# Convergences of Combinatorial Ricci Flows to Degenerated Circle Packings in
Hyperbolic Background Geometry

Guangming Hu, Sicheng Lu, Dong Tan, Youliang Zhong and Puchun Zhou

###### Abstract

This paper investigates a kind of degenerated circle packings in hyperbolic
background geometry. A main problem is whether a prescribed total geodesic
curvature data can be realized by a degenerated circle packing or not. We
fully characterize the sufficient and necessary conditions and show the
uniqueness. Furthermore, we introduce the combinatoral Ricci flow to find the
desired degenerated circle packed surface, analougus to the methods of Chow-
Luo [6] and Takatsu [23].

Mathematics Subject Classification (2020): 52C25, 52C26, 53A70.

##  1 Introduction

###  1.1 Background

In the last few decades, the research of discrete conformal structures on
polyhedral surfaces has been widely concerned, which is a discrete simulation
corresponding to conformal structures on smooth surfaces. The circle packings
on polyhedral surfaces can be regarded as a naturally discrete problem of
conformal structures. In 1936, Koebe [15] showed the projective rigidity of
circle packings on the Riemann sphere. In 1970, Andreev [2] rediscovered this
rigidity theorem. In 1976, Thurston [25] investigated circle packings on both
Euclidean and hyperbolic closed surfaces as a method to explore hyperbolic
3-manifolds.

The variational principle and combinatorial Ricci flow play an impormtant role
in the study of circle packing. Colin de Verdiere [7] utilized the variational
principle to reconfirm the existence and uniqueness of Thurston’s circle
packing theorem. This principle inspires a wealth of remarkable works on
polyhedron surfaces and related topics, among which are [4, 16, 22, 17, 18].
However, the proof of existence part is non-constructive. Chow-Luo introduced
the combinatorial Ricci flow in [6] to control the evolution of geodesic
curvature, giving a new proof of the existence. This is an analogue of
Hamilton’s Ricci flow on surfaces [13]. Their result can be regarded as an
algorithm for finding the desired circle packing that converges exponentially
fast. The sufficient and necessary conditions for convergence are also given.

The settings of variational principle and combinatorial Ricci flow in
Euclidean and hyperbolic geometry can not be applied directly to spherical
geometry, due to the breaking of convexity [19]. Recently, Nie [20] discovered
a new convex functional where geodesic curvature is replaced by the total
geodesic curvature, and obtains the rigidity of circle pattern in spherical
background geometry. The fifth author and his collaborators introduce the
combinatorial Ricci flow in [10] and obtains convergence results similar to
Chow-Luo. The first author and his collaborators extend the study of total
geodesic curvature to hyperbolic geometry [3]. The circles in the packing are
allowed to be usual circles, horocycles and hypercycles. In general, the
sufficient and necessary condition for the existence and rigidity of circle
packing or pattern and the convergence of combinatorial Ricci flow in various
settings are established.

Now it is a natural question to ask: what is the asymptotic behavior of the
combinatorial Ricci flow when the sufficient and necessary conditions are not
valid and take critical values; can we describe it geometrically? This problem
is firstly proposed by Chow-Luo in [6]. Takatsu [23] addresses this problem in
the original setting of Chow-Luo, based on an infinitesimal description of
degenerated circle patterns. In this paper, we try to answer a similar
question under the setting of [3] and give a geometric interpretation of
degenerated generalized circle packed surfaces. These degenerated structures
provide a partial boundary of the space of generalized circle packed surfaces.
And a convergence result is obtained, which allows the combinatorial Ricci
flow converge to a boundary point. Similar results on circle patterns in
spherical background geometry have been obtained by partial authors of this
paper in [14].

To state our main results, we first clarify some basic settings.

###  1.2 Generalized hyperbolic circle packing and realization problem

Let $S$ be a connected closed surface and $G$ be a simple triangulation of
$S$. Denote by $V$, $E$, $F$ the sets of vertices, edges and triangles of $G$,
respectively. For simplicity, we usually identify $V$ as the set
$\\{1,2,\cdots,\lvert V\rvert\\}.$

A generalized circle packed surface is obtained by gluing generalized
hyperbolic triangles along their edges according to the triangulation $G$,
such that the circle packing on adjacent triangles matches (See Definition
2.1, 2.4). The triangulation $G$ is called the nerve of this circle packed
surface. A surface is completely determined by the geodesic curvatures of the
circles centered at the vertices, which can be encoded as a vector
$\vec{k}:=(k_{v})_{v\in V}\in\mathbb{R}^{V}$. Hence the space of all
generalized circle packed surfaces with nerve $G$ is identified with the first
quadrant $\Omega_{k}:=\mathbb{R}_{>0}^{V}$.

Finding a structure with prescribed geometric quantities is a key problem in
the study of polyhedral surfaces. Analogous to the classical setting of
prescribed Gaussian curvatures at vertices, the problem of prescribed total
geodesic curvatures of vertices in hyperbolic background geometry is studied
in [3].

For any non-empty subset $I\subset V$, let $F_{I}\subset F$ be the set of
triangles with at least one vertex in $I$, and define a function $\eta^{I}$ on
$\mathbb{R}_{\geq 0}^{V}$ as

(1.1) $\eta^{I}(\vec{T}):=\pi\big{\lvert}F_{I}\big{\rvert}-\sum_{i\in I}T_{i}$

for any $\vec{T}:=(T_{v})_{v\in V}\in\mathbb{R}_{\geq 0}^{V}$. Denote by
$\Omega_{T}$ the bounded open polyhedron

(1.2) $\Omega_{T}:=\left\\{\ \vec{T}\in\mathbb{R}^{V}_{>0}\ \left|\
\eta^{I}(\vec{T})>0,\ \forall\ \textrm{non-empty subset }\ I\subset V\right.\
\right\\}\ .$

It is shown in [3] that there exists a generalized circle packed surface
realizing the prescribed total geodesic curvature
$\vec{T}\in\mathbb{R}_{>0}^{V}$ if and only if $\vec{T}\in\Omega_{T}$. This
result is obtained by showing that the map

(1.3) $\begin{split}\Phi:\Omega_{k}&\to\mathbb{R}_{>0}^{V}\\\
\vec{k}&\mapsto(T_{v}(\vec{k}))_{v\in V}\end{split}$

recording the total geodesic curvature of circles is actually a homeomorphism
onto $\Omega_{T}$ (Also see Theorem 2.7). Furthermore, the combinatorial Ricci
flow is introduced to find the desired surface with the prescribed total
geodesic curvature.

Since $\Omega_{T}$ admits an apparent boundary in Euclidean space, one may
naturally ask the following problem:

Problem  Can the point on the boundary $\partial\Omega_{T}$ be realized as the
prescribed total geodesic curvature of some geometric surface? How to find it?

In this paper, we will give a partially affirmative answer to the problem.

Denote by $\overline{\Omega}_{k}:=\mathbb{R}_{\geq 0}^{V}$ and

(1.4) $\widehat{\Omega}_{T}:=\left\\{\ \vec{T}\in\mathbb{R}^{V}_{\geq 0}\
\left|\ \eta^{I}(\vec{T})>0,\ \forall\ \textrm{non-empty }\ I\subset
V\right.\right\\}.$

Let

$\partial\Omega_{k}:=\overline{\Omega}_{k}\setminus\Omega_{k}\quad\text{and}\quad\partial_{0}\Omega_{T}:=\widehat{\Omega}_{T}\setminus\Omega_{T}$

where the subscript $0$ emphasizes the boundary $\partial_{0}\Omega_{T}$ is a
partial boundary rather that the whole boundary of $\Omega_{T}$. A point in
$\partial\Omega_{k}$ corresponds to the unique degenerated generalized circle
packed surface (See Proposition 3.9), on which the total geodesic curvature of
packed circles are still well-defined. It is the geometric object for our
realization problem.

######  Theorem 1.1.

For any $\vec{T}\in\partial_{0}\Omega_{T}$, there exists a unique degenerated
generalized circle packed surface realizing the prescribed total geodesic
curvature $\vec{T}$.

Similar to Theorem 2.7, Theorem 1.1 is obtained by showing that the map
recording the total geodesic curvature is a homeomorphism. This map admits a
layered structure. For any non-empty $I\subset V$, the _stratum_
$\partial^{I}\Omega_{k}$ (or $\partial_{0}^{I}\Omega_{T}$ resp.) of the
boundary $\partial\Omega_{k}$ (or $\partial_{0}\Omega_{T}$ resp.) indexed by
$I$ is the subset of elements satisfying

(1.5) $k_{i}=0\ (\text{or }T_{i}=0\text{ resp.})\iff i\in I.$

By denoting

$\partial^{\varnothing}\Omega_{k}=\Omega_{k}\quad\text{and}\quad\partial_{0}^{\varnothing}\Omega_{T}=\Omega_{T},$

we have the following decomposition

(1.6) $\overline{\Omega}_{k}=\bigcup_{I\subset
V}\partial^{I}\Omega_{k}\quad\text{and}\quad\widehat{\Omega}_{T}=\bigcup_{I\subset
V}\partial_{0}^{I}\Omega_{T}.$

Theorem 1.1 is then derived from the following result on continuous extension
of $\Phi$.

######  Theorem 1.2.

The homeomorphism $\Phi:\Omega_{k}\to\Omega_{T}$ extends continuously to a
homeomorphism $\widehat{\Phi}:\overline{\Omega}_{k}\to\widehat{\Omega}_{T}$,
with $\widehat{\Phi}(\partial^{I}\Omega_{k})=\partial_{0}^{I}\Omega_{T}$ for
any $I\subset V$.

Note that by our convention, the result of [3] for $\Phi$ is the case where
$I=\varnothing$.

######  Remark 1.3.

Since the degenerated surfaces allow some hypercycles to become geodesics,
they may be disconnected (see Remark 3.7). These surfaces are also closed to
_crowned surfaces_ , which arise as the complement of a general geodesic
measured lamination on a hyperbolic surface. See [5, 12] for more details
about crowned surfaces.

###  1.3 Combinatorial Ricci flow

To find the actual degenerated generalized circle packed surface with
prescribed total geodesic curvature $\vec{T}\in\partial_{0}\Omega_{T}$, we
take advantage of the combinatorial Ricci flow, which is similar to the work
of Chow-Luo [6]. The flow is even defined for degenerated surfaces on certain
strata.

Given possibly empty $J\subset I\subset V$ and $\vec{T}=(T_{v})_{v\in
V}\in\partial_{0}^{I}\Omega_{T}$, the _combinatorial Ricci flow_ on
$\partial^{J}\Omega_{k}$ is defined by

(1.7)
$\frac{\mathrm{d}k_{v}(t)}{\mathrm{d}t}=-\big{(}T_{v}(\vec{k}(t))-T_{v}\big{)}\cdot
k_{v}(t),\ \quad\text{for any }v\in V\setminus J.$

Here $\vec{k}(t)=(k_{v}(t))_{v\in V\setminus J}$ is the flow ray in
$\partial^{J}\Omega_{k}$ over $t\geq 0$, with initial data $\vec{k}(0)$.

######  Theorem 1.4.

For any $\vec{T}\in\partial_{0}^{I}\Omega_{T}$, $J\subset I$ and
$\vec{k}(0)\in\partial^{J}\Omega_{k}$, the flow ray $\vec{k}(t)$ of (1.7)
converges to $\widehat{\Phi}^{-1}(\vec{T})\in\partial^{I}\Omega_{k}$ as
$t\to+\infty$.

Note that $\partial^{I}\Omega_{k}$ (or $\partial_{0}^{I}\Omega_{T}$ resp.)
lies on the boundary of $\partial^{J}\Omega_{k}$ (or
$\partial_{0}^{J}\Omega_{T}$ resp.) whenever $J\subset I$. See Figure 1 for a
comprehensive illustration of the flow rays in Theorem 1.4.

It has been shown in [3] that for any
$\vec{T}\in\Omega_{T}=\partial^{\varnothing}\Omega_{T}$ , the flow ray of
(1.7) through any initial data $\vec{k}(0)\in\Omega_{k}$ converges
exponentially fast to the point $\Phi^{-1}(\vec{T})$. The following corollary
of Theorem 1.4, by taking $J=\varnothing$, extends the result to the partial
boundary $\partial_{0}\Omega_{T}$ of $\Omega_{T}$, allowing flow rays in the
interior converge at the boundary.

######  Corollary 1.5.

For any $\vec{T}\in\partial_{0}\Omega_{T}$ and $\vec{k}(0)\in\Omega_{k}$, the
flow ray from $\vec{k}(0)$ diverges in $\Omega_{k}$. However, it converges in
$\overline{\Omega}_{k}=\mathbb{R}_{\geq 0}^{V}$ as $t$ tends to $+\infty$ and
the limit point is exactly
$\widehat{\Phi}^{-1}(\vec{T})\in\partial\Omega_{k}$.

Figure 1: A schematic picture of flow rays. The limit point $K$, marked as a
diamond, lies on $\partial^{\\{1,3\\}}\Omega_{k}$, which is a common boundary
of two disjoint strata
$\partial^{\\{1\\}}\Omega_{k},\partial^{\\{3\\}}\Omega_{k}$. The thin lines
lie completely in lower-dimensional strata, while the thick lines are in
$\Omega_{k}$.

Organization This paper is organized as follows. In Section 2, we review the
settings of three-circle configuration and generalized circle packed surfaces.
In Section 3, we introduce degenerated hyperbolic circle packed surfaces. In
Section 4, we show the continuous extension of the homeomorphism
$\Phi:\mathbb{R}_{>0}^{V}\to\Omega_{T}$, which finishes the proof of Theorem
1.2. In Section 5, we prove the convergence of a combinatorial Ricci flow with
prescribed total geodesic curvatures on the partial boundary of
$\partial_{0}\Omega_{T}$. The Appendix A provides some detailed computations
on hyperbolic geometry used here, especially in Section 3 and 4.

##  2 The hyperbolic circle packed surfaces

###  2.1 Three-circle configuration

In this section, we draw upon some fundamental concepts on the hyperbolic
geometry. Also see the Appendix for more quantitative computations.

A _generalized circle_ in $\mathbb{H}^{2}$ is a circle of constant geodesic
curvature with respect to the hyperbolic metric. A generalized circle is
either a circle, a horocycle or a hypercycle. To generalise center from
circles, we define the _center of a horocycle_ as the unique limit point on
the boundary and the _center of a hypercycle_ as the geodesic connecting its
two limit points on the boundary (See Figure 2). Using the hyperboloid and
Klein model for $\mathbb{H}^{2}$, generalized circles have a more quantative
uniform definition (See Lemma A.2).

We will focus on arcs of generalized circles. The inner angle of a hypercycle
arc is defined as follows: Let $\Gamma$ be a hypercycle with geodesic axis
$\gamma$ and radius $r$, and $C$ be an arc on $\Gamma$ with endpoints
$c_{1},c_{2}$. Let the shortest distance projection of $C$ onto the geodesic
$\gamma$ be $\alpha$. Then the hyperbolic length of $\alpha$ is defined to be
the (generalized) _inner angle_ $\theta$ of arc $C$. See the left of Figure 2.
The relations among the radii and absolute geodesic curvature of generalized
circle, and the inner angle and the length of arc of the generalized circle,
are summarized in Table 1.

Figure 2: A circle, a hypercycle and a horocycle on $\mathbb{H}$.

######  Definition 2.1 (circle packed generalized triangle).

Let $C_{1},C_{2},C_{3}$ be three generalized circles externally tangent to
each other, arranged counterclockwise in $\mathbb{H}^{2}$.

  1. 1.

For $i\neq j\in\\{1,2,3\\}$, the _edge_ $e_{ij}=(v_{i}v_{j})$ is the geodesic
connecting the centers of $C_{i}$ and $C_{j}$, passing through the point
tangency. When some of them is hypercycle, the edge must be perpendicular to
its center geodesic.

  2. 2.

For $i\in\\{1,2,3\\}$, the _vertex_ $v_{i}$ is the center of $C_{i}$ if
$C_{i}$ is a circle or a horocycle. If $C_{i}$ is a hypercycle, the vertex
$v_{i}$ is defined as the geodesic segment between the two edges
$(v_{i}v_{j})$ and $(v_{i}v_{k})$ where $j\neq k$ are indexes other than $i$.

  3. 3.

A _generalized (geodesic) triangle_ , denoted by $\Delta$, is the closed
hyperbolic polygon bounded by the three vertices $v_{1},v_{2},v_{3}$ and three
edges $e_{12},e_{23},e_{31}$.

