Angle structure on general hyperbolic $3$-manifolds

Let us recall the definitions of the ideal tetrahedra and partially truncated tetrahedra first.

Denote $\mathbb{H}^{3}\subset RP^{3}$ by the open unite ball representing the hyperbolic 3-space via the projective Klein model. Following the terminology in [4], we have

Given that the hyperideal tetrahedron has been extensively studied and used by several mathematicians, we would like to emphasize that the definition of a hyperideal tetrahedron varies from author to author. Some authors require that all four vertices of a hyperideal tetrahedron must be hyperideal vertices, i.e. points outside the closure of $\mathbb{H}^{3}$, e.g. [27]. It is called strictly hyperideal by Schlenker [39].

The concept of ideal tetrahedron, hyperideal tetrahedron and partially truncated tetrahedra of 1-3 type can be generalized to polyhedra case easily.

Similarly, one can define the truncation of a hyperideal tetrahedron (or polyhedron resp.), which is truncated at all hyperideal vertices, i.e. points outside the closure of $\mathbb{H}^{3}$. After the truncating, one get a partially truncated tetrahedron (or polyhedron resp.).

Similar to the role of Epstein-Penner’s decomposition [7] played in [17], we developed a theory of decomposition of general hyperbolic 3-manifolds, which will paly essential role in the proof of our main theorem. The theory says any hyperbolic 3-manifold admits a geometric polyhedral decomposition.

Theorem 2.7 may be considered as a combination of Epstein-Penner’s decomposition [7] and Kojima’s decomposition [28]. However, to some extent, our conclusion is slightly stronger, and only ideal polyhedra and 1-$k$ type polyhedra appear in our decomposition. It is worth emphasizing that there are several advantages to choosing type 1-3 tetrahedra and type 1-$k$ polyhedra. Firstly, for polyhedral decomposition, it is more suitable for the algorithm. Secondly, it is very suitable for the proof of the main theorem in this paper, which can highlight the essence of the proof, and can also greatly simplify the expression, making the details clearer and more readable. See Section 3-4 for details.

Strictly speaking, the ideal tetrahedron, as well as the hyperideal tetrahedron, have no real existent vertices. But for the sake of convenience, we often think that they all have four vertices. For example, a truncated type 1-3 tetrahedral $\sigma$ has three real vertices, which are the three vertices of a hyperbolic triangle obtained by truncating a strictly hyperideal vertex, and four non-existent vertices, of which three are ideal vertices and one is a strictly hyperideal vertex. But we often think that these four unreal vertices are all the vertices of $\sigma$, while those three real vertices are not considered vertices of $\sigma$.

Additionally, it should be noted that the geometry of ideal tetrahedra, hyperideal tetrahedra, and truncated tetrahedra is hyperbolic. For each ideal tetrahedron, it corresponds to a topological tetrahedron with no geometry (i.e., on the boundary of a topological 3-ball, four marked vertices are selected, and each of the two marked points is connected to a marked edge, which does not intersect each other). It can even be considered equivalently that these topological tetrahedra are combinatorial tetrahedra with only combinatorial structures, where combinatorial structures refer to vertices, edges, faces, and connection relationships between them. We can also think of it as corresponding to a type of Euclidean tetrahedra, which may not be the same in shape, but have the same combinatorial structure. Similarly, each truncated hyperbolic tetrahedron corresponds to a combinatorial tetrahedron and a type of truncated Euclidean tetrahedron. For example, the following Figure 2 depicts the correspondence between a tetrahedron with only one truncated vertex (left) and a tetrahedron with four truncated vertices (right). If a (truncated) Euclidean tetrahedron $\sigma$ corresponds to a (truncated) hyperbolic tetrahedron according to the above rules, then a vertex $v$ in $\sigma$ is called an ideal vertex or a hyperideal vertex, defined according to its corresponding vertex type in the (truncated) hyperbolic tetrahedron.

The surface generated by truncating a vertex is called an external surface. The remaining faces are called internal faces, or in other words, faces containing at least one ideal vertex are called internal faces. In Euclidean tetrahedra and combinatorial tetrahedra, external and internal faces can be defined similarly. The edges in the external faces are called the external edges, and the remaining edges are called the internal edges. Equivalently, the edges that contain at least one ideal vertex are called the internal edges. Thus, each tetrahedron, whether it is an ideal tetrahedron or a truncated tetrahedron, always has exactly six internal edges. Below we provide the precise meaning of an ideal triangulation.

Note that the quotient of external faces in $X$ constitutes a triangulation of $\partial M$. The $1$-simplices ($2$-simplices resp.) in $M\setminus\partial M$ are called internal edges (faces resp.) of $\mathcal{T}$.

