# Generalizing Andreev’s Theorem via circle patterns

Ze Zhou
School of Mathematical Sciences  Shenzhen University  Shenzhen 518060  China
<EMAIL_ADDRESS>
###### Abstract

In this paper we derive an extended Circle Pattern Theorem that allows obtuse
overlap angles. As a consequence, we characterize a broader subclass of
compact convex hyperbolic polyhedra which possess possibly obtuse dihedral
angles and thus generalize Andreev’s Theorem. For proofs of these results, we
establish a discrete analog of the "weak solution/regularity theory" scheme.

Mathematics Subject Classifications (2020): 52C26, 51M20, 57M15.

##  1 Introduction

A circle pattern $\mathcal{P}$ on the Riemann sphere $\hat{\mathbb{C}}$ is a
collection of closed disks. The contact graph $G(\mathcal{P})$ of
$\mathcal{P}$ is the graph having a vertex for each disk, and having an edge
between the vertices $u$ and $w$ if the corresponding disks $D_{u},D_{w}$
intersect each other. For each edge $e=[u,w]$ of $G(\mathcal{P})$, we have the
overlap angle $\Theta(e)\in[0,\pi)$. We refer the readers to Stephenson’s
monograph [32] for basic background on circle patterns.

Let us consider the following questions: Given a graph $G$ and a function
$\Theta:E\to[0,\pi)$ defined on the set of edges, is there a circle pattern
with contact graph isomorphic to $G$ and overlap angles given by $\Theta$? If
so, to what extent is the pattern unique? This problem is well posed under the
condition that $0\leq\Theta\leq\pi/2$. In this case, by Thurston’s
interpretation [33] of Andreev’s Theorem [2] (see also [30]), the problem has
a complete answer when $G$ is the $1$-skeleton of a triangulation of the
sphere. Mention that the special case of vanishing angles was due to Koebe
[21]. The result is now known as the Koebe-Andreev-Thurston Theorem (or KAT
Theorem for short), which plays a fundamental role in approximating conformal
maps [28] and hyperbolization of Haken 3-mainifolds [25]. For an account of
history of KAT Theorem together with an extensive list of references, see the
recent article of Philip Bowers [8].

During the past decades, there was fruitful research relating to KAT Theorem.
These include the works of Marden-Rodin [23], Beardon-Stephenson [5], de
Verdi$\mathrm{\grave{e}}$re [11], Bowers-Stephenson [9], Chow-Luo [10],
Bobenko-Springborn [6] and many others. For circle patterns with possibly
obtuse overlap angles, whether similar results hold is a longstanding problem
(see, e.g., [9, 17]). Rivin [27], Bao-Bonahon [3], Bobenko-Springborn [6] and
Schlenker [31] made significant progresses, mainly restricting to ideal and
hyperideal settings. Recently, Ge-Hua-Zhou [15] and Jiang-Luo-Zhou [19]
derived some generalizations of Marden-Rodin Theorem [23], but their results
gave only "weak solutions" that did not rule out extraneous overlaps. To the
best of our knowledge, so far a fully direct generalization of the KAT Theorem
that allows obtuse overlap angles remains open.

In this paper we try to attack the problem. We say a circle pattern
$\mathcal{P}=\\{D(v)\\}_{v\in V}$ on the Riemann sphere $\hat{\mathbb{C}}$ is
irreducible if $\cup_{v\in A}D(v)\subsetneq\hat{\mathbb{C}}$ for every strict
subset $A\subsetneq V$. Below is our main result.

######  Theorem 1.1.

Let $T$ be a triangulation of the sphere with more than four vertices. Suppose
that $\Theta:E\to(0,\pi)$ is a function (where $E$ denotes the set of edges of
$T$) satisfying the conditions below:

  * $\mathrm{\mathbf{(c1)}}$

If $e_{1},e_{2}$ form a simple arc with non-adjacent endpoints, then
$\Theta(e_{1})+\Theta(e_{2})\leq\pi$, and one of the inequalities is strict
when $T$ is the boundary of a double tetrahedron.

  * $\mathrm{\mathbf{(c2)}}$

If $e_{1},e_{2},e_{3}$ form the boundary of a triangle of $T$, then
$\sum_{\mu=1}^{3}\Theta(e_{\mu})>\pi$, and
$\Theta(e_{1})+\Theta(e_{2})<\Theta(e_{3})+\pi$,
$\Theta(e_{2})+\Theta(e_{3})<\Theta(e_{1})+\pi$,
$\Theta(e_{3})+\Theta(e_{1})<\Theta(e_{2})+\pi$.

  * $\mathrm{\mathbf{(c3)}}$

If $e_{1},e_{2},e_{3}$ form a simple closed curve separating the vertices of
$T$, then $\sum_{\mu=1}^{3}\Theta(e_{\mu})<\pi$.

  * $\mathrm{\mathbf{(c4)}}$

If $e_{1},e_{2},e_{3},e_{4}$ form a simple closed curve separating the
vertices of $T$, then $\sum_{\mu=1}^{4}\Theta(e_{\mu})<2\pi$.

Then there exists an irreducible circle pattern $\mathcal{P}$ on the Riemann
sphere $\hat{\mathbb{C}}$ with contact graph isomorphic to the $1$-skeleton of
$T$ and overlap angles given by $\Theta$. Furthermore, $\mathcal{P}$ is unique
up to linear and anti-linear fractional maps.

######  Remark 1.2.

Condition $\mathrm{\mathbf{(c1)}}$ is motivated by an angle relation (see
Lemma 4.4) and is mainly applied to avoiding extraneous overlaps. Condition
$\mathrm{\mathbf{(c2)}}$ is related to inner angles of spherical triangles.
Notice that $\mathrm{\mathbf{(c1)}},\mathrm{\mathbf{(c2)}}$ exclude some
extreme cases mentioned by Bowers-Stephenson [9, p. 214]. Meanwhile,
conditions $\mathrm{\mathbf{(c3)}},\mathrm{\mathbf{(c4)}}$ are parallel to the
conditions in the original KAT Theorem.

In light of Thurston’s interpretation [33, Chap. 13], Theorem 1.1 leads to the
following generalization of Andreev’s Theorem [2]. Mention that the rigidity
part has already been proved by Rivin-Hodgson [26, Corollary 4.6]. For an
abstract polyhedron $P$, recall that a prismatic $k$-circuit $\Gamma$ is a
simple closed curve formed of $k$ edges of the dual complex $P^{\ast}$ such
that all of the endpoints of the edges of $P$ intersected by $\Gamma$ are
distinct.

######  Theorem 1.3.

Let $P$ be an abstract trivalent polyhedron with more than four faces. Suppose
that $\Theta:\mathcal{E}\to(0,\pi)$ is a function (where $\mathcal{E}$ denotes
the set of edges of $P$) such that the following conditions hold:

  * $\mathrm{\mathbf{(a1)}}$

Whenever $\Gamma$ is a simple arc formed by two edges of $P^{\ast}$ such that
$\Gamma$ has non-adjacent endpoints and intersects distinct edges
$e_{1},e_{2}$ of $P$, then $\Theta(e_{1})+\Theta(e_{2})\leq\pi$, and one of
the inequalities is strict if $P$ is the triangular prism.

  * $\mathrm{\mathbf{(a2)}}$

Whenever three distinct edges $e_{1},e_{2},e_{3}$ of $P$ meet at a vertex,
then $\sum_{\mu=1}^{3}\Theta(e_{\mu})>\pi$, and
$\Theta(e_{1})+\Theta(e_{2})<\Theta(e_{3})+\pi$,
$\Theta(e_{2})+\Theta(e_{3})<\Theta(e_{1})+\pi$,
$\Theta(e_{3})+\Theta(e_{1})<\Theta(e_{2})+\pi$.

  * $\mathrm{\mathbf{(a3)}}$

Whenever $\Gamma$ is a prismatic 3-circuit intersecting edges
$e_{1},e_{2},e_{3}$, then $\sum_{\mu=1}^{3}\Theta(e_{\mu})<\pi$.

  * $\mathrm{\mathbf{(a4)}}$

Whenever $\Gamma$ is a prismatic 4-circuit intersecting edges
$e_{1},e_{2},e_{3},e_{4}$, then $\sum_{\mu=1}^{4}\Theta(e_{\mu})<2\pi$.

Then there exists a compact convex hyperbolic polyhedron $Q$ combinatorially
equivalent to $P$ with dihedral angles given by $\Theta$. Furthermore, $Q$ is
unique up to isometries of $\mathbb{H}^{3}$.

######  Remark 1.4.

One may expect to describe the space of dihedral angles of compact convex
hyperbolic polyhedra of a given combinatorial type. Unfortunately, as
indicated by Díaz [13, 14] and Roeder [29], this involves non-linear
conditions that are hard to verify.

To acquire Theorem 1.1, we will introduce a series of configurations spaces
which consist of circle patterns satisfying finer and finer properties. In
this way we reduce the proof to showing the associated test map is surjective,
which can be solved via topological degree theory. For this purpose, one needs
to check the test map is proper and thus the topological degree is well-
defined. To be specific, it suffices to deduce that under suitable conditions
a sequence of circle patterns can produce a limit pattern with required
properties.

Let $W$ denote the set of functions $\Theta:E\to(0,\pi)$ satisfying conditions
$\mathrm{\mathbf{(c1)}}-\mathrm{\mathbf{(c4)}}$. Precisely, we will show the
following Normal Family Theorem for circle patterns.

######  Theorem 1.5.

Let $\\{\mathcal{P}_{n}\\}$ be a sequence of irreducible circle patterns on
$\hat{\mathbb{C}}$, each of which has contact graph isomorphic to the
$1$-skeleton of a triangulation $T$ of the sphere with more than four
vertices. Then $\\{\mathcal{P}_{n}\\}$ (modulo linear and anti-linear
fractional maps) contains a subsequence convergent to an irreducible circle
pattern with the same contact graph if the corresponding overlap angle
function sequence $\\{\Theta_{n}\\}$ stays in a compact subset of $W$.