  4. 4.

The three generalized circles, together with the generalized geodesic
triangle, is called a _three-circle configuration_. The pattern inside the
closed polygon is called a _circle packed generalized triangle_.

  5. 5.

A circle packed generalized triangle is called _of curvature
$(k_{1},k_{2},k_{3})$_ if the geodesic curvature of $C_{i}$ is constantly
$k_{i}$ for $i=1,2,3$.

######  Definition 2.2 (Dual triangle).

Given a three-circle configuration with generalized triangle $\Delta$, let
$t_{ij}$ be the tangent point of $C_{i},C_{j}$ on the edge
$e_{ij}=(v_{i}v_{j})$. The _dual triangle_ $\Delta^{*}$ is a non-geodesic
triangle inside $\Delta$, bounded by the three circles, with vertices
$t_{12},t_{23},t_{31}$.

See Figure 3 as an example of all the concepts involved.

| Radius |  | Absolute value of
---
geodesic curvature
Arc length
Circle | $0<r<\infty$ | $k=\coth r$ | $l=\theta\sinh r$
Horocycle | $r=\infty$ | $k=1$ | —–
Hypercycle | $0<r<\infty$ | $k=\tanh r$ | $l=\theta\cosh r$
Table 1:  The relationship of radius, absolute value of geodesic curvature and
arc length.

######  Lemma 2.3.

Given a triple $\vec{k}=(k_{1},k_{2},k_{3})\in\mathbb{R}_{>0}^{3}$, there
exists a unique three-circle configuration (or circle packed triangle) (up to
isometry) such that the geodesic curvature of $C_{i}$ is exactly $k_{i}$ for
$i=1,2,3$.

###### Proof.

Let $C_{1}$ and $C_{2}$ be two tangent circles of curvature $k_{1}$ and
$k_{2}$. Denote by $t_{1,2}$ the tangent point of $C_{1}$ and $C_{2}$. Let $C$
be a circle of curvature $k$ which passes $t_{1,2}$ and is perpendicular to
both $C_{1}$ and $C_{2}$. Take $C_{3}$ as the circle of curvature $k_{3}$
tangent to $C_{1}$ and perpendicular to $C$. Consider the intersection number
$I(k)$ of $C_{2}$ and $C_{3}$ which is an increasing function of $k$. There
exist $k_{2}>k_{1}>1$ such that the intersection number $I(k_{1})=0$ and
$I(k_{2})=2$. Then there exists a unique $k_{0}\in(k_{1},k_{2})$ such that the
intersection number $I(k_{0})$ is equal to $1$. The lemma follows. ∎

Figure 3: A typical three-circle configuration and a circle packed generalized
triangle. The triangle is of curvature $(k_{1},k_{2},k_{3})$ with
$0<k_{1}<1=k_{2}<k_{3}$, hence the vertex $v_{1}$ is a geodesic segment while
$v_{2}$ is an ideal vertex. The shaded region is its dual triangle.

###  2.2 Generalized circle packed hyperbolic surface

Let $S$ be a oriented compact surface and $G$ be a simple triangulation.
Denote by $V$, $E$ and $F$ the set of vertices, edges and faces of $G$
respectively. All $f\in F$ are counterclockwise oriented, with respect to the
orientation of $S$. An edge $e\in E$ and a positively oriented triangle $f\in
F$ are also represented by their vertices as $e=(ij)$ and $f=(ijk)$.

######  Definition 2.4.

Given $(S,G)$ as above, for any $\vec{k}=(k_{v})_{v\in V}\in\Omega_{k}$ we
construct a hyperbolic surface as follows:

  * •

For any $f=(uvw)\in F(G)$, realize $f$ as a circle packed generalized triangle
$\Delta_{f}$ of curvature $(k_{u},k_{v},k_{w})\in\Omega_{k}^{\Delta}$.

  * •

If $f_{1},f_{2}\in F$ share the same edge $e=(ij)\in E$, glue the two
generalized triangles $\Delta_{f_{1}},\Delta_{f_{2}}$ along the edges
corresponding to $e$ by isometry, such that the vertices of two dual triangles
on these edges coincide and the tangent points of two circles on these edges
coincide.

  * •

The surface obtained by gluing all edges of $\Delta_{f}$’s as above is called
a _generalized circle packed surface_ with nerve $G$ and curvature $\vec{k}$.

  * •

On a generalized circle packed surface, for any $v\in V$, all circle arcs
centered at vertex $v$ form a closed circle $C_{v}$. The collection
$\\{C_{v}\\}_{v\in V}$ is called a _generalized circle packing_ on that
surface.

By the construction, all generalized circle packed surfaces with nerve $G$ are
parameterized by $\Omega_{k}$. Hence when the surface and triangulation
$(S,G)$ is given, we will simply represent a surfaces by a point
$\vec{k}\in\Omega_{k}$.

######  Remark 2.5.

We omit all the adjective “generalized” from now on for simplicity.

######  Remark 2.6.

For any $\vec{k}=(k_{v})_{v\in V}\in\Omega_{k}$, there is a partition of $V$
given by

$V_{B}:=\\{\ i\in V\ \lvert\ 0<k_{i}<1\ \\},\quad V_{P}:=\\{\ i\in V\ \lvert\
k_{i}=1\ \\},\quad V_{C}:=\\{\ i\in V\ \lvert\ k_{i}>1\ \\}.$

Then topologically, the circle packed surface with curvature $\vec{k}$ is
obtained by removing $\lvert V_{P}\rvert$ points and $\lvert V_{B}\rvert$ open
disks, all disjoint. Geometrically, the surface contains $\lvert V_{B}\rvert$
closed geodesic boundaries, $\lvert V_{P}\rvert$ cusps and $\lvert
V_{C}\rvert$ conical points (possibly of cone angle $2\pi$), and is smooth on
the complementary. See an example in Figure 4.

Figure 4: A typical generalized circle packed surface, where $G$ is the
tetrahedron triangulation of the sphere. The surface has one cusp
corresponding to $v_{1}$, a totally geodesic boundary corresponding to
$v_{4}$, and two conical points.

For $\vec{k}=(k_{v})_{v\in V}\in\Omega_{k}$ and $f=({u}{v}{w})\in F$, let
$l_{v}^{f}(\vec{k})$ be the hyperbolic length of the circle arc $C_{v}^{f}$
centered at vertex $i$ in the triangle $\Delta_{f}$ of curvature
$(k_{u},k_{v},k_{w})$. The _total geodesic curvatures_ of arc $C_{v}^{f}$ is
defined as

(2.1) $T_{v}^{f}(\vec{k}):=l_{v}^{f}(\vec{k})\cdot k_{v}.$

Recall that $F_{\\{v\\}}$ is the set of triangles containing $v\in V$ as one
of its vertex. Then the _total geodesic curvatures_ of $C_{v}$ on the circle
packed surface with curvature $\vec{k}$ is defined as

(2.2) $T_{v}(\vec{k}):=\sum_{f\in F_{\\{v\\}}}T_{v}^{f}(\vec{k}).$

The following result tells us that all circle packed surfaces with given nerve
$G$ can be reparametrized by total geodesic curvatures.

######  Theorem 2.7 ([3]).

The image of the map $\Phi$ defined by (2.2) and (1.3) is precisely
$\Omega_{T}$ in (1.2). And $\Phi:\Omega_{k}\to\Omega_{T}$ is a homeomorphism.

##  3 The degenerated hyperbolic circle packed surfaces

To precisely define the points in $\partial\Omega_{k}$, we introduce
generalized circles with zero geodesic curvature. We begin with three-circle
configurations, allowing some of the generalized circles to be geodesics, and
then glue these configurations together to form hyperbolic surfaces analogous
to hyperbolic circle-packed surfaces.

###  3.1 Degenerated circle packed triangle

Degenerated three-circle configurations are defined similarly to three-circle
configurations in Definition 2.1.

######  Definition 3.1.

A _degenerated three-circle configuration_ is a three-circle configuration
where some generalized circles degenerate into total geodesics of
$\mathbb{H}^{2}$. The pattern obtained in (4) of Definition 2.1 is called a
_degenerated circle-packed triangle_ , with edges possibly degenerating to
ideal points. The _dual triangle_ is defined similarly, possibly having ideal
vertices.

######  Lemma 3.2.

Given $\vec{k}=(k_{1},k_{2},k_{3})\in\partial\mathbb{R}_{\geq 0}^{3}$, there
exists a unique (up to isometry) degenerated three-circle configuration (or
degenerated circle-packed triangle) such that the geodesic curvature of
$C_{i}$ is exactly $k_{i}$ for $i=1,2,3$.

###### Proof.

It is sufficient to assume that $k_{1}\leq k_{2}\leq k_{3}$. The case that
$0=k_{1}<k_{2}\leq k_{3}$ follows from the same argument of Lemma 2.3.

Consider the case that $0=k_{1}=k_{2}<k_{3}$. Let $C_{1}$ be the geodesic
joining $-1$ and $1$ in $\mathbb{H}^{2}$, and let $C_{2}$ be the geodesic
joining $-1$ and $-i$ in $\mathbb{H}^{2}$. For $s\in(-1,1)$, let $C_{3}(s)$ be
the circle of curvature $k_{3}$ tangent to $C_{1}$ at $s$. The intersection
number $I(s)$ between $C_{2}$ and $C_{3}(s)$ is a decreasing function of
$s\in(-1,1)$. There exist $-1<s_{1}<s_{2}<1$ such that $I(s_{1})=2$ and
$I(s_{2})=0$. Then there exists $s_{0}\in(s_{1},s_{2})$ such that
$I(s_{0})=1$. The lemma holds for $k_{1}=k_{2}=0$.

If $k_{1}=k_{2}=k_{3}=0$, the degenerated circle-packed triangle is an ideal
hyperbolic triangle.

∎

Lemma 3.2 is illustrated in Figure 5.

Figure 5: Consider degenerate three-circle configurations and circle-packed
triangles, where one, two, or three circles degenerate into geodesics. Note
that the case where all $k_{i}=0$ is distinct from the case where all
$k_{i}=1$. In the former, the dual triangle coincides with the generalized
triangle itself.

###  3.2 Pairing of half-edges

To define a degenerate circle-packed surface, we introduce topological
information about the triangulation $G$ to describe the gluing process along
degenerate edges. These edges may be ideal points at the boundary of infinity.

######  Definition 3.3 (Half-edge).

Let $f=(u,v,w)\in F$ be a positively oriented triangle of $G$. The oriented
edge $e^{f}=(u,v)$ is called a _half-edge_ of edge $e=(uv)\in E$ of $f$.

If $g=(v,u,x)\in F$ is the other positively oriented triangle sharing the edge
$e=(uv)$, then $e^{g}=(v,u)$ is another half-edge of $e$ of $g$. Each edge
$e\in E$ has two half-edges in two adjacent triangles, with reversed
orientations. Denote by $\widetilde{E}^{+}$ the set of all half-edges of $G$.
There is a natural 2-to-1 map

(3.1) $\pi_{E}:\widetilde{E}^{+}\to E$

defined by forgetting its orientation.

######  Definition 3.4 (Pairing of half-edges).

Let $\sigma_{G}:\widetilde{E}^{+}\to\widetilde{E}^{+}$ be the natural
involution reversing orientations of half-edges. That is, for any
$(u,v)\in\tilde{E}^{+}$,

$\sigma_{G}(u,v):=(v,u).$

If $\sigma_{G}(e^{f})$ is a half-edge inside $g\in F$, then $e$ is a common
edge of $f,g$.

###  3.3 Degenerated circle packed surface

In this subsection, we generalize circle-packed surfaces (Definition 2.4) to
degenerate circle-packed surfaces (Definition 3.6). Set $\vec{k}=(k_{v})_{v\in
V}\in\overline{\Omega}_{k}=\mathbb{R}_{\geq 0}^{V}$. For any $f=(u,v,w)\in F$,
let $\Delta_{f}$ be a (degenerate) circle-packed triangle of curvatures
$(k_{u},k_{v},k_{w})\in\overline{\Omega}_{k}^{\Delta}:=\mathbb{R}_{\geq
0}^{3}$.

######  Definition 3.5.

For $e=(uv)\in E$ with $k_{u}=k_{v}=0$, let $e^{f}=(u,v)$ be a half-edge of
edge $e$ of $f$. A neighborhood around the ideal point from the degenerate
edge $(uv)$ of $\Delta_{f}$ is called the _spike corresponding to the half-
edge_ $e^{f}$.

For $e=(uv)\in E$ and $f,f^{\prime}\in F(e)$, we glue $\Delta_{f}$ and
$\Delta_{f^{\prime}}$ as follows:

  * •

If $k_{u}=k_{v}=0$, the corresponding half-edges
$\pi_{E}^{-1}(e)=\\{e^{f},e^{g}\\}$ of $f,g$ degenerate to ideal points, and
the pairing of half-edges $\sigma_{G}$ forms a pairing of these spikes.

  * •

Otherwise, if at least one of $k_{u}$ or $k_{v}$ is not zero, we glue
$\Delta_{f}$ and $\Delta_{f^{\prime}}$ as in Definition 2.4.

Let

$E^{0}(\vec{k}):=\\{\ e=(uv)\ \lvert\ k_{u}=k_{v}=0\ \\}.\ $

The lift $\pi_{E}^{-1}(E^{0}(\vec{k}))$ of $E^{0}(\vec{k})$ under projection
$\pi_{E}^{-1}$ (3.1) is a subset of $\tilde{E}^{+}$.

######  Definition 3.6.

For $\vec{k}\in\partial\Omega_{k}$, the _degenerated circle packed surface_
corresponding to $\vec{k}$ is defined as a (possibly disconnected) hyperbolic
surface obtained by gluing circle packed triangles as before, together with a
pairing $\sigma_{G}$ of $\pi_{E}^{-1}(E^{0}(\vec{k}))$.

######  Remark 3.7.

See Figure 6 for a typical example of a degenerated surface which is
disconnected. Since each spike is paired with another by
$\sigma_{G}(\vec{k})$, this hyperbolic surface can be considered "virtually
connected": entering one spike leads to another paired spike. The "tangent
point at infinity" resembles a "wormhole" connecting the two triangles.

A new type of boundary component arises if circle packed surface is
degenerated.

######  Definition 3.8.

A _crowned boundary component_ of a hyperbolic surface $X$ is an "unordered"
sequence $(g_{1},\dots,g_{n})$, where $g_{i}$ is a complete geodesic boundary
of $X$, satisfying:

  * •

$g_{n}$ is tangent to $g_{1}$;

  * •

$g_{i}$ is tangent to $g_{i+1}$ for $i=1,\dots,n-1$.

The term "unordered" means that the indices can be cycled, i.e., any cyclic
permutation of $\\{1,\dots,n\\}$ represents the same sequence.

######  Proposition 3.9.

For each $\vec{k}\in\partial\Omega_{k}$, it can be uniquely (up to isometry)
realized as a degenerated circle packed surface as in Definition 3.6. A
boundary component of $S\setminus E^{0}(\vec{k})$ corresponds to a crowned
boundary component of $\vec{k}$.

###### Proof.

For any $f=(u,v,w)\in F$, by Lemmas 2.3 and 3.2, there exists a unique
(degenerate) circle-packed triangle (up to isometry) of curvatures
$(k_{u},k_{v},k_{w})\in\overline{\Omega}_{k}^{\Delta}:=\mathbb{R}_{\geq
0}^{3}$. Two spikes are paired if and only if they correspond to the same
edge. Combining the uniqueness of isometries for gluing (possibly degenerated)
circle packed triangles (Definitions 2.4 and 3.6), there is a unique
degenerated circle packed surface realizing $\vec{k}$.

Consider a boundary component $\beta$ of $S\setminus E^{0}(\vec{k})$. Choose
an orientation on $\beta$ such that $S\setminus E^{0}(\vec{k})$ is on the left
of $\beta$. Write $\beta$ as a sequence $(\tilde{e}_{1},\cdots,\tilde{e}_{n})$
of half-edges satisfying:

  * •

The orientation of $\tilde{e}_{i}$ matches the chosen orientation of $\beta$
for $i=1,\cdots,n$.

  * •

The tail of $\tilde{e}_{n}$ is the head of $\tilde{e}_{1}$.

