# Rigidity and deformation of discrete conformal structures on polyhedral
surfaces

Xu Xu

###### Abstract

Discrete conformal structure on polyhedral surfaces is a discrete analogue of
the conformal structure on smooth surfaces, which includes tangential circle
packing, Thurston’s circle packing, inversive distance circle packing and
vertex scaling as special cases and generalizes them to a very general
context. Glickenstein [39] conjectured the rigidity of discrete conformal
structures on polyhedral surfaces, which includes Luo’s conjecture on the
rigidity of vertex scaling [57] and Bowers-Stephenson’s conjecture on the
rigidity of inversive distance circle packings [6] on polyhedral surfaces as
special cases. In this paper, we prove Glickenstein’s conjecture using a
variational principle.

We further study the deformation of discrete conformal structures on
polyhedral surfaces by combinatorial curvature flows. It is proved that the
combinatorial Ricci flow for discrete conformal structures, which is a
generalization of Chow-Luo’s combinatorial Ricci flow for circle packings [7]
and Luo’s combinatorial Yamabe flow for vertex scaling [57], could be extended
to exist for all time and the extended combinatorial Ricci flow converges
exponentially fast for any initial data if the discrete conformal structure
with prescribed combinatorial curvature exists. This confirms another
conjecture of Glickenstein [39] on the convergence of the combinatorial Ricci
flow and provides an effective algorithm for finding discrete conformal
structures with prescribed combinatorial curvatures.

The relationship of discrete conformal structures on polyhedral surfaces and
$3$-dimensional hyperbolic geometry is also discussed. As a result, we obtain
some new convexities of the co-volume functions for some generalized
$3$-dimensional hyperbolic tetrahedra.

MSC (2020): 52C25; 52C26

Keywords:  Rigidity; Combinatorial Ricci flow; Discrete conformal structure;
Polyhedral surfaces

###### Contents

  1. 1 Introduction
    1. 1.1 Polyhedral surfaces, discrete conformal structures and the main rigidity results
    2. 1.2 Combinatorial curvature flows for discrete conformal structures on polyhedral surfaces
    3. 1.3 Relationships with $3$-dimensional hyperbolic geometry
    4. 1.4 Basic ideas of the proof of Theorem 1.1
    5. 1.5 The organization of the paper
  2. 2 Euclidean discrete conformal structures
    1. 2.1 Admissible space of Euclidean discrete conformal structures for a triangle
    2. 2.2 Negative semi-definiteness of the Jacobian matrix in the Euclidean case
    3. 2.3 Rigidity of Euclidean discrete conformal structures
  3. 3 Hyperbolic discrete conformal structures
    1. 3.1 Admissible space of hyperbolic discrete conformal structures for a triangle
    2. 3.2 Negative definiteness of the Jacobian matrix in the hyperbolic case
    3. 3.3 Rigidity of hyperbolic discrete conformal structures
  4. 4 Deformation of discrete conformal structures
  5. 5 Relationships of discrete conformal structures on polyhedral surfaces and 3-dimensional hyperbolic geometry
    1. 5.1 Construction of discrete conformal structures via generalized hyperbolic tetrahedra
    2. 5.2 Convexities of co-volume functions of generalized hyperbolic tetrahedra
  6. 6 Open problems
    1. 6.1 Convergence of discrete conformal structures to the Riemann mapping
    2. 6.2 Discrete uniformization theorems for discrete conformal structures
    3. 6.3 Convergence of combinatorial curvature flows with surgery

##  1 Introduction

Conformal structure on Riemannian manifolds is one of the most important
geometric structures that has been extensively studied in differential
geometry, which defines Riemannian metrics pointwisely by scalar functions
defined on the manifolds. Discrete conformal structure on polyhedral manifolds
is a discrete analogue of the conformal structure on Riemannian manifolds,
which assigns the discrete metrics by scalar functions defined on the
vertices. There have been many research activities on different types of
discrete conformal structures on manifolds since the work of Thurston [81],
including the tangential circle packing, Thurston’s circle packing, inversive
distance circle packing and vertex scaling on surfaces, sphere packing and
Thurston’s sphere packing on $3$-dimensional manifolds and others. Most of the
existing discrete conformal structures were invented and studied individually
in the past. The generic notion of discrete conformal structures on polyhedral
surfaces was introduced recently independently by Glickenstein [38] and
Glickenstein-Thomas [41] from Riemannian geometry perspective and by Zhang-
Guo-Zeng-Luo-Yau-Gu [96] from Bobenko-Pinkall-Springborn’s observation [3] on
the relationships of vertex scaling on polyhedral surfaces and $3$-dimensional
hyperbolic geometry, which includes the discrete conformal structures from
different types of circle packings and vertex scaling on polyhedral surfaces
as special cases and generalizes them to a very general context.

In this paper, we study the geometry of generic discrete conformal structures
on polyhedral surfaces. The global rigidity of discrete conformal structures
on polyhedral surfaces is proved, which unifies and generalizes the known
rigidity results for different types of circle packings and vertex scaling on
polyhedral surfaces. We further use the combinatorial curvature flows,
including the combinatorial Ricci flow and combinatorial Calabi flow, to study
the deformation of discrete conformal structures on polyhedral surfaces. To
handle the potential singularities along the combinatorial Ricci flow, we
extend the combinatorial curvature by constants. It is proved that the
solution of combinatorial Ricci flow could be extended to exist for all time
and the extended combinatorial Ricci flow converges exponentially fast if the
discrete conformal structure with prescribed combinatorial curvature exists.
The longtime behavior of combinatorial Calabi flow is also discussed. The
combinatorial curvature flows provide effective algorithms for finding
discrete conformal structures with prescribed combinatorial curvatures.
Motivated by Bobenko-Pinkall-Spingborn’s observation [3] and Zhang-Guo-Zeng-
Luo-Yau-Gu’s work [96], we further discuss the relationship of discrete
conformal structures on polyhedral surfaces and $3$-dimensional hyperbolic
geometry. A natural geometric interpolation of the structure conditions on the
weights used in the main results in terms of $3$-dimensional hyperbolic
geometry is given. As a corollary, some new convexities of co-volume functions
for some generalized hyperbolic tetrahedra in $\mathbb{H}^{3}$ are obtained.

###  1.1 Polyhedral surfaces, discrete conformal structures and the main
rigidity results

Polyhedral surface is a discrete analogue of Riemannian surface. Suppose
$(M,\mathcal{T})$ is a connected closed triangulated surface with a
triangulation $\mathcal{T}$, which is the quotient of a finite disjoint union
of triangles by identifying all the edges of triangles in pair by
homeomorphism. We use $V,E,F$ to denote the set of vertices, edges and faces
in the triangulation $\mathcal{T}$ respectively. For simplicity, we will use
one index to denote a vertex (such as $i\in V$ or $v_{i}\in V$), two indices
to denote an edge (such as $\\{ij\\}\in E$) and three indices to denote a
triangle (such as $\\{ijk\\}\in F$). Denote the set of positive real numbers
as $\mathbb{R}_{>0}$ and $|V|=N$.

######  Definition 1 ([62]).

A polyhedral surface $(M,\mathcal{T},l)$ with background geometry $\mathbb{G}$
($\mathbb{G}=\mathbb{E}^{2},\mathbb{H}^{2}$ or $\mathbb{S}^{2}$) is a
triangulated surface $(M,\mathcal{T})$ with a map
$l:E\rightarrow\mathbb{R}_{>0}$ such that any face $\\{ijk\\}\in F$ could be
embedded in $\mathbb{G}$ as a nondegenerate triangle with edge lengths
$l_{ij},l_{ik},l_{jk}$ given by $l$. $l:E\rightarrow\mathbb{R}_{>0}$ is called
a Euclidean (hyperbolic or spherical respectively) polyhedral metric if
$\mathbb{G}=\mathbb{E}^{2}$ ($\mathbb{G}=\mathbb{H}^{2}$ or
$\mathbb{G}=\mathbb{S}^{2}$ respectively).

The nondegenerate condition for the face $\\{ijk\\}\in F$ in Definition 1 is
equivalent to the edge lengths $l_{ij},l_{ik},l_{jk}$ satisfy the triangle
inequalities ($l_{ij}+l_{ik}+l_{jk}<2\pi$ additionally if
$\mathbb{G}=\mathbb{S}^{2}$). Intuitively, a polyhedral surface with
background geometry $\mathbb{G}$ ($\mathbb{G}=\mathbb{E}^{2},\mathbb{H}^{2}$
or $\mathbb{S}^{2}$) could be obtained by gluing triangles in $\mathbb{G}$
isometrically along the edges in pair. For polyhedral surfaces, there may
exist conic singularities at the vertices, which could be described by
combinatorial curvature. The combinatorial curvature is a map
$K:V\rightarrow(-\infty,2\pi)$ that assigns the vertex $i\in V$ $2\pi$ less
the sum of inner angles at $i$, i.e.

$K_{i}=2\pi-\sum_{\\{ijk\\}\in F}\theta_{i}^{jk},$ (1.1)

where $\theta_{i}^{jk}$ is the inner angle at $i$ in the triangle
$\\{ijk\\}\in F$ of the polyhedral surface $(M,\mathcal{T},l)$. The
combinatorial curvature satisfies the following discrete Guass-Bonnet formula
([7] Proposition 3.1)

$\sum_{i\in V}K_{i}=2\pi\chi(M)-\lambda\text{Area}(M),$ (1.2)

where $\lambda=-1,0,1$ for
$\mathbb{G}=\mathbb{H}^{2},\mathbb{E}^{2},\mathbb{S}^{2}$ respectively and
$\text{Area}(M)$ denotes the area of the surface $M$.

######  Definition 2 ([41, 96]).

Suppose $(M,\mathcal{T})$ is a triangulated connected closed surface and
$\varepsilon:V\rightarrow\\{-1,0,1\\}$, $\eta:E\rightarrow\mathbb{R}$ are two
weights defined on the vertices and edges respectively with
$\eta_{ij}=\eta_{ji}$. A discrete conformal structure on the weighted
triangulated surface $(M,\mathcal{T},\varepsilon,\eta)$ with background
geometry $\mathbb{G}$ is a map $f:V\rightarrow\mathbb{R}$ such that

(1)

the edge length $l_{ij}$ for the edge $\\{ij\\}\in E$ is given by

$\displaystyle
l_{ij}=\sqrt{\varepsilon_{i}e^{2f_{i}}+\varepsilon_{j}e^{2f_{j}}+2\eta_{ij}e^{f_{i}+f_{j}}}$
(1.3)

for $\mathbb{G}=\mathbb{E}^{2}$,

$\displaystyle
l_{ij}=\cosh^{-1}\left(\sqrt{(1+\varepsilon_{i}e^{2f_{i}})(1+\varepsilon_{j}e^{2f_{j}})}+\eta_{ij}e^{f_{i}+f_{j}}\right)$
(1.4)

for $\mathbb{G}=\mathbb{H}^{2}$ and

$\displaystyle
l_{ij}=\cos^{-1}\left(\sqrt{(1-\varepsilon_{i}e^{2f_{i}})(1-\varepsilon_{j}e^{2f_{j}})}-\eta_{ij}e^{f_{i}+f_{j}}\right)$
(1.5)

for $\mathbb{G}=\mathbb{S}^{2}$;

(2)

the edge length function $l:E\rightarrow\mathbb{R}_{>0}$ defined by (1.3),
(1.4), (1.5) is a Euclidean, hyperbolic and spherical polyhedral metric on
$(M,\mathcal{T})$ respectively.

The weight $\varepsilon:V\rightarrow\\{-1,0,1\\}$ is called the scheme
coefficient and $\eta:E\rightarrow\mathbb{R}$ is called the discrete conformal
structure coefficient.

Two discrete conformal structures defined on the same weighted triangulated
surface $(M,\mathcal{T},\varepsilon,\eta)$ with the same background geometry
$\mathbb{G}$ is said to be conformally equivalent.