Let us give an outline of the proof of this theorem. We first endow
$\hat{\mathbb{C}}$ with the spherical metric and assume each $\mathcal{P}_{n}$
satisfies some suitable normalization conditions. We easily see
$\\{\mathcal{P}_{n}\\}$ contains a subsequence convergent to a pre-pattern
$\mathcal{P}_{\infty}$ in certain compact extension space. It remains to
verify that $\mathcal{P}_{\infty}$ satisfies the following properties:

  * $\mathrm{\mathbf{(p1)}}$

Each pre-disk in $\mathcal{P}_{\infty}$ is neither the entire sphere nor a
point;

  * $\mathrm{\mathbf{(p2)}}$

$\mathcal{P}_{\infty}$ has contact graph isomorphic to the $1$-skeleton of
$T$. Namely, no extraneous overlap is created in $\mathcal{P}_{\infty}$;

  * $\mathrm{\mathbf{(p3)}}$

$\mathcal{P}_{\infty}$ is irreducible.

To assert $\mathrm{\mathbf{(p1)}}$, we turn to an estimate based on the
irreducibility assumption and an analog of the Ring Lemma (see Lemma 4.5 and
Lemma 4.6). With respect to $\mathrm{\mathbf{(p2)}}$, as pointed out by
Bowers-Stephenson [9, p. 214], it is one of the main obstructions for
extending the KAT Theorem. Fortunately, our approach reduces the problem to
showing there is no "tangential type" of extraneous overlaps, which can be
settled through some results relating such extraneous overlaps to violations
of conditions $\mathrm{\mathbf{(c1)}}-\mathrm{\mathbf{(c4)}}$ (see Lemma 4.4
and Lemma 4.7). Finally, $\mathrm{\mathbf{(p3)}}$ follows from
$\mathrm{\mathbf{(p2)}}$ together with some elementary facts (see Lemma 3.2
and Lemma 4.3). To summarize, the whole proof can be seen in Section 5.

Notice that the above strategy is parallel to the "weak solution/regularity
theory" scheme in PDE theory, where $\mathcal{P}_{\infty}$ plays a similar
role to a weak solution, some relevant lemmas play similar roles to prior
estimates, and configuration spaces play similar roles to Sobolev spaces.

The paper is organized as follows. In next section, assuming Theorem 1.5, we
give a quick proof of Theorem 1.1 by using configuration spaces and
topological degree theory. In Section 3, we derive Theorem 1.3 from Theorem
1.1. To this end, we actually prove a theorem answering a question on which
circle patterns induce compact hyperbolic polyhedra (see Theorem 3.1). In
Section 4, we establish several lemmas which serve as "prior estimates" for
circle patterns. In Section 5, we prove Theorem 1.5. The last section is an
appendix which complements the proofs of two lemmas in Section 4.

Statement. We mention that Theorem 1.1 and Theorem 1.3 grow out from an early
version of the preprint [35], where the author investigated two classes of
circle patterns through different geometry models. To improve readability, we
decide to divide the former manuscript into two articles. Specifically, in the
latest version of the preprint [35], we restrict to circle patterns that
possess at least one interstice. For patterns without any interstice, these
are considered in the current paper.

##  2 Circle patterns from the viewpoint of manifolds

In this section we shall give a quick proof of Theorem 1.1 assuming Theorem
1.5. For this purpose, we introduce several configuration spaces which
parameterize circle patterns with finer and finer properties. Mention that
such a global viewpoint has been rooted in the works of Bauer-Stephenson-
Wegert [4], Zhou [34], Bowers-Bowers-Pratt [7] and Connelly-Gortler [12] in
consideration of computational mechanisms (see [4, 12]) or understanding of
local rigidity (see [34, 7]).

To begin with, we endow $\hat{\mathbb{C}}$ with the following Riemannian
metric

$\mathrm{ds}=\frac{2|\mathrm{d}z|}{1+|z|^{2}}.$

Note that $(\hat{\mathbb{C}},\mathrm{ds})$ is isometric to the unit sphere
$\mathbb{S}^{2}$ in $\mathbb{R}^{3}$. In what follows we shall not distinguish
$(\hat{\mathbb{C}},\mathrm{ds})$ and $\mathbb{S}^{2}$ for the sake of
simplicity.

Recall that $T$ is a triangulation of the sphere with more than four vertices.
Let $V,E,F$ denote the sets of vertices, edges and triangles of $T$,
respectively. Set $M=\hat{\mathbb{C}}^{|V|}\times(0,\pi)^{|V|}$. Then $M$ is a
smooth manifold parameterizing the space of patterns of $|V|$ disks on
$(\hat{\mathbb{C}},\mathrm{ds})$. Since $T$ is a triangulation, it is trivial
to see

$2|E|=3|F|.$

Combining with Euler’s Formula, we have

$\displaystyle\dim(M)$ $\displaystyle=3|V|\,$
$\displaystyle=|E|+(2|E|-3|F|)+3\big{(}|V|-|E|+|F|\big{)}$
$\displaystyle=|E|+6.$

A point
$(\mathbf{z},\mathbf{r})=(z_{1},\cdots,z_{|V|},r_{1},\cdots,r_{|V|})\in M$ is
called a configuration, since it assigns each vertex $v_{i}\in V$ a closed
disk $D(v_{i})$, where $D(v_{i})$ is centered at $z_{i}$ and is of radius
$r_{i}$. For an edge $e=[v_{i},v_{j}]\in E$, the inversive distance
$I(e,\mathbf{z},\mathbf{r})$ is

$\frac{\cos r_{i}\cos r_{j}-\cos d(z_{i},z_{j})}{\sin r_{i}\sin r_{j}},$

where $d(z_{i},z_{j})$ denotes the spherical distance between $z_{i}$ and
$z_{j}$.

Let us introduce several subspaces of $M$. The first subspace $M_{E}\subset M$
is the set of configurations subject to

$-1<(e,\mathbf{z},\mathbf{r})<1$

for every $e\in E$. In this case, we can then compute the overlap angle
$\Theta(e,\mathbf{z},\mathbf{r})$ via the following formula:

$\Theta(e,\mathbf{z},\mathbf{r})=\arccos I(e,\mathbf{z},\mathbf{r}).$

Next we define $M_{\Theta}\subset M_{E}$ to be the subspace of configurations
with overlap angle functions satisfying conditions
$\mathrm{\mathbf{(c2)}}-\mathrm{\mathbf{(c4)}}$ and the following condition:

  * $\mathrm{\mathbf{(x1)}}$

If $e_{1},e_{2}$ form a simple arc with non-adjacent endpoints, then
$\Theta(e_{1})+\Theta(e_{2})<\pi$.

In addition, let $M_{G}\subset M_{\Theta}$ consist of configurations with the
following property:

  * $\mathrm{\mathbf{(x2)}}$

Whenever the distinct vertices $v_{\alpha},v_{\beta}$ are non-adjacent, then
$D(v_{\alpha})\cap D(v_{\beta})=\emptyset$.

Thus each $(\mathbf{z},\mathbf{r})\in M_{G}$ gives a circle pattern with
contact graph isomorphic to the $1$-skeleton of $T$. In addition, let
$M_{IG}\subset M_{G}$ consist of configurations subject to the irreducible
property:

  * $\mathrm{\mathbf{(x3)}}$

For any strict subset $A\subsetneq V$, $\cup_{v_{i}\in
A}D(v_{i})\subsetneq\hat{\mathbb{C}}$.

Note that $M_{E},M_{\Theta},M_{G},M_{IG}$ are open subsets of $M$ and thus are
smooth manifolds.

Let $v_{a},v_{b},v_{c}$ be the vertices of a triangle of $T$ and let
$e_{a},e_{b},e_{c}$ be the edges opposite to $v_{a},v_{b},v_{c}$,
respectively. We use $M_{IG}^{\star}\subset M_{IG}$ to represent the subspace
of configurations satisfying the following normalization conditions:

  * $\mathrm{\mathbf{(x4)}}$

$z_{a}=0$, $z_{b}>0$, $\operatorname{Im}(z_{c})>0$;

  * $\mathrm{\mathbf{(x5)}}$

$r_{a}=r_{b}=r_{c}=\pi/2$.

Let $W_{0}$ be the set of functions $\Theta:E\to(0,\pi)$ satisfying conditions
$\mathrm{\mathbf{(x1)}}$ and $\mathrm{\mathbf{(c2)}}-\mathrm{\mathbf{(c4)}}$.
We define the test map as follows:

$\displaystyle f:$ $\displaystyle M^{\star}_{IG}$
$\displaystyle\longrightarrow$ $\displaystyle\quad\quad\quad\,\;W_{0}$
$\displaystyle(\mathbf{z},\mathbf{r})$ $\displaystyle\longmapsto$
$\displaystyle\big{(}\Theta(e_{1},\mathbf{z},\mathbf{r}),\Theta(e_{2},\mathbf{z},\mathbf{r}),\cdots\big{)}.$

To derive Theorem 1.1, we need to demonstrate that $f$ is a surjective map.
For this purpose, we will make the use of topological degree theory. Note that
one needs to prove nothing if $W_{0}$ is an empty set. So from now on we focus
on the case that $W_{0}\neq\emptyset$. We first observe the following fact.

######  Lemma 2.1.

If $W_{0}\neq\emptyset$, then $W_{0}\cap(0,\pi/2)^{|E|}\neq\emptyset$.
Furthermore, $W_{0}\cap(0,\pi/2)^{|E|}$ has positive measure in
$(0,\pi)^{|E|}$.