  * •

The tail of $\tilde{e}_{i}$ is the head of $\tilde{e}_{i+1}$ for
$i=1,\cdots,n-1.$

By Definition 3.5, a half-edge $e_{i}$ corresponds to a spike if and only if
its projection $\pi_{E}(e_{i})$ is contained in $E^{0}(\vec{k})$. For
$i=1,\cdots,n-1$, let $g_{i}$ (resp. $g_{n}$) be the common complete geodesic
component of the spikes corresponding to $e_{i}$ and $e_{i+1}$ (resp. $e_{n}$
and $e_{1}$). By Definition 3.8, the sequence $(g_{1},\cdots,g_{n})$
represents a crowned boundary of the hyperbolic surface. ∎

Figure 6: A typical local picture of a degenerated circle-packed surface is
shown. Here, $k_{i}=0$ for $i=1,2,3,4,5$ and $k_{i}>0$ for $i=0,6,7,8$. All
bold circles represent hyperbolic geodesic boundaries. The shaded region,
which acts as a connected component of the degenerated surface, is a complete
hyperbolic surface with one crowned boundary and one closed geodesic boundary.

######  Remark 3.10.

A hyperbolic surface with a crowned boundary is called a crowned hyperbolic
surface. One motivation for studying crowned hyperbolic surfaces is that the
metric completion of the complement of a geodesic lamination on a hyperbolic
surface can result in a crowned hyperbolic surface. Gupta applied crowned
hyperbolic surfaces to construct limiting harmonic maps [12]. Tamburelli
provided a correspondence between wild globally hyperbolic maximal anti-de
Sitter structures and pairs of crowned hyperbolic surfaces [24].

##  4 Extended homeomorphism

In this section, we prove Theorem 1.2 which extends
$\Phi:\Omega_{k}\to\Omega_{T}$ as a homeomorphism
$\widehat{\Phi}:\overline{\Omega}_{k}\to\widehat{\Omega}_{T}$. We first obtain
the homeomorphic extensions of the maps $\Phi^{\Delta}$ for generalized
triangles (Proposition 4.10). Next, we show that the extended map
$\widehat{\Phi}$ is a homeomorphism on every stratum of the boundary
(Proposition 4.11). Hence $\widehat{\Phi}$ is a continuous bijection. Finally
we show that its inverse map is also continuous.

###  4.1 Extended homeomorphisms for generalized triangles

It was shown in [3] that the map

(4.1) $\begin{split}\Phi^{\Delta}\ :\ \Omega_{k}^{\Delta}\ &\to\
\Omega_{T}^{\Delta}\subset\mathbb{R}_{>0}^{3}\\\
(k_{1},k_{2},k_{3})&\mapsto(T_{1},T_{2},T_{3})\end{split}$

recording the total geodesic curvature of arcs in a triangle is a
homeomorphism, where

$\Omega_{T}^{\Delta}:=\\{\ (T_{1},T_{2},T_{3})\in\mathbb{R}_{>0}^{3}\ \lvert\
T_{1}+T_{2}+T_{3}<\pi\ \\}\ .$

In this subsection, we show that it extends continuously to a homeomorphism
from $\overline{\Omega}_{k}^{\Delta}:=\mathbb{R}_{\geq 0}^{3}$ to
$\widehat{\Omega}_{T}^{\Delta}$ (Proposition 4.10), where
$\widehat{\Omega}_{T}^{\Delta}:=\\{\ (T_{1},T_{2},T_{3})\in\mathbb{R}_{\geq
0}^{3}\ \lvert\ T_{1}+T_{2}+T_{3}<\pi\ \\}$ is a partial closure of
$\Omega_{T}^{\Delta}$, similar to (1.4).

The first ingredient is the following precise formula for the map
$\Phi^{\Delta}$.

######  Theorem 4.1.

For a circle packed triangle with curvature $(k_{1},k_{2},k_{3})$, let $T_{i}$
be the total geodesic curvature of the circle arc centered at vertex $v_{i},\
i=1,2,3$. Then for any cyclic re-ordering $(i,j,k)$ of $(1,2,3)$, when none of
$k_{1},k_{2},k_{3}$ is $1$, we have

$T_{i}=T_{i}(k_{1},k_{2},k_{3})=\frac{\theta_{i}k_{i}}{\sqrt{\lvert
1-k_{i}^{2}\rvert}}\ ,$

where

(4.2)
$\frac{2+(k_{i}k_{j}+k_{j}k_{k}+k_{k}k_{i})-k_{i}^{2}}{(k_{i}+k_{j})(k_{i}+k_{k})}=\left\\{\begin{array}[]{ll}\cos\theta_{i}&,\
k_{i}>1\\\ \cosh\theta_{i}&,\ k_{i}<1\end{array}\right.\ .$

Furthermore, the extension of the map given by

$(k_{1},k_{2},k_{3})\mapsto\big{(}T_{1}(k_{1},k_{2},k_{3}),T_{2}(k_{1},k_{2},k_{3}),T_{3}(k_{1},k_{2},k_{3})\big{)}$

on $\Omega_{k}^{\Delta}=\mathbb{R}^{3}_{>0}$ is a $C^{1}$-map which is
identical to the map $\Phi^{\Delta}$ of (4.1).

In particular, for $k_{i}=1$, we have

$T_{i}=\frac{2}{\sqrt{(k_{j}+1)(k_{k}+1)}}\ .$

######  Corollary 4.2.

Let $C_{P}$ be the circle passing through the points of tangency a circle
packed triangle with curvature $(k_{1},k_{2},k_{3})$. Then the geodesic
curvature $k_{P}$ of $C_{P}$ is given by

(4.3) $k_{P}^{2}=k_{1}k_{2}+k_{2}k_{3}+k_{3}k_{1}+1.$

Note that the $C_{P}$ in Corollary 4.2 is orthogonal to three circles. In [3],
$C_{P}$ plays an important role in the relations for geometric quantities of a
triangle.

######  Remark 4.3.

We will proved Theorem 4.1 and Corollary 4.2 in Appendix A, by applying the
Klein model of hyperbolic plane.

With the help of the explicit formula in Theorem 4.1, we obtain the asymptotic
behavior of total geodesic curvatures for triangles as follows. For any subset
$I\subset\\{1,2,3\\}$, the strata $\partial^{I}\Omega_{k}^{\Delta}$ and
$\partial_{0}^{I}\Omega_{T}^{\Delta}$ are defined similarly as (1.5).

######  Proposition 4.4.

The homeomorphism $\Phi^{\Delta}:\Omega_{k}^{\Delta}\to\Omega_{T}^{\Delta}$
extends to a continuous map
$\widehat{\Phi}^{\Delta}:\overline{\Omega}_{k}^{\Delta}\rightarrow\widehat{\Omega}_{T}^{\Delta}$,
with the image of $\partial^{I}\Omega_{k}^{\Delta}$ lying in
$\partial_{0}^{I}\Omega_{T}^{\Delta}$ for any $I\subset\\{1,2,3\\}$.

###### Proof.

For any $(k_{1}^{*},k_{2}^{*},k_{3}^{*})\in\partial^{I}\Omega_{k}^{\Delta}$
and any sequence
$\\{(k^{n}_{1},k^{n}_{2},k^{n}_{3})\\}_{n=1}^{\infty}\subset\Omega_{k}^{\Delta}$
converging to $(k_{1}^{*},k_{2}^{*},k_{3}^{*})$, we claim that there exists a
unique $T_{i}^{*}\in[0,\pi)$ with
$T_{i}(k_{1}^{n},k_{2}^{n},k_{3}^{n})\rightarrow T_{i}^{*}$ for $i=1,2,3$. The
equation (4.2) implies that there exists $T_{i}^{*}\in(0,\pi)$ such that
$T_{i}(k_{1}^{n},k_{2}^{n},k_{3}^{n})\rightarrow T_{i}^{*}$ for any $i\notin
I$. With loss of generality, we assume that $0<k_{i}^{n}<1$ since
$k_{i}^{n}\to 0^{+}$ for $i\in I$.

Then

$\cosh\theta_{i}^{n}=1+\frac{2(1-(k_{i}^{n})^{2})}{(k_{i}^{n}+k_{j}^{n})(k_{i}^{n}+k_{k}^{n})}\leq
1+\frac{2(1-(k_{i}^{n})^{2})}{(k_{i}^{n})^{2}}\leq\frac{2}{(k_{i}^{n})^{2}}\
.$

Combining the fact that $\theta_{i}^{n}\leq 2\ln\frac{2}{k_{i}^{n}}$, we have

$\theta_{i}^{n}k_{i}^{n}\leq 2k_{i}^{n}\ln\frac{2}{k_{i}^{n}}\to
0\quad\textrm{as\ }n\to\infty,$

and

$\lim_{n\rightarrow\infty}T_{i}(k_{1}^{n},k_{2}^{n},k_{3}^{n})=0.$

We obtain the claim by taking $T_{i}^{*}$ as $0$ for $i\in I$.

By applying the Gauss-Bonnet formula on the dual triangle $\Delta^{*}$ of the
limiting degenerated circle packed triangle, we have

(4.4)
$T_{1}(k_{1}^{*},k_{2}^{*},k_{3}^{*})+T_{2}(k_{1}^{*},k_{2}^{*},k_{3}^{*})+T_{3}(k_{1}^{*},k_{2}^{*},k_{3}^{*})=\pi-\mathrm{Area}(\Delta^{*}).$

Since the dual triangle $\Delta^{*}$ never vanishes, $\sum_{i\notin
I}T_{i}(k_{1}^{*},k_{2}^{*},k_{3}^{*})<\pi$. Therefore, the image of
$\partial^{I}\Omega_{k}^{\Delta}$ is contained in
$\partial_{0}^{I}\Omega_{T}^{\Delta}$. ∎

Next, we show that the map
$\widehat{\Phi}^{\Delta}:\overline{\Omega}_{k}^{\Delta}\rightarrow\widehat{\Omega}_{T}^{\Delta}$
is a homeomorphism by variational principle. Similar to the method used in
[3], we need the following result.

######  Lemma 4.5 ([8], Lemma 5.4).

Let $W:\mathbb{R}^{n}\to\mathbb{R}$ be a smooth strictly convex function with
positively definite Hessian matrix, then $\triangledown
W:\Omega\to\mathbb{R}^{n}$ is a smooth embedding.

The desired convex function is obtained by the integral of a closed 1-form.
For a proper (possibly empty) subset $I$ of $\\{1,2,3\\}$, let

(4.5) $\begin{split}\ln:\quad\partial^{I}\Omega_{k}^{\Delta}\
&\longrightarrow\quad\mathbb{R}^{3-\lvert I\rvert}\\\
(k_{1},k_{2},k_{3})&\longmapsto\vec{S}=(S_{j})_{j\notin I}\end{split}$

be the diffeomorphism defined on the boundary stratum, where $S_{j}:=\ln
k_{j}$ for $j\notin I$. Define a 1-form on $\mathbb{R}^{3-\lvert I\rvert}$ as

(4.6) $\omega_{\Delta}^{I}:=\sum_{j\notin
I}T_{j}(e^{S_{1}},e^{S_{2}},e^{S_{3}})\mathrm{d}S_{j},$

where $e^{S_{i}}$ is appointed to be zero if $i\in I$.

######  Lemma 4.6.

For any proper $I\subset\\{1,2,3\\}$, the 1-form $\omega_{\Delta}^{I}$ is
closed on $\mathbb{R}^{3-\lvert I\rvert}$.

###### Proof.

For $I=\varnothing$, it has been shown in [3, Lemma 2.8].

If $I$ is consisted of two elements, then $\mathbb{R}^{3-|I|}$ is $\mathbb{R}$
and $\omega_{\Delta}^{I}$ is naturally closed on $\mathbb{R}$.

It remains to consider the case that $I$ has only one element. Without loss of
generality, set $I=\\{3\\}$. Let $l_{i}:=T_{i}/k_{i}$ be the length of the
arcs for $i=1,2,3$. We have

$\frac{\partial T_{1}}{\partial S_{2}}=\frac{\partial T_{1}}{\partial
k_{2}}\frac{\mathrm{d}k_{2}}{\mathrm{d}S_{2}}=k_{2}\frac{\partial
T_{1}}{\partial k_{2}}=k_{1}k_{2}\frac{\partial l_{1}}{\partial
k_{2}}\quad\textrm{and }\quad\frac{\partial T_{2}}{\partial
S_{1}}=k_{1}k_{2}\frac{\partial l_{2}}{\partial k_{1}}\ .$

It sufficies to prove that $\frac{\partial l_{1}}{\partial
k_{2}}=\frac{\partial l_{2}}{\partial k_{1}}$. This can be done by a direct
computation and we left it at the last of Appendix A.2. ∎

Now for any proper $I\subset\\{1,2,3\\}$, pick a base point
$\vec{S}_{I}\in\mathbb{R}^{3-|I|}.$ By Lemma 4.6 and Theorem 4.1, the integral
of the 1-form

(4.7)
$\mathcal{E}^{I}(\vec{S}):=\int_{\vec{S}_{I}}^{\vec{S}}\omega_{\Delta}^{I}$

is a well-defined $C^{2}$-function on $\mathbb{R}^{3-\lvert I\rvert}$. We call
it the _energy function_ of index $I$.

By generalizing [3, Lemma 2.11] from empty set $I$ to proper set $I$, we state
the following lemma. We omit the proof, since it is similar to [3, Lemma 2.11]
.

######  Lemma 4.7.

For any proper $I\subset\\{1,2,3\\}$, $\mathcal{E}^{I}$ is strictly convex on
$\mathbb{R}^{3-\lvert I\rvert}$.

Now all the conditions of Lemma 4.5 are satisfied. To give the exact image of
the extended map $\widehat{\Phi}^{\Delta}$, we need the following asymptotic
behaviour.

######  Lemma 4.8.

Let $I\subset\\{1,2,3\\}$ be non-empty. If $k_{i}\to+\infty$ for any $i\in I$
and $k_{i}$ is bounded from above for any $i\notin I$, then $\sum_{i\in
I}T_{i}(k_{1},k_{2},k_{3})\to\pi$.

###### Proof.

If $I=\\{i\\}$, by (4.2),

$\cos\theta_{i}=\frac{2/k_{i}^{2}+(k_{j}+k_{k})/k_{i}+k_{j}k_{k}/k_{i}^{2}-1}{(1+k_{j}/k_{i})(1+k_{k}/k_{i})}\to-1\quad\textrm{as}\
k_{i}\to+\infty.$

Hence $\theta_{i}\to\pi$ and then $T_{i}\to\pi$.

If $\lvert I\rvert\geq 2$, there are at least two of $k_{i}$’s tend to
$+\infty$. By Corollary 4.2, we have

$k_{P}^{2}=k_{1}k_{2}+k_{2}k_{3}+k_{3}k_{1}+1\to+\infty\ .$

Hence the circle $C_{P}$ collapses to a point, and so does the dual triangle
$\Delta^{*}$ which lies inside the dual triangle. By Gauss-Bonnet formula,
$T_{1}+T_{2}+T_{3}=\pi-\textrm{Area}(\Delta^{*})\to\pi$. If $\lvert
I\rvert=2$, we may assume $I=\\{1,2\\}$ and by (4.2) again we have

$\cos\theta_{3}\ \textrm{or}\
\cosh\theta_{3}=\frac{2/k_{1}k_{2}+1+k_{3}/k_{1}+k_{3}/k_{2}-k_{3}^{2}/k_{1}k_{2}}{(1+k_{3}/k_{1})(1+k_{3}/k_{2})}\to
1\quad\textrm{as}\ k_{1},k_{2}\to+\infty.$

Hence $T_{3}\to 0$ in this case, and $T_{1}+T_{2}\to\pi$. ∎

Note that the limit of individual $T_{i}$ may not always exist.

######  Proposition 4.9.

For any $I\subset\\{1,2,3\\}$, $\widehat{\Phi}^{\Delta}$ maps
$\partial^{I}\Omega_{k}^{\Delta}$ homeomorphically onto
$\partial_{0}^{I}\Omega_{T}^{\Delta}$.

###### Proof.

Obviously, $\widehat{\Phi}^{\Delta}$ maps $(0,0,0)$ to $(0,0,0)$. Let
$\vec{e}_{1}=(\pi,0,0),\vec{e}_{2}=(0,\pi,0),\vec{e}_{3}=(0,0,\pi)$ be the
other three vertices of $\Omega_{T}^{\Delta}$.