######  Remark 1.

The relationships of the discrete conformal structure in Definition 2 and the
existing special types of discrete conformal structures are contained in Table
1.

Scheme | $\varepsilon_{i}$ | $\varepsilon_{j}$ | $\eta_{ij}$
---|---|---|---
Tangential circle packing | $+1$ | $+1$ | $+1$
Thurston’s circle packing | $+1$ | $+1$ | $(-1,1]$
Inversive distance circle packing | $+1$ | $+1$ | $(-1,+\infty)$
Vertex scaling | $0$ | $0$ | $(0,+\infty)$
Discrete conformal structure | $\\{+1,0,-1\\}$ | $\\{+1,0,-1\\}$ | $(-1,+\infty)$
Table 1: Relationships of different discrete conformal structures

By Table 1, the tangential circle packing is a special case of Thurston’s
circle packing and Thurston’s circle packing is a special case of inversive
distance circle packing. For simplicity, we unify all these three types of
circle packings as inversive distance circle packing in the following. By
Table 1 again, the discrete conformal structure in Definition 2 contains
inversive distance circle packing and vertex scaling as special cases.
Furthermore, the discrete conformal structure in Definition 2 contains the
mixed type of discrete conformal structures with $\varepsilon_{i}=0$ for some
vertices $i\in V$ and $\varepsilon_{j}=1$ for the other vertices $j\in V$.
There have been lots of research activities on special types of discrete
conformal structures on polyhedral manifolds. For inversive distance circle
packings on surfaces, please refer to [1, 2, 4, 5, 6, 7, 8, 13, 17, 18, 19,
21, 23, 24, 25, 28, 29, 30, 31, 32, 33, 38, 40, 41, 46, 47, 50, 51, 52, 53,
54, 61, 67, 68, 74, 76, 79, 80, 81, 85, 88, 95, 97, 98] and others. For vertex
scaling on surfaces, please refer to [3, 12, 22, 44, 43, 45, 57, 64, 66, 73,
77, 78, 83, 84, 89, 91, 92, 99] and others. There are also some research
activities on tangential sphere packing and Thurston’s sphere packings on
$3$-dimensional manifolds, please refer to [9, 20, 26, 27, 35, 36, 38, 40, 48,
49, 72, 80, 86] and others. In the following, when we mention the discrete
conformal structure on polyhedral surfaces, it is referred to the generic
discrete conformal structure in Definition 2 unless otherwise declared.

######  Remark 2.

Definition 2 of discrete conformal structure on polyhedral surfaces was
introduced independently simultaneously by Glickenstein-Thomas [41] and by
Zhang-Guo-Zeng-Luo-Yau-Gu [96]. Glickenstein-Thomas’s approach for defining
discrete conformal structure on polyhedral surfaces is from Riemannian
geometry perspective, where they used the notion of partial edge length
introduced by Glickenstein [38, 40] and required that the deformation of
discrete conformal structure depends in a reasonable form of the partial edge
length. The notion of partial edge length ensures the existence of some
geometric structures on the Poincaré dual of the triangulation and the
conditions on the deformation of discrete conformal structure is a discrete
analogue of the fact that smooth conformal change depends only on the scalar
function defined on the manifolds. It is a remarkable result of Glickenstein-
Thomas [41] that this definition of discrete conformal structure could be
classified, which has the form presented in Definition 2 with
$\varepsilon_{i}$ replaced by an arbitrary constant $\alpha_{i}\in\mathbb{R}$.
As pointed out by Thomas ([80] page 53), one can reparameterize discrete
conformal structures so that $\alpha_{i}\in\\{-1,0,1\\}$ while keeping the
induced polyhedral metrics invariant, which gives rise to Definition 2. Zhang-
Guo-Zeng-Luo-Yau-Gu’s approach [96] for defining discrete conformal structure
was motivated by Bobenko-Pinkall-Springborn’s important observation [3] on the
relationships of vertex scaling on polyhedral surfaces and $3$-dimensional
hyperbolic geometry. They defined the edge lengths by embedding the triangle
in a generalized tetrahedron in the extended hyperbolic $3$-space. In this
approach, Zhang-Guo-Zeng-Luo-Yau-Gu [96] explicitly constructed all kinds of
discrete conformal structures contained in Definition 2. The advantage of this
approach is that it gives explicit geometric interpretations to the discrete
Ricci energies for different kinds of special discrete conformal structures on
polyhedral surfaces, which will be further discussed in Section 5.

A basic problem in discrete conformal geometry is to understand the
relationships between the discrete conformal structure and its curvature. We
prove the following result on rigidity of discrete conformal structures on
polyhedral surfaces, which confirms two conjectures of Glickenstein in [39].

######  Theorem 1.1.

Suppose $(M,\mathcal{T},\varepsilon,\eta)$ is a weighted triangulated
connected closed surface with the weights $\varepsilon:V\rightarrow\\{0,1\\}$
and $\eta:E\rightarrow\mathbb{R}$ satisfying

$\displaystyle\varepsilon_{s}\varepsilon_{t}+\eta_{st}>0,\ \
\forall\\{st\\}\in E$ (1.6)

and

$\displaystyle\varepsilon_{q}\eta_{st}+\eta_{qs}\eta_{qt}\geq 0,\ \
\\{q,s,t\\}=\\{i,j,k\\}$ (1.7)

for any triangle $\\{ijk\\}\in F$.

(a)

A Euclidean discrete conformal structure $f:V\rightarrow\mathbb{R}$ on
$(M,\mathcal{T},\varepsilon,\eta)$ is determined by its combinatorial
curvature $K:V\rightarrow\mathbb{R}$ up to a vector
$c(1,1,\cdots,1),c\in\mathbb{R}$.

(b)

A hyperbolic discrete conformal structure $f:V\rightarrow\mathbb{R}$ on
$(M,\mathcal{T},\varepsilon,\eta)$ is determined by its combinatorial
curvature $K:V\rightarrow\mathbb{R}$.

######  Remark 3.

If $\varepsilon_{i}=1$ for all $i\in V$, Theorem 1.1 is reduced to the
rigidity of inversive distance circle packing on surfaces obtained by Guo
[46], Luo [61] and the author [85, 88], which was conjectured by Bowers-
Stephenson [6]. If $\varepsilon_{i}=0$ for all $i\in V$, Theorem 1.1 is
reduced to the rigidity of vertex scaling on surfaces obtained by Luo [57] and
Bobenko-Pinkall-Springborn [3], the global rigidity of which was conjectured
by Luo [57]. Theorem 1.1 unifies these results and further contains the case
of mixed type that $\varepsilon_{i}=1$ for some vertices $i\in
V_{1}\neq\emptyset$ and $\varepsilon_{j}=0$ for the other vertices $j\in
V\setminus V_{1}\neq\emptyset$. The local rigidity for discrete conformal
structures on polyhedral surfaces was previously obtained by Glickenstein [40]
and Glickenstein-Thomas [41] under a condition that the discrete conformal
structure induces a well-centered geometric center for each triangle in the
triangulation, which is not easy to check. The local rigidity for some
subcases of Theorem 1.1 was also previously obtained by Guo-Luo [47] from
Thurston’s viewpoint with the standard cosine law replaced by different cosine
laws in hyperbolic geometry.

###  1.2 Combinatorial curvature flows for discrete conformal structures on
polyhedral surfaces

Finding discrete conformal structures with prescribed combinatorial curvatures
on polyhedral surfaces is an important problem in discrete conformal geometry,
which has lots of theoretical and practical applications [7, 95].
Combinatorial curvature flow is an effective approach for handling this
problem, which was pioneered by Chow-Luo’s work [7] on combinatorial Ricci
flow for Thurston’s circle packing on polyhedral surfaces. For discrete
conformal structures on polyhedral surfaces, the combinatorial Ricci flow was
introduced by Zhang-Guo-Zeng-Luo-Yau-Gu [96]. For simplicity, set

$\displaystyle u_{i}=f_{i}$ (1.8)

for any $i\in V$ in the Euclidean background geometry and

$\displaystyle u_{i}=\left\\{\begin{array}[]{ll}f_{i},&\hbox{ if
$\varepsilon_{i}=0$,}\\\
\frac{1}{2}\log\left|\frac{\sqrt{1+\varepsilon_{i}e^{2f_{i}}}-1}{\sqrt{1+\varepsilon_{i}e^{2f_{i}}}+1}\right|,&\hbox{
if $\varepsilon_{i}\neq 0$,}\end{array}\right.$ (1.9)

for the hyperbolic background geometry. For simplicity,
$u:V\rightarrow\mathbb{R}$ is also called a discrete conformal structure in
the following.

######  Definition 3 ([96]).

Suppose $(M,\mathcal{T},\varepsilon,\eta)$ is a weighted triangulated
connected closed surface with weights $\varepsilon:V\rightarrow\\{-1,0,1\\}$
and $\eta:E\rightarrow\mathbb{R}$. The combinatorial Ricci flow for discrete
conformal structures on polyhedral surfaces is defined as

$\displaystyle\frac{du_{i}}{dt}=-K_{i}$ (1.10)

for Euclidean and hyperbolic background geometry.

The normalized combinatorial Ricci flow for the discrete conformal structures
with Euclidean background geometry is

$\displaystyle\frac{du_{i}}{dt}=K_{av}-K_{i},$ (1.11)

where $K_{av}=\frac{2\pi\chi(M)}{N}$ is the average curvature.

######  Remark 4.

There have been lots of researches on the combinatorial Ricci flow on two and
three dimensional manifolds. For the combinatorial Ricci flow for circle
packings on polyhedral surfaces, please refer to [7, 21, 23, 24, 25, 28, 29,
31, 32, 33, 85] and others. For the combinatorial Yamabe flow for vertex
scaling on polyhedral surfaces, please refer to [22, 43, 44, 57, 89, 92] and
others. There are also some research activities for combinatorial Ricci flow
and combinatorial Yamabe flow on $3$-dimensional manifolds, please refer to
[11, 14, 15, 20, 27, 35, 36, 26, 58, 90, 87, 94] and others.

######  Remark 5.

The combinatorial Ricci flow in Definition 3 unifies the known form of
combinatorial Ricci flow or combinatorial Yamabe flow for different special
discrete conformal structures on polyhedral surfaces. If $\varepsilon_{i}=1$
for all $i\in V$, the combinatorial Ricci flow in Definition 3 is reduced to
Chow-Luo’s combinatorial Ricci flow for circle packings on polyhedral surfaces
in [7, 23, 25]. If $\varepsilon_{i}=0$ for all $i\in V$, the combinatorial
Ricci flow in Definition 3 is reduced to Luo’s combinatorial Yamabe flow for
vertex scalings on polyhedral surfaces in [57]. The combinatorial Ricci flow
in Definition 3, introduced by Zhang-Guo-Zeng-Luo-Yau-Gu [96], unifies Chow-
Luo’s combinatorial Ricci flow and Luo’s combinatorial Yamabe flow and
generalizes them to a very general context. Specially, it includes the mixed
type case that $\varepsilon_{i}=1$ for some vertices $i\in V_{1}\neq\emptyset$
and $\varepsilon_{j}=0$ for the other vertices $j\in V\setminus
V_{1}\neq\emptyset$. The combinatorial Ricci flow in Definition 3 can also be
defined for spherical background geometry with $u_{i}$ satisfying
$\frac{\partial f_{i}}{\partial u_{i}}=\sqrt{1-\varepsilon_{i}e^{2f_{i}}}$.
Please refer to [41, 96] for more details. In this paper, we focus on the
cases of Euclidean and hyperbolic background geometry.

The combinatorial Ricci flows (1.10) and (1.11) may develop singularities,
which correspond to the triangles in the polyhedral surfaces degenerate along
the flows. To handle the potential singularities of the combinatorial Ricci
flow (1.10) and (1.11), we extend the combinatorial curvature by constants and
then extend the combinatorial Ricci flow through the singularities. We have
the following result on the longtime existence and convergence for the
solution of extended combinatorial Ricci flow, which confirms a conjecture of
Glickenstein [39] on the convergence rate of combinatorial Ricci flow and
provides an effective algorithm for finding discrete conformal structures with
prescribed combinatorial curvatures.