###### Proof.

Choose $\Theta_{\ast}\in W_{0}$. For $s\in(0,1]$, set

$\Theta_{s}=s\Theta_{\ast}+\frac{(1-s)\pi}{3}.$

Taking $s_{0}$ sufficiently close to zero, we have $\Theta_{s_{0}}\in
W_{0}\cap(0,\pi/2)^{|E|}$, which yields

$W_{0}\cap(0,\pi/2)^{|E|}\neq\emptyset.$

For $\delta>0$ , let $U_{\delta}$ consist of functions $\Theta:E\to(0,\pi)$
satisfying

$\Theta_{s_{0}}(e)<\Theta(e)<\Theta_{s_{0}}(e)+\delta,\;\;\;\forall e\in E.$

When $\delta$ is sufficiently small, it is easy to see $U_{\delta}\subset
W_{0}\cap(0,\pi/2)^{|E|}$. Hence $W_{0}\cap(0,\pi/2)^{|E|}$ has positive
measure in $(0,\pi)^{|E|}$. ∎

Recall that the test map $f$ is smooth. Applying Sard’s Theorem [16, 18, 24,
20], one can find a regular value $\widetilde{\Theta}\in U_{\delta}\subset
W_{0}\cap(0,\pi/2)^{|E|}$ of the map $f$. In light of Thurston’s
interpretation [33, Chap. 13], the following lemma is a consequence of
Andreev’s Theorem [2]. See also the work of Bowers-Stephenson [9].

######  Lemma 2.2.

There exists an irreducible circle pattern $\widetilde{\mathcal{P}}$ on the
Riemann sphere $\hat{\mathbb{C}}$ with contact graph isomorphic to the
$1$-skeleton of $T$ and overlap angles given by $\widetilde{\Theta}$.
Moreover, $\widetilde{\mathcal{P}}$ is unique up to linear and anti-linear
fractional maps.

Notice that $\widetilde{\mathcal{P}}$ can be transformed into a normalized one
through a unique linear or anti-linear fractional map, which indicates the
following lemma.

######  Lemma 2.3.

Suppose $W_{0}\neq\emptyset$. Then $M^{\star}_{IG}\neq\emptyset$. In addition,
$M^{\star}_{IG}$ is a manifold of dimension $|E|$.

Since $W_{0}$ is an open subset of $(0,\pi)^{|E|}$, it follows that

$\dim(M^{\star}_{IG})=\dim(W_{0})=|E|.$

Meanwhile, assuming Theorem 1.5, we assert the properness of the test map.

######  Lemma 2.4.

Under the hypothesis that $W_{0}\neq\emptyset$, the map $f:M^{\star}_{IG}\to
W_{0}$ is proper.

###### Proof.

For any compact subset $\Omega$ of $W_{0}$, we need to check that
$f^{-1}(\Omega)$ is compact. It then suffices to show any sequence
$\\{(\mathbf{z}_{n},\mathbf{r}_{n})\\}$ in $f^{-1}(\Omega)$ contains a
convergent subsequence. Note that every $(\mathbf{z}_{n},\mathbf{r}_{n})$
gives a normalized irreducible circle patterns $\mathcal{P}_{n}$ on
$\hat{\mathbb{C}}$ with contact graph isomorphic to the $1$-skeleton of $T$
and overlap angle function staying in the compact subset $\Omega$. Applying
Theorem 1.5, we immediately conclude the lemma. ∎

By manifold theory (see, e.g., [16, 18, 24, 20]), the topological degree of
$f$ is well-defined. What is more, we have the following property.

######  Proposition 2.5.

Suppose $W_{0}\neq\emptyset$. Then $\deg(f)=\pm 1$.

###### Proof.

Because $W_{0}$ is connected, we can choose an arbitrary regular value of $f$
to calculate the degree. In particular, we have

$\deg(f)=\sum_{(\mathbf{z},\mathbf{r})\in
f^{-1}(\widetilde{\Theta})}\operatorname{sgn}\left(\operatorname{det}(Df(\mathbf{z},\mathbf{r}))\right),$

where $Df(\mathbf{z},\mathbf{r})$ is the Jacobian matrix of $f$ at
$(\mathbf{z},\mathbf{r})$. Meanwhile, it follows from Lemma 2.2 that
$f^{-1}(\widetilde{\Theta})$ contains only one point, which yields
$\deg(f)=\pm 1$. ∎

The proposition below is thus straightforward.

######  Proposition 2.6.

The map $f:M^{\star}_{IG}\to W_{0}$ is surjective.

We are now ready to prove the main theorem.

###### Proof of Theorem 1.1 (assuming Theorem 1.5).

For each positive integer $n$, set

$\Theta_{n}=\frac{n\Theta}{n+1}+\frac{\pi}{3(n+1)}.$

Clearly, $\Theta_{n}\in W_{0}\subset W$. Owing to Proposition 2.6, there
exists an irreducible circle pattern $\mathcal{P}_{n}$ on $\hat{\mathbb{C}}$
realizing the data $(T,\Theta_{n})$. By Theorem 1.5, we can pick up a
subsequence of $\\{\mathcal{P}_{n}\\}$ convergent to an irreducible circle
pattern $\mathcal{P}_{\infty}$ with contact graph isomorphic to the
$1$-skeleton of $T$. Moreover, it is easy to see $\mathcal{P}_{\infty}$ has
overlap angles given by $\Theta$, which asserts the existence part of the
theorem. Finally, the rigidity part follows from the rigidity of the
corresponding compact convex hyperbolic polyhedron, which will be proved in
Theorem 1.3. ∎

##  3 Generalizing Andreev’s Theorem

To prove Theorem 1.3, we recall the correspondence between circle patterns and
hyperbolic polyhedra observed by Thurston [33, Chap. 13]. Specifically, given
a circle pattern $\mathcal{P}=\\{D(v)\\}_{v\in V}$ on
$\hat{\mathbb{C}}=\partial\mathbb{H}^{3}$ with contact graph isomorphic to the
$1$-skeleton of a triangulation of $T$, we associate each disk $D(v)$ with a
half-space $H(v)\subset\mathbb{H}^{3}$ that equals to the closed convex hull
of ideal points in $\hat{\mathbb{C}}\setminus
D(v)\subset\partial\mathbb{H}^{3}$. This gives rise to a map sending each
pattern $\mathcal{P}$ to a convex set $Q(\mathcal{P})\subset\mathbb{H}^{3}$,
where $Q({\mathcal{P}})=\cap_{v\in V}H(v)$. Moreover, if $Q(\mathcal{P})$ is a
compact polyhedron, then it has boundary decomposition dual to $T$ and
dihedral angles equal to the corresponding overlap angles of $\mathcal{P}$.

Unfortunately, $Q(\mathcal{P})$ is possibly not a compact polyhedron. A
natural problem asks which circle patterns induce such polyhedra. Here we give
a partial answer by showing the following theorem.

######  Theorem 3.1.

Let $T$ be a triangulation of the sphere with more than four vertices and let
$\mathcal{P}$ be a circle pattern on $\hat{\mathbb{C}}$ with contact graph
isomorphic to the $1$-skeleton of $T$ and overlap angle function satisfying
condition $\mathrm{\mathbf{(c2)}}$. Assume the following condition holds:

  * $\mathrm{\mathbf{(cr)}}$

Whenever the edges
$[v_{\alpha},v_{\beta}],[v_{\beta},v_{\gamma}],[v_{\gamma},v_{\alpha}]$ form a
simple closed curve separating the vertices of $T$, then $D(v_{\alpha})\cap
D(v_{\beta})\cap D(v_{\gamma})=\emptyset$.

Then $Q(\mathcal{P})$ is a compact convex hyperbolic polyhedron with boundary
decomposition dual to $T$ and dihedral angles equal to the corresponding
overlap angles of $\mathcal{P}$.

###### Proof.

For each $v\in V$, set $\Pi(v)=\partial_{\mathbb{H}^{3}}H(v)$. Let
$v_{i},v_{j},v_{k}$ be the vertices of a triangle of $T$ and let
$e_{i},e_{j},e_{k}$ be the edges opposite to $v_{i},v_{j},v_{k}$,
respectively. We first claim $\Pi(v_{i}),\Pi(v_{j}),\Pi(v_{k})$ intersect at a
point in $\mathbb{H}^{3}$, which is reduced to checking the Gram matrix
$J(v_{i},v_{j},v_{k})$ of the geodesic planes
$\Pi(v_{i}),\Pi(v_{j}),\Pi(v_{k})$ is positive-definite. Note that
$J(v_{i},v_{j},v_{k})$ is equal to

$\left[\begin{array}[]{ccc}1&-\cos\theta_{i}&-\cos\theta_{j}\\\
-\cos\theta_{i}&1&-\cos\theta_{k}\\\
-\cos\theta_{j}&-\cos\theta_{k}&1\end{array}\right],$

where $\theta_{\mu}=\Theta(e_{\mu})$ for $\mu=i,j,k$. Due to Sylvester’s
criterion, we need to prove the $m$-th leading principal minor of
$J(v_{i},v_{j},v_{k})$ is positive for $m=1,2,3$. The cases that $m=1,2$ are
straightforward. For $m=3$, it is equivalent to asserting the determinant of
$J(v_{i},v_{j},v_{k})$ is also positive. Indeed, a routine calculation shows
the determinant is equal to

$-4\cos\dfrac{\theta_{i}+\theta_{j}+\theta_{k}}{2}\cos\dfrac{\theta_{i}+\theta_{j}-\theta_{k}}{2}\cos\dfrac{\theta_{j}+\theta_{k}-\theta_{i}}{2}\cos\dfrac{\theta_{k}+\theta_{i}-\theta_{j}}{2},$

Under condition $\mathrm{\mathbf{(c2)}}$, the above expression is positive,
which asserts the claim.