By definition, an angle structure on a single topological, or combinatorial ideal tetrahedron $\sigma$ assigns to each edge of $\sigma$ a positive number, called dihedral angle, such that the sum of the angles associated to the three edges meeting in a vertex is $\pi$ for each vertex of the tetrahedron. After a simple calculation, it can be found that the dihedral angles of opposite sides are equal. For a truncated tetrahedron with at least one hyperideal vertex, the definition of an angle structure is a little different. The main difference is that the sum of the dihedral angles associated to the three edges meeting in a strictly hyperideal vertex (or equivalently, a truncated face) is less than $\pi$. Combining the above two definitions, the concept of an angle structure can be extended to the most general 3-dimensional pseudo-manifolds with ideal triangulations, see [11], [19], [21], [24], [26], [27] for instance.

In the above definition, if all the dihedral angles are allowed to be taken in the closed interval $[0,\pi]$, it is said to be a semi-angle structure. By the work of Bao-Bonahon [4] ( or see [27] and [39]), any angle structure on $(M,\mathcal{T})$ endows each single 3-simplex $\sigma_{i}$ with a hyperbolic geometry, making it an ideal tetrahedron or hyperideal truncated tetrahedron according to its type. However, it must be pointed out that due to the possible differences in edge lengths (even after choosing decorations), these hyperbolic tetrahedra generally cannot be glued together to obtain a hyperbolic structure on $M$.

Let $\mathcal{T}$ be an ideal triangulation on a topological 3-manifold $M$ with 3-simplex $\sigma_{1},...,\sigma_{n}$.

For each $1\leq i\leq n$, there are seven types of normal disks on $\sigma_{i}$, of which 4 are normal triangles, corresponding to 4 vertices; 3 normal quadrilaterals, corresponding to 3 sets of opposing edges. For example, Figure 3 shows a special normal triangle, as well as an example of a special normal quadrilateral.

Recall $n$ is the number of all types of tetrahedra in the triangulation $\mathcal{T}$. We fix an ordering of all normal disc types $(q_{1},...,q_{3n},t_{1},...,t_{4n})$ in $\mathcal{T}$, where $q_{i}$ denotes a normal quadrilateral type and $t_{j}$ a normal triangle type. For any given normal surface $F$, its normal coordinate $\overline{F}=(x_{1},...,x_{3n},y_{1},...,y_{4n})$ is a vector in $\mathbb{R}^{7n}$, where $x_{i}$ is the number of normal discs of type $q_{i}$ in $F$, and $y_{j}$ is the number of normal discs of type $t_{j}$ in $F$. A properly embedded normal surface $F$ is uniquely determined up to normal isotopy by its normal coordinate ([26], [40]). $(x_{1},...,x_{3n})$ is called the quadrilateral coordinate of $F$.

For a normal disk $D$, let $b(D)$ be the number of normal arcs in $D\cap\partial M$. If $t$ is a normal triangle in $F$ meeting a particular tetrahedra at edges $e_{i}$ with edge valence $d_{i}$, $1\leq i\leq 3$, then its contribution to the Euler characteristic of $F$ is taken to be

If $q$ is a normal quadrilateral in $F$ meeting a particular tetrahedra at edges $e_{i}$ with edge valence $d_{i}$, $1\leq i\leq 4$, then its contribution to the Euler characteristic of $F$ is taken to be

Then the generalized Euler characteristic function $\chi^{*}:\mathbb{R}^{7n}\rightarrow\mathbb{R}$ is defined as

where $q_{i}$ and $t_{j}$ are the normal quadrilaterals and normal triangles in $F$ respectively. The generalized Euler characteristic function is linear, and coincides with the classical Euler characteristic for any embedded or immersed normal surface $F$ in $(M,\mathcal{T})$. See Section 2.6, [26] for details.

There are some linear constraints between the normal coordinate of a normal surface $F$. Let $f$ be an internal face shared by two tetrahedra, say $\sigma_{i}$ and $\sigma_{j}$. Let $a$ be a normal isotopy class of a normal arc in $f$. There are two types of normal disks in $\sigma_{i}$, one is type $x_{i}$ for normal triangles, the other is type $y_{j}$ for normal quadrilaterals, such that when restricted to $f$, the corresponding normal arcs are of the same type as $a$. Similarly, there are two types of normal disks in $\sigma_{j}$, one is type $x_{i^{\prime}}$ for normal triangles, the other is type $y_{j^{\prime}}$ for normal quadrilaterals with similar properties. Thus, the normal coordinate $\overline{F}$ satisfies a linear equation at the face $f$ as follows, which is called the compatibility equation:

Denote $\mathcal{C}(M,\mathcal{T})$ by the set of all vectors $(x_{1},...,x_{3n},y_{1},...,y_{4n})\in\mathbb{R}^{7n}$ satisfying (2.2) at any internal face. For this linear space, Kang-Rubinstein [19] once introduced a basis

where $n$ is the number of $3$-simplex in $\mathcal{T}$ and $m$ is the number of internal edges in $\mathcal{T}$. For concrete meanings of $W_{\sigma_{i}}$ and $W_{e_{j}}$, see [19] and [26]. Then for any $s\in\mathcal{C}(M,\mathcal{T})$, there exists a unique coordinate $(\omega_{1},...,\omega_{n},z_{1},...,z_{m})$ such that