For any proper $I\subset\\{1,2,3\\}$, by the definition of energy function
$\mathcal{E}^{I}$ in (4.6) and (4.7), we have

$\frac{\partial\mathcal{E}^{I}}{\partial
S_{j}}=T_{j}(e^{S_{1}},e^{S_{2}},e^{S_{3}})\ ,\quad\textrm{for all }j\notin I\
.$

By Lemma 4.5 and Lemma 4.7, $\nabla\mathcal{E}^{I}$ is a smooth embedding of
$\mathbb{R}^{3-\lvert I\rvert}$. Composing with the diffeomorphism $\ln$,
$\widehat{\Phi}^{\Delta}$ is a homeomorphism of
$\partial^{I}\Omega_{k}^{\Delta}$ onto its image.

Now for any $j\notin I$, let
$\\{(k_{1}^{n},k_{2}^{n},k_{3}^{n})\\}_{n=1}^{\infty}$ be a sequence in
$\partial^{I}\Omega_{k}^{\Delta}$ satisfying the first case of Lemma 4.8.
Together with Guass-Bonnet formula, we have

$\lim_{n\to\infty}\widehat{\Phi}^{\Delta}(k_{1}^{n},k_{2}^{n},k_{3}^{n})=\vec{e_{j}}\
.$

It is a basic fact that the gradient of a strictly convex $C^{1}$-function on
$\mathbb{R}^{n}$ is a homeomorphism from $\mathbb{R}^{n}$ to a convex domain
in $\mathbb{R}^{n}$. Thus,
$\widehat{\Phi}^{\Delta}\left(\partial^{I}\Omega_{k}^{\Delta}\right)$ must
contain the open convex hull of $(0,0,0)$ and $\vec{e}_{j}$ for $j\notin I$.
But this is exactly $\partial_{0}^{I}\Omega_{T}^{\Delta}$. Hence, with
Proposition 4.4,
$\widehat{\Phi}^{\Delta}\left(\partial^{I}\Omega_{k}^{\Delta}\right)=\partial_{0}^{I}\Omega_{T}^{\Delta}$
and the map is a homeomorphism. ∎

######  Proposition 4.10.

The extended map
$\widehat{\Phi}^{\Delta}:\overline{\Omega}_{k}^{\Delta}\to\widehat{\Omega}_{T}^{\Delta}$
is a homeomorphism.

###### Proof.

By Proposition 4.4 and 4.9, $\widehat{\Phi}^{\Delta}$ is a continuous
bijection from $\overline{\Omega}_{k}^{\Delta}$ to
$\widehat{\Omega}_{T}^{\Delta}$. We only need to show that the inverse map
$(\widehat{\Phi}^{\Delta})^{-1}$ is also continuous. Let
$\\{\vec{T}^{n}\\}_{n=1}^{+\infty}$ be a sequence in
$\widehat{\Omega}_{T}^{\Delta}$ that converges to some
$\vec{T}^{*}\in\partial_{0}^{I}\Omega_{T}^{\Delta}$. Let $\vec{k}^{n}$ and
$\vec{k}^{*}$ be the preimage of $\vec{T}^{n}$ and $\vec{T}^{*}$ under
$\widehat{\Phi}^{\Delta}$, respectively. Then we only need to show that
$\lim_{n\to\infty}\vec{k}^{n}=\vec{k}^{*}$.

We first show that $\\{\vec{k}^{n}\\}_{n=1}^{\infty}$ is bounded in
$\overline{\Omega}_{k}^{\Delta}=\mathbb{R}_{\geq 0}^{3}$. If not, by passing
to a subsequence, we may assume that $\\{\vec{k}^{n}\\}$ satisfy the condition
of Lemma 4.8 for some non-empty $J\subset\\{1,2,3\\}$. Then
$T_{1}^{*}+T_{2}^{*}+T_{3}^{*}=\lim(T_{1}^{n}+T_{2}^{n}+T_{3}^{n})=\pi$ by
Lemma 4.8. This contradicts the assumption
$\lim\widehat{\Phi}^{\Delta}(\vec{k}_{n})=\vec{T}^{*}\in\partial_{0}^{I}\Omega_{T}^{\Delta}$
that requires $T_{1}^{*}+T_{2}^{*}+T_{3}^{*}<\pi$.

If $\lim\vec{k}^{n}\neq\vec{k}^{*}$, since $\\{\vec{k}^{n}\\}_{n=1}^{+\infty}$
is bounded, there exists a subsequence $\\{\vec{k}^{n_{i}}\\}_{i=1}^{+\infty}$
such that

$\lim_{n_{i}\to\infty}\vec{k}^{n_{i}}=\vec{k^{\prime}}\neq\vec{k}^{*}\ .$

Since $\widehat{\Phi}^{\Delta}$ is continuous, we have

$\widehat{\Phi}^{\Delta}(\vec{k^{\prime}})=\lim_{n_{i}\to\infty}\widehat{\Phi}^{\Delta}(\vec{k}^{n_{i}})=\lim_{n_{i}\to\infty}\vec{T}^{n_{i}}=\vec{T}^{*}\
.$

However, $\widehat{\Phi}^{\Delta}(\vec{k}^{*})=\vec{T}^{*}$ by assumption.
This contradicts the bijectivity of $\widehat{\Phi}^{\Delta}$. Hence
$(\widehat{\Phi}^{\Delta})^{-1}$ is continuous. This finishes the proof
completely. ∎

###  4.2 Homeomorphism in every stratum

Now we turn to circle packed surfaces. The results are essentially obtained by
summing up local results for three-circle configurations.

By Proposition 4.4 and the Gauss-Bonnet formula, the homeomorphism
$\Phi:\Omega_{k}\to\Omega_{T}$ extends continuously to
$\widehat{\Phi}:\overline{\Omega}_{k}\to\widehat{\Omega}_{T}$, with
$\widehat{\Phi}(\partial^{I}\Omega_{k})\subset\partial_{0}^{I}\Omega_{T}$ for
any non-empty $I\subset V$. In this subsection, we obtain the following
proposition, which is a global version of Proposition 4.9. The proof is
inspired by the work of Bobenko-Springborn [4].

######  Proposition 4.11.

For any $I\subseteq V$, the restricted map
$\widehat{\Phi}:\partial^{I}\Omega_{k}\to\partial^{I}_{0}\Omega_{T}$ is a
homeomorphism.

######  Remark 4.12.

If $I=V$, $\widehat{\Phi}$ maps the origin point to the origin point. By
taking $I=\varnothing$, Proposition 4.11 implies Theorem 2.7. This subsection
also provides an alternative proof.

Proposition 4.11 will be proved as following:

Denote $v\in f$ if $v\in V$ is a vertex of triangle $f\in F$. For a fixed
proper set $I\subset V$, let $\overline{V}:=V\setminus I$ and
$\overline{F}:=F_{V\setminus I}$. Note that $f\in\overline{F}$ if and only if
$f$ contains at least one vertex outside $I$. Since $I\neq V$, both
$\overline{V}$ and $\overline{F}$ are non-empty. Let

(4.8) $\ln:\partial^{I}\Omega_{k}\to\Omega^{I}_{S}:=\mathbb{R}^{\overline{V}}$

be the diffeomorphism of the stratum defined by $\vec{k}=(k_{i})_{i\in
V}\mapsto\vec{S}=(\ln k_{i})_{i\notin I}$. We will define a set
$\mathcal{CO}^{I}$, called “coherent system” (Definition 4.15), with a natural
embedding $\iota:\Omega_{S}^{I}\hookrightarrow\mathcal{CO}^{I}$ (4.12) and a
projection $\Psi:\mathcal{CO}^{I}\to\partial_{0}^{I}\Omega_{T}$ (4.13), such
that on each stratum $\partial^{I}\Omega_{k}$, the map $\widehat{\Phi}$ is
expressed as a composition

(4.9)
${\partial^{I}\Omega_{k}}$${\Omega_{S}^{I}}$${\mathcal{C}O^{I}}$${\partial_{0}^{I}\Omega_{T}\
.}$$\scriptstyle{\ln}$$\scriptstyle{\cong}$$\widehat{\Phi}=\Psi\circ\iota\circ\ln$$\scriptstyle{\iota}$$\scriptstyle{\Psi}$

Proposition 4.11 is then deduced from the following two propositions, which
will be proved in §4.2.1 and §4.2.2 respectively.

######  Proposition 4.13.

$\widehat{\Phi}$ maps $\partial^{I}\Omega_{k}$ homeomorphically onto
$\Psi(\mathcal{CO}^{I})$.

######  Proposition 4.14.

$\Psi(\mathcal{CO}^{I})=\partial_{0}^{I}\Omega_{T}$.

####  4.2.1 Coherent system and image set

######  Definition 4.15.

The _coherent system_ is a subset of $\mathbb{R}_{\geq 0}^{3\lvert F\rvert}$
defined as

(4.10) $\mathcal{CO}:=\bigg{\\{}\ \left(T_{v}^{f}\right)_{v\in f}\
\bigg{\lvert}\ T_{u}^{f}+T_{v}^{f}+T_{w}^{f}<\pi\ \textrm{ for any }f=(uvw)\in
F\ \bigg{\\}}\ .$

And for any $I\subseteq V$, let

(4.11) $\mathcal{CO}^{I}:=\bigg{\\{}\ \left(T_{v}^{f}\right)_{v\in
f}\in\mathcal{CO}\ \bigg{\lvert}\ T_{v}^{f}=0\textrm{ if and only if }v\in I\
\bigg{\\}}\ .$

Geometrically, a point in $\mathcal{CO}$ realizes each $f$ as a possibly
degenerated triangle, so that the total geodesic curvature of circle arc
centered at $v\in f$ is exactly $T_{v}^{f}$ if $T_{v}^{f}>0$, and is geodesic
if $T_{v}^{f}=0$. These triangles do not need to match along the edges.

We follow notations in (2.1), (2.2) and (4.8). There is a natural embedding

(4.12)
$\begin{split}\iota:\quad\Omega_{S}^{I}\quad&\longrightarrow\quad\mathcal{CO}^{I}\\\
\vec{S}=\ln\vec{k}&\longmapsto\left(T_{v}^{f}(\vec{k})\right)_{v\in
f}\end{split}$

where $\vec{k}\in\partial^{I}\Omega_{k}$, and there exists a natural
projection

(4.13)
$\begin{split}\Psi:\quad\mathcal{CO}\quad&\longrightarrow\quad\widehat{\Omega}_{T}\\\
\left(T^{f}_{v}\right)_{v\in f}&\longmapsto\left(\sum_{f\in
F_{\\{v\\}}}T_{v}^{f}\right)_{v\in V}\end{split}$

by summing up at vertices. Note that
$\Psi(\mathcal{CO}^{I})\subseteq\partial_{0}^{I}\Omega_{T}$ by the Gauss-
Bonnet formula.

In order to prove Proposition 4.13 which stated that $\widehat{\Phi}$ is a
surjective homeomorphism from $\partial^{I}\Omega_{k}$ to
$\Psi(\mathcal{CO}^{I})$, we need the following result. We say
$f:\mathbb{R}^{n}\to\mathbb{R}$ is _coercive_ if $f(x)\to+\infty$ as
$\|x\|\to+\infty$.

######  Lemma 4.16 ([11], Lemma 4.6).

Suppose $f(x)$ is a $C^{2}$-smooth, strictly convex function on
$\mathbb{R}^{n}$. The following two statements are equivalent:

  1. (1)

$f$ has a unique minimum;

  2. (2)

$f$ is coercive.

###### Proof of Proposition 4.13.

It is easy to see that
$\widehat{\Phi}=\Psi\circ\iota\circ\ln:\partial^{I}\Omega_{k}\to\partial_{0}^{I}\Omega_{T}$.
Hence $\widehat{\Phi}(\partial^{I}\Omega_{k})\subset\Psi(\mathcal{CO}^{I})$.

Now we fix a point $\vec{T}=(T_{v})_{v\in V}$ in $\Psi(\mathcal{CO}^{I})$,
with a preimage $(T_{v}^{f})_{v\in f}$ in $\mathcal{CO}^{I}$. For a triangle
$f=(uvw)\in\overline{F}$, we can define a closed 1-form $\omega_{f}^{I}$ on
$\mathbb{R}^{\\{u,v,w\\}\setminus I}$ as (4.6). It naturally induces a closed
1-form on $\Omega_{S}^{I}=\mathbb{R}^{V\setminus
I}=\mathbb{R}^{\overline{V}}$. However, we will modify this 1-form to obtain
global results. Define another 1-form

(4.14)
$\widetilde{\omega}_{f}^{I}:=\omega_{f}^{I}-\left(T_{u}^{f}\mathrm{d}S_{u}+T_{v}^{f}\mathrm{d}S_{v}+T_{w}^{f}\mathrm{d}S_{w}\right)$

on $\mathbb{R}^{\\{u,v,w\\}\setminus I}$, where $T_{i}^{f}\mathrm{d}S_{i}$ is
defined to be zero when $i\in I$. Then let

(4.15)
$\widetilde{\mathcal{E}}^{I}_{f}:=\int_{\vec{S}_{0}^{I}}^{\vec{S}^{I}}\widetilde{\omega}_{f}^{I}.$

Finally, define the energy function as

(4.16)
$\widetilde{\mathcal{E}}^{I}:=\sum_{f\in\overline{F}}\widetilde{\mathcal{E}}^{I}_{f}.$

By (4.6) and the definitions above, when $\ln\vec{k}=\vec{S}\in\Omega_{S}^{I}$
and $v\in\overline{V}$,

(4.17) $\frac{\partial\widetilde{\mathcal{E}}^{I}}{\partial
S_{v}}(\vec{S})=\sum_{f\in\overline{F}}\frac{\partial\widetilde{\mathcal{E}}^{I}_{f}}{\partial
S_{v}}(\vec{S})=\sum_{f\in
F_{\\{v\\}}}\left(T_{v}^{f}(\vec{k})-T_{v}^{f}\right)=T_{v}(\vec{k})-\left(\sum_{f\in
F_{\\{v\\}}}T_{v}^{f}\right)=T_{v}(\vec{k})-T_{v}\ .$

In other words,
$\nabla\widetilde{\mathcal{E}}^{I}(\ln\vec{k})=\widehat{\Phi}(\vec{k})-\vec{T}$
as a vector-valued function on $\partial^{I}\Omega_{k}$.

Claim: $\widetilde{\mathcal{E}}^{I}$ is a strictly convex and coercive
function on $\Omega_{S}^{I}=\mathbb{R}^{\overline{V}}$.

Hence $\widetilde{\mathcal{E}}^{I}$ contains a unique minimum point
$\vec{S}_{0}=\ln(\vec{k}_{0})$ in $\Omega_{S}^{I}$ by Lemma 4.16, and
$\frac{\partial\widetilde{\mathcal{E}}^{I}}{\partial S_{v}}(\vec{S}_{0})=0$
for any $v\in\overline{V}$. On the other hand, by (4.17) we have
$\frac{\partial\widetilde{\mathcal{E}}^{I}}{\partial
S_{v}}(\vec{S}_{0})=T_{v}(\vec{k}_{0})-T_{v}$. This implies
$\widehat{\Phi}(\vec{k}_{0})=\vec{T}=\Psi((T_{v}^{f})_{v\in f})$. Therefore,
we have $\Psi(\mathcal{CO}^{I})=\widehat{\Phi}(\partial^{I}\Omega_{k})$.

Since $\widetilde{\mathcal{E}}^{I}$ is strictly convex from
$\mathbb{R}^{\overline{V}}$, by Lemma 4.5, $\nabla\widetilde{\mathcal{E}}^{I}$
is a smooth embedding. By (4.17), $\nabla\widetilde{\mathcal{E}}^{I}$ is
simply differed from the restriction of $\widehat{\Phi}$ by a constant
translation. Therefore, $\widehat{\Phi}$ also maps the stratum
homeomorphically onto its image. ∎

###### Proof of Claim.

Let $f=(uvw)\in\overline{F}$ as before. By Lemma 4.7, as a $C^{2}$-smooth
function on $\mathbb{R}^{\\{u,v,w\\}\setminus I}$,
$\widetilde{\mathcal{E}}^{I}_{f}$ is the sum of a strictly convex function and
a linear function, hence strictly convex again. Since $(T_{v}^{f})_{v\in
f}\in\mathcal{CO}^{I}$, there exists
$\vec{k}_{f}\in\overline{\Omega}_{k}^{\Delta}$ such that
$\widehat{\Phi}_{T}^{\Delta}(\vec{k}_{f})=(T^{f}_{u},T^{f}_{v},T^{f}_{w})$, by
Proposition 4.10. Let
$\vec{S}_{f}:=\ln(\vec{k}_{f})\in\mathbb{R}^{\\{u,v,w\\}\setminus I}$, then
$\vec{S}_{f}$ is a zero of the 1-form $\widetilde{\omega}_{f}^{I}$. By Lemma
4.16, $\widetilde{\mathcal{E}}^{I}_{f}$ is coercive when regarded as a
function on $\mathbb{R}^{\\{u,v,w\\}\setminus I}$. It also has a unique
minimum.