######  Theorem 1.2.

Suppose $(M,\mathcal{T},\varepsilon,\eta)$ is a weighted triangulated
connected closed surface with the weights $\varepsilon:V\rightarrow\\{0,1\\}$
and $\eta:E\rightarrow\mathbb{R}$ satisfying the structure conditions (1.6)
and (1.7).

(a)

The solution of normalized combinatorial Ricci flow (1.11) in the Euclidean
background geometry and the solution of the combinatorial Ricci flow (1.10) in
the hyperbolic background geometry could be extended to exist for all time for
any initial discrete conformal structure on
$(M,\mathcal{T},\varepsilon,\eta)$.

(b)

The solution of the extended combinatorial Ricci flow is unique for any
initial discrete conformal structure.

(c)

If there exists a Euclidean discrete conformal structure with constant
combinatorial curvature on $(M,\mathcal{T},\varepsilon,\eta)$, the solution of
the extended normalized Euclidean combinatorial Ricci flow converges
exponentially fast for any initial Euclidean discrete conformal structure; If
there exists a hyperbolic discrete conformal structure with zero combinatorial
curvature on $(M,\mathcal{T},\varepsilon,\eta)$, the solution of the extended
hyperbolic combinatorial Ricci flow converges exponentially fast for any
initial hyperbolic discrete conformal structure.

######  Remark 6.

If $\varepsilon_{i}=1$ for all $i\in V$, the result in Theorem 1.2 is reduced
to the convergence result for Chow-Luo’s combinatorial Ricci flow for
Thurston’s circle packing obtained in [7, 21] and for inversive distance
circle packing obtained in [23, 25]. If $\varepsilon_{i}=0$ for all $i\in V$,
the result in Theorem 1.2 is reduced to the convergence result for Luo’s
combinatorial Yamabe flow for vertex scaling obtained in [22]. The results in
Theorem 1.2 further contain the case of mixed type that $\varepsilon_{i}=1$
for some vertices $i\in V_{1}\neq\emptyset$ and $\varepsilon_{j}=0$ for the
other vertices $j\in V\setminus V_{1}\neq\emptyset$. The idea of extension to
handle the singularities of the combinatorial Ricci flow comes from Bobenko-
Pinkall-Springborn [3] and Luo [61]. There is another approach in [43, 44] to
extend the combinatorial Yamabe flow for vertex scaling on polyhedral
surfaces, in which the singularities are resolved by doing surgery along the
flow by edge flipping when the triangulation is not Delaunay in the polyhedral
metric along the flow. In this approach, the condition on the existence of
discrete conformal structure with constant combinatorial curvature in Theorem
1.2 is removed in [43, 44].

Combinatorial Calabi flow is another effective combinatorial curvature flow
for finding discrete conformal structures with prescribed combinatorial
curvatures on polyhedral surfaces, which was introduced by Ge [17] (see also
[18]) for Thurston’s Euclidean circle packings. Since then, the combinatorial
Calabi flow was extensively studied, see [16, 19, 28, 30, 33, 34, 56, 87, 99]
and others. The combinatorial Calabi flow for discrete conformal structures on
polyhedral surfaces in Definition 2 is defined as follows.

######  Definition 4.

Suppose $(M,\mathcal{T},\varepsilon,\eta)$ is a weighted triangulated
connected closed surface with weights $\varepsilon:V\rightarrow\\{-1,0,1\\}$
and $\eta:E\rightarrow\mathbb{R}$. The combinatorial Calabi flow for discrete
conformal structures on polyhedral surfaces is defined as

$\displaystyle\frac{du_{i}}{dt}=\Delta K_{i}$ (1.12)

for Euclidean and hyperbolic background geometry, where
$\Delta=-\frac{\partial(K_{1},\cdots,K_{N})}{\partial(u_{1},\cdots,u_{N})}$ is
the combinatorial Laplace operator for the discrete conformal structure.

######  Remark 7.

The combinatorial Calabi flow introduced in Definition 4 unifies the known
form of combinatorial Calabi flow for different special discrete conformal
structures on polyhedral surfaces. If $\varepsilon_{i}=1$ for all the vertices
$i\in V$, the combinatorial Calabi flow (1.12) in Definition 4 is reduced to
the combinatorial Calabi flow for circle packings on polyhedral surfaces
introduced in [17, 18, 30]. If $\varepsilon_{i}=0$ for all the vertices $i\in
V$, the combinatorial Calabi flow (1.12) in Definition 4 is reduced to the
combinatorial Calabi flow for vertex scaling on polyhedral surfaces introduced
in [17, 99]. The combinatorial Calabi flow in Definition 4 further contains
the mixed type case that $\varepsilon_{i}=1$ for some vertices $i\in
V_{1}\neq\emptyset$ and $\varepsilon_{j}=0$ for the other vertices $j\in
V\setminus V_{1}\neq\emptyset$. Similar to the combinatorial Ricci flow, the
combinatorial Calabi flow (1.12) could also be defined for spherical
background geometry with $u_{i}$ satisfying $\frac{\partial f_{i}}{\partial
u_{i}}=\sqrt{1-\varepsilon_{i}e^{2f_{i}}}$.

The combinatorial Calabi flow is a negative gradient flow of the combinatorial
Calabi energy $\mathcal{C}=\frac{1}{2}\sum_{i=1}^{N}K_{i}^{2}$. We have the
following result on the longtime behavior of combinatorial Calabi flow (1.12).

######  Theorem 1.3.

Suppose $(M,\mathcal{T},\varepsilon,\eta)$ is a weighted triangulated
connected closed surface with the weights $\varepsilon:V\rightarrow\\{0,1\\}$
and $\eta:E\rightarrow\mathbb{R}$ satisfying the structure conditions (1.6)
and (1.7).

(a)

If the solution of the combinatorial Calabi flow (1.12) converges to a
nondegenerate discrete conformal structure $u^{*}$, then $u^{*}$ has constant
combinatorial curvature.

(b)

If there exists a discrete conformal factor $u^{*}$ with constant
combinatorial curvature ($\frac{2\pi\chi(M)}{N}$ for Euclidean background
geometry and $0$ for hyperbolic background geometry), then there exists a real
number $\delta>0$ such that if the initial value $u(0)$ of the combinatorial
Calabi flow (1.12) satisfies $||u(0)-u^{*}||<\delta$, the solution of the
combinatorial Calabi flow (1.12) exists for all time and converges
exponentially fast to $u^{*}$.

######  Remark 8.

In the case of Thurston’s circle packing, the combinatorial Calabi flow (1.12)
does not develop singularities and is proved to converge exponentially fast to
Thurston’s circle packing metrics with constant combinatorial curvature.
Please refer to [17, 18, 19, 30] for details. In the case of vertex scaling,
the combinatorial Calabi flow (1.12) may develop singularities. However, by
doing surgery along the combinatorial Calabi flow (1.12) by edge flipping
introduced in [43, 44], it is proved [99] that the solution of combinatorial
Calabi flow with surgery for vertex scaling exists for all time and converges
exponentially fast for any initial discrete conformal factor. For generic
initial discrete conformal structure in Definition 2, the global convergence
of the combinatorial Calabi flow (1.12) is not known up to now.

One can also study the parameterized combinatorial curvature
$R_{\alpha,i}=\frac{K_{i}}{e^{\alpha u_{i}}}$ for discrete conformal structure
in Definition 2 with $\alpha\in\mathbb{R}$ and $u:V\rightarrow\mathbb{R}$
defined by (1.8) and (1.9), which was introduced in [28, 31, 32, 33] for
circle packings and in [89, 92] for vertex scaling on polyhedral surfaces. The
global rigidity of parameterized combinatorial curvature $R_{\alpha}$ with
respect to the discrete conformal structure was established in [93], where the
author further introduced the combinatorial $\alpha$-Ricci flow and
combinatorial $\alpha$-Calabi flow for discrete conformal structures to study
the prescribed combinatorial $\alpha$-curvature problem.

###  1.3 Relationships with $3$-dimensional hyperbolic geometry

Motivated by Bobenko-Pinkall-Springborn’s observation [3] on the relationships
of vertex scaling on polyhedral surfaces and $3$-dimensional hyperbolic
geometry, Zhang-Guo-Zeng-Luo-Yau-Gu [96] constructed all the discrete
conformal structures contained in Definition 2 via generalized $3$-dimensional
hyperbolic tetrahedra. The basic idea is to construct a generalized hyperbolic
tetrahedron $T_{Oijk}$ with the vertices $O,v_{i},v_{j},v_{k}$ ideal or hyper-
ideal. $O$ is ideal when we study the Euclidean discrete conformal structure
and hyper-ideal when we study the hyperbolic background geometry. The vertex
$v_{s}\in\\{v_{i},v_{j},v_{k}\\}$ is hyper-ideal if $\varepsilon_{s}=1$ and
ideal if $\varepsilon_{s}=0$. In the case that
$v_{s}\in\\{v_{i},v_{j},v_{k}\\}$ is hyper-ideal, the line segment $Ov_{s}$
has nonempty intersection with $\mathbb{H}^{3}$ in the Klein model. In this
way, for each pair $v_{s},v_{t}$ of $v_{i},v_{j},v_{k}$, a weight $\eta_{st}$
can be naturally assigned via the signed edge length of $\\{v_{s}v_{t}\\}$. In
the Euclidean background geometry, the edge lengths of the intersection
triangle $T_{ijk}\cap H_{O}$ is given by (1.3), where $H_{O}$ is the
horosphere attached to the ideal vertex $O$ and $f_{s}$ is minus of the signed
edge length $l_{Ov_{s}}$ with $s\in\\{i,j,k\\}$. The case for hyperbolic
background geometry is similar. By truncating the generalized hyperbolic
tetrahedron $T_{Oijk}$ with horospheres or hyperbolic planes, we can attach it
with a finite hyperbolic polyhedron $P$, the volume and co-volume of which are
functions of $l_{Ov_{i}},l_{Ov_{j}},l_{Ov_{k}}$ and
$\eta_{ij},\eta_{ik},\eta_{jk}$. For the details of the construction of
$T_{Oijk}$ and assignments of $\eta_{ij},\eta_{ik},\eta_{jk}$, please refer to
Section 5.

We find that the structure condition (1.7) is a direct consequence of the
cosine laws for generalized hyperbolic triangles. This partially answers a
question of Gortler [42]. Using the geometric explanation of the discrete
conformal structures in terms of $3$-dimensional hyperbolic geometry, we
further obtain some new convexities of co-volume functions of generalized
hyperbolic tetrahedra in extended $3$-dimensional hyperbolic space.

######  Theorem 1.4.

Suppose $T=\\{Oijk\\}$ is a generalized tetrahedron constructed above.

(a)

The weights $\eta_{ij},\eta_{ik},\eta_{jk}$ on the edges
$\\{ij\\},\\{ik\\},\\{jk\\}$ satisfy the structure conditions (1.6) and (1.7).

(b)

The co-volume of the generalized tetrahedron $T=\\{Oijk\\}$ defined by (5.2)
with fixed weights $\eta_{ij},\eta_{ik},\eta_{jk}$ is a convex function of the
signed edge lengths $l_{Ov_{i}},l_{Ov_{j}},l_{Ov_{k}}$.

###  1.4 Basic ideas of the proof of Theorem 1.1

The proof for the rigidity of discrete conformal structures on triangulated
surfaces involves a variational principle introduced by Colin de Verdière [13]
for tangential circle packings on triangulated surfaces. The variational
principle on triangulated surfaces and triangulated $3$-manifolds has been
extensively studied in [3, 4, 7, 9, 10, 46, 48, 49, 55, 57, 59, 60, 61, 62,
65, 71, 76, 85, 86, 88, 97] and others.