Meanwhile, by Steinitz’s Theorem, we have an abstract compact polyhedron
$Q(T)$ with boundary $\partial Q(T)$ being a cellular decomposition dual to
$T$. We now reduce the theorem to showing there exists a homeomorphism between
$\partial Q(T)$ and $\partial_{\mathbb{H}^{3}}Q(\mathcal{P})$. For closed
subsets $A,B$ of certain topological space, recall Leibniz’s Rule:

$\partial(A\cap B)=(\partial A\cap B)\cup(A\cap\partial B).$

It follows that

$\partial_{\mathbb{H}^{3}}Q(\mathcal{P})\,=\,\partial_{\mathbb{H}^{3}}\left(\cap_{v\in
V}H(v)\right)\,=\,\cup_{v\in V}\mathcal{F}(v),$

where

$\mathcal{F}(v)=\big{(}\cap_{u\in V,u\neq
v}H(u)\big{)}\cap\partial_{\mathbb{H}^{3}}H(v)=\big{(}\cap_{u\in V,u\neq
v}H(u)\big{)}\cap\Pi(v).$ (3.1)

For each $u\in V\setminus\\{v\\}$ which is not adjacent to $v$, we claim

$\Pi(v)\subset H(u).$ (3.2)

Otherwise, $\partial D(v)\cap D(u)\neq\emptyset$, which contradicts that
$D(v),D(u)$ are disjoint. Combining (3.1) and (3.2), we derive

$\mathcal{F}(v)=\big{(}\cap_{\mu=1}^{d}H(u_{\mu})\big{)}\cap\Pi(v),$ (3.3)

where $u_{1},\cdots,u_{d}$, in anticlockwise order, are all adjacent vertices
of $v$.

Let $\sum(v)$ be the face of $\partial Q(T)$ dual to $v$. Then $\sum(v)$ is a
topological polygon marked by triangles $f_{1},\cdots,f_{d}$ of $T$, where
$f_{\tau}$ has vertices $v,u_{\tau},u_{\tau+1}$ for $\tau=1,\cdots,d$ (set
$u_{d+1}=u_{1}$). We will show there exists a mark-preserving homeomorphism
between $\sum(v)$ and $\mathcal{F}(v)$. To this end, let
$\partial_{\Pi}\mathcal{F}(v)$ denote the boundary of $\mathcal{F}(v)$ in the
geodesic plane $\Pi(v)$. Applying Leibniz’s Rule to (3.3), we obtain

$\partial_{\Pi}\mathcal{F}(v)=\cup_{\mu=1}^{d}\left(\big{(}\,\cap_{\eta=1,\eta\neq\mu}^{d}H(u_{\eta})\,\big{)}\cap\Pi(u_{\mu})\cap\Pi(v)\right).$
(3.4)

When $\eta\neq\mu-1,\mu,\mu+1$, note that

$\Pi(u_{\mu})\cap\Pi(v)\subset H(u_{\eta}).$

Otherwise, $\partial D(u_{\mu})\cap\partial D(v)\cap
D(u_{\eta})\neq\emptyset$, which contradicts (3.5) in the following Lemma 3.2.
We thus simplify formula (3.4) as

$\partial_{\Pi}\mathcal{F}(v)=\cup_{\mu=1}^{d}\left(H(u_{\mu-1})\cap
H(u_{\mu+1})\cap\Pi(u_{\mu})\cap\Pi(v)\right):=\cup_{\mu=1}^{d}\mathcal{L}_{\mu}.$

It is easy to see $\mathcal{L}_{\mu}$ is a geodesic segment with endpoints
$q(f_{\mu-1}),q(f_{\mu})$, where

$q(f_{\tau})=\Pi(u_{\tau})\cap\Pi(u_{\tau+1})\cap\Pi(v)$

for $\tau=1,\cdots,d$. That means $\mathcal{F}(v)$ is a convex polygon with
vertices $q(f_{1}),\cdots,q(f_{d})$. Hence the map

$(f_{1},\cdots,f_{d})\longmapsto(q(f_{1}),\cdots,q(f_{d}))$

induces a mark-preserving homeomorphism between $\sum(v)$ and
$\mathcal{F}(v)$.

Using the Pasting Lemma, we thus obtain a global homeomorphism between
$\partial Q(T)$ and $\partial Q(\mathcal{P})$, which implies $Q(\mathcal{P})$
is a compact convex hyperbolic polyhedron with boundary decomposition dual to
$T$. Finally, the property that $Q(\mathcal{P})$ has dihedral angles equal to
corresponding overlap angles of $\mathcal{P}$ is straightforward. ∎

The following lemma, required in the above proof, indicates some properties
behind the conditions of Theorem 3.1.

######  Lemma 3.2.

Let $T,\mathcal{P}$ be as in Theorem 3.1. If $v_{i},v_{j},v_{k}$ are distinct
vertices of $T$ such that

$D(v_{i})\cap D(v_{j})\cap D(v_{k})\neq\emptyset,$ (3.5)

then $v_{i},v_{j},v_{k}$ are the vertices of a triangle of $T$. As a result,

$D(v_{i})\cap D(v_{j})\cap D(v_{k})\cap D(v_{l})=\emptyset$ (3.6)

for any four distinct vertices $v_{i},v_{j},v_{k},v_{l}$ of $T$.

###### Proof.

First we claim any two distinct vertices among $v_{i},v_{j},v_{k}$ are
adjacent. Otherwise, without loss of generality, suppose $v_{i}$ is not
adjacent to $v_{j}$. Then $D(v_{i})\cap D(v_{j})=\emptyset$, which contradicts
(3.5). We further claim $v_{i},v_{j},v_{k}$ are the vertices of a triangle of
$T$. Otherwise, the edges $[v_{i},v_{j}],[v_{j},v_{k}],[v_{k},v_{i}]$ form a
simple closed curve separating the vertices of $T$. By condition
$\mathrm{\mathbf{(cr)}}$, this also leads a contradiction to (3.5). Finally,
remembering $T$ is not the boundary of a tetrahedron, we immediately derive
(3.6) from (3.5). ∎

Before the proof of Theorem 1.3, we recall also the following fact which will
relate the irreducibility assumption to condition $\mathrm{\mathbf{(cr)}}$.

######  Lemma 3.3.

Let $D_{i},D_{j},D_{k}$ be three mutually intersecting disks on
$\hat{\mathbb{C}}$ with overlap angles
$\Theta_{ij},\Theta_{jk},\Theta_{ki}\in[0,\pi)$. Suppose
$\mathbb{D}_{i}\cup\mathbb{D}_{j}\cup\mathbb{D}_{k}\subsetneq\hat{\mathbb{C}}$,
where $\mathbb{D}_{\mu}$ denotes the interior of $D_{\mu}$ for $\mu=i,j,k$. If
$\Theta_{ij}+\Theta_{jk}+\Theta_{ki}<\pi$, then

$D_{i}\cap D_{j}\cap D_{k}=\emptyset.$

We are now ready to prove the general Andreev’s Theorem.

###### Proof of Theorem 1.3.

Let $T$ be the triangulation of the sphere dual to the boundary decomposition
of $P$ and let $\Theta^{\ast}:E\to(0,\pi)$ be the function assigning each edge
of $T$ with the angle given by $\Theta$ at the dual edge. Since
$\Theta:\mathcal{E}\to(0,\pi)$ satisfies conditions
$\mathrm{\mathbf{(a1)}}-\mathrm{\mathbf{(a4)}}$, it is easy to see
$\Theta^{\ast}:E\to(0,\pi)$ satisfies conditions
$\mathrm{\mathbf{(c1)}}-\mathrm{\mathbf{(c4)}}$. By Theorem 1.1, there exists
an irreducible circle pattern $\mathcal{P}$ realizing $(T,\Theta^{\ast})$.

Suppose $v_{\alpha},v_{\beta},v_{\gamma}$ are three vertices such that
$[v_{\alpha},v_{\beta}],[v_{\beta},v_{\gamma}],[v_{\gamma},v_{\alpha}]$ form a
simple closed curve separating the vertices of $T$. Because $\mathcal{P}$ is
irreducible and $T$ contains more than four vertices, we have
$D(v_{\alpha})\cup D(v_{\beta})\cup D(v_{\gamma})\subsetneq\hat{\mathbb{C}}$.
Recalling $\Theta^{\ast}:E\to(0,\pi)$ satisfies conditions
$\mathrm{\mathbf{(c3)}}$, we then check condition $\mathrm{\mathbf{(cr)}}$ by
Lemma 3.3. The existence part of the theorem now follows from Theorem 3.1.
Finally, the rigidity part is a straightforward consequence of the following
Theorem 3.4. ∎

The theorem below was proved by Rivin-Hodgson [26, Corollary 4.6] via
analogous arguments used by Cauchy in the proof of his rigidity theorem [1]
for compact convex polyhedra in Euclidean 3-space. In fact, a similar proof
can be also seen in the work of Roeder-Hubbard-Dunbar [30, Proposition 4.1].

######  Theorem 3.4 (Rivin-Hodgson).

Compact convex polyhedra in $\mathbb{H}^{3}$ with trivalent vertices are
determined up to isometries by their combinatorics and dihedral angles.

##  4 Prior estimates for circle patterns

As mentioned before, we regard configurations spaces as discrete analogs of
Sobolev spaces. To prove Theorem 1.5, it is necessary to establish some prior
estimates relating different configurations spaces. For quickly detecting the
ideas, the readers may focus on the core information of these estimates at
first and revisit the proofs in details after getting the big picture of the
paper.

We begin with several lemmas which serve as local estimates. Mention that some
relevant results have already appeared in the works of Thurston [33], Marden-
Rodin [23], Bowers-Stephenson [9], Chow-Luo [10] and others.