However, as a $C^{2}$-function on $\Omega_{S}^{I}$,
$\widetilde{\mathcal{E}}^{I}_{f}$ is merely convex, and coercive along certain
directions. More precisely, let $H^{I}_{f}$ be the
$\lvert\overline{V}\rvert\times\lvert\overline{V}\rvert$ Hessian of
$\widetilde{\mathcal{E}}^{I}_{f}$, and
$\vec{S}=(S_{i})_{i\in\bar{V}}\in\mathbb{R}^{\overline{V}}$ be a column
vector. We have the following four statements:

  1. (1)

for any non-zero vector $\vec{S}\in\mathbb{R}^{\overline{V}}$, we have
${\vec{S}}^{T}H^{I}_{f}\vec{S}\geq 0$;

  2. (2)

if $S_{i}\neq 0$ for some $i\in\\{u,v,w\\}\setminus I$, then
${\vec{S}}^{T}H^{I}_{f}\vec{S}>0$;

  3. (3)

$\widetilde{\mathcal{E}}^{I}_{f}\geq m_{f}$ for some constant $m_{f}$;

  4. (4)

if $\lvert S_{i}\rvert\to+\infty$ for some $i\in\\{u,v,w\\}\setminus I$, then
$\widetilde{\mathcal{E}}^{I}_{f}(S)\to+\infty$.

Now let $H^{I}$ be the
$\lvert\overline{V}\rvert\times\lvert\overline{V}\rvert$ Hessian of
$\widetilde{\mathcal{E}}^{I}$. For any non-zero vector
$\vec{S}\in\mathbb{R}^{\overline{V}}$, choose $j\in\overline{V}$ with
$S_{j}\neq 0$. There exists at least one $f_{0}\in
F_{\\{j\\}}\subset\overline{F}$. By statements (i) and (ii) listed above,

$\vec{S}^{T}H^{I}\vec{S}=\sum_{f\in\overline{F}}\vec{S}^{T}H^{I}_{f}\vec{S}\geq\vec{S}^{T}H^{I}_{f_{0}}\vec{S}>0\
.$

This implies $H^{I}$ is again positively definite and
$\widetilde{\mathcal{E}}^{I}$ is strictly convex.

To prove that $\widetilde{\mathcal{E}}^{I}$ is coercive on
$\mathbb{R}^{\overline{V}}$, it sufficies to show that, for any $A>0$, there
exists $B>0$ such that $\tilde{\mathcal{E}}^{I}(\vec{S})>A$ holds for every
$\vec{S}\in\mathbb{R}^{\bar{V}}$ with $\|\vec{S}\|_{\infty}>B$. By statement
(iv) listed above and the finiteness of $|F|$, for any large $A>0$, there is
some constant $B>0$ such that

for all $f=(uvw)\in\overline{F}$, whenever $\lvert S_{i}\rvert>B$ for some
$i\in\\{u,v,w\\}\setminus I$, we have

$\widetilde{\mathcal{E}}^{I}_{f}(\vec{S})>A-\sum_{g\neq
f\in\overline{F}}m_{g}\ .$

Therefore, for $\vec{S}\in\mathbb{R}^{\overline{V}}$ with
$\|\vec{S}\|_{\infty}>B$, there exists a vertex $j\in\overline{V}$ such that
$\lvert S_{j}\rvert>B$ and $f_{0}\in F_{\\{j\\}}\subset\overline{F}$. By the
choice of $B$ and the lower bounds for $\widetilde{\mathcal{E}}^{I}_{f}$’s, we
have

$\widetilde{\mathcal{E}}^{I}=\widetilde{\mathcal{E}}^{I}_{f_{0}}+\sum_{f_{0}\neq
f\in\overline{F}}\widetilde{\mathcal{E}}^{I}_{f}>\left(A-\sum_{f_{0}\neq
g\in\overline{F}}m_{g}\right)+\sum_{f_{0}\neq f\in\overline{F}}m_{f}=A\ .$

∎

####  4.2.2 Feasible flow and surjectivity

To obtain Proposition 4.14 which implies that
$\Psi:\mathcal{CO}^{I}\to\partial_{0}^{I}\Omega_{T}$ is actually surjective,
we have to solve linear equations with inequality constrains. This turns out
to be a problem of constructing a balanced flow on an oriented graph, which is
a linear programming. The argument here is similar to [4, §4.3].

For a fixed proper set $I\subset V$, let $A(I)$ be the set of inner corners at
a vertex outside $I$, or equivalently,

$A(I):=\bigg{\\{}\ (v,f)\in(V\setminus I)\times F\ \bigg{\lvert}\ v\in f\
\bigg{\\}}\subset\overline{V}\times\overline{F}\ .$

Let $\vec{M}=(M_{v})_{v\in V}$ be a fixed point in
$\partial_{0}^{I}\Omega_{T}$. An oriented graph $(N,X,\mathcal{I})$ associated
to the triangulation $G$ and $I$ is defined as following.

• The vertices set
$N:=\overline{V}\sqcup\overline{F}\sqcup\\{\mbox{\scalebox{1.3}{$\boxtimes$}}\\}$,
where  $\boxtimes$ is an extra virtual vertex.

• There are three types of oriented edges in $X$.

(1). For any $v\in\overline{V}$, there is an oriented edge $x_{v}$ from
$\boxtimes$ to $v$.

(2). For any $(v,f)\in A(I)$, there is an oriented edge $x_{v,f}$ from $v$ to
$f$.

(3). For any $f\in\overline{F}$, there is an oriented edge $x_{f}$ from $f$ to
$\boxtimes$ .

Next, assign each oriented edge $x\in X$ a closed interval
$\mathcal{I}_{x}=[a_{x},b_{x}]$ of $[-\infty,+\infty]$

• For any $v\in\overline{V}$, $\mathcal{I}_{x_{v}}:=[M_{v},M_{v}]$.

• For any $(v,f)\in A(I)$, $\mathcal{I}_{x_{v,f}}:=[\delta,+\infty)$.

• For any $f\in\overline{F}$,
$\mathcal{I}_{x_{f}}:=(-\infty,\pi-\varepsilon]$.

Here $\delta,\varepsilon$ are two sufficiently small positive constants
(depending on $\vec{M}\in\partial_{0}^{I}\Omega_{T}$) to be determined. This
graph faithfully represents the linear equations and constrains.

######  Definition 4.17.

A _feasible flow_ of $(N,X,\mathcal{I})$ is a function $\phi:X\to\mathbb{R}$
such that

  1. 1.

it is balanced at each vertex $n\in N$, namely

$\sum_{x\in in(n)}\phi(x)=\sum_{y\in ex(n)}\phi(y),\quad\forall n\in N\ ;$

  2. 2.

it is constrained by the intervals $\mathcal{I}_{x}$ on each edge $x\in X$,
namely

$a_{x}\leq\phi(x)\leq b_{x},\quad\forall x\in X\ .$

For $n\in N$, $in(n)$ is the set of oriented edges ending at $n$, and $ex(n)$
is the set of oriented edges starting from $n$.

See Figure 7 as an example of the graph $(N,X,\mathcal{I})$. It is easy to
check that a feasible flow is equivalent to a solution of
$\Psi(\vec{T})=\vec{M}\in\partial_{0}^{I}\Omega_{T}$, with
$\vec{T}\in\mathcal{CO}^{I}$.

######  Lemma 4.18.

For a given proper subset $I\subset V$ and $\vec{M}=(M_{v})_{v\in
V}\in\partial_{0}^{I}\Omega_{T}$, there exists some $\vec{T}=(T_{v}^{f})_{v\in
f}\in\mathcal{CO}^{I}$ with $\Psi(\vec{T})=\vec{M}$ if and only if there
exists a feasible flow $\phi$ of $(N,X,\mathcal{I})$ associated to $G$ and $I$
for suitable $\varepsilon,\delta>0$ defined as above.

Figure 7: Left: $G$ is the tetrahedron triangulation of the sphere. $F_{i}$ is
the face opposite to vertex $v_{i}$. Right: the oriented graph
$(N,X,\mathcal{I})$ associated with $G$ and $I:=\\{v_{4}\\}$. The top row
shows the closed intervals assigned to the edges. The two  $\boxtimes$ ’s are
identified as the same vertex.

The following Feasible Flow Theorem gives a condition under which feasible
flows exist. See [4, §4.3], [9, Ch. II, §3]

######  Theorem 4.19 (Feasible Flow Theorem).

There is a feasible flow of $(N,X,\mathcal{I})$ if and only if

(4.18) $\sum_{x\in ex(N^{\prime})}b_{x}\geq\sum_{y\in in(N^{\prime})}a_{y}$

for any non-empty proper subset $N^{\prime}\subset N$, where $ex(N^{\prime})$
(resp. $in(N^{\prime})$) is the set of oriented edges that start from (resp.
end at) a vertex in $N^{\prime}$ but end at (resp. start from) a vertex
outside $N^{\prime}$.

######  Proposition 4.20.

For any given proper subset $I\subset V$ and $\vec{M}=(M_{v})_{v\in
V}\in\partial_{0}^{I}\Omega_{T}$, there are suitable $\delta,\varepsilon>0$
such that the graph $(N,X,\mathcal{I})$ associated with $G$ and $I$ has a
feasible flow.

###### Proof.

It sufficies to check the Feasible Flow Theorem for every $N^{\prime}\subset
N$.

Case (A). $\mbox{\scalebox{1.3}{$\boxtimes$}}\in N^{\prime}$.

Case (A.1). If there exists $f\in\overline{F}\setminus N^{\prime}$, then
$x_{f}\in in(N^{\prime})$ and $a_{x_{f}}=-\infty$. (4.18) holds.

Case (A.2). If $\overline{F}\subset N^{\prime}$, then the properness of
$N^{\prime}$ implies that $\overline{V}\setminus N^{\prime}$ is not empty.
Then $ex(N^{\prime})$ consists of $x_{v}$ with $v\in\overline{V}\setminus
N^{\prime}$, and $in(N^{\prime})$ consists of $x_{v,f}$ with
$v\in\overline{V}\setminus N^{\prime}$ and $(v,f)\in A(I)$. Hence (4.18)
requires that

$\sum_{v\in\overline{V}\setminus N^{\prime}}M_{v}\geq\sum_{(v,f)\in
A(I),v\in\overline{V}\setminus N^{\prime}}\delta\ .$

This will be satisfied if we let

(4.19)
$\delta<\frac{\min_{v\in\overline{V}}\\{M_{v}\\}}{\max_{v\in\overline{V}}\lvert
F_{v}\rvert}\ .$

Case (B). $\mbox{\scalebox{1.3}{$\boxtimes$}}\notin N^{\prime}$. Let
$V^{\prime}:=\overline{V}\cap N^{\prime}$ and $F^{\prime}:=\overline{F}\cap
N^{\prime}$.

Case (B.1). If $V^{\prime}=\varnothing$, then
$N^{\prime}=F^{\prime}\neq\varnothing$. We have $ex(N^{\prime})=\\{x_{f}\lvert
f\in F^{\prime}\\}$, $in(N^{\prime})=\\{x_{v,f}\lvert f\in F^{\prime},(v,f)\in
A(I)\\}$. Now (4.18) requires $(\pi-\varepsilon)\lvert
F^{\prime}\rvert\geq\delta\lvert in(N^{\prime})\rvert$. Note that $\lvert
in(N^{\prime})\rvert\leq 3\lvert F^{\prime}\rvert$. Hence we only need to
require $\pi>\varepsilon+3\delta$, or

(4.20) $0<\delta,\varepsilon<\pi/4\ .$

Case (B.2). If there exists some $v\in V^{\prime}$. If there further exists
some $f\in F_{v}\setminus N^{\prime}$, then $x_{v,f}\in ex(N^{\prime})$ and
$b_{x_{v,f}}=+\infty$, satisfying (4.18). So we may assume
$F_{V^{\prime}}\subset N^{\prime}$. On the other hand, if there is some $f\in
F^{\prime}$ with $v\notin V^{\prime}$ for all $(v,f)\in A(I)$, then by the
choice of $\delta,\varepsilon$ in Case (B.1), (4.18) is locally satisfied at
$f$. Hence we only need to consider the case where
$F^{\prime}=F_{V^{\prime}}$. Now that $ex(N^{\prime})=\\{x_{f}\lvert f\in
F_{V^{\prime}}\\}$. $in(N^{\prime})$ contains two types of edges:
$x_{v^{\prime}}$ with $v^{\prime}\in V^{\prime}$, and $x_{v,f}$ with
$v\in\overline{V}\setminus N^{\prime},f\in F_{V^{\prime}},(v,f)\in A(I)$. Note
that $f\in F_{V^{\prime}}$ contains at least one vertex in $V^{\prime}$.
Therefore $\\#\\{x_{v,f}\lvert v\in\overline{V}\setminus N^{\prime},f\in
F_{V^{\prime}}\\}\leq 2\lvert F_{V^{\prime}}\rvert$. Now that

$\displaystyle\sum_{x\in ex(N^{\prime})}b_{x}$
$\displaystyle=(\pi-\varepsilon)\lvert F_{V^{\prime}}\rvert\ ,$
$\displaystyle\sum_{y\in in(N^{\prime})}a_{y}$
$\displaystyle=\sum_{v^{\prime}\in
V^{\prime}}M_{v^{\prime}}+\delta\cdot\\#\left\\{x_{v,f}\left\lvert
v\in\overline{V}\setminus N^{\prime},f\in
F_{V^{\prime}}\right.\right\\}\leq\sum_{v^{\prime}\in
V^{\prime}}M_{v^{\prime}}+2\delta\lvert F_{V^{\prime}}\rvert\ .$

Hence we only need to require that $(\pi-\varepsilon)\lvert
F_{V^{\prime}}\rvert>\sum_{v^{\prime}\in
V^{\prime}}M_{v^{\prime}}+2\delta\lvert F_{V^{\prime}}\rvert$ for any non-
empty $V^{\prime}\subset\overline{V}$, which is equivalent to, by (1.1),

$\eta^{V^{\prime}}(M)>(\varepsilon+2\delta)\lvert F_{V^{\prime}}\rvert\ .$

Since $M\in\partial_{0}^{I}\Omega_{T}\subset\widehat{\Omega}_{T}$,
$\eta^{V^{\prime}}(M)>0$ for all non-empty $V^{\prime}\subset V$ by (1.4).
Hence

(4.21)
$\min_{V^{\prime}\subsetneq\overline{V}}\left\\{\frac{\eta^{V^{\prime}}(M)}{\left\lvert
F_{V^{\prime}}\right\rvert}\right\\}$

is positive, and the choice for $\varepsilon+2\delta$ always exists.

Combining (4.19), (4.20), (4.21), the choice of positive real numbers
$\delta,\varepsilon$ always exists. ∎

###### Proof of Proposition 4.14.

The existence of $\Psi$-preimages of any vector of
$\partial_{0}^{I}\Omega_{T}$ follows from Lemma 4.18 and Proposition 4.20. ∎

###  4.3 Global homeomorphism

We give the proof of Theorem 1.2 as follows.