The key point using the variational principle to prove the rigidity of
discrete conformal structures on polyhedral surfaces is constructing a
globally defined convex function with the combinatorial curvature as its
gradient, which could be reduced to constructing a concave function of
discrete conformal structures on a triangle with inner angles as its gradient.
There are two approaches to construct the concave function for discrete
conformal structures on a triangle. The first approach is used to study the
rigidity of vertex scaling on polyhedral surfaces. This was accomplished by
Bobenko-Pinkall-Springborn [3] using the volume functions for some generalized
hyperbolic tetrahedra, which are concave functions of the dihedral angles and
could be extended to be defined globally. This approach depends on the
explicit forms and concavity of the volume functions of generalized hyperbolic
tetrahedra in terms of dihedral angles, which were obtained by Rivin [71] for
ideal tetrahedra and by Leibon [55] for generalized hyperbolic tetrahedra with
one hyper-ideal vertex and three ideal vertices. For the other cases, the
concave functions with inner angles as gradient were constructed using the
volume functions of generalized hyperbolic tetrahedra by Zhang-Guo-Zeng-Luo-
Yau-Gu [96]. However, the concavities of volume functions of generalized
hyperbolic tetrahedra in these cases are not fully understood. See [75] for
some partial results. The second approach is used to study the rigidity of
inversive distance circle packings on polyhedral surfaces. This was
accomplished by Guo [46], Luo [61] and the author [85, 88] by defining the
concave function for a triangle as the integral of a closed $1$-form defined
by inner angles on the admissible space of inversive distance circle packing
metrics. The difficulty for this approach is to prove that the integral is a
well-defined locally concave function and could be extended to be a globally
defined concave function.

In this paper, we adopt the second approach. The idea comes from a new proof
[88] of Bowers-Stephenson’s conjecture on the rigidity of inversive distance
circle packing on surfaces. By solving the global version of triangle
inequalities for a triangle using the geometric center introduced by
Glickenstein [38, 40], we obtain an explicit characterization for the
admissible space of discrete conformal structures on the triangle, which is
homotopy equivalent to $\mathbb{R}^{3}$ and therefore simply connected. This
implies the Ricci energy function, defined as the integral of a closed
$1$-form of the inner angles on the admissible space of discrete conformal
structures for a triangle, is well-defined. By the continuity of eigenvalues
of the hession matrix of the Ricci energy function, the concavity of the Ricci
energy function is reduced to find a discrete conformal structure with
negative definite hession matrix. This is accomplished by introducing the
parameterized admissible space of discrete conformal structures and choosing
some “good” point in the parameterized admissible space. As the Ricci energy
function constructed by the two approaches are the same up to some constant,
the concavity of the Ricci energy function for a triangle proved here
conversely implies the convexity of the co-volume functions of some
generalized hyperbolic tetrahedra with some edge lengths or dihedral angles
fixed. The extension of Ricci energy function to be defined globally follows
from Luo’s generalization [61] of Bobenko-Pinkall-Springborn’s extension in
[3]. The rigidity of discrete conformal structure on polyhedral surfaces
follows from the concavity of the extended Ricci energy function for a
triangle.

The main results obtained in this paper could be generalized to compact
triangulated surfaces with boundary without any further difficulty by doubling
the surface across the boundary. For simplicity, we will not state the
paralleling results for compact triangulated surfaces with boundary.

###  1.5 The organization of the paper

In Section 2, we study the Euclidean discrete conformal structures on
polyhedral surfaces and prove a generalization of Theorem 1.1 (a). In Section
3, we study the hyperbolic discrete conformal structures on polyhedral
surfaces and prove a generalization of Theorem 1.1 (b). In Section 4, we study
the combinatorial Ricci flow and combinatorial Calabi flow for discrete
conformal structures on polyhedral surfaces and prove generalizations of
Theorem 1.2 and Theorem 1.3. In Section 5, we discuss the relationships of
discrete conformal structures on polyhedral surfaces and $3$-dimensional
hyperbolic geometry and prove Theorem 1.4. In Section 6, we discuss some open
problems and conjectures.

Acknowledgements
The paper was motivated by the ideas developed when the author was visiting
Professor Feng Luo at Rutgers University. The author thanks Professor Feng Luo
for his invitation to Rutgers University and the communications during and
after the visit. The author thanks Siqi Zhang for pointing out an improvement
of Lemma 2.5. The research of the author is supported by Fundamental Research
Funds for the Central Universities under Grant no. 2042020kf0199.

##  2 Euclidean discrete conformal structures

###  2.1 Admissible space of Euclidean discrete conformal structures for a
triangle

In this subsection, we study the admissible space $\Omega_{ijk}^{E}(\eta)$ of
nondegenerate Euclidean discrete conformal structures for a triangle
$\sigma=\\{ijk\\}\in F$ with the edge lengths $l_{ij},l_{ik},l_{jk}$ defined
by (1.3) and the weights $\varepsilon:V\rightarrow\\{0,1\\}$ and
$\eta:E\rightarrow\mathbb{R}$ satisfying the structure conditions (1.6) and
(1.7), which is defined to be the set of Euclidean discrete conformal
structures with the edge lengths $l_{ij},l_{ik},l_{jk}$ satisfying the
triangle inequalities.

Note that (1.3) uniquely defines a positive number for
$\varepsilon:V\rightarrow\\{0,1\\}$ and $f:V\rightarrow\mathbb{R}$ under the
structure condition (1.6). For the triangle $\\{ijk\\}\in F$, the positive
edge lengths $l_{ij},l_{ik},l_{jk}$ satisfy the triangle inequalities

$\displaystyle l_{ij}<l_{ik}+l_{jk},\ l_{ik}<l_{ij}+l_{jk},\
l_{jk}<l_{ij}+l_{ik}$ (2.1)

if and only if

$\displaystyle 0<$
$\displaystyle(l_{ij}+l_{ik}+l_{jk})(l_{ij}+l_{ik}-l_{jk})(l_{ij}-l_{ik}+l_{jk})(-l_{ij}+l_{ik}+l_{jk})$
(2.2) $\displaystyle=$ $\displaystyle
2l_{ij}^{2}l_{ik}^{2}+2l_{ij}^{2}l_{jk}^{2}+2l_{ik}^{2}l_{jk}^{2}-l_{ij}^{4}-l_{ik}^{4}-l_{jk}^{4}.$

For simplicity of notations, set

$\displaystyle r_{i}=e^{f_{i}},\ \forall i\in V.$ (2.3)

Sometimes we also call $r\in\mathbb{R}^{V}_{>0}$ as a Euclidean discrete
conformal structure. Then the edge length $l_{ij}$ in the Euclidean background
geometry, i.e. (1.3), is given by

$\displaystyle
l_{ij}=\sqrt{\varepsilon_{i}r_{i}^{2}+\varepsilon_{j}r_{j}^{2}+2\eta_{ij}r_{i}r_{j}}.$
(2.4)

Submitting (2.4) into (2.2), by direct calculations, we have

$\displaystyle(l_{ij}+l_{ik}+l_{jk})(l_{ij}+l_{ik}-l_{jk})(l_{ij}-l_{ik}+l_{jk})(-l_{ij}+l_{ik}+l_{jk})$
$\displaystyle=$ $\displaystyle
4r_{i}^{2}r_{j}^{2}r_{k}^{2}[(\varepsilon_{i}\varepsilon_{j}-\eta_{ij}^{2})r_{k}^{-2}+(\varepsilon_{i}\varepsilon_{k}-\eta_{ik}^{2})r_{j}^{-2}+(\varepsilon_{j}\varepsilon_{k}-\eta_{jk}^{2})r_{i}^{-2}$
$\displaystyle+2(\varepsilon_{k}\eta_{ij}+\eta_{ik}\eta_{jk})r_{i}^{-1}r_{j}^{-1}+2(\varepsilon_{j}\eta_{ik}+\eta_{ij}\eta_{jk})r_{i}^{-1}r_{k}^{-1}+2(\varepsilon_{i}\eta_{jk}+\eta_{ij}\eta_{ik})r_{j}^{-1}r_{k}^{-1}].$

Set

$\displaystyle\kappa_{i}=r_{i}^{-1},\kappa_{j}=r_{j}^{-1},\kappa_{k}=r_{k}^{-1},$
(2.5)
$\displaystyle\gamma_{i}=\varepsilon_{i}\eta_{jk}+\eta_{ij}\eta_{ik},\gamma_{j}=\varepsilon_{j}\eta_{ik}+\eta_{ij}\eta_{jk},\gamma_{k}=\varepsilon_{k}\eta_{ij}+\eta_{ik}\eta_{jk},$
(2.6)

and

$\displaystyle Q^{E}=$
$\displaystyle(\varepsilon_{j}\varepsilon_{k}-\eta_{jk}^{2})\kappa_{i}^{2}+(\varepsilon_{i}\varepsilon_{k}-\eta_{ik}^{2})\kappa_{j}^{2}+(\varepsilon_{i}\varepsilon_{j}-\eta_{ij}^{2})\kappa_{k}^{2}+2\kappa_{i}\kappa_{j}\gamma_{k}+2\kappa_{i}\kappa_{k}\gamma_{j}+2\kappa_{j}\kappa_{k}\gamma_{i}.$
(2.7)

Then

$\displaystyle\gamma_{i}\geq 0,\gamma_{j}\geq 0,\gamma_{k}\geq 0$ (2.8)

by the structure condition (1.7) and

$\displaystyle(l_{ij}+l_{ik}+l_{jk})(l_{ij}+l_{ik}-l_{jk})(l_{ij}-l_{ik}+l_{jk})(-l_{ij}+l_{ik}+l_{jk})=4r_{i}^{2}r_{j}^{2}r_{k}^{2}Q^{E}.$

As a consequence of the arguments above, we have the following result.

######  Lemma 2.1.

Suppose $(M,\mathcal{T},\varepsilon,\eta)$ is a weighted triangulated surface
with the weights $\varepsilon:V\rightarrow\\{0,1\\}$ and
$\eta:E\rightarrow\mathbb{R}$. $\\{ijk\\}\in F$ is a topological triangle in
the triangulation. Then the positive edge lengths $l_{ij},l_{ik},l_{jk}$
defined by (1.3) satisfy the triangle inequalities if and only if $Q^{E}>0$.

Set

$\displaystyle h_{i}=$
$\displaystyle(\varepsilon_{j}\varepsilon_{k}-\eta_{jk}^{2})\kappa_{i}+\kappa_{j}\gamma_{k}+\kappa_{k}\gamma_{j},$
(2.9) $\displaystyle h_{j}=$
$\displaystyle(\varepsilon_{i}\varepsilon_{k}-\eta_{ik}^{2})\kappa_{j}+\kappa_{i}\gamma_{k}+\kappa_{k}\gamma_{i},$
$\displaystyle h_{k}=$
$\displaystyle(\varepsilon_{i}\varepsilon_{j}-\eta_{ij}^{2})\kappa_{k}+\kappa_{i}\gamma_{j}+\kappa_{j}\gamma_{i}.$

Then by Lemma 2.1, $r=(r_{i},r_{j},r_{k})\in\mathbb{R}^{3}_{>0}$ is a
degenerate Euclidean discrete conformal structure for the triangle $\\{ijk\\}$
if and only if

$Q^{E}=\kappa_{i}h_{i}+\kappa_{j}h_{j}+\kappa_{k}h_{k}\leq 0,$

which implies that at least one of $h_{i},h_{j},h_{k}$ is nonpositive. We
further have the following result on the signs of $h_{i},h_{j},h_{k}$.

######  Lemma 2.2.

Suppose $(M,\mathcal{T},\varepsilon,\eta)$ is a weighted triangulated surface
with the weights $\varepsilon:V\rightarrow\\{0,1\\}$ and
$\eta:E\rightarrow\mathbb{R}$ satisfying the structure conditions (1.6) and
(1.7). $\\{ijk\\}\in F$ is a topological triangle of the triangulated surface.
For any $r=(r_{i},r_{j},r_{k})\in\mathbb{R}^{3}_{>0}$, there is no subset
$\\{s,t\\}\subset\\{i,j,k\\}$ such that $h_{s}\leq 0$ and $h_{t}\leq 0$.