######  Lemma 4.1.

Suppose $\Theta_{ij},\Theta_{jk},\Theta_{ki}\in(0,\pi)$ are three angles
satisfying

$\Theta_{ij}+\Theta_{jk}+\Theta_{ki}>\pi,\;\;\Theta_{ij}+\Theta_{jk}<\Theta_{ki}+\pi,\;\;\Theta_{jk}+\Theta_{ki}<\Theta_{ij}+\pi,\;\;\Theta_{ki}+\Theta_{ij}<\Theta_{jk}+\pi.$

For any three reals $r_{i},r_{j},r_{k}\in(0,\pi)$, there exists a
configuration of three intersecting disks $D_{i},D_{j},D_{k}$ on
$(\hat{\mathbb{C}},\mathrm{ds})$, unique up to isometries, having radii
$r_{i},r_{j},r_{k}$ and meeting in overlap angles
$\Theta_{ij},\Theta_{jk},\Theta_{ki}$. Moreover, the centers of these disks
determine a spherical triangle.

Figure 1: Three-circle configuration

The proof is analogous to the hyperbolic and Euclidean versions established by
Zhou [34] and Jiang-Luo-Zhou [19], but the details are somewhat complicated.
We postpone it to the appendix. Meanwhile, we observe the following simple
fact.

######  Lemma 4.2.

Let $D_{i},D_{j},D_{k}$ be as in Lemma 4.1. Then $\partial D_{i}\cap\partial
D_{j}\cap\partial D_{k}=\emptyset$.

######  Lemma 4.3.

Let $D_{i},D_{j},D_{k}$ be three intersecting disks on
$(\hat{\mathbb{C}},\mathrm{ds})$ with radii $r_{i},r_{j},r_{k}\in(0,\pi/2]$
and overlap angles $\Theta_{ij},\Theta_{jk},\Theta_{ki}\in[0,\pi)$. If
$\Theta_{ij}+\Theta_{jk}+\Theta_{ki}<\pi$, then

$D_{i}\cap D_{j}\cap D_{k}=\emptyset.$

###### Proof.

Let $\mathbb{D}_{i},\mathbb{D}_{j},\mathbb{D}_{k}$ be the interiors of
$D_{i},D_{j},D_{k}$, respectively. Since $r_{i},r_{j}\in(0,\pi/2]$, it is easy
to see the set $\hat{\mathbb{C}}\setminus(\mathbb{D}_{i}\cup\mathbb{D}_{j})$
contains a pair of antipodal points. Because $r_{k}\leq\pi/2$, one of the
antipodal points can not be covered by $\mathbb{D}_{k}$. That means
$\mathbb{D}_{i}\cup\mathbb{D}_{j}\cup\mathbb{D}_{k}\subsetneq\hat{\mathbb{C}}$.
Applying Lemma 3.3, we obtain $D_{i}\cap D_{j}\cap D_{k}=\emptyset$. ∎

######  Lemma 4.4.

Let $D_{i},D_{j},D_{k}$ be three mutually intersecting disks on
$\hat{\mathbb{C}}$ with overlap angles
$\Theta_{ij},\Theta_{jk},\Theta_{ki}\in[0,\pi)$. Suppose $D_{i}\cap
D_{j}\subset D_{k}$. Then

$\Theta_{jk}+\Theta_{ki}\geq\Theta_{ij}+\pi.$

In particular, when $D_{i},D_{j}$ intersect at only one point, we have

$\Theta_{jk}+\Theta_{ki}\geq\pi,$

where the equality holds if and only if $D_{i}\cap D_{j}=\partial
D_{i}\cap\partial D_{j}\subset\partial D_{k}$.

###### Proof.

The lemma is straightforward after mapping $\partial D_{i},\partial D_{j}$
into intersecting lines through a suitable Möbius transformation. See Figure
2. ∎

Figure 2: Angle relation

Next we introduce some global estimates. Unless otherwise stated, in what
follows we always assume $T$ is a triangulation of the sphere with more than
four vertices. For any disk $D(v)$ on $(\hat{\mathbb{C}},\mathrm{ds})$, we use
$r(D(v))$ to denote the radius of $D(v)$.

######  Lemma 4.5.

Let $\mathcal{P}=\\{D(v)\\}_{v\in V}$ be an irreducible circle pattern on
$(\hat{\mathbb{C}},\mathrm{ds})$ with contact graph isomorphic to the
$1$-skeleton of $T$ and overlap angle function $\Theta:E\to(0,\pi)$ satisfying
conditions $\mathrm{\mathbf{(c2)}}$ and $\mathrm{\mathbf{(c3)}}$. Suppose
$v_{a},v_{b},v_{c}$ are the vertices of a triangle of $T$. If

$r(D(v_{a}))=r(D(v_{b}))=r(D(v_{c}))=\pi/2,$

then

$r(D(v))<\pi/2,\quad\forall\,v\in V\setminus\\{v_{a},v_{b},v_{c}\\}.$

###### Proof.

Assume on the contrary that there exists $v_{0}\in
V\setminus\\{v_{a},v_{b},v_{c}\\}$ such that

$r(D(v_{0}))\geq\pi/2.$

Then $D(v_{0})\cap D(v_{\mu})\neq\emptyset$ for $\mu=a,b,c$, which yields

$D(v_{0})\cap D(v_{a})\cap D(v_{b})\cap D(v_{c})\neq\emptyset.$

However, similar arguments to the proofs of Theorem 1.3 and Lemma 3.2 imply

$D(v_{0})\cap D(v_{a})\cap D(v_{b})\cap D(v_{c})=\emptyset.$

It is a contradiction. We thus finish the proof. ∎

The following is an analog of the Ring Lemma of Rodin-Sullivan [28]. Remind
the Euclidean version in the case that $0<\Theta\leq\pi/2$ was obtained by He
[17, Lemma 7.1].

######  Lemma 4.6.

Let $\mathcal{P}$ be an irreducible circle pattern on
$(\hat{\mathbb{C}},\mathrm{ds})$ with contact graph isomorphic to the
$1$-skeleton of $T$ and overlap angle function $\Theta:E\to(0,\pi)$ staying in
a compact subset $\Omega\subset W$. Suppose $r(D(v))\leq\pi/2$ for all $v\in
V$. Then there exists a positive constant $C$ depending only on $T$ and
$\Omega$ such that

$r(D(v))\leq Cr(D(u))$

for each pair of adjacent vertices $v,u\in V$.

The lemma relies on subtle analysis with the proof postponed to the appendix.

A circle pattern $\mathcal{P}=\\{D(v)\\}_{v\in V}$ on
$(\hat{\mathbb{C}},\mathrm{ds})$ is said to be $T$-type if there is a geodesic
triangulation of $(\hat{\mathbb{C}},\mathrm{ds})$ isotopic to $T$ with
vertices given by the centers of the disks in $\mathcal{P}$. In what follows
we write the interior of each closed disk $D(v)$ as $\mathbb{D}(v)$.

######  Lemma 4.7.

Let $\mathcal{P}=\\{D(v)\\}_{v\in V}$ be a $T$-type circle pattern on
$(\hat{\mathbb{C}},\mathrm{ds})$ with overlap angle function
$\Theta:E\to(0,\pi)$ satisfying condition $\mathrm{\mathbf{(c2)}}$. Suppose
$v_{\alpha},v_{\beta}\in V$ is a pair of non-adjacent vertices. For any $p\in
D(v_{\alpha})\cap D(v_{\beta})$, then there exists $v_{\eta}\in
V\setminus\\{v_{\alpha},v_{\beta}\\}$ such that $p\in\mathbb{D}(v_{\eta})$.

###### Proof.

As in Figure 3, suppose $u_{1},\cdots,u_{d}$, in anticlockwise order, are all
adjacent vertices of $v_{\alpha}$. For $\mu=1,\cdots,d$, let
$l(v_{\alpha}u_{\mu})$ be the geodesic ray starting from $v_{\alpha}$ along
the direction to $u_{\mu}$ and let
$\operatorname{Sec}(v_{\alpha}u_{\mu}u_{\mu+1})\subset D(v_{\alpha})$ be the
circular sector between $l(v_{\alpha}u_{\mu})$ and $l(v_{\alpha}u_{\mu+1})$
(set $u_{d+1}=u_{1}$). Under condition $\mathrm{\mathbf{(c2)}}$, we are ready
to see

$\operatorname{Sec}(v_{\alpha}u_{\mu}u_{\mu+1})\subset\Delta(v_{\alpha}u_{\mu}u_{\mu+1})\cup\mathbb{D}(u_{\mu})\cup\mathbb{D}(u_{\mu+1}).$

where $\Delta(v_{\alpha}u_{\mu}u_{\mu+1})$ denotes the geodesic triangle with
vertices $v_{\alpha},u_{\mu},u_{\mu+1}$. Hence

$D(v_{\alpha})=\cup_{\mu=1}^{d}\operatorname{Sec}(v_{\alpha}u_{\mu}u_{\mu+1})\subset\big{(}\cup_{\mu=1}^{d}\mathbb{D}(u_{\mu})\big{)}\cup\big{(}\cup_{\mu=1}^{d}\Delta(v_{\alpha}u_{\mu}u_{\mu+1})\big{)}.$

Set
$\operatorname{K}(v_{\alpha})=\cup_{\mu=1}^{d}\Delta(v_{\alpha}u_{\mu}u_{\mu+1})$
and
$\operatorname{\mathbb{K}}(v_{\alpha})=\operatorname{K}(v_{\alpha})\setminus\partial\operatorname{K}(v_{\alpha})$.
Notice that

$\partial\operatorname{K}(v_{\alpha})\subset\cup_{\mu=1}^{d}\mathbb{D}(u_{\mu}).$