###### Proof of Theorem 1.2.

According to Proposition 4.11, we have shown that

$\widehat{\Phi}:\partial^{I}\mathbb{R}_{>0}^{V}\to\partial_{0}^{I}\Omega_{T}$

is a homeomorphism for every $I\subset V$. Then $\widehat{\Phi}$ is bijective.
Since $\widehat{\Phi}$ is continuous, to show that it is an actual
homeomorphism, we need to show that the inverse map is also continuous. Denote
by $\\{\vec{T}^{n}\\}_{n=1}^{+\infty}$ a sequence in $\widehat{\Omega}_{T}$
converging to some $\vec{T}^{*}\in\partial_{0}^{I}\Omega_{T}$. Set
$\vec{k}^{n}$ and $\vec{k}^{*}$ the preimage of $\vec{T}^{n}$ and
$\vec{T}^{*}$ under $\widehat{\Phi}$, respectively. Similar to the proof of
Proposition 4.10, we only need to show that $\\{\vec{k}^{n}\\}_{n=1}^{\infty}$
is bounded in $\overline{\Omega}_{k}$. If not, by passing to a subsequence, we
can assume that there exists a vertex subset $J\subset V$ such that $\lim
k_{j}^{n}=+\infty$ for all $j\in J$, and $k_{i}^{n}$ is bounded from above for
all $i\notin J$. The set $F_{J}$ can be decomposed as $F_{J}=F_{J_{1}}\sqcup
F_{J_{2}}\sqcup F_{J_{3}}$, where $F_{J_{m}}$ is the set of faces having
exactly $m$ vertex in $J$ ($m=1,2,3$). By Lemma 4.8, for $f\in F_{J_{m}}$, as
$n\to\infty$,

$\sum_{v\in f,v\in J}T_{v}^{f}(\vec{k}^{n})\to\pi$

and the sum contains exactly $m$ terms. So as $n\to\infty$,

$\sum_{j\in J}T_{j}(\vec{k}^{n})=\sum_{f\in F_{J}}\sum_{v\in f,v\in
J}T_{v}^{f}(\vec{k}^{n})\to\pi\lvert F_{J}\rvert\ .$

This contradicts the assumption
$\lim\vec{T}^{n}=\vec{T}^{*}\in\partial_{0}^{I}\Omega_{T}$ that requires
$\eta^{J}(\vec{T}^{*})=\pi\lvert F_{J}\rvert-\sum_{j\in
J}T_{j}(\vec{k}^{*})>0$ in (1.2). Hence $\widehat{\Phi}^{-1}$ is continuous
and Theorem 1.2 is proved. ∎

##  5 Combinatorial Ricci flow with prescribed total geodesic curvatures

In this section, we prove Theorem 1.4, which states the convergence of the
combinatorial Ricci flow (1.7) with prescribed total geodesic curvatures
$\vec{T}\in\partial_{0}^{I}\Omega_{T}$.

Let $J\subset I\subset V$ as before. Recall that for any
$\vec{k}\in\partial^{J}\Omega_{k}$,
$\vec{S}:=\ln\vec{k}\in\Omega_{S}^{I}=\mathbb{R}^{V\setminus J}$ is defined as
$(\ln k_{i})_{i\notin J}$. By this change of variables, the flow (1.7)
transforms to

(5.1)
$\displaystyle\frac{d}{dt}S_{v}(t)=-\left(T_{v}(\vec{S}(t))-T_{v}\right),\quad\text{for
any }v\in V\setminus J,$

which is the gradient flow of the potential function
$\widetilde{\mathcal{E}}^{I}$ (4.16) on $\Omega^{I}_{S}$. Here, $\vec{k}(t)$
is regarded as a vector-valued function of $\vec{S}$.

######  Lemma 5.1.

For any $\vec{k}(0)\in\partial^{J}\Omega_{k}$, the solution of the flow (5.1)
with the initial value $\vec{S}(0):=\ln\vec{k}(0)$ exists for all time
$t\in[0,\infty)$ and is unique.

###### Proof.

Since $T_{v}$ is a $C^{1}$ function of $\vec{S}$, by Cauchy-Lipschitz Theorem,
the flow (5.1) exists a unique solution $\vec{S}(t)$ for some
$t\in[0,\varepsilon)$. Since $\widehat{\Omega}_{T}$ is a bounded region in
$\mathbb{R}^{V}$,

$\left|\frac{dS_{v}}{dt}\right|=\left|T_{v}(\vec{k}(t))-T_{v}\right|$

is always bounded. Hence, the flow (5.1) exists for all $t\in[0,\infty)$. The
uniqueness of the flow can be easily deduced from the classical theory of
ordinary differential equations. ∎

In the spirit of the work of Takatsu [23], we introduce the important lemma
for estimating the gradient of a convex function along its gradient flow, see
e.g. [1, Theorem 4.3.2] .

######  Lemma 5.2.

Let $h:\mathbb{R}^{n}\rightarrow\mathbb{R}$ be a $C^{2}-$smooth convex
function. Suppose that $\vec{\xi}:[0,\infty)\rightarrow\mathbb{R}^{n}$ is a
gradient flow of $h.$ Then for any $\tau>0$ and
$\vec{\xi}^{*}\in\mathbb{R}^{n}$, we have

$\displaystyle\left\lvert\nabla
h(\vec{\xi}(\tau))\right\rvert^{2}\leq\left\lvert\nabla
h(\vec{\xi}^{*})\right\rvert^{2}+\frac{1}{\tau^{2}}\left\lvert\vec{\xi}^{*}-\vec{\xi}(0)\right\rvert^{2}.$

With the help of Theorem 1.2, Proposition 4.11 and Lemma 5.2, we can prove the
convergence of the combinatorial Ricci flow.

###### Proof of Theorem 1.4.

By Lemma 5.1, there exists a unique solution $\vec{S}(t)$ to the flow (5.1)
with initial value $\vec{S}(0):=\ln\vec{k}(0)\in\partial^{J}\Omega_{k}$ and
$t\in[0,\infty)$. Let $\vec{k}(t)$ be the preimage of $\vec{S}(t)$ by the
diffeomorphism $\ln:\partial^{J}\Omega_{k}\to\partial^{J}\Omega_{S}$. Then
$\vec{k}(t)$ is a solution of (1.7) with initial value $\vec{k}(0)$. By
Proposition 4.11, let
$\vec{k}^{*}:=\widehat{\Phi}^{-1}(\vec{T})\in\partial^{I}\Omega_{k}$.

It remains to show that $\vec{k}(t)\rightarrow\vec{k}^{*}$ as $t\to+\infty$.
We define an auxiliary curve $\vec{k}^{A}(t)\ (t\geq 0)$ in
$\partial^{J}\Omega_{k}$ as

$\vec{k}_{v}^{A}(t):=\begin{cases}1/t,&\text{if }v\in I\backslash J,\\\
k_{v}^{*},&\text{if }v\notin I.\end{cases}$

By Theorem 1.2, we have

(5.2) $\displaystyle\lim_{t\to\infty}T_{v}(\vec{k}^{A}(t))=T_{v}$

holds for all $v\in V$. Let $\vec{S}^{A}(t):=\ln\vec{k}^{A}(t)$ be the image
of this curve in $\Omega^{J}_{S}$ under the change of variables. By Lemma 5.2
and the fact that $\vec{S}(t)$ is the gradient flow of
$\widetilde{\mathcal{E}}^{I}$, we have

(5.3)
$\left\lvert\nabla\widetilde{\mathcal{E}}^{I}(\vec{S}(t))\right\rvert^{2}\leq\left\lvert\nabla\widetilde{\mathcal{E}}^{I}(\vec{S}^{A}(t))\right\rvert^{2}+\frac{1}{t^{2}}\left\lvert\vec{S}^{A}(t)-\vec{S}(0)\right\rvert^{2},\quad\forall
t>0.$

Consider the convergence of RHS of (5.3). Recall (4.17) that

$\left\lvert\nabla\widetilde{\mathcal{E}}^{I}(\vec{S}(t))\right\rvert^{2}=\sum_{v\in
V}(T_{v}(\vec{k}(t))-T_{v})^{2}\quad\text{and}\quad\left\lvert\nabla\widetilde{\mathcal{E}}^{I}(\vec{S}^{A}(t))\right\rvert^{2}=\sum_{v\in
V}(T_{v}(\vec{k}^{A}(t))-T_{v})^{2}.$

By the convergence (5.2), we have the first term of RHS of (5.3) tends to $0$
as $t$ tends to infinity. The convergence of the second term of RHS of (5.3)
follows from the observation that

$\displaystyle\left\lvert\vec{S}^{A}(t)-\vec{S}(0)\right\rvert^{2}$
$\displaystyle=O(|\ln t|^{2}).$

By applying convergences above to (5.3), we have

$\sum_{v\in
V}(T_{v}(\vec{k}(t))-T_{v})^{2}=\left\lvert\nabla\widetilde{\mathcal{E}}^{I}(\vec{S}(t))\right\rvert^{2}$

tends to $0$ as $t$ tends to infinity, that is,

$\displaystyle\lim_{t\rightarrow\infty}\widehat{\Phi}(\vec{k}(t))=\vec{T}\in\partial_{0}^{I}\Omega_{T}.$

Combining Theorem 1.2, we finish the proof. ∎

##  6 Acknowledgments

All the authors sincerely express their gratitude to Xin Nie for his abundant
and generous communication on the topic of circle packing and related fields.

Guangming Hu is supported by NSFC (No. 12101275) and Natural Science Research
Start-up Foundation of Recruiting Talents of Nanjing University of Posts and
Telecommunications (No. NY224040). Sicheng Lu is supported by China
Postdoctoral Science Foundation (Certificate Number 2024M753071). Dong Tan is
supported by the NSFC (No. 12001122, 12271533, 12361014) and the special
grants for Guangxi Ba Gui Scholars. Youliang Zhong is partially supported by
NSFC (No. 12271174) and Guangdong Basic and Applied Basic Research Foundation
(No. 2023A1515010929), and Guangzhou Science and Technology Program (No.
202201010160). Puchun Zhou is supported by Shanghai Science and Technology
Program (Project No. 22JC1400100).

##  Appendix A Hyperbolic calculations

We prove Theorem 4.1, Corollary 4.2 and part of Lemma 4.6 in this appendix.

###  A.1 The hyperboloid and Klein model of hyperbolic plane

We first recall some basic concepts and formulae for hyperbolic plane. See
[21, Chapter 3], [25, Chapter 2] for more details.

The _Lorentzian 3-space_ $\mathbb{E}^{2,1}$ is the vector space
$\mathbb{R}^{3}$ endowed with the Lorentzian inner product

(A.1)
$\langle\boldsymbol{x},\boldsymbol{y}\rangle:=x_{1}y_{1}+x_{2}y_{2}-x_{3}y_{3}.$

A vector $\boldsymbol{x}\in\mathbb{E}^{2,1}$ is said to be _time-like, space-
like and light-like_ if $\langle\boldsymbol{x},\boldsymbol{x}\rangle$ is
negative, positive and zero. The _light cone_ is the set of all light-like
vectors in $\mathbb{E}^{2,1}$.

The hyperbolic plane is the subset

$\mathbb{H}:=\big{\\{}\ \boldsymbol{x}\in\mathbb{E}^{2,1}\ \lvert\
\langle\boldsymbol{x},\boldsymbol{x}\rangle=-1,\ x_{3}>0\ \big{\\}}.$

The Lorentzian inner product induces a Riemannian metric on $\mathbb{H}$.
Geodesics in $\mathbb{H}$ are the intersection of $\mathbb{H}$ with planes
through the origin of $\mathbb{E}^{2,1}$.

Every time-like $\boldsymbol{x}\in\mathbb{E}^{2,1}$ can be normalized as
$\bar{\boldsymbol{x}}:=\boldsymbol{x}/\sqrt{-\langle\boldsymbol{x},\boldsymbol{x}\rangle}\in\mathbb{H}$.
Every light-like $\boldsymbol{x}\in\mathbb{E}^{2,1}$ can be normalized as
$\bar{\boldsymbol{x}}:=(x_{1},x_{2},1)$ with $x_{1}^{2}+x_{2}^{2}=1$. And
every space-like $\boldsymbol{x}\in\mathbb{E}^{2,1}$ can be normalized as
$\bar{\boldsymbol{x}}:=\boldsymbol{x}/\sqrt{\langle\boldsymbol{x},\boldsymbol{x}\rangle}$.
It is also associated with a _dual geodesic_ $\boldsymbol{x}^{\perp}:=\\{\
\boldsymbol{y}\in\mathbb{H}\ \lvert\
\langle\boldsymbol{x},\boldsymbol{y}\rangle=0\ \\}$. In this case, every point
$\boldsymbol{y}\in\boldsymbol{x}^{\perp}$ determines a geodesic lying on the
plane through $\boldsymbol{x},\boldsymbol{y}$ and the origin. This geodesic is
always perpendicular to $\boldsymbol{x}^{\perp}$.

The hyperbolic distance is strongly related to the Lorentzian inner product.
This induces a simple and uniform expression for generalized circles.

######  Proposition A.1.

Let $\boldsymbol{x},\boldsymbol{y}$ be two different points in
$\mathbb{E}^{2,1}$.

  1. 1.

For time-like $\boldsymbol{x},\boldsymbol{y}$, the hyperbolic distance between
the normalized vectors
$\bar{\boldsymbol{x}},\bar{\boldsymbol{y}}\in\mathbb{H}$ is given by

$\cosh
d(\bar{\boldsymbol{x}},\bar{\boldsymbol{y}})=-\frac{\langle\boldsymbol{x},\boldsymbol{y}\rangle}{\sqrt{\langle\boldsymbol{x},\boldsymbol{x}\rangle\langle\boldsymbol{y},\boldsymbol{y}\rangle}}.$

  2. 2.

For time-like $\boldsymbol{x}$ and space-like $\boldsymbol{y}$, the distance
between $\bar{\boldsymbol{x}}\in\mathbb{H}$ and the dual geodesic
$\boldsymbol{y}^{\perp}$ is given by

$\sinh
d(\bar{\boldsymbol{x}},\boldsymbol{y}^{\perp})=\pm\frac{\langle\boldsymbol{x},\boldsymbol{y}\rangle}{\sqrt{-\langle\boldsymbol{x},\boldsymbol{x}\rangle\langle\boldsymbol{y},\boldsymbol{y}\rangle}}.$

The signature is negative if and only if the plane through
$\boldsymbol{y}^{\perp}$ and the origin separates
$\boldsymbol{x},\boldsymbol{y}$.

  3. 3.

Let $\boldsymbol{x},\boldsymbol{y}$ be space-like.

     * •

If the plane through $\boldsymbol{x},\boldsymbol{y}$ and the origin intersects
$\mathbb{H}$, then the two dual geodesics
$\boldsymbol{x}^{\perp},\boldsymbol{y}^{\perp}$ are disjoint and the distance
between them is given by

$\cosh
d(\boldsymbol{x}^{\perp},\boldsymbol{y}^{\perp})=\pm\frac{\langle\boldsymbol{x},\boldsymbol{y}\rangle}{\sqrt{\langle\boldsymbol{x},\boldsymbol{x}\rangle\langle\boldsymbol{y},\boldsymbol{y}\rangle}}.$

The choice of signature is similar to the previous case.

     * •

If the plane through $\boldsymbol{x},\boldsymbol{y}$ and the origin is
disjoint from $\mathbb{H}$, then the two dual geodesics
$\boldsymbol{x}^{\perp},\boldsymbol{y}^{\perp}$ intersect and the intersection
angle between them is given by

$\cos\theta(\boldsymbol{x}^{\perp},\boldsymbol{y}^{\perp})=\pm\frac{\langle\boldsymbol{x},\boldsymbol{y}\rangle}{\sqrt{\langle\boldsymbol{x},\boldsymbol{x}\rangle\langle\boldsymbol{y},\boldsymbol{y}\rangle}}.$

     * •

If the plane through $\boldsymbol{x},\boldsymbol{y}$ and the origin is tangent
to the light cone, then the two dual geodesics
$\boldsymbol{x}^{\perp},\boldsymbol{y}^{\perp}$ intersect at the light cone
and
$\langle\boldsymbol{x},\boldsymbol{y}\rangle/\sqrt{\langle\boldsymbol{x},\boldsymbol{x}\rangle\langle\boldsymbol{y},\boldsymbol{y}\rangle}=\pm
1$.

  4. 4.

A horocycle in $\mathbb{H}$ is given by

$C(\boldsymbol{v},t):=\\{\ \boldsymbol{x}\in\mathbb{H}\ \lvert\
\langle\boldsymbol{x},\boldsymbol{v}\rangle=t<0\ \\},$

where $\boldsymbol{v}$ is a light-like vector normalized as $v_{3}=1$ and $t$
is a negative constant.

######  Lemma A.2.

Every generalized circle is a set

$C(\boldsymbol{c},R):=\left\\{\ \boldsymbol{x}\in\mathbb{H}\ \big{\lvert}\
\langle\boldsymbol{x},\boldsymbol{c}\rangle=R\ \right\\}$

determined by a vector $\boldsymbol{c}\in\mathbb{E}^{2,1}$ and a real number
$R\in\mathbb{R}$. We call $C(\boldsymbol{c},R)$ a generalized circle centered
at $\boldsymbol{c}$.