Proof. Without loss of generality, assume $h_{i}\leq 0$, $h_{j}\leq 0$. Then
by the definition of $h_{i},h_{j}$ in (2.9), we have

$\displaystyle\kappa_{j}\gamma_{k}+\kappa_{k}\gamma_{j}\leq(\eta_{jk}^{2}-\varepsilon_{j}\varepsilon_{k})\kappa_{i},\
\
\kappa_{i}\gamma_{k}+\kappa_{k}\gamma_{i}\leq(\eta_{ik}^{2}-\varepsilon_{i}\varepsilon_{k})\kappa_{j},$

which implies

$\displaystyle\eta_{jk}-\varepsilon_{j}\varepsilon_{k}\geq 0,\
\eta_{ik}-\varepsilon_{i}\varepsilon_{k}\geq 0$ (2.10)

and

$\displaystyle\gamma_{k}^{2}-(\eta_{jk}^{2}-\varepsilon_{j}\varepsilon_{k})(\eta_{ik}^{2}-\varepsilon_{i}\varepsilon_{k})=-\varepsilon_{i}\varepsilon_{j}\varepsilon_{k}+\varepsilon_{k}\eta_{ij}^{2}+\varepsilon_{j}\varepsilon_{k}\eta_{ik}^{2}+\varepsilon_{i}\varepsilon_{k}\eta_{jk}^{2}+2\varepsilon_{k}\eta_{ij}\eta_{ik}\eta_{jk}\leq
0$ (2.11)

by the structure conditions (1.6) and (1.7), where the property
$\varepsilon_{k}^{2}=\varepsilon_{k}$ for the weight
$\varepsilon:V\rightarrow\\{0,1\\}$ is used. Set
$F=-\varepsilon_{i}\varepsilon_{j}\varepsilon_{k}+\varepsilon_{k}\eta_{ij}^{2}+\varepsilon_{j}\varepsilon_{k}\eta_{ik}^{2}+\varepsilon_{i}\varepsilon_{k}\eta_{jk}^{2}+2\varepsilon_{k}\eta_{ij}\eta_{ik}\eta_{jk}.$
Then $F\leq 0$ by (2.11). On the other hand,

$\displaystyle F=$
$\displaystyle\varepsilon_{k}(\varepsilon_{i}\eta_{jk}-\varepsilon_{j}\eta_{ik})^{2}+(\varepsilon_{i}\varepsilon_{j}+\eta_{ij})[2\varepsilon_{k}\eta_{ik}\eta_{jk}+\varepsilon_{k}(\eta_{ij}-\varepsilon_{i}\varepsilon_{j})]$
(2.12) $\displaystyle=$
$\displaystyle\varepsilon_{k}(\varepsilon_{i}\eta_{jk}-\varepsilon_{j}\eta_{ik})^{2}+(\varepsilon_{i}\varepsilon_{j}+\eta_{ij})[\varepsilon_{k}(\varepsilon_{k}\eta_{ij}+\eta_{ik}\eta_{jk})+\varepsilon_{k}(\eta_{ik}\eta_{jk}-\varepsilon_{i}\varepsilon_{j})]$
$\displaystyle=$
$\displaystyle\varepsilon_{k}(\varepsilon_{i}\eta_{jk}-\varepsilon_{j}\eta_{ik})^{2}$
$\displaystyle+(\varepsilon_{i}\varepsilon_{j}+\eta_{ij})[\varepsilon_{k}\gamma_{k}+\varepsilon_{k}(\eta_{ik}-\varepsilon_{i}\varepsilon_{k})(\eta_{jk}-\varepsilon_{j}\varepsilon_{k})+\varepsilon_{i}\varepsilon_{k}(\eta_{jk}-\varepsilon_{j}\varepsilon_{k})+\varepsilon_{j}\varepsilon_{k}(\eta_{ik}-\varepsilon_{i}\varepsilon_{k})]$
$\displaystyle\geq$ $\displaystyle 0$

by (2.10), $\varepsilon_{i},\varepsilon_{j},\varepsilon_{k}\in\\{0,1\\}$ and
the structure conditions (1.6) and (1.7). Therefore, $F=0$.

In the case of $\varepsilon_{k}=0$, by the structure condition (1.6), we have

$\displaystyle\varepsilon_{i}\varepsilon_{j}+\eta_{ij}>0,\eta_{ik}>0,\eta_{jk}>0.$
(2.13)

By $h_{i}\leq 0$, $h_{j}\leq 0$ and $\varepsilon_{k}=0$, we have
$\eta_{ik}\eta_{jk}\kappa_{j}+\gamma_{j}\kappa_{k}\leq\eta_{jk}^{2}\kappa_{i}$
and
$\eta_{ik}\eta_{jk}\kappa_{i}+\gamma_{i}\kappa_{k}\leq\eta_{ik}^{2}\kappa_{j}$.
This implies

$\displaystyle\gamma_{i}=\varepsilon_{i}\eta_{jk}+\eta_{ij}\eta_{ik}=0,\
\gamma_{j}=\varepsilon_{j}\eta_{ik}+\eta_{ij}\eta_{jk}=0,$ (2.14)

which implies $\varepsilon_{i}\varepsilon_{j}-\eta_{ij}^{2}=0$ by (2.13). Note
that
$\varepsilon_{i}\varepsilon_{j}-\eta_{ij}^{2}=(\varepsilon_{i}\varepsilon_{j}-\eta_{ij})(\varepsilon_{i}\varepsilon_{j}+\eta_{ij})$.
Combining with (2.13) again, we have
$\varepsilon_{i}\varepsilon_{j}-\eta_{ij}=0$. By (2.13) once again, we have
$\varepsilon_{i}\varepsilon_{j}>0$, which implies
$\varepsilon_{i}=\varepsilon_{j}=\eta_{ij}=1$. However, in this case, (2.14)
implies $\eta_{ik}+\eta_{jk}=0$, which contradicts (2.13).

In the case of $\varepsilon_{k}=1$, by $F=0$ and (2.12), we have

$\displaystyle\varepsilon_{i}\eta_{jk}-\varepsilon_{j}\eta_{ik}=\eta_{ij}+\eta_{ik}\eta_{jk}=(\eta_{ik}-\varepsilon_{i})(\eta_{jk}-\varepsilon_{j})=\varepsilon_{i}(\eta_{jk}-\varepsilon_{j})=\varepsilon_{j}(\eta_{ik}-\varepsilon_{i})=0,$
(2.15)

which implies $\eta_{ik}=\varepsilon_{i}$ or $\eta_{jk}=\varepsilon_{j}$. By
$\varepsilon_{k}=1$ and the structure condition (1.6), we have

$\displaystyle\varepsilon_{i}\varepsilon_{j}+\eta_{ij}>0,\varepsilon_{i}+\eta_{ik}>0,\varepsilon_{j}+\eta_{jk}>0.$
(2.16)

If $\eta_{ik}=\varepsilon_{i}$, (2.16) implies $\varepsilon_{i}=\eta_{ik}=1$,
which further implies $\eta_{jk}=\varepsilon_{j}$ by (2.15). By (2.16) again,
we have $\varepsilon_{j}=\eta_{jk}=1$, which implies
$\eta_{ij}+\varepsilon_{i}\varepsilon_{j}=\eta_{ij}+\eta_{ik}\eta_{jk}=0$ by
(2.15). This contradicts (2.16). The same arguments also apply to the case
$\eta_{jk}=\varepsilon_{j}$.

In summary, there exists no subset $\\{s,t\\}\subset\\{i,j,k\\}$ such that
$h_{s}\leq 0$ and $h_{t}\leq 0$. $\square$

As a direct corollary of Lemma 2.2, we have the following stronger result on
the signs on $h_{i},h_{j},h_{k}$ for degenerate Euclidean discrete conformal
structures on the triangle $\\{ijk\\}\in F$.

######  Corollary 2.3.

Suppose $(M,\mathcal{T},\varepsilon,\eta)$ is a weighted triangulated surface
with the weights $\varepsilon:V\rightarrow\\{0,1\\}$ and
$\eta:E\rightarrow\mathbb{R}$ satisfying the structure conditions (1.6) and
(1.7). $\\{ijk\\}\in F$ is a topological triangle in the triangulated surface.
If $r=(r_{i},r_{j},r_{k})\in\mathbb{R}^{3}_{>0}$ is a degenerate Euclidean
discrete conformal structure for the triangle $\\{ijk\\}\in F$, then one of
$h_{i},h_{j},h_{k}$ is negative and the others are positive.

Proof. As $r=(r_{i},r_{j},r_{k})\in\mathbb{R}^{3}_{>0}$ is a degenerate
Euclidean discrete conformal structure for the triangle $\\{ijk\\}\in F$, we
have $Q^{E}=\kappa_{i}h_{i}+\kappa_{j}h_{j}+\kappa_{k}h_{k}\leq 0$ by Lemma
2.1, which implies at least one of $h_{i},h_{j},h_{k}$ is nonpositive. Without
loss of generality, assume $h_{i}\leq 0$. Then Lemma 2.2 implies that
$h_{j}>0,h_{k}>0$. If $h_{i}=0$, combining with $h_{j}>0,h_{k}>0$, we have
$Q^{E}=\kappa_{i}h_{i}+\kappa_{j}h_{j}+\kappa_{k}h_{k}>0$, which contradicts
$Q^{E}=\kappa_{i}h_{i}+\kappa_{j}h_{j}+\kappa_{k}h_{k}\leq 0$. Therefore,
$h_{i}<0$, $h_{j}>0$, $h_{k}>0$. $\square$

######  Remark 9.

Corollary 2.3 has an interesting geometric explanation as follows. For a
nondegenerate Euclidean discrete conformal structure for the triangle
$\\{ijk\\}\in F$, there exists a geometric center $C_{ijk}$ for the triangle
$\\{ijk\\}$ ([40], Proposition 4), which has the same power distance to the
vertices $i,j,k$. Please refer to Figure 1.

Figure 1: Signed distances of geometric center

Here the power distance of a point $p$ to the vertex $i$ is defined to be
$\pi_{p}(i)=d^{2}(i,p)-\varepsilon_{i}r_{i}^{2}$, where $d(i,p)$ is the
Euclidean distance between $p$ and the vertex $i$. Denote $h_{jk,i}$ as the
signed distance of the geometric center $C_{ijk}$ to the edge $\\{jk\\}$,
which is defined to be positive if the geometric center $C_{ijk}$ is on the
same side of the line determined by $\\{jk\\}$ as the triangle $\\{ijk\\}$ and
negative otherwise (or zero if the geometric center $C_{ijk}$ is on the line).
Projections of the geometric center $C_{ijk}$ to the edges
$\\{ij\\},\\{ik\\},\\{jk\\}$ give rise to the geometric centers of these
edges, which are denoted by $C_{ij},C_{ik},C_{jk}$ respectively. The signed
distance $d_{ij}$ of $C_{ij}$ to the vertex $i$ is defined to be positive if
$C_{ij}$ is on the same side as $j$ along the line determined by $\\{ij\\}$
and negative otherwise (or zero if $C_{ij}$ is the same as $i$). $d_{ji}$ is
defined similarly. Note that $d_{ij}+d_{ji}=l_{ij}$ and $d_{ij}\neq d_{ji}$ in
general. For nondegenerate Euclidean discrete conformal structures, we have
the following formulas [38, 40, 41]

$\displaystyle h_{jk,i}=\frac{d_{ji}-d_{jk}\cos\theta_{j}}{\sin\theta_{j}},\ \
d_{ij}=\frac{\varepsilon_{i}r_{i}^{2}+\eta_{ij}r_{i}r_{j}}{l_{ij}},$

where $\theta_{j}$ is the inner angle at the vertex $j$ of the Euclidean
triangle $\\{ijk\\}$. By direct calculations,

$\displaystyle
h_{jk,i}=\frac{r_{i}^{2}r_{j}^{2}r_{k}^{2}}{Al_{jk}}\kappa_{i}h_{i},$ (2.17)

where $A=l_{ij}l_{ik}\sin\theta_{i}$. Corollary 2.3 implies that the geometric
center $C_{ijk}$ for a nondegenerate Euclidean discrete conformal structure
does not lie in some region in the plane determined by the triangle as it
tends to be degenerate. Note that $h_{i},h_{j},h_{k}$ is defined for all
$(r_{i},r_{j},r_{k})\in\mathbb{R}^{3}_{>0}$, while
$h_{ij,k},h_{ik,j},h_{jk,i}$ are defined only for nondegenerate discrete
conformal structures.