Consequently,

$D(v_{\alpha})\subset\big{(}\cup_{\mu=1}^{d}\mathbb{D}(u_{\mu})\big{)}\cup\operatorname{\mathbb{K}}(v_{\alpha}).$
(4.1)

Figure 3: The central disk is covered by the union of linked triangles and
adjacent disks

We now assume, by contradiction, that there exists $p_{0}\in D(v_{\alpha})\cap
D(v_{\beta})$ which does not belong to any open disks of $\mathcal{P}$ except
for $\mathbb{D}(v_{\alpha}),\mathbb{D}(v_{\beta})$. Particularly,

$p_{0}\notin\cup_{\mu=1}^{d}\mathbb{D}(u_{\mu}).$

Taking (4.1) into consideration, one obtains

$p_{0}\in\operatorname{\mathbb{K}}(v_{\alpha}).$

Similarly, we define $\operatorname{\mathbb{K}}(v_{\beta})$ and check that

$p_{0}\in\operatorname{\mathbb{K}}(v_{\beta}).$

Thus
$\operatorname{\mathbb{K}}(v_{\alpha})\cap\operatorname{\mathbb{K}}(v_{\beta})$
is non-empty, which occurs if and only if there exists an edge between
$v_{\alpha}$ and $v_{\beta}$. This contradicts that $v_{\alpha},v_{\beta}$ is
a pair of non-adjacent vertices. ∎

##  5 Normal Family Theorem for circle patterns

We are now ready to give the proof of the Normal Family Theorem based on
estimates in the above section. In fact, our idea is very similar to the
regularization process used in PDE theory.

###### Proof of Theorem 1.5.

Let $v_{a},v_{b},v_{c}$ be the vertices of a triangle of $T$. Using suitable
linear and anti-linear fractional maps, we may assume each pattern
$\mathcal{P}_{n}=\\{D_{n}(v)\\}_{v\in V}$ satisfies

$r(D_{n}(v_{a}))=r(D_{n}(v_{b}))=r(D_{n}(v_{c}))=\pi/2.$ (5.1)

According to Lemma 4.5, we derive

$r(D_{n}(v))\leq\pi/2,\quad\forall\,v\in V.$

Meanwhile, we claim there exists $\delta\in(0,\pi/2]$ such that

$r(D_{n}(v))\geq\delta,\quad\forall\,v\in V.$

Otherwise, by passing to a subsequence if necessary, there exists $v_{0}\in V$
such that

$r(D_{n}(v_{0}))\to 0.$

From the general Ring Lemma (Lemma 4.6), it follows that

$r(D_{n}(v))\to 0,\quad\forall\,v\in V,$

which contradicts (5.1).

Therefore, $\\{\mathcal{P}_{n}\\}$ is parameterized by a sequence in
$\hat{\mathbb{C}}^{|V|}\times[\delta,\pi/2]^{|V|}$. Taking a subsequence if
necessary, we may assume $\\{\mathcal{P}_{n}\\}$ converges to a circle pattern
$\mathcal{P}_{\infty}=\\{D_{\infty}(v)\\}_{v\in V}$ on
$(\hat{\mathbb{C}},\mathrm{ds})$ with the following property:

$\delta\leq r(D_{\infty}(v))\leq\pi/2,\quad\forall\,v\in V.$

Recalling that $\\{\Theta_{n}\\}$ stays in a compact subset of $W$, we check
the overlap angle function $\Theta_{\infty}$ of $\mathcal{P}_{\infty}$
satisfies conditions $\mathrm{\mathbf{(c1)}}-\mathrm{\mathbf{(c4)}}$. By Lemma
4.1, the geodesic triangle of centers for each face of $T$ is well-defined,
which indicates $\mathcal{P}_{\infty}$ is a $T$-type circle pattern.

To derive $\mathcal{P}_{\infty}$ has the same contact graph with
$\mathcal{P}_{n}$, it suffices to show
$D_{\infty}(v_{\alpha}),D_{\infty}(v_{\beta})$ are disjoint for each pair of
non-adjacent vertices $v_{\alpha},v_{\beta}\in V$. Assume on the contrary
$D_{\infty}(v_{\alpha})\cap D_{\infty}(v_{\beta})\neq\emptyset$. It is easy to
see the disks $D_{\infty}(v_{\alpha}),D_{\infty}(v_{\beta})$ are externally
tangent. Otherwise, $D_{\infty}(v_{\alpha})\cap D_{\infty}(v_{\beta})$
contains interior points, which implies $D_{n}(v_{\alpha})\cap
D_{n}(v_{\beta})$ also contains interior points for $n$ sufficiently large.
This contradicts the assumptioin that $D_{n}(v_{\alpha}),D_{n}(v_{\beta})$ are
disjoint. Now there are two cases to distinguish:

Figure 4: If the disks $D_{\infty}(v_{\alpha}),D_{\infty}(v_{\beta})$ touch at
a point, Lemma 4.4 implies the overlap angle function violates condition
$\mathrm{\mathbf{(c1)}}$

  * $(i)$

$D_{\infty}(v_{\alpha})\cap D_{\infty}(v_{\beta})$ consists of a unique point
$p$. Recall that $\mathcal{P}_{\infty}$ is a $T$-type circle pattern with
overlap angle function $\Theta_{\infty}$ satisfying condition
$\mathrm{\mathbf{(c2)}}$. Because of Lemma 4.7, there exists $v_{\eta}\in
V\setminus\\{v_{\alpha},v_{\beta}\\}$ such that

$p\in\mathbb{D}_{\infty}(v_{\eta}),$

which implies both $D_{\infty}(v_{\alpha})\cap D_{\infty}(v_{\eta})$ and
$D_{\infty}(v_{\beta})\cap D_{\infty}(v_{\eta})$ contain interior points. We
claim $v_{\eta}$ is an adjacent vertex of both $v_{\alpha}$ and $v_{\beta}$.
Otherwise, either $D_{\infty}(v_{\alpha}),D_{\infty}(v_{\eta})$ or
$D_{\infty}(v_{\beta}),D_{\infty}(v_{\eta})$ is a pair of externally tangent
disks, which leads to contradictions. Notice that

$D_{\infty}(v_{\alpha})\cap
D_{\infty}(v_{\beta})=\\{p\\}\subset\mathbb{D}_{\infty}(v_{\eta}).$

Lemma 4.4 gives

$\Theta_{\infty}([v_{\alpha},v_{\eta}])+\Theta_{\infty}([v_{\eta},v_{\beta}])>\pi,$

which violates condition $\mathrm{\mathbf{(c1)}}$.

  * $(ii)$

$D_{\infty}(v_{\alpha})\cap D_{\infty}(v_{\beta})=\partial
D_{\infty}(v_{\alpha})=\partial D_{\infty}(v_{\beta})$. Namely,
$\mathbb{D}_{\infty}(v_{\beta})$ is the complement of
$D_{\infty}(v_{\alpha})$. In case that every $v_{\tau}\in
V\setminus\\{v_{\alpha},v_{\beta}\\}$ is an adjacent vertex of both
$v_{\alpha}$ and $v_{\beta}$, then $T$ is the boundary of an $m$-gonal
bipyramid. Apparently,

$\Theta_{\infty}([v_{\alpha},v_{\tau}])+\Theta_{\infty}([v_{\tau},v_{\beta}])=\pi.$

If $m=3$, this violates condition $\mathrm{\mathbf{(c2)}}$. If $m\geq 4$, we
can find edges $e_{1},e_{2},e_{3},e_{4}$ forming a simple closed curve
separating the vertices of $T$ such that

$\sum\nolimits_{\mu=1}^{4}\Theta(e_{\mu})=2\pi,$

which violates condition $\mathrm{\mathbf{(c4)}}$.

It remains to consider the case that there exists $v_{\tau_{0}}\in
V\setminus\\{v_{\alpha},v_{\beta}\\}$ that is non-adjacent to $v_{\alpha}$ or
$v_{\beta}$. Without loss of generality, suppose $v_{\tau_{0}}$ is not
adjacent to $v_{\alpha}$. Then the disks
$D_{\infty}(v_{\tau_{0}}),D_{\infty}(v_{\alpha})$ are externally tangent,
which implies

$\mathbb{D}_{\infty}(v_{\tau_{0}})\subset\hat{\mathbb{C}}\setminus
D_{\infty}(v_{\alpha})=\mathbb{D}_{\infty}(v_{\beta}).$

As a result,

$\mathbb{D}_{\infty}(v_{\tau_{0}})\cap\mathbb{D}_{\infty}(v_{\beta})=\mathbb{D}_{\infty}(v_{\tau_{0}}).$

On the other hand, when $v_{\tau_{0}},v_{\beta}$ is a pair of non-adjacent
vertices, we have

$\mathbb{D}_{\infty}(v_{\tau_{0}})\cap\mathbb{D}_{\infty}(v_{\beta})=\emptyset.$

When $v_{\tau_{0}},v_{\beta}$ is a pair of adjacent vertices, we know
$\Theta_{\infty}([v_{\tau_{0}},v_{\beta}])\in(0,\pi)$, which yields

$\mathbb{D}_{\infty}(v_{\tau_{0}})\cap\mathbb{D}_{\infty}(v_{\beta})\subsetneq\mathbb{D}_{\infty}(v_{\tau_{0}}).$

No matter which case occurs, this leads to contradictions.