###### Proof.

A circle is the set of points that are equidistant from the center point,
while a hypercycle is defined as the set of points equidistant from a center
geodesic. This lemma is simply a rephrasing of origin definitions, using
results in Proposition A.1. ∎

The normalized vector in $\mathbb{E}^{2,1}$ behaves discontinuously when
switching from time- or space-like to light-like. We shall pass to the Klein
model for a solution and a more consistent computation. Let $K:=\\{\
\boldsymbol{x}\in\mathbb{E}^{2,1}\ \lvert\ x_{3}=1\ \\}$. The _Klein model_
for hyperbolic plane is obtained by mapping $\mathbb{H}$ to $K$ through a
projection centered at the origin of $\mathbb{E}^{2,1}$. Then $\mathbb{H}$ is
mapped to the unit disk $\mathbb{D}_{K}:=\\{\ (x_{1},x_{2},1)\ \lvert\
x_{1}^{2}+x_{2}^{2}<1\ \\}$ of $K$, endowed with the push-forward metric.
Essentially the whole space $\mathbb{E}^{2,1}$ projects to a real projective
plane. However, under some suitable settings, the plane $K$ is enough for our
usage.

Geodesics in $\mathbb{D}_{K}$ are simply straight lines inside the disk. And
every point $\boldsymbol{x}$ outside the closed unit disk gives rise to a dual
geodesic $\boldsymbol{x}^{\perp}$ as before. This geodesic is the line that
connects the points of tangency of the two tangent lines to $\mathbb{D}_{K}$
at point $\boldsymbol{x}$. Formulae in Proposition A.1 are still available.
And the center of a generalized circle in Lemma A.2 can be chosen anywhere on
$K$.

Figure 8: The relations between hyperbolic distance and Lorentzian inner
product in various case of Proposition A.1, shown in Klein model.

######  Lemma A.3.

The geodesic curvature of the generalized circle centered at
$\boldsymbol{c}=(c_{1},c_{2},1)\in K$ and passing through
$\boldsymbol{z}=(z_{1},z_{2},1)\in\mathbb{D}_{K}$, with
$\langle\boldsymbol{c},\boldsymbol{z}\rangle<0$, is

(A.2)
$k=\frac{-\langle\boldsymbol{c},\boldsymbol{z}\rangle}{\sqrt{\langle\boldsymbol{c},\boldsymbol{z}\rangle^{2}-\langle\boldsymbol{c},\boldsymbol{c}\rangle\langle\boldsymbol{z},\boldsymbol{z}\rangle}}\
.$

The formula still holds when $\boldsymbol{c}$ is light-like.

###### Proof.

For circles and hypercycles, the formula is obtained by Proposition A.1 and
the relations between radius and curvature in Table 1. For horocycles, the
center $\boldsymbol{c}$ is light-like and
$\langle\boldsymbol{c},\boldsymbol{c}\rangle=0$. By the assumption
$\langle\boldsymbol{c},\boldsymbol{z}\rangle<0$, the right-hand-side of (A.2)
is $1$. ∎

######  Proposition A.4.

Let $\Gamma$ be an arc of a generalized circle of geodesic curvature $k$, with
end points $\boldsymbol{x},\boldsymbol{y}\in\mathbb{D}_{K}$. Let
$d=d(\boldsymbol{x},\boldsymbol{y})>0$ and $\theta$ be the generalized inner
angle of $\Gamma$. We require $\theta\leq\pi$ when $k>1$. Then the length $l$
and the total geodesic curvature $T$ of $\Gamma$ is given by

(A.3) $l(k,d)=\frac{\theta}{\sqrt{\lvert 1-k^{2}\rvert}},\quad
T(k,d)=\frac{k\theta}{\sqrt{\lvert 1-k^{2}\rvert}}$

for all $k>0,k\neq 1$ and $d>0$, where

(A.4) $1+(1-k^{2})(\cosh d-1)=\left\\{\begin{array}[]{ll}\cos\theta&,\
\textrm{if}\ k>1\\\ \cosh\theta&,\ \textrm{if}\ k<1\end{array}\right.\ .$

Furthermore, the formula extends over $k=1$ with continuous first order
derivative. In particular, when $k=1$ and $0<d<+\infty$,

(A.5) $l(1,d)=T(1,d)=\sqrt{2(\cosh d-1)}\ .$

Hence both $l(k,d)$ and $T(k,d)$ are $C^{1}$-functions over
$\mathbb{R}_{>0}^{2}$.

###### Proof.

(A.3) is simply the definition. Also see Table 1.

When $k>1$, let $\boldsymbol{c}$ be the center of the circle. By hyperbolic
cosine law in the triangle with vertices
$\boldsymbol{x},\boldsymbol{y},\boldsymbol{c}$, $\cosh
d=\cosh^{2}r-\sinh^{2}r\cos\theta\ .$ Then (A.4) for the $k>1$ case is
obtained by direct substituting.

When $k<1$, the arc $\Gamma$, the center geodesic and the two radii at
$\boldsymbol{x},\boldsymbol{y}$ bound a quadrilateral with 2 right angles.
Then we have $\cosh d=\cosh^{2}r\cosh\theta-\sinh^{2}r\ .$ The formula is
obtained again by direct substituting.

When $k=1$, (A.5) can be obtained by passing to the upper half-plane model of
hyperbolic plane, with the horocycle mapped to a horizontal line.

To prove the continuity, we need the following Taylor series when $x\to
0^{+}$:

(A.6) $\displaystyle\arccos(1-x)$
$\displaystyle=\sqrt{2x}+\frac{x^{3/2}}{6\sqrt{2}}+o(x^{2})\ ,$
$\displaystyle\mathrm{arccosh}(1+x)$
$\displaystyle=\sqrt{2x}-\frac{x^{3/2}}{6\sqrt{2}}+o(x^{2})\ .$

For simplicity and future application, let

(A.7) $D:=\cosh d-1\quad\textrm{and}\quad K:=1-k^{2}\ .$

Then $D^{\prime}(d)=\sinh d=\sqrt{D(D+2)}$.

Whenever $d$ converges in $\mathbb{R}_{>0}$, $D$ also converges in
$\mathbb{R}_{>0}$ and we have

$\displaystyle\lim_{k\to 1^{+}}\frac{\arccos\big{(}1+KD\big{)}}{\sqrt{-K}}$
$\displaystyle=\lim_{K\to 0^{-}}\frac{\sqrt{-2KD}}{\sqrt{-K}}=\sqrt{2D},$
$\displaystyle\lim_{k\to 1^{-}}\frac{\arccos\big{(}1+KD\big{)}}{\sqrt{K}}$
$\displaystyle=\lim_{K\to 0^{+}}\frac{\sqrt{2KD}}{\sqrt{K}}=\sqrt{2D}.$

Thus $l(k,d)$ is continuous on $\mathbb{R}_{>0}^{2}$, and so does $T(k,d)$.

It can be shown by direct computation that when $k\neq 1$,

$\frac{\partial l}{\partial d}=\sqrt{\frac{2+D}{2+KD}}$

and it extends continuously over $k=1$. The following would also be useful:

(A.8) $\frac{\partial l}{\partial D}=\frac{1}{\sqrt{D(2+KD)}}\ .$

The continuity of $\partial l/\partial k$ requires detailed computation. When
$k>1$,

$\frac{\partial l}{\partial
k}=\frac{2kD}{-K\sqrt{D(2+KD)}}-\frac{k\arccos(1+KD)}{-K\sqrt{-K}}.$

By (A.6),

$\displaystyle\lim_{k\to 1^{+}}\frac{\partial l}{\partial k}$
$\displaystyle=\lim_{K\to
0^{-}}\left(\frac{2\sqrt{D}}{-K\sqrt{2+KD}}-\frac{\sqrt{-2KD}+(-KD)^{3/2}/6\sqrt{2}}{-K\sqrt{-K}}\right)$
$\displaystyle=\lim_{K\to
0^{-}}\frac{\sqrt{2D}}{-K}\left(\frac{\sqrt{2}}{\sqrt{2+KD}}-1\right)-\frac{D\sqrt{D}}{6\sqrt{2}}=\frac{D\sqrt{D}}{3\sqrt{2}}\
.$

When $k<1$,

$\frac{\partial l}{\partial k}=\frac{k\
\mathrm{arccosh}(1+KD)}{K\sqrt{K}}-\frac{2kD}{K\sqrt{D(2+KD)}}\ .$

By (A.6),

$\displaystyle\lim_{k\to 1^{-}}\frac{\partial l}{\partial k}$
$\displaystyle=\lim_{K\to
0^{+}}\left(\frac{\sqrt{2KD}-(KD)^{3/2}/6\sqrt{2}}{K\sqrt{K}}-\frac{2\sqrt{D}}{K\sqrt{2+KD}}\right)$
$\displaystyle=\lim_{K\to
0^{+}}\frac{\sqrt{2D}}{K}\left(1-\frac{\sqrt{2}}{\sqrt{2+KD}}\right)-\frac{D\sqrt{D}}{6\sqrt{2}}=\frac{D\sqrt{D}}{3\sqrt{2}}\
.$

Hence $\frac{\partial l}{\partial k}$ is continuous at $k=1$. These show that
$l=l(k,d)$ is a $C^{1}$-function on $\mathbb{R}_{>0}^{2}$, and so does
$T=T(k,d)=k\cdot l(k,d)$. ∎

###  A.2 Three-circle configuration

Now we turn to the three-circle configuration and the circle packed triangle,
using the hyperboloid and Klein model this time.

We start with the hyperboloid model. Suppose in a three-circle configuration,
for $i=1,2,3$, the circle $C_{i}$ is given by $C(\boldsymbol{c_{i}},R_{i})$ as
Lemma A.2, $\boldsymbol{c_{i}}\in\mathbb{E}^{2,1},\ R_{i}\in\mathbb{R}$. With
the help of dual geodesic, in the definition of (possibly degenerated)
generalized circle packed triangle, when some $C_{i}$ is actually a hypercycle
(or a geodesic), the vertex $\boldsymbol{v_{i}}$ is a segment on the dual
geodesic $\boldsymbol{c_{i}}^{\perp}$. An edge $e_{ij}$ is the geodesic
segment realizing the inner product
$\langle\boldsymbol{\bar{c}_{i}},\boldsymbol{\bar{c}_{i}}\rangle$ as Figure 8.
This segment may have infinite or zero length.

Before passing to the Klein model, we need some normalization. Since the dual
triangle always has non-empty interior (even for degenerated ones in Section
3), we can apply a hyperbolic isometry so that $(0,0,1)$ lies in the interior
of dual triangle. Under such setting, the boundary of circle packed triangle
never passes $(0,0,1)$ and all centers $\boldsymbol{c_{i}}$ will not lie on
the horizontal plane $\\{x_{3}=0\\}\subset\mathbb{E}^{2,1}$. Hence everything
can be projected to the plane $K$, and the origin $(0,0,1)$ always lies in the
interior of a triangle. We will stick to this normalization for circle packed
triangles form now on.

Now let $\boldsymbol{x_{i}}$ be the projection of $\boldsymbol{c_{i}}$ in $K$.
Since the three circles are externally tangent to each other, and with the
normalization condition, we have
$\langle\boldsymbol{x_{i}},\boldsymbol{x_{j}}\rangle<0$ for any $i\neq j$. The
vertices of the dual triangle
$\boldsymbol{t_{12}},\boldsymbol{t_{23}},\boldsymbol{t_{31}}$ must lie in
$\mathbb{D}_{K}$. Let $(i,j,k)$ be a cyclic re-ordering of $(1,2,3)$. By our
setting, $\boldsymbol{t_{ij}}$ lies inside the straight line segment
connecting $\boldsymbol{x_{i}}$ and $\boldsymbol{x_{j}}$ , hence

$\exists\ \lambda_{k}^{i},\lambda_{k}^{j}\in(0,1)\ \textrm{with}\
\lambda_{k}^{i}+\lambda_{k}^{j}=1,\ \textrm{such that}\
\boldsymbol{t_{ij}}=\lambda_{k}^{i}\boldsymbol{x_{i}}+\lambda_{k}^{j}\boldsymbol{x_{j}}$

as vectors in $K\subset\mathbb{E}^{2,1}$. See Figure 9 for the settings. We
will compute required quantities by studying the relations between these
coefficients.

Let $c_{ij}:=\langle\boldsymbol{x_{i}},\boldsymbol{x_{j}}\rangle$ for all
$i,j\in\\{1,2,3\\}$, and define $\delta_{ij}:=c_{ij}^{2}-c_{ii}c_{jj}$ when
$i\neq j$. Then when $i\neq j$, $c_{ij}<0$ by assumption. By the condition
that the straight line segment connecting $\boldsymbol{x_{i}}$ and
$\boldsymbol{x_{j}}$ intersects $\mathbb{D}_{K}$, it can be shown that
$\delta_{ij}>0$.

Figure 9: The same circle packed triangle as Figure 3 but viewed in the Klein
model. Note that circles are general conics in Klein model. The origin of
$\mathbb{D}_{K}$ is assumed to be contained in the dual triangle. All used
notations are gathered on the right.

Before the computation, note the symmetry of the settings of three-circle
configuration. Any formula stated for a cyclic re-ordering of $(1,2,3)$ would
still hold if any two indices are exchanged. This is called _the symmetry
principle_ for short.

It is remarkable that all the settings above and computations below are
available even when some $k_{i}=1$. That is the core reason for choosing the
Klein model and using a proof in the flavor of projective geometry.

###### Proof of Theorem 4.1.

We would use Proposition A.4 to convert the computation of $T_{i}$ into the
computation of distance between two points of tangency
$\boldsymbol{t_{ki}},\boldsymbol{t_{ij}}$.

Since $\boldsymbol{t_{ki}},\boldsymbol{t_{ij}}\in\mathbb{D}_{K}$, by
Proposition A.1.1, $\cosh
d(\boldsymbol{t_{ki}},\boldsymbol{t_{ij}})=-\frac{\langle\boldsymbol{t_{ki}},\boldsymbol{t_{ij}}\rangle}{\sqrt{\langle\boldsymbol{t_{ki}},\boldsymbol{t_{ki}}\rangle\langle\boldsymbol{t_{ij}},\boldsymbol{t_{ij}}\rangle}}$.
So we need to compute $\langle\boldsymbol{t_{ki}},\boldsymbol{t_{ij}}\rangle$
and $\langle\boldsymbol{t_{ki}},\boldsymbol{t_{ki}}\rangle$.