Given a normal disk $d$ in some $\sigma_{i}$, if $d$ is a $k$-sided polygon, and intersects the internal edges $e_{i1},...,e_{ik}$ of some particular $\sigma_{i}$, then its combinatorial area $A(d)$ is defined to be

Luo-Tillmann ([26], Lemma 15) got the following relationship between $\chi^{*}$ and $\chi^{(A)}$:

If $x=(x_{1},...,x_{k})\in\mathbb{R}^{k}$, we use $x>0$ ($x\geq 0$, $x<0$, $x\leq 0$ resp.) to mean that all components of $x_{i}$ are positive (non-negative, negative, non-positive resp.). To approach our main results, we need the following duality result from linear programming, which is known as Farkas’s lemma, and can be found, for instance, in [42]. In the following lemma, vectors in $\mathbb{R}^{k}$ and $\mathbb{R}^{l}$ are considered to be column vectors.

By Theorem 2.7, we obtain the following (topological) ideal triangulation of $M$:

Note at each internal edge in the ideal triangulation $\mathcal{T}$, the value of its dihedral angle is in $[0,\pi]$, hence we obtain a naturally semi-angle structure on $\mathcal{T}$.

Let $\mathcal{T}$ be the ideal triangulation derived in the previous section, and all its (flat) ideal or partially truncated tetrahedra are $\sigma_{1},...,\sigma_{n}$. Suppose $\alpha$ is an angle structure on $(M,\mathcal{T})$. For each $\sigma_{i}$ and each internal edge $e_{ij}~{}(1\leq j\leq 6)$ in $\sigma_{i}$, consider the dihedral angle $\alpha_{ij}=\alpha(e_{ij})$ at $e_{ij}$ as a variable of the equations in Farkas’ Lemma 2.15, see Figure 6.

At the four normal triangles $t_{i}^{1}$, $...$, $t_{i}^{4}$ of $\sigma_{i}$, we have the following equations:

If $\sigma_{i}$ is a partially truncated tetrahedron, then only one of its $a_{i}^{k}$ is less then $\pi$, and the the remaining three are equal to $\pi$. If $\sigma_{i}$ is an (flat) ideal tetrahedron, then all its $a_{i}^{k}$ are equal to $\pi$. The elements of $\{a_{i}^{k}|1\leq i\leq n,1\leq k\leq 4\}$ are divided into two classes, one of them satisfies $\{a_{i_{1}}^{k_{1}}<\pi\}$ with the corresponding normal triangles denoted by $\{t_{i_{1}}^{k_{1}}\}$, the other satisfies $\{a_{i_{2}}^{k_{2}}=\pi\}$ with the corresponding normal triangles denoted by $\{t_{i_{2}}^{k_{2}}\}$.

Recall $n$ is the number of $3$-simplex in $\mathcal{T}$ and $m$ is the number of internal edges in $\mathcal{T}$. At each internal edge $e_{j}$ of $\mathcal{T}$, where $1\leq j\leq m$, there holds:

where the sum traverses all the tetrahedra surround the edge $e_{j}$.

Consider the vector

where $a_{i}^{k}$ ($i=1,...,n$, $k=1,...,4$) comes from (3.1) and $b_{j}=2\pi$ ($j=1,...,m$) comes from (3.2). Writing the system of equations (3.1) and (3.2) in a matrix form as

Consider the transpose $B^{T}$ of $B$, which is also the dual of $B$. It has one variable $h_{i}^{k}$ for each normal triangle type $t_{i}^{k}$ and one variable $z_{j}$ for each internal edge $e_{j}$. If we denote the variables corresponding to $B^{T}$ by $(h,z)=(...,h_{i}^{k},...,...,z_{j},...)$, where $i=1,...,n$, $k=1,...,4$ and $j=1,...,m$, then $B^{T}(h,z)^{T}=(...,z_{j}+h_{i}^{k}+h_{i}^{l},...)^{T}$.

The following useful formula (i.e., Formula (4.9) in [26]) belongs to Luo-Tillmann:

where $W_{\omega,z}=\sum\limits_{i=1}^{n}\omega_{i}W_{\sigma_{i}}+\sum\limits_{j=1}^{m}z_{j}W_{e_{j}}$, and the summation runs over all internal edges in the whole triangulation $\mathcal{T}$.

Note the vector $(a,b)$ looks like this

where $a_{i_{1}}^{k_{1}}<\pi$. In order to obtain an angle structure on $(M,\mathcal{T})$, we only need to prove that there exists a vector $(a,b)=(...,a_{i_{1}}^{k_{1}},...,\pi,...,2\pi,...)$ such that

Further using the third part of Farkas’s lemma, i.e., Lemma 2.15, we just need to show that for all $(h,z)$ with $B^{T}(h,z)^{T}\neq 0$ and $B^{T}(h,z)^{T}\leq 0$, one has $(h,z)\cdot(a,b)<0$.

By the third part of Farkas’s Lemma 2.15,

Hence $(M,\mathcal{T})$ admits an angle structure, and Proposition 1.2 is proved.