We thus prove $\mathcal{P}_{\infty}$ has contact graph isomorphic to the
$1$-skeleton of $T$. To check $\mathcal{P}_{\infty}$ is irreducible, we need
to show

$\cup_{w\in A}D_{\infty}(w)\subsetneq\hat{\mathbb{C}}$

for each strict subset $A$ of $V$. Recall that $r(D_{\infty}(v))\leq\pi/2$ for
every $v\in V$. By Lemma 4.3, we assert $\mathcal{P}_{\infty}$ satisfies
condition $\mathrm{\mathbf{(cr)}}$. Choose a triangle of $T$ with vertices
$v_{i},v_{j}\in V$, $v_{k}\in V\setminus A$. For each $w\in
V\setminus\\{v_{i},v_{j},v_{k}\\}$, it follows from Lemma 3.2 that

$D_{\infty}(v_{i})\cap D_{\infty}(v_{j})\cap D_{\infty}(v_{k})\cap
D_{\infty}(w)=\emptyset.$

In particular, the point $q=\partial D_{\infty}(v_{i})\cap\partial
D_{\infty}(v_{j})\cap D_{\infty}(v_{k})$ does not belong to any disks in
$\mathcal{P}_{\infty}$ except for
$D_{\infty}(v_{i}),D_{\infty}(v_{j}),D_{\infty}(v_{k})$. Hence there is a
neighborhood $N_{q}$ of $q$ such that

$N_{q}\cap D_{\infty}(w)=\emptyset,\quad\forall\,w\in
V\setminus\\{v_{i},v_{j},v_{k}\\}.$

Setting $O_{q}=N_{q}\setminus(D_{\infty}(v_{i})\cup D_{\infty}(v_{j}))$, we
obtain

$\big{(}\cup_{w\in A}D_{\infty}(w)\big{)}\cap O_{q}=\emptyset.$

Consequently,

$\cup_{w\in A}D_{\infty}(w)\subset\hat{\mathbb{C}}\setminus
O_{q}\subsetneq\hat{\mathbb{C}},$

which concludes $\mathcal{P}_{\infty}$ is irreducible. ∎

Combining Theorem 1.5 and Theorem 3.1, we further deduce the following Normal
Family Theorem for hyperbolic polyhedra. Let $\Lambda$ denote the set of
dihedral angle functions $\Theta:\mathcal{E}\to(0,\pi)$ satisfying conditions
$\mathrm{\mathbf{(a1)}}-\mathrm{\mathbf{(a4)}}$.

######  Theorem 5.1.

Let $\\{Q_{n}\\}$ be a sequence of compact convex polyhedra in
$\mathbb{H}^{3}$, each of which is combinatorially equivalent to an abstract
trivalent polyhedron $P$ with more than four faces. Then $\\{Q_{n}\\}$ (modulo
isometries of $\mathbb{H}^{3}$) contains a subsequence convergent to a compact
convex polyhedron in $\mathbb{H}^{3}$ with the same combinatorial type if the
corresponding dihedral angle function sequence $\\{\Theta_{n}\\}$ stays in a
compact subset of $\Lambda$.

######  Remark 5.2.

A more direct approach to this theorem might be following the works of Andreev
[2], Rivin-Hodgson [26] and Roeder-Hubbard-Dunbar [30]. In particular, most
arguments in [2, 30] remain valid in our situation. The only substantial
modification might be showing the property that all faces of the limit
polyhedron are still parallelograms. Toward this end, Andreev [2] and Roeder-
Hubbard-Dunbar [30] made the use of Gauss-Bonnet Formula, while our
observation (Lemma 4.4) indicates that condition $\mathrm{\mathbf{(a1)}}$ is
approximately adequate for this purpose. Finally, we mention that Theorem 5.1
can provide an alternative proof of the general Andreev’s Theorem.

##  6 Appendix

This section is devoted to proofs of Lemma 4.1 and Lemma 4.6.

###### Proof of Lemma 4.1.

Set

$l_{ij}=\arccos(\cos r_{i}\cos r_{j}-\cos\Theta_{ij}\sin r_{i}\sin r_{j})$
(6.1)

and $l_{jk},l_{ki}$ similarly. We reduce the proof to showing the following
inequalities:

$l_{ij}+l_{jk}>l_{ki},\;\;\;l_{jk}+l_{ki}>l_{ij},\;\;\;l_{ki}+l_{ij}>l_{jk},\;\;\;l_{ij}+l_{jk}+l_{ki}<2\pi.$

Equivalently, let us verify

$\displaystyle\sin\dfrac{l_{ij}+l_{jk}+l_{ki}}{2}\sin\dfrac{l_{ij}+l_{jk}-l_{ki}}{2}\sin\dfrac{l_{jk}+l_{ki}-l_{ij}}{2}\sin\dfrac{l_{ki}+l_{ij}-l_{jk}}{2}$
(6.2) $\displaystyle=$ $\displaystyle\dfrac{\sin^{2}l_{ij}\sin^{2}l_{jk}-(\cos
l_{ij}\cos l_{jk}-\cos l_{ki})^{2}}{4}>0.$

To simplify notations, for $\mu=i,j,k$, set $a_{\mu}=\cos r_{\mu}$,
$x_{\mu}=\sin r_{\mu}$. Combining (6.1) and (6.2), we need to demonstrate

$\displaystyle\sin^{2}\Theta_{ij}x^{2}_{i}x^{2}_{j}+\sin^{2}\Theta_{jk}x_{j}^{2}x^{2}_{k}+\sin^{2}\Theta_{ki}x^{2}_{k}x_{i}^{2}-(2+2\cos\Theta_{ij}\cos\Theta_{jk}\cos\Theta_{ki})x_{i}^{2}x_{j}^{2}x_{k}^{2}$
(6.3)
$\displaystyle\;\;+2\lambda_{jk}^{i}a_{j}a_{k}x_{j}x_{k}x_{i}^{2}+2\lambda_{ki}^{j}a_{k}a_{i}x_{k}x_{i}x_{j}^{2}+2\lambda_{ij}^{k}a_{i}a_{j}x_{i}x_{j}x_{k}^{2}>0,$

where

$\lambda_{jk}^{i}=\cos\Theta_{jk}+\cos\Theta_{ki}\cos\Theta_{ij}.$

Since $a^{2}_{\mu}+x^{2}_{\mu}=1$, an equivalent form of (6.3) is

$\begin{split}&\sin^{2}\Theta_{ij}a_{k}^{2}x^{2}_{i}x^{2}_{j}+\sin^{2}\Theta_{jk}a_{i}^{2}x_{j}^{2}x^{2}_{k}+\sin^{2}\Theta_{ki}a_{j}^{2}x^{2}_{k}x_{i}^{2}+\zeta_{ijk}x_{i}^{2}x_{j}^{2}x_{k}^{2}\\\
&\quad+2\lambda^{i}_{jk}a_{j}a_{k}x_{j}x_{k}x_{i}^{2}+2\lambda^{j}_{ki}a_{k}a_{i}x_{k}x_{i}x_{j}^{2}+2\lambda^{k}_{ij}a_{i}a_{j}x_{i}x_{j}x_{k}^{2}>0,\end{split}$
(6.4)

where

$\zeta_{ijk}=1-\cos^{2}\Theta_{ij}-\cos^{2}\Theta_{jk}-\cos^{2}\Theta_{ki}-2\cos\Theta_{ij}\cos\Theta_{jk}\cos\Theta_{ki}.$

Observe that $\Theta_{jk},\Theta_{ki},\Theta_{ij}$ form the inner angles of a
spherical triangle. Let $\phi_{i},\phi_{j},\phi_{k}$ denote the lengths of
sides opposite to $\Theta_{jk},\Theta_{ki},\Theta_{ij}$, respectively. The
second spherical law of cosines gives

$\lambda^{i}_{jk}=\cos\Theta_{jk}+\cos\Theta_{ki}\cos\Theta_{ij}=\cos\phi_{i}\sin\Theta_{ki}\sin\Theta_{ij}.$
(6.5)

Set

$s_{i}=\sin\Theta_{jk}a_{i}x_{j}x_{k},\;\;\;s_{j}=\sin\Theta_{ki}a_{j}x_{k}x_{i},\;\;\;s_{k}=\sin\Theta_{ij}a_{k}x_{i}x_{j}.$

Inserting (6.5) into (6.4), we reduce the proof to checking

$s_{i}^{2}+s_{j}^{2}+s_{k}^{2}+2\cos\phi_{i}s_{j}s_{k}+2\cos\phi_{j}s_{k}s_{i}+2\cos\phi_{k}s_{i}s_{j}+\zeta_{ijk}x_{i}^{2}x_{j}^{2}x_{k}^{2}>0.$

For this purpose, completing the square gives

$\displaystyle
s_{i}^{2}+s_{j}^{2}+s_{k}^{2}+2\cos\phi_{i}s_{j}s_{k}+2\cos\phi_{j}s_{k}s_{i}+2\cos\phi_{k}s_{i}s_{j}$
$\displaystyle=$
$\displaystyle(s_{i}+\cos\phi_{j}s_{k}+\cos\phi_{k}s_{j})^{2}+\sin^{2}\phi_{j}s^{2}_{k}+\sin^{2}\phi_{k}s^{2}_{j}+2(\cos\phi_{i}-\cos\phi_{j}\cos\phi_{k})s_{j}s_{k}$
$\displaystyle\geq$
$\displaystyle\sin^{2}\phi_{j}s^{2}_{k}+\sin^{2}\phi_{k}s^{2}_{j}+2(\cos\phi_{i}-\cos\phi_{j}\cos\phi_{k})s_{j}s_{k}.$

