solid tori with an arbitrary conical singularity along the core. Our method is to decompose the torus into tetrahedra and maximize a volume functional.

Andreev's Theorem on hyperbolic polyhedra

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 …

量子$6j$-符号的渐近性与广义双曲四面体

Asymptotics of quantum $6j$-symbols and generalized hyperbolic tetrahedra

该研究在Turaev-Viro不变量的容许性条件下，揭示量子$6j$-符号的6元组对应四类几何结构的二面角，证明多数情形下其指数增长率等于广义双曲四面体体积，且体积可负导致序列指数衰减，填补了相关认知空白。

部分截断三角形与四面体的三角学

Trigonometry of partially truncated triangles and tetrahedra

本文利用双曲面模型和洛伦兹几何，建立了具有理想顶点的四边形和五边形的双曲余弦与正弦定理，并证明了部分截断四面体的横向长度仅取决于内部边长集合，且给出了界限。

$b$-$6j$符号的渐近行为与反de Sitter四面体

Asymptotics of $b$-$6j$ symbols and anti-de Sitter tetrahedra

本文研究了b-6j符号的渐近行为及其与反de Sitter四面体体积的关系，并揭示了其与双曲几何和共形场论的联系。

Gauss images of hyperbolic cusps with convex polyhedral boundary

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 Rivin-Hodgson 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…

Volumes and degeneration -- on a conjecture of J. W. Milnor

In his paper "On the Schlafli differential equality", J. Milnor conjectured that the volume of n-dimensional hyperbolic and spherical simplices, as a function of the dihedral angles, extends continuously to the closure of the space of allowable angles (``The continuity conjecture''), and furthermore, the limit at a boundary point is equal to 0 if and only if the point lies in the closure of the space of angles of Euclidean tetrahedra (``the Vanishing Conjecture''). A proof of the Continuity Conjecture was given by F. Luo -- Luo's argument uses Kneser's formula for the volume together with some…