Now we can solve the admissible space of Euclidean discrete conformal
structures for a triangle $\\{ijk\\}\in F$, which shows that the admissible
space of Euclidean discrete conformal structures for the triangle
$\\{ijk\\}\in F$ with the weights satisfying the structure conditions (1.6)
and (1.7) is simply connected. First, we have the following result.

######  Lemma 2.4.

Suppose $(M,\mathcal{T},\varepsilon,\eta)$ is a weighted triangulated surface
with the weights $\varepsilon:V\rightarrow\\{0,1\\}$ and
$\eta:E\rightarrow\mathbb{R}$ satisfying the structure conditions (1.6) and
(1.7). $\\{ijk\\}\in F$ is a topological triangle in the triangulated surface.
If we further have $\varepsilon_{j}\varepsilon_{k}-\eta_{jk}^{2}\geq 0$,
$\varepsilon_{i}\varepsilon_{k}-\eta_{ik}^{2}\geq 0$ and
$\varepsilon_{i}\varepsilon_{j}-\eta_{ij}^{2}\geq 0$, then the admissible
space $\Omega^{E}_{ijk}(\eta)$ of Euclidean discrete conformal structure in
the parameter $r$ is $\mathbb{R}^{3}_{>0}$ and then simply connected.

Proof. By Lemma 2.1, we just need to prove that for any
$r=(r_{i},r_{j},r_{k})\in\mathbb{R}^{3}_{>0}$, we have $Q^{E}>0$. If
$\varepsilon_{j}\varepsilon_{k}-\eta_{jk}^{2}\geq 0$,
$\varepsilon_{i}\varepsilon_{k}-\eta_{ik}^{2}\geq 0$ and
$\varepsilon_{i}\varepsilon_{j}-\eta_{ij}^{2}\geq 0$, then we have $Q^{E}\geq
0$ by the definition (2.7) of $Q^{E}$ and the structure condition (1.7). If
$Q^{E}=0$ in this case, then
$\eta_{jk}^{2}=\varepsilon_{j}\varepsilon_{k},\eta_{ik}^{2}=\varepsilon_{i}\varepsilon_{k},\eta_{ij}^{2}=\varepsilon_{i}\varepsilon_{j}$.
By the structure condition (1.6), we have
$\eta_{jk}=\varepsilon_{j}\varepsilon_{k},\eta_{ik}=\varepsilon_{i}\varepsilon_{k},\eta_{ij}=\varepsilon_{i}\varepsilon_{j}$.
Combining with the structure condition (1.6) again, we have
$\varepsilon_{i}=\varepsilon_{j}=\varepsilon_{k}=\eta_{ij}=\eta_{ik}=\eta_{jk}=1$,
which implies
$Q^{E}=4\kappa_{i}\kappa_{j}+4\kappa_{i}\kappa_{k}+4\kappa_{j}\kappa_{k}>0$
for any $r=(r_{i},r_{j},r_{k})\in\mathbb{R}^{3}_{>0}$. Therefore, the
admissible space $\Omega^{E}_{ijk}(\eta)=\mathbb{R}^{3}_{>0}$, which is simply
connected. $\square$

By Lemma 2.4, we just need to study the case that at least one of
$\varepsilon_{j}\varepsilon_{k}-\eta_{jk}^{2}$,
$\varepsilon_{i}\varepsilon_{k}-\eta_{ik}^{2}$,
$\varepsilon_{i}\varepsilon_{j}-\eta_{ij}^{2}$ is negative. Suppose
$r=(r_{i},r_{j},r_{k})\in\mathbb{R}^{3}_{>0}$ is a degenerate Euclidean
discrete conformal structure for this case. Then $Q^{E}\leq 0$ by Lemma 2.1.
By Proposition 2.3, one of $h_{i},h_{j},h_{k}$ is negative and the other two
are positive. Without loss of generality, assume $h_{i}<0$, $h_{j}>0,h_{k}>0$.
By the definition (2.9) of $h_{i}$ and the structure condition (1.7), we have
$(\eta_{jk}^{2}-\varepsilon_{j}\varepsilon_{k})\kappa_{i}>\gamma_{k}\kappa_{j}+\gamma_{j}\kappa_{k}\geq
0,$ which implies $\eta_{jk}^{2}-\varepsilon_{j}\varepsilon_{k}>0$.

Taking $Q^{E}$ as a quadratic function of $\kappa_{i},\kappa_{j},\kappa_{k}$.
Then $Q^{E}\leq 0$ is equivalent to

$\displaystyle A_{i}\kappa_{i}^{2}+B_{i}\kappa_{i}+C_{i}\geq 0,$ (2.18)

where

$\displaystyle A_{i}=$
$\displaystyle\eta_{jk}^{2}-\varepsilon_{j}\varepsilon_{k}>0,$ (2.19)
$\displaystyle B_{i}=$
$\displaystyle-2(\gamma_{k}\kappa_{j}+\gamma_{j}\kappa_{k})\leq 0,$
$\displaystyle C_{i}=$
$\displaystyle(\eta_{ik}^{2}-\varepsilon_{i}\varepsilon_{k})\kappa_{j}^{2}+(\eta_{ij}^{2}-\varepsilon_{i}\varepsilon_{j})\kappa_{k}^{2}-2\kappa_{j}\kappa_{k}\gamma_{i}.$

######  Lemma 2.5.

Suppose $(M,\mathcal{T},\varepsilon,\eta)$ is a weighted triangulated surface
with the weights $\varepsilon:V\rightarrow\\{0,1\\}$ and
$\eta:E\rightarrow\mathbb{R}$ satisfying the structure conditions (1.6).
$\\{ijk\\}\in F$ is a topological triangle in the triangulated surface. If
$A_{i}=\eta_{jk}^{2}-\varepsilon_{j}\varepsilon_{k}>0$, then the discriminant
$\Delta_{i}=B_{i}^{2}-4A_{i}C_{i}$ for (2.18) is positive.

Proof. By direct calculations, we have

$\displaystyle\Delta_{i}=4(\varepsilon_{j}\kappa_{k}^{2}+\varepsilon_{k}\kappa_{j}^{2}+2\eta_{jk}\kappa_{j}\kappa_{k})(\varepsilon_{i}\eta_{jk}^{2}+\varepsilon_{j}\eta_{ik}^{2}+\varepsilon_{k}\eta_{ij}^{2}+2\eta_{ij}\eta_{ik}\eta_{jk}-\varepsilon_{i}\varepsilon_{j}\varepsilon_{k}).$
(2.20)

By the structure condition (1.6), the sign of $\Delta_{i}$ is determined by
the term
$\varepsilon_{i}\eta_{jk}^{2}+\varepsilon_{j}\eta_{ik}^{2}+\varepsilon_{k}\eta_{ij}^{2}+2\eta_{ij}\eta_{ik}\eta_{jk}-\varepsilon_{i}\varepsilon_{j}\varepsilon_{k}$
in (2.20), which is symmetric in $i,j,k$. If one of
$\varepsilon_{i},\varepsilon_{j},\varepsilon_{k}$ is zero, say
$\varepsilon_{i}=0$, we have $\eta_{ij}>0$ and $\eta_{ik}>0$ by the structure
condition (1.6), which implies

$\varepsilon_{i}\eta_{jk}^{2}+\varepsilon_{j}\eta_{ik}^{2}+\varepsilon_{k}\eta_{ij}^{2}+2\eta_{ij}\eta_{ik}\eta_{jk}-\varepsilon_{i}\varepsilon_{j}\varepsilon_{k}=(\varepsilon_{j}\eta_{ik}-\varepsilon_{k}\eta_{ij})^{2}+2\eta_{ij}\eta_{ik}(\eta_{jk}+\varepsilon_{j}\varepsilon_{k})>0.$

If $\varepsilon_{i}=\varepsilon_{j}=\varepsilon_{k}=1$, we have

$\varepsilon_{i}\eta_{jk}^{2}+\varepsilon_{j}\eta_{ik}^{2}+\varepsilon_{k}\eta_{ij}^{2}+2\eta_{ij}\eta_{ik}\eta_{jk}-\varepsilon_{i}\varepsilon_{j}\varepsilon_{k}=\eta_{jk}^{2}+\eta_{ik}^{2}+\eta_{ij}^{2}+2\eta_{ij}\eta_{ik}\eta_{jk}-1.$

The positivity of this term under the condition $A_{i}=\eta_{jk}^{2}-1>0$ has
been proved in Lemma 2.3 in [88]. For completeness, we present a proof here.
If $\eta_{ij}\geq 0$ and $\eta_{ik}\geq 0$, then
$\eta_{jk}^{2}+\eta_{ik}^{2}+\eta_{ij}^{2}+2\eta_{ij}\eta_{ik}\eta_{jk}-1\geq\eta_{jk}^{2}-1>0$
by assumption. If $\eta_{ij}<0$, then $\eta_{ij}\in(-1,0)$ by the structure
condition (1.6). This implies
$\eta_{jk}^{2}+\eta_{ik}^{2}+\eta_{ij}^{2}+2\eta_{ij}\eta_{ik}\eta_{jk}-1=(\eta_{ik}+\eta_{ij}\eta_{jk})^{2}+(\eta_{jk}^{2}-1)(1-\eta_{ij}^{2})>0$.
The same argument applies to $\eta_{ik}<0$. Therefore, $\Delta_{i}>0$ by the
equation (2.20). $\square$

By the proof of Lemma 2.5, we have the following direct corollary.

######  Corollary 2.6.

Suppose $(M,\mathcal{T},\varepsilon,\eta)$ is a weighted triangulated surface
with the weights $\varepsilon:V\rightarrow\\{0,1\\}$ and
$\eta:E\rightarrow\mathbb{R}$ satisfying the structure conditions (1.6).
$\\{ijk\\}\in F$ is a topological triangle in the triangulated surface. If one
of
$\eta_{jk}^{2}-\varepsilon_{j}\varepsilon_{k},\eta_{ik}^{2}-\varepsilon_{i}\varepsilon_{k},\eta_{ij}^{2}-\varepsilon_{i}\varepsilon_{j}$
is positive, then the term

$G=\varepsilon_{i}\eta_{jk}^{2}+\varepsilon_{j}\eta_{ik}^{2}+\varepsilon_{k}\eta_{ij}^{2}+2\eta_{ij}\eta_{ik}\eta_{jk}-\varepsilon_{i}\varepsilon_{j}\varepsilon_{k}$
(2.21)

is positive.

######  Remark 10.

One can also take $Q^{E}$ as a quadratic function of $\kappa_{j}$ or
$\kappa_{k}$ and define $\Delta_{j}$, $\Delta_{k}$ similarly. By symmetry,
under the same conditions as that in Lemma 2.5, if
$\eta_{ik}^{2}-\varepsilon_{i}\varepsilon_{k}>0$, then $\Delta_{j}>0$ and if
$\eta_{ij}^{2}-\varepsilon_{i}\varepsilon_{j}>0$, then $\Delta_{k}>0$.