Using the spherical law of cosines, we obtain

$\cos\phi_{i}-\cos\phi_{j}\cos\phi_{k}=\cos\Theta_{jk}\sin\phi_{j}\sin\phi_{k}.$

It follows that

$\displaystyle
s_{i}^{2}+s_{j}^{2}+s_{k}^{2}+2\cos\phi_{i}s_{j}s_{k}+2\cos\phi_{j}s_{k}s_{i}+2\cos\phi_{k}s_{i}s_{j}$
$\displaystyle\geq$
$\displaystyle\sin^{2}\phi_{j}s^{2}_{k}+\sin^{2}\phi_{k}s^{2}_{j}+2\cos\Theta_{jk}\sin\phi_{j}\sin\phi_{k}s_{j}s_{k}$
$\displaystyle=$
$\displaystyle(\sin\phi_{j}s_{k}+\cos\Theta_{jk}\sin\phi_{k}s_{j})^{2}+\sin^{2}\Theta_{jk}\sin^{2}\phi_{k}s^{2}_{j}$
$\displaystyle\geq$
$\displaystyle\sin^{2}\phi_{k}\sin^{2}\Theta_{jk}\sin^{2}\Theta_{ki}a_{j}^{2}x_{k}^{2}x_{i}^{2},$

Meanwhile, a routine computation yields

$\displaystyle\zeta_{ijk}$
$\displaystyle=1-\cos^{2}\Theta_{jk}-\cos^{2}\Theta_{ki}-\cos^{2}\Theta_{ij}-2\cos\Theta_{jk}\cos\Theta_{ki}\cos\Theta_{ij}$
$\displaystyle=\sin^{2}\Theta_{jk}\sin^{2}\Theta_{ki}-(\cos\Theta_{ij}+\cos\Theta_{jk}\cos\Theta_{ki})^{2}$
$\displaystyle=\sin^{2}\Theta_{jk}\sin^{2}\Theta_{ki}-\cos^{2}\phi_{k}\sin^{2}\Theta_{jk}\sin^{2}\Theta_{ki}$
$\displaystyle=\sin^{2}\phi_{k}\sin^{2}\Theta_{jk}\sin^{2}\Theta_{ki}.$

As a result,

$\displaystyle
s_{i}^{2}+s_{j}^{2}+s_{k}^{2}+2\cos\phi_{i}s_{j}s_{k}+2\cos\phi_{j}s_{k}s_{i}+2\cos\phi_{k}s_{i}s_{j}+\zeta_{ijk}x_{i}^{2}x_{j}^{2}x_{k}^{2}$
$\displaystyle\geq$
$\displaystyle\sin^{2}\phi_{k}\sin^{2}\Theta_{jk}\sin^{2}\Theta_{ki}x_{k}^{2}x_{i}^{2}$
$\displaystyle>$ $\displaystyle 0.$

We thus finish the proof. ∎

###### Proof of Lemma 4.6.

We mainly follow the idea of He [17, Lemma 7.1]. Assume the lemma is not true.
For each positive integer $n$, then there exists a circle pattern
$\mathcal{P}_{n}=\\{D_{n}(v)\\}_{v\in V}$ on $(\hat{\mathbb{C}},\mathrm{ds})$
with contact graph isomorphic to the $1$-skeleton of $T$ and overlap angle
function $\Theta_{n}\in\Omega$ such that

$r(D_{n}(v_{\alpha}))>n\times r(D_{n}(u_{\beta}))$

for a pair of adjacent vertices $v_{\alpha},u_{\beta}\in V$. What is more,
$\mathcal{P}_{n}$ is subject to the requirement that $r(D_{n}(v))\leq\pi/2$
for all $v\in V$. By passing to a subsequence if necessary, we may assume for
each vertex $v\in V$, $\lim_{n\to\infty}r(D_{n}(v))$ exists in $[0,\pi/2]$.

Let $V^{+}$ be the set of vertices $v\in V$ for which
$\lim_{n\to\infty}r(D_{n}(v))>0$ and let $V^{0}=V\setminus V+$. Obviously, we
have $u_{\beta}\in V^{0}$. Meanwhile, we claim $V^{+}\neq\emptyset$.
Otherwise, $r(D_{n}(v))\to 0$ for all $v\in V$. It follows that the diameter
of $(\hat{\mathbb{C}},\mathrm{ds})\cong\mathbb{S}^{2}$ tends to zero, which
leads to a contradiction. Since $V^{+},V^{0}$ is a pair of complementary
nonempty subsets of $V$, there exist two adjacent vertices belonging to
$V^{+}$ and $V^{0}$, respectively. Without loss of generality, we assume
$v_{\alpha}\in V^{+}$, $u_{\beta}\in V^{0}$.

Passing to a subsequence again, we may further assume that for each $v\in
V^{+}$, the sequence of disks $D_{n}(v)$ converges to some disk
$D_{\infty}(v)$ in $(\hat{\mathbb{C}},\mathrm{ds})$. In addition, we use
$\Theta_{\infty}([u,w])$ to denote the overlap angle between the disks
$D_{\infty}(u),D_{\infty}(w)$ when $u,w\in V^{+}$ is a pair of adjacent
vertices. Since $\\{\Theta_{n}\\}$ stays in the compact subset $\Omega\subset
W$, we derive $\Theta_{\infty}$ satisfies conditions
$\mathrm{\mathbf{(c1)}}-\mathrm{\mathbf{(c4)}}$ when the related overlap
angles exist.

Notice that there exist at least two vertices in $V^{+}$ adjacent to
$v_{\alpha}$. Let $u_{1},\cdots,u_{d}$, in anticlockwise order, be all
adjacent vertices of $v_{\alpha}$. Without loss of generality, we may assume
$u_{1}\in V^{+}$, $u_{2}=u_{\beta}\in V_{0}$. Set

$j=\min\\{\,l\,|\,l>2,\,u_{l}\in V^{+}\,\\}.$

Then $u_{j}\in V^{+}$ and $u_{i}\in V^{0}$ for $i=2,\cdots,j-1$. One is ready
to see the circles $\partial D_{\infty}(u_{1}),\partial D_{\infty}(u_{j})$
have an intersection point in $\partial D_{\infty}(v_{\alpha})$, which yields

$\partial D_{\infty}(u_{1})\cap\partial D_{\infty}(u_{j})\cap\partial
D_{\infty}(v_{\alpha})\neq\emptyset.$ (6.6)

We claim $u_{1},u_{j}$ is a pair of non-adjacent vertices. Otherwise, either
$v_{\alpha},u_{1},u_{j}$ are the vertices of a triangle of $T$ or the edges
$[v_{\alpha},u_{1}],[u_{1},u_{j}],[u_{j},v_{\alpha}]$ form a simple closed
curve separating the vertices of $T$. In the first case, recall that
$\Theta_{\infty}$ satisfies conditions $\mathrm{\mathbf{(c2)}}$. It follows
from Lemma 4.2 that

$\partial D_{\infty}(u_{1})\cap\partial D_{\infty}(u_{j})\cap\partial
D_{\infty}(v_{\alpha})=\emptyset,$

which leads to a contradiction. In the second case, note that
$r(D_{\infty}(v))\leq\pi/2$ for all $v\in V$. Because $\Theta_{\infty}$
satisfies conditions $\mathrm{\mathbf{(c3)}}$, Lemma 4.3 gives

$D_{\infty}(u_{1})\cap D_{\infty}(u_{j})\cap
D_{\infty}(v_{\alpha})=\emptyset,$

which also contradicts (6.6).

Since $u_{1},u_{j}$ is a pair of non-adjacent vertices, the disks
$D_{\infty}(u_{1}),D_{\infty}(u_{j})$ are externally tangent. Therefore,
either $D_{\infty}(u_{1}),D_{\infty}(u_{j})$ intersect at a unique point
$p\in\partial D_{\infty}(v_{\alpha})$ or $D_{\infty}(u_{1}),D_{\infty}(u_{j})$
share the boundary. In the first case, Lemma 4.4 indicates

$\Theta_{\infty}([v_{\alpha},u_{1}])+\Theta_{\infty}([u_{j},v_{\alpha}])=\pi.$
(6.7)

In the second case, one immediately derives (6.7)

Meanwhile, suppose $w_{1}=v_{\alpha},w_{2}=u_{j-1},\cdots,w_{m}$, in
anticlockwise order, are all adjacent vertices of $u_{j}$. Similarly, we can
obtain $w_{k}\in V^{+}$ which is not adjacent to $w_{1}=v_{\alpha}$ such that
$w_{i}\in V^{0}$ for $i=2,\cdots,k-1$. We claim $w_{k}\neq u_{1}$. Otherwise,
$w_{k},v_{\alpha}$ is a pair of adjacent vertices, which leads to a
contradiction.

It follows that the vertices $u_{1},w_{k}$ are either adjacent or non-
adjacent. In the first case, similar arguments to formula (6.7) give

$\Theta_{\infty}([u_{1},w_{k}])+\Theta_{\infty}([w_{k},u_{j}])=\pi.$ (6.8)

Combing (6.7) and (6.8), we obtain

$\Theta_{\infty}([v_{\alpha},u_{1}])+\Theta_{\infty}([u_{1},w_{k}])+\Theta_{\infty}([w_{k},u_{j}])+\Theta_{\infty}([u_{j},v_{\alpha}])=2\pi,$

which violates condition $\mathrm{\mathbf{(c4)}}$, considering that the edges
$[v_{\alpha},u_{1}],[u_{1},w_{k}],[w_{k},u_{j}],[u_{j},v_{\alpha}]$ form a
simple closed curve separating the vertices of $T$. In the second case, it is
easy to see both the disks $D_{\infty}(u_{j}),D_{\infty}(w_{k})$ are
externally tangent to $D_{\infty}(u_{1})$ at a common point, which implies

$\Theta_{\infty}([u_{j},w_{k}])=\pi.$

This contradicts the assumption that $\\{\Theta_{n}\\}$ stays in the compact
subset $\Omega\subset W$. ∎

Acknowledgements. The author would like to thank Steven Gortler for a great
many helpful conversations and encouragement. He also thanks both the
Institute for Computational and Experimental Research Mathematics (ICERM) at
Brown University and the organizers for hosting the workshop on circle
packings and geometric rigidity that indirectly led to the development of this
paper. Finally, he thanks NSFC (No.11601141 and No.11631010) for financial
supports.

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