It can be shown by a direct computation that
$\langle\boldsymbol{x_{i}},\boldsymbol{t_{ki}}\rangle^{2}-\langle\boldsymbol{x_{i}},\boldsymbol{x_{i}}\rangle\langle\boldsymbol{t_{ki}},\boldsymbol{t_{ki}}\rangle=(\lambda_{j}^{k})^{2}\delta_{ik}$.
Since $\boldsymbol{t_{ki}}$ lies on the circle centered at
$\boldsymbol{x_{i}}$, by Lemma A.3, we have

(A.9)
$\langle\boldsymbol{x_{i}},\boldsymbol{t_{ki}}\rangle=-\lambda_{j}^{k}k_{i}\sqrt{\delta_{ik}}\
.$

Applying this formula and the symmetry principle, we have

(A.10)
$\langle\boldsymbol{t_{ki}},\boldsymbol{t_{ki}}\rangle=\lambda_{j}^{i}\langle\boldsymbol{x_{i}},\boldsymbol{t_{ki}}\rangle+\lambda_{j}^{k}\langle\boldsymbol{x_{k}},\boldsymbol{t_{ki}}\rangle=-\lambda_{j}^{i}\lambda_{j}^{k}\sqrt{\delta_{ik}}(k_{i}+k_{k})\
.$

Note that
$\langle\boldsymbol{x_{i}},\boldsymbol{t_{ki}}\rangle=\lambda_{j}^{i}c_{ii}+\lambda_{j}^{k}c_{ik}$.
Then (A.9) implies
$\lambda_{j}^{k}(k_{i}\sqrt{\delta_{ik}}+c_{ik})=-\lambda_{j}^{i}c_{ii}$.
Since $\lambda_{j}^{i}\lambda_{j}^{k}\neq 0$, by the symmetry principle, we
have
$(c_{ik}+k_{i}\sqrt{\delta_{ik}})(c_{ik}+k_{k}\sqrt{\delta_{ik}})=c_{ii}c_{kk}$.
Substituting $\delta_{ik}=c_{ik}^{2}-c_{ii}c_{kk}$ into the above equation and
simplifying, we obtain

(A.11) $-c_{ik}=\frac{1+k_{i}k_{k}}{k_{i}+k_{k}}\sqrt{\delta_{ik}}\ .$

On the other hand, both $\boldsymbol{t_{ki}},\boldsymbol{t_{ij}}$ lie on the
circle centered at $\boldsymbol{x_{i}}$. By Lemma A.2,
$\frac{\langle\boldsymbol{x_{i}},\boldsymbol{t_{ki}}\rangle^{2}}{\langle\boldsymbol{t_{ki}},\boldsymbol{t_{ki}}\rangle}=\frac{\langle\boldsymbol{x_{i}},\boldsymbol{t_{ij}}\rangle^{2}}{\langle\boldsymbol{t_{ij}},\boldsymbol{t_{ij}}\rangle}$.
A direct computation gives
$\frac{\langle\boldsymbol{x_{i}},\boldsymbol{t_{ki}}\rangle^{2}}{\langle\boldsymbol{t_{ki}},\boldsymbol{t_{ki}}\rangle}=\frac{(\lambda_{j}^{k})^{2}\delta_{ik}}{\langle\boldsymbol{t_{ki}},\boldsymbol{t_{ki}}\rangle}-c_{ii}$.
Hence we have
$\frac{(\lambda_{j}^{k})^{2}\delta_{ik}}{\langle\boldsymbol{t_{ki}},\boldsymbol{t_{ki}}\rangle}=\frac{(\lambda_{k}^{j})^{2}\delta_{ij}}{\langle\boldsymbol{t_{ij}},\boldsymbol{t_{ij}}\rangle}$.
Using (A.10), we get

(A.12)
$\frac{\lambda_{k}^{j}\lambda_{j}^{i}\sqrt{\delta_{ij}}}{k_{i}+k_{j}}=\frac{\lambda_{j}^{k}\lambda_{k}^{i}\sqrt{\delta_{ik}}}{k_{i}+k_{k}}\
.$

Now that all quantities in the above are non-zero, by a cyclic product of the
formula, we get the following Ceva type formula:

(A.13)
$\lambda_{i}^{j}\lambda_{j}^{k}\lambda_{k}^{i}=\lambda_{i}^{k}\lambda_{k}^{j}\lambda_{j}^{i}\
.$

Finally we are able to compute $\cosh
d(\boldsymbol{t_{ki}},\boldsymbol{t_{ij}})$. By (A.10) and (A.12),

$\displaystyle\langle\boldsymbol{t_{ki}},\boldsymbol{t_{ki}}\rangle\langle\boldsymbol{t_{ij}},\boldsymbol{t_{ij}}\rangle$
$\displaystyle=\lambda_{j}^{k}\lambda_{j}^{i}\sqrt{\delta_{ik}}(k_{i}+k_{k})\
\lambda_{k}^{i}\lambda_{k}^{j}\sqrt{\delta_{ij}}(k_{i}+k_{j})$
$\displaystyle=[\lambda_{j}^{k}\lambda_{k}^{i}\sqrt{\delta_{ik}}(k_{i}+k_{j})]^{2}\
.$

And

$\displaystyle\langle\boldsymbol{t_{ij}},\boldsymbol{t_{ki}}\rangle$
$\displaystyle=\lambda_{k}^{i}\langle\boldsymbol{x_{i}},\boldsymbol{t_{ki}}\rangle+\lambda_{k}^{j}\lambda_{j}^{i}c_{ij}+\lambda_{k}^{j}\lambda_{j}^{k}c_{jk}$
$\displaystyle=-\lambda_{k}^{i}\lambda_{j}^{k}k_{i}\sqrt{\delta_{ik}}-\lambda_{k}^{j}\lambda_{j}^{i}\frac{1+k_{i}k_{j}}{k_{i}+k_{j}}\sqrt{\delta_{ij}}+\lambda_{k}^{j}\lambda_{j}^{k}c_{jk}\quad(\textrm{By
(\ref{Appeq:xitj}) and (\ref{Appeq:cij})})$
$\displaystyle=-\lambda_{k}^{i}\lambda_{j}^{k}k_{i}\sqrt{\delta_{ik}}-\frac{\lambda_{j}^{k}\lambda_{k}^{i}\sqrt{\delta_{ik}}}{k_{i}+k_{k}}(1+k_{i}k_{j})+\lambda_{k}^{j}\lambda_{j}^{k}c_{jk}\quad(\textrm{By
(\ref{Appeq:sqrdelta})})$
$\displaystyle=-\lambda_{j}^{k}\lambda_{k}^{i}\sqrt{\delta_{ik}}\left(k_{i}+\frac{1+k_{i}k_{j}}{k_{i}+k_{k}}\right)+\lambda_{k}^{j}\lambda_{j}^{k}c_{jk}\
.$

Hence

$\frac{\langle\boldsymbol{t_{ki}},\boldsymbol{t_{ij}}\rangle}{\sqrt{\langle\boldsymbol{t_{ki}},\boldsymbol{t_{ki}}\rangle\langle\boldsymbol{t_{ij}},\boldsymbol{t_{ij}}\rangle}}=-\frac{k_{i}+\frac{1+k_{i}k_{j}}{k_{i}+k_{k}}}{k_{i}+k_{j}}+\frac{\lambda_{k}^{j}c_{jk}}{\lambda_{k}^{i}\sqrt{\delta_{ik}}(k_{i}+k_{j})}\
.$

By (A.13) and (A.12),

$\frac{\lambda_{k}^{j}}{\lambda_{k}^{i}}=\frac{\lambda_{i}^{j}\lambda_{j}^{k}}{\lambda_{j}^{i}\lambda_{i}^{k}}=\frac{\sqrt{\delta_{ik}}}{\sqrt{\delta_{jk}}}\frac{k_{k}+k_{j}}{k_{k}+k_{i}}\
.$

Together with (A.11),

$\frac{\lambda_{k}^{j}c_{jk}}{\lambda_{k}^{i}\sqrt{\delta_{ik}}(k_{i}+k_{j})}=\frac{c_{jk}}{(k_{i}+k_{j})}\frac{1}{\sqrt{\delta_{jk}}}\frac{k_{k}+k_{j}}{k_{k}+k_{i}}=\frac{-(1+k_{j}k_{k})}{(k_{i}+k_{j})(k_{i}+k_{k})}\
.$

Then we have

(A.14) $\cosh
d(\boldsymbol{t_{ki}},\boldsymbol{t_{ij}})=-\frac{\langle\boldsymbol{t_{ki}},\boldsymbol{t_{ij}}\rangle}{\sqrt{\langle\boldsymbol{t_{ki}},\boldsymbol{t_{ki}}\rangle\langle\boldsymbol{t_{ij}},\boldsymbol{t_{ij}}\rangle}}=1+\frac{2}{(k_{i}+k_{j})(k_{i}+k_{k})}\
.$

The desired formula (4.2) is obtained from the above and Proposition A.4.

The smoothness is obtained by the chain rules. Consider $(i,j,k)=(1,2,3)$ as
an example. Let $l$ be the function of two variables in Proposition A.4, and
$D$ be defined as (A.7). Now $D$ is also a function of all $k_{i}$’s, so we
have

$\frac{\partial l_{1}}{\partial k_{1}}(k_{1},k_{2},k_{3})=\frac{\partial
l}{\partial k}(k_{1},D_{1})+\frac{\partial l}{\partial
D}(k_{1},D_{1})\cdot\frac{\partial D_{1}}{\partial k_{1}}(k_{1},k_{2},k_{3}),$

where $D_{1}=2/(k_{1}+k_{2})(k_{1}+k_{3})$ from (A.14). Similarly,

$\frac{\partial l_{1}}{\partial k_{2}}(k_{1},k_{2},k_{3})=\frac{\partial
l}{\partial k}(k_{1},D_{1})\cdot\frac{\partial D_{1}}{\partial
k_{2}}(k_{1},k_{2},k_{3}).$

We have shown the continuity of $\frac{\partial l}{\partial k}$ and
$\frac{\partial l}{\partial D}$ in the proof of Proposition A.4, and $D_{1}$
is a smooth function on $\mathbb{R}^{3}$ by (A.14). Hence, $l_{1}$ is a
$C^{1}$-function of all $k_{i}$’s. And so does the total geodesic curvature
$T_{1}=k_{1}\cdot l_{1}$.

∎

###### Proof of Corollary 4.2.

We shall use the relations in the bigon intersected by two orthogonal circles.
See [3, Lemma 2.7].

When $k_{1}>1$, the relation between the inner angle $\theta_{1}$ and
curvature $k_{1},k_{P}$ is given by

$\cot\frac{\theta_{1}}{2}=\frac{k_{P}}{\sqrt{k_{1}^{2}-1}}\ .$

By (4.2),
$\cos\theta_{1}=1+\frac{2(1-k_{1}^{2})}{(k_{1}+k_{2})(k_{1}+k_{3})}$. Then the
desired formula (4.3) is obtained by the trigonometric identity $\cos
2\theta=(\cot^{2}\theta-1)/(\cot^{2}\theta+1)$.

When $k_{1}<1$, we have

$\coth\frac{\theta_{1}}{2}=\frac{k_{P}}{\sqrt{1-k_{1}^{2}}}$

and the identity $\cosh 2\theta=(\coth^{2}\theta+1)/(\coth^{2}\theta-1)$. The
computation is similar.

When $k_{1}=1$, we shall consider the length of the horocycle arc and we have

$l_{1}=T_{1}=\frac{2}{k_{P}}\ .$

Together with the formula for $T_{1}$ when $k_{1}=1$, we have
$k_{P}^{2}=(k_{2}+1)(k_{3}+1)$. This shows (4.3) still holds when some of the
$k_{i}$’s is $1$. ∎

###### Proof of Lemma 4.6, continued.

By the proof of Proposition A.4, we have

$\frac{\partial l}{\partial
D}=\frac{1}{\sqrt{D(2+KD)}}\quad\textrm{and}\quad\frac{\partial l}{\partial
k}=\frac{k}{K}\left(l-2\sqrt{\frac{D}{2+KD}}\right)\ .$

When $k_{3}=0$ and $k_{1},k_{2}>0$, by (A.14),

$D_{1}:=\cosh d(t_{12},t_{31})-1=\frac{2}{k_{1}(k_{1}+k_{2})},\quad
D_{2}:=\cosh d(t_{12},t_{23})-1=\frac{2}{k_{2}(k_{1}+k_{2})}$

are always well-defined. Hence the above formulae still hold for $k_{3}=0$
when taking partial derivative with respect to $k_{1}$ or $k_{2}$ on
$\partial^{\\{3\\}}\Omega_{k}^{\Delta}$. By the chain rules,

$\frac{\partial l_{1}}{\partial k_{2}}=\frac{\partial l}{\partial
D}(k_{1},D_{1})\cdot\frac{\partial D_{1}}{\partial
k_{2}}=\frac{1}{\sqrt{D_{1}\left(2+(1-k_{1}^{2})D_{1}\right)}}\frac{-D_{1}}{k_{1}+k_{2}}=\frac{-1}{(k_{1}+k_{2})\sqrt{1+k_{1}k_{2}}}\
,$ $\frac{\partial l_{2}}{\partial k_{1}}=\frac{\partial l}{\partial
D}(k_{2},D_{2})\cdot\frac{\partial D_{2}}{\partial
k_{1}}=\frac{1}{\sqrt{D_{2}\left(2+(1-k_{2}^{2})D_{2}\right)}}\frac{-D_{2}}{k_{1}+k_{2}}=\frac{-1}{(k_{1}+k_{2})\sqrt{1+k_{1}k_{2}}}\
.$

Hence $\frac{\partial l_{1}}{\partial k_{2}}=\frac{\partial l_{2}}{\partial
k_{1}}$. And we prove that $T_{1}\mathrm{d}S_{1}+T_{2}\mathrm{d}S_{2}$ is
closed. ∎

## References

  * [1] L. Ambrosio, N. Gigli, and G. Savaré, _Gradient flows in metric spaces and in the space of probability measures,_ 2nd ed., Lectures in Mathematics ETH Zürich, Birkh äuser Verlag, Basel, 2008
  * [2] E. M. Andreev, _Convex polyhedra in Lobachevsky spaces._ (Russian) Mat. Sb. (N.S.) 81 (123), 1970, 445–478.
  * [3] Te Ba, Guangming Hu and Yu Sun, _Circle packings and total geodesic curvatures in hyperbolic background geometry,_ preprint arxiv:2307.13572
  * [4] A. I. Bobenko, B. A. Springborn, _Variational principles for circle patterns and Koebe’s theorem,_ Trans. Amer. Math. Soc. 356 (2004), 659–689.
  * [5] A. J. Casson and S. A. Bleiler, _Automorphisms of surfaces after Nielsen and Thurston_ , London Mathematical Society Student Texts, vol. 9, Cambridge University Press, Cambridge, 1988.
  * [6] B. Chow and F. Luo, _Combinatorial Ricci flows on surfaces_ , J. Differential Geom. 63, (2003), 97–129.
  * [7] Y. Colin de Verdiere, _Un principe variationnel pour les empilements de cercles_ , Invent. Math. 104, 1991, 655–669.
  * [8] J. Dai, X. D. Gu and F. Luo, _Variational principles for discrete surfaces_ , Advanced Lectures in Mathematics 4, Higher Education Press, Beijing, 2008.
  * [9] L. R. Ford, Jr. and D. R. Fulkerson, _Flows in networks,_ Princeton University Press, Princeton, NJ, 1962.
  * [10] H. Ge, B. Hua, and P. Zhou, _A combinatorial curvature flow in spherical background geometry._ J. Funct. Anal., 286(7), 2024, 110335.
  * [11] H. Ge and X. Xu, _On a combinatorial curvature for surfaces with inversive distance circle packing metrics,_ J. Funct. Anal. 275 (2018), No. 3, 523–558.
  * [12] S. Gupta, _Limits of harmonic maps and crowned hyperbolic surfaces_ , Trans. Amer. Math. Soc. 372 (11), 2019, 7573–7596.
  * [13] R. S. Hamilton, _The Ricci flow on surfaces,_ Mathematics and general relativity (Santa Cruz, 1986), Contemp. Math., 71, 1988, 237–262.
  * [14] G. Hu, Z. Lei, Y. Sun, and P. Zhou, _The Convergence of Prescribed Combinatorial Ricci Flows for Total Geodesic Curvatures in Spherical Background Geometry,_ Arxiv:2404.05982.
  * [15] P. Koebe, _Kontaktprobleme der konformen Abbildung,_ Abh. S$\ddot{a}$chs. Akad. Wiss. Leipzig Math.-Natur. 88 (1936), 141-164.
  * [16] G. Leibon,_Characterizing the Delaunay decompositions of compact hyperbolic surfaces,_ Geom. Topol., 6, 2002, 361–391.
  * [17] F. Luo, _A characterization of spherical polyhedral surfaces,_ J. Differential Geom., 74, 2006, 407–424.
  * [18] F. Luo, _On Teichmüller spaces of surfaces with boundary,_ Duke Math. J., 139, 2007, 463–482.
  * [19] F. Luo. _Rigidity of polyhedral surfaces I,_ J. Differential Geom., 96(2), 2014, 241–302.
  * [20] X. Nie, _On circle patterns and spherical conical metrics_ , Proc. Amer. Math. Soc., 152(02), 2024, 843–853.
  * [21] J. G. Ratcliffe, _Foundations of Hyperbolic Manifolds,_ Springer-Verlag, Tokyo, 1994.
  * [22] I. Rivin, _Euclidean structures on simplicial surfaces and hyperbolic volume,_ Ann. of Math., 139(2), 1994, 553–580.
  * [23] A. Takatsu, _Convergence of combinatorial Ricci flows to degenerate circle patterns,_ Trans. Amer. Math. Soc., 372(11), 2019, 7597-7617.
  * [24] A. Tamburelli, _Wild Globally Hyperbolic Maximal Anti–de Sitter Structures_ , J. Lond. Math. Soc. (2) (103), 2021, 198–221.
  * [25] W. P. Thurston, _Geometry and topology of 3-manifolds,_ Princeton lecture notes, 1976.

Guangming Hu

College of Science, Nanjing University of Posts and Telecommunications,
Nanjing, 210003, P.R. China. Email address<EMAIL_ADDRESS>
Sicheng Lu

The Institute of Geometry and Physics, University of Science and Technology of
China, Hefei, 230026, P. R. China  Email address<EMAIL_ADDRESS>
Dong Tan

Guangxi Center for Mathematical Research, Guangxi University, 530004, Nanning,
China Email address<EMAIL_ADDRESS>
Youliang Zhong

School of Mathematics, South China University of Technology, 510641,
Guangzhou, P. R. China.  Email address<EMAIL_ADDRESS>
Puchun Zhou

School of Mathematical Sciences, Fudan University, Shanghai, 200433, P.R.
China  Email address<EMAIL_ADDRESS>