######  Theorem 2.7.

Suppose $(M,\mathcal{T},\varepsilon,\eta)$ is a weighted triangulated closed
connected surface with the weights $\varepsilon:V\rightarrow\\{0,1\\}$ and
$\eta:E\rightarrow\mathbb{R}$ satisfying the structure conditions (1.6) and
(1.7). $\\{ijk\\}\in F$ is a topological triangle in the triangulated surface.
Then the admissible space $\Omega_{ijk}^{E}(\eta)$ of nondegerate Euclidean
discrete conformal structures for the triangle $\\{ijk\\}\in F$ is nonempty
and simply connected with analytical boundary components. Furthermore, the
admissible space $\Omega_{ijk}^{E}(\eta)$ in $r$ could be written as

$\displaystyle\Omega_{ijk}^{E}(\eta)=\mathbb{R}^{3}_{>0}\setminus\sqcup_{\alpha\in\Lambda}V_{\alpha},$

where
$\Lambda=\\{q\in\\{i,j,k\\}|A_{q}=\eta_{st}^{2}-\varepsilon_{s}\varepsilon_{t}>0,\\{q,s,t\\}=\\{i,j,k\\}\\}$,
$\sqcup_{\alpha\in\Lambda}V_{\alpha}$ is a disjoint union of $V_{\alpha}$ and
$V_{\alpha}$ is a closed region in $\mathbb{R}^{3}_{>0}$ bounded by an
analytical function defined on $\mathbb{R}^{2}_{>0}$.

Proof. We solve the admissible space of nondegenerate Euclidean discrete
conformal structures for the triangle $\\{ijk\\}\in F$ by giving a precise
description of the space of degenerate Euclidean discrete conformal
structures.

Suppose $(r_{i},r_{j},r_{k})\in\mathbb{R}^{3}_{>0}$ is a degenerate Euclidean
discrete conformal structure for the triangle $\\{ijk\\}\in F$. Then we have
$Q^{E}=\kappa_{i}h_{i}+\kappa_{j}h_{j}+\kappa_{k}h_{k}\leq 0$ by Lemma 2.1. By
Corollary 2.3, one of $h_{i},h_{j},h_{k}$ is negative and the other two are
positive. Without loss of generality, assume $h_{i}<0$ and $h_{j}>0,h_{k}>0$,
which implies $A_{i}=\eta_{jk}^{2}-\varepsilon_{j}\varepsilon_{k}>0$ by (2.9)
and the structure condition (1.7). Taking $Q^{E}\leq 0$ as a quadratic
inequality of $\kappa_{i}$. Then the solution of $Q^{E}\leq 0$, i.e.
$A_{i}\kappa_{i}^{2}+B_{i}\kappa_{i}+C_{i}\geq 0$, is

$\displaystyle\kappa_{i}\geq\frac{-B_{i}+\sqrt{\Delta_{i}}}{2A_{i}}\
\text{or}\ \kappa_{i}\leq\frac{-B_{i}-\sqrt{\Delta_{i}}}{2A_{i}}$

by Lemma 2.5. Note that

$\displaystyle
2A_{i}\kappa_{i}+B_{i}=2(\eta_{jk}^{2}-\varepsilon_{j}\varepsilon_{k})\kappa_{i}-2(\gamma_{k}\kappa_{j}+\gamma_{j}\kappa_{k})=-2h_{i},$
(2.22)

we have $\kappa_{i}>\frac{-B_{i}}{2A_{i}}\geq 0$ by $h_{i}<0$ and
$A_{i}=\eta_{jk}^{2}-\varepsilon_{j}\varepsilon_{k}>0$, which implies the
solution $(r_{i},r_{j},r_{k})\in\mathbb{R}^{3}_{>0}$ of $Q^{E}\leq 0$ with
$h_{i}<0,h_{j}>0,h_{k}>0$ should be
$\kappa_{i}\geq\frac{-B_{i}+\sqrt{\Delta_{i}}}{2A_{i}}$. Therefore,
$\mathbb{R}^{3}_{>0}\setminus\Omega_{ijk}^{E}(\eta)\subset\cup_{\alpha\in\Lambda}V_{\alpha}$,
where
$\Lambda=\\{q\in\\{i,j,k\\}|A_{q}=\eta_{st}^{2}-\varepsilon_{s}\varepsilon_{t}>0,\\{q,s,t\\}=\\{i,j,k\\}\\}$,

$\displaystyle
V_{i}=\\{(r_{i},r_{j},r_{k})\in\mathbb{R}^{3}_{>0}|\kappa_{i}\geq\frac{-B_{i}+\sqrt{\Delta_{i}}}{2A_{i}}\\}=\\{(r_{i},r_{j},r_{k})\in\mathbb{R}^{3}_{>0}|r_{i}\leq\frac{2A_{i}}{-B_{i}+\sqrt{\Delta_{i}}}\\}$
(2.23)

and $V_{j},V_{k}$ are defined similarly.

Conversely, suppose
$(r_{i},r_{j},r_{k})\in\cup_{\alpha\in\Lambda}V_{\alpha}\subseteq\mathbb{R}^{3}_{>0}$.
Without loss of generality, assume $(r_{i},r_{j},r_{k})\in V_{i}$ and
$A_{i}=\eta_{jk}^{2}-\varepsilon_{j}\varepsilon_{k}>0$. Then
$\kappa_{i}\geq\frac{-B_{i}+\sqrt{\Delta_{i}}}{2A_{i}}$, which is equivalent
to $2A_{i}\kappa_{i}+B_{i}\geq\sqrt{\Delta_{i}}$ by $A_{i}>0$. Taking the
square of both sides gives $A_{i}\kappa_{i}^{2}+B_{i}\kappa_{i}+C_{i}\geq 0$,
which is equivalent to $Q^{E}\leq 0$. Therefore,
$\cup_{\alpha\in\Lambda}V_{\alpha}\subset\mathbb{R}^{3}_{>0}\setminus\Omega_{ijk}^{E}(\eta).$
In summary,
$\cup_{\alpha\in\Lambda}V_{\alpha}=\mathbb{R}^{3}_{>0}\setminus\Omega_{ijk}^{E}(\eta)$.

Suppose there exists some $(r_{i},r_{j},r_{k})\in\mathbb{R}^{3}_{>0}$ with
$(r_{i},r_{j},r_{k})\in V_{\alpha}\cap V_{\beta}$ for some
$\alpha,\beta\in\\{i,j,k\\}$. Without loss of generality, assume
$(r_{i},r_{j},r_{k})\in V_{i}\cap V_{j}$. Then $A_{i}>0,A_{j}>0$, which
implies $\Delta_{i}>0,\Delta_{j}>0$ by Lemma 2.5 and Remark 10. By
$(r_{i},r_{j},r_{k})\in V_{i}$, we have
$\kappa_{i}\geq\frac{-B_{i}+\sqrt{\Delta_{i}}}{2A_{i}}$, which implies
$h_{i}=-\frac{1}{2}(2A_{i}\kappa_{i}+B_{i})\leq-\frac{1}{2}\sqrt{\Delta_{i}}<0$
by $A_{i}>0$ and (2.22). Corollary 2.3 further implies
$h_{i}<0,h_{j}>0,h_{k}>0$. The same arguments applies to
$(r_{i},r_{j},r_{k})\in V_{j}$, which shows that $h_{j}<0,h_{i}>0,h_{k}>0$.
This is a contradiction. Therefore, $V_{\alpha}\cap V_{\beta}=\emptyset$ for
$\forall\alpha,\beta\in\Lambda$. This implies
$\Omega_{ijk}^{E}(\eta)=\mathbb{R}^{3}_{>0}\setminus\sqcup_{\alpha\in\Lambda}V_{\alpha}$,
which is homotopy equivalent to $\mathbb{R}^{3}_{>0}$ and then simply
connected. $\square$

######  Remark 11.

By the proof of Theorem 2.7, if $V_{i}$ defined by (2.23) is nonempty, then
$h_{i}<0,h_{j}>0,h_{k}>0$ for $(r_{i},r_{j},r_{k})\in V_{i}$.

We further introduce the following parameterized admissible space of
nondegenerate Euclidean discrete conformal structures for the triangle
$\\{ijk\\}\in F$

$\displaystyle\Omega_{ijk}^{E}=\\{(r_{i},r_{j},r_{k},\eta_{ij},\eta_{ik},\eta_{jk})\in\mathbb{R}^{3}_{>0}\times\mathbb{R}^{3}|\eta\
\text{satisfies}\ (\ref{structure condition 1}),(\ref{structure condition 2})\
\text{and }(r_{i},r_{j},r_{k})\in\Omega_{ijk}^{E}(\eta)\\}.$

The parameterized admissible space $\Omega_{ijk}^{E}$ contains some points
with good properties.

######  Lemma 2.8.

The point $(r_{i},r_{j},r_{k},\eta_{ij},\eta_{ik},\eta_{jk})=(1,1,1,1,1,1)$ is
contained in $\Omega_{ijk}^{E}$. Furthermore, $h_{i}>0,h_{j}>0,h_{k}>0$ at
this point.

Proof. It is straight forward to check that
$(\eta_{ij},\eta_{ik},\eta_{jk})=(1,1,1)$ satisfies the structure conditions
(1.6) and (1.7). By the definition (2.9) of $h_{i},h_{j},h_{k}$, we have

$\displaystyle h_{i}=$
$\displaystyle(\varepsilon_{j}\varepsilon_{k}-\eta_{jk}^{2})\kappa_{i}+\kappa_{j}\gamma_{k}+\kappa_{k}\gamma_{j}=\varepsilon_{j}\varepsilon_{k}+\varepsilon_{j}+\varepsilon_{k}+1>0,$
$\displaystyle h_{j}=$
$\displaystyle(\varepsilon_{i}\varepsilon_{k}-\eta_{ik}^{2})\kappa_{j}+\kappa_{i}\gamma_{k}+\kappa_{k}\gamma_{i}=\varepsilon_{i}\varepsilon_{k}+\varepsilon_{i}+\varepsilon_{k}+1>0,$
$\displaystyle h_{k}=$
$\displaystyle(\varepsilon_{i}\varepsilon_{j}-\eta_{ij}^{2})\kappa_{k}+\kappa_{i}\gamma_{j}+\kappa_{j}\gamma_{i}=\varepsilon_{i}\varepsilon_{j}+\varepsilon_{i}+\varepsilon_{j}+1>0$

at $(r_{i},r_{j},r_{k},\eta_{ij},\eta_{ik},\eta_{jk})=(1,1,1,1,1,1)$, which
implies $Q^{E}=\kappa_{i}h_{i}+\kappa_{j}h_{j}+\kappa_{k}h_{k}>0$. Therefore,
$(1,1,1,1,1,1)\in\Omega_{ijk}^{E}$ by Lemma 2.1. $\square$

Theorem 2.7 have the following corollary on the parameterized admissible space
$\Omega_{ijk}^{E}$.

######  Corollary 2.9.

Suppose $(M,\mathcal{T},\varepsilon,\eta)$ is a weighted triangulated surface
with the weights $\varepsilon:V\rightarrow\\{0,1\\}$ and
$\eta:E\rightarrow\mathbb{R}$ satisfying the structure conditions (1.6) and
(1.7). $\\{ijk\\}\in F$ is a topological triangle in the triangulated surface.
Then the parameterized admissible space $\Omega_{ijk}^{E}$ for the triangle
$\\{ijk\\}\in F$ is connected.

Proof. Set

$\displaystyle\Gamma=\\{(\eta_{ij},\eta_{ik},\eta_{jk})\in\mathbb{R}^{3}|(\eta_{ij},\eta_{ik},\eta_{jk})\
\text{satisfies}\ (\ref{structure condition 1}),(\ref{structure condition
2})\\}$

as the space of parameters. We shall prove that $\Gamma$ is connected, from
which the connectivity of $\Omega_{ijk}^{E}$ follows by Theorem 2.7 and the
continuity of $Q$ as a function of
$(r_{i},r_{j},r_{k},\eta_{ij},\eta_{ik},\eta_{jk})\in\mathbb{R}^{3}_{>0}\times\mathbb{R}^{3}$.

It is obviously that $\mathbb{R}^{3}_{>0}\subset\Gamma$, which is connected.
We will show that any point in $\Gamma$ could be connected to
$\mathbb{R}^{3}_{>0}$ by paths in $\Gamma$. As the boundary of
$\mathbb{R}^{3}_{>0}$ is connected to $\mathbb{R}^{3}_{>0}$, we just need to
consider the case that some component of
$(\eta_{ij},\eta_{ik},\eta_{jk})\in\Gamma$ is negative. Without loss of
generality, assume $\eta_{ij}<0$, then $\varepsilon_{i}=\varepsilon_{j}=1$ by
the structure condition $\eta_{ij}+\varepsilon_{i}\varepsilon_{j}>0$.

In the case of $\varepsilon_{k}=0$, the structure conditions (1.6), (1.7) are
equivalent to

$\displaystyle 1+\eta_{ij}>0,\eta_{ik}>0,\eta_{jk}>0$ (2.24)

and

$\displaystyle\eta_{jk}+\eta_{ij}\eta_{ik}\geq
0,\eta_{ik}+\eta_{ij}\eta_{jk}\geq 0,\eta_{ik}\eta_{jk}\geq 0.$ (2.25)

If $(\eta_{ij},\eta_{ik},\eta_{jk})\in\Gamma$ and $\eta_{ij}<0$, it is
straightforward to check that $(t\eta_{ij},\eta_{ik},\eta_{jk})$ satisfies
(2.24) and (2.25) for any $t\in[0,1]$. This implies
$(t\eta_{ij},\eta_{ik},\eta_{jk})\in\Gamma$, $\forall t\in[0,1]$, which is a
path connecting $(\eta_{ij},\eta_{ik},\eta_{jk})$ and $\mathbb{R}^{3}_{\geq
0}$. Therefore, $\Gamma$ is connected.

In the case of $\varepsilon_{k}=1$, the structure conditions (1.6), (1.7) are
equivalent to

$\displaystyle 1+\eta_{ij}>0,1+\eta_{ik}>0,1+\eta_{jk}>0$ (2.26)

and

$\displaystyle\eta_{jk}+\eta_{ij}\eta_{ik}\geq
0,\eta_{ik}+\eta_{ij}\eta_{jk}\geq 0,\eta_{ij}+\eta_{ik}\eta_{jk}\geq 0.$
(2.27)

In this case, the connectivity of $\Gamma$ has been proved in [88]. For
completeness, we present the proof here. By the structure conditions (2.26)
and (2.27), we have $\eta_{ij}+\eta_{ik}\geq 0$, $\eta_{ij}+\eta_{jk}\geq 0$,
$\eta_{ik}+\eta_{jk}\geq 0$, which implies at most one of
$\eta_{ij},\eta_{ik},\eta_{jk}$ is negative. By the assumption that
$\eta_{ij}<0$, we have $\eta_{ik}\geq 0,\eta_{jk}\geq 0$. It is
straightforward to check that $(t\eta_{ij},\eta_{ik},\eta_{jk})$ satisfies
(2.26) and (2.27) for any $t\in[0,1]$. This implies
$(t\eta_{ij},\eta_{ik},\eta_{jk})\in\Gamma$, $\forall t\in[0,1]$, which is a
path connecting $(\eta_{ij},\eta_{ik},\eta_{jk})$ and $\mathbb{R}^{3}_{\geq
0}$. Therefore, $\Gamma$ is connected. $\square$

###  2.2 Negative semi-definiteness of the Jacobian matrix in the Euclidean
case

Suppose $\\{ijk\\}$ is a nondegenerate Euclidean triangle with edge lengths
given by (1.3) and $\theta_{i},\theta_{j},\theta_{k}$ are the inner angles at
the vertices $i,j,k$ in the triangle respectively. Set $u_{i}=f_{i}=\ln
r_{i}$.

######  Lemma 2.10 ([38, 41, 96]).

Suppose $(M,\mathcal{T},\varepsilon,\eta)$ is a weighted triangulated surface
with the weights $\varepsilon:V\rightarrow\\{0,1\\}$ and
$\eta:E\rightarrow\mathbb{R}$. Suppose
$(r_{i},r_{j},r_{k})\in\mathbb{R}^{3}_{>0}$ is a nondegenerate Euclidean
discrete conformal structure for the topological triangle $\\{ijk\\}\in F$.
Then

$\displaystyle\frac{\partial\theta_{i}}{\partial
u_{j}}=\frac{\partial\theta_{j}}{\partial
u_{i}}=\frac{r_{i}^{2}r_{j}^{2}r_{k}^{2}}{Al_{ij}^{2}}[(\varepsilon_{i}\varepsilon_{j}-\eta_{ij}^{2})\kappa_{k}^{2}+\gamma_{i}\kappa_{j}\kappa_{k}+\gamma_{j}\kappa_{i}\kappa_{k}]=\frac{r_{i}^{2}r_{j}^{2}r_{k}}{Al_{ij}^{2}}h_{k}$
(2.28)

and

$\displaystyle\frac{\partial\theta_{i}}{\partial
u_{i}}=-\frac{\partial\theta_{i}}{\partial
u_{j}}-\frac{\partial\theta_{i}}{\partial u_{k}},$ (2.29)

where $A=l_{ij}l_{ik}\sin\theta_{i}$.

Proof. By the chain rules, we have

$\displaystyle\frac{\partial\theta_{i}}{\partial u_{j}}=$
$\displaystyle\frac{\partial\theta_{i}}{\partial l_{jk}}\frac{\partial
l_{jk}}{\partial u_{j}}+\frac{\partial\theta_{i}}{\partial
l_{ik}}\frac{\partial l_{ik}}{\partial
u_{j}}+\frac{\partial\theta_{i}}{\partial l_{ij}}\frac{\partial
l_{ij}}{\partial u_{j}}.$ (2.30)

By the derivative cosine law ([7], Lemma A1), we have

$\displaystyle\frac{\partial\theta_{i}}{\partial
l_{jk}}=\frac{l_{jk}}{A},\frac{\partial\theta_{i}}{\partial
l_{ik}}=\frac{-l_{jk}\cos\theta_{k}}{A},\frac{\partial\theta_{i}}{\partial
l_{ij}}=\frac{-l_{jk}\cos\theta_{j}}{A},$ (2.31)

where $A=l_{ik}l_{jk}\sin\theta_{k}$. By the definition (2.4) of
$l_{ij},l_{ik},l_{jk}$ in $r_{i},r_{j},r_{k}$, we have

$\displaystyle\frac{\partial l_{jk}}{\partial
u_{j}}=\frac{\varepsilon_{j}r_{j}^{2}+\eta_{jk}r_{j}r_{k}}{l_{jk}},\ \
\frac{\partial l_{ik}}{\partial u_{j}}=0,\ \ \frac{\partial l_{ij}}{\partial
u_{j}}=\frac{\varepsilon_{j}r_{j}^{2}+\eta_{ij}r_{i}r_{j}}{l_{ij}}.$ (2.32)

Submitting (2.31) and (2.32) into (2.30), we have

$\displaystyle\frac{\partial\theta_{i}}{\partial u_{j}}=$
$\displaystyle\frac{l_{jk}}{A}\cdot\frac{\varepsilon_{j}r_{j}^{2}+\eta_{jk}r_{j}r_{k}}{l_{jk}}+\frac{-l_{jk}\cos\theta_{j}}{A}\cdot\frac{\varepsilon_{j}r_{j}^{2}+\eta_{ij}r_{i}r_{j}}{l_{ij}}$
(2.33) $\displaystyle=$
$\displaystyle\frac{1}{2Al_{ij}^{2}}[2(\varepsilon_{j}r_{j}^{2}+\eta_{jk}r_{j}r_{k})l_{ij}^{2}+(l_{ik}^{2}-l_{ij}^{2}-l_{jk}^{2})(\varepsilon_{j}r_{j}^{2}+\eta_{ij}r_{i}r_{j})]$
$\displaystyle=$
$\displaystyle\frac{r_{i}^{2}r_{j}^{2}r_{k}^{2}}{Al_{ij}^{2}}[(\varepsilon_{i}\varepsilon_{j}-\eta_{ij}^{2})\kappa_{k}^{2}+\gamma_{i}\kappa_{j}\kappa_{k}+\gamma_{j}\kappa_{i}\kappa_{k}]$
$\displaystyle=$
$\displaystyle\frac{r_{i}^{2}r_{j}^{2}r_{k}}{Al_{ij}^{2}}h_{k},$

where the cosine law is used in the second line and the definition (2.4) of
edge lengths is used in the third line. As the last line of (2.33) is
symmetric in $i$ and $j$, we have $\frac{\partial\theta_{i}}{\partial
u_{j}}=\frac{\partial\theta_{j}}{\partial u_{i}}$. Similarly, we have
$\frac{\partial\theta_{i}}{\partial u_{k}}=\frac{\partial\theta_{k}}{\partial
u_{i}}$. $\frac{\partial\theta_{i}}{\partial
u_{i}}=-\frac{\partial\theta_{i}}{\partial
u_{j}}-\frac{\partial\theta_{i}}{\partial u_{k}}$ follows from
$\theta_{i}+\theta_{j}+\theta_{k}=\pi$, $\frac{\partial\theta_{i}}{\partial
u_{j}}=\frac{\partial\theta_{j}}{\partial u_{i}}$ and
$\frac{\partial\theta_{i}}{\partial u_{k}}=\frac{\partial\theta_{k}}{\partial
u_{i}}$. $\square$

######  Remark 12.

The symmetry $\frac{\partial\theta_{i}}{\partial
u_{j}}=\frac{\partial\theta_{j}}{\partial u_{i}}$ for discrete conformal
structures in Lemma 2.10 was previously obtained by Glickenstein [38],
Glickenstein-Thomas [41] and Zhang-Guo-Zeng-Luo-Yau-Gu [96]. Here we give a
proof by direct calculations for completeness. Combining (2.17) and Lemma
2.10, we have [38, 41, 96]

$\displaystyle\frac{\partial\theta_{i}}{\partial
u_{j}}=\frac{h_{ij,k}}{l_{ij}},$ (2.34)

which provides a nice geometric explanation for the derivative
$\frac{\partial\theta_{i}}{\partial u_{j}}$.

######  Remark 13.

By (2.28), (2.29) and Remark 11, if $(\eta_{ij},\eta_{ik},\eta_{jk})$ defined
on the triangle $\\{ijk\\}$ satisfies the structure conditions (1.6) and (1.7)
and $(r_{i},r_{j},r_{k})\in\Omega_{ijk}^{E}(\eta)$ tends to a point
$(\overline{r}_{i},\overline{r}_{j},\overline{r}_{k})\in\partial V_{i}$ with
$V_{i}\neq\emptyset$, we have $\frac{\partial\theta_{i}}{\partial
u_{j}}\rightarrow+\infty$, $\frac{\partial\theta_{i}}{\partial
u_{k}}\rightarrow+\infty$ and $\frac{\partial\theta_{i}}{\partial
u_{i}}\rightarrow-\infty$.

Lemma 2.10 shows that the Jacobian matrix

$\displaystyle\Lambda^{E}_{ijk}:=\frac{\partial(\theta_{i},\theta_{j},\theta_{k})}{\partial(u_{i},u_{j},u_{k})}=\left(\begin{array}[]{ccc}\frac{\partial\theta_{i}}{\partial
u_{i}}&\frac{\partial\theta_{i}}{\partial
u_{j}}&\frac{\partial\theta_{i}}{\partial u_{k}}\\\
\frac{\partial\theta_{j}}{\partial u_{i}}&\frac{\partial\theta_{j}}{\partial
u_{j}}&\frac{\partial\theta_{j}}{\partial u_{k}}\\\
\frac{\partial\theta_{k}}{\partial u_{i}}&\frac{\partial\theta_{k}}{\partial
u_{j}}&\frac{\partial\theta_{k}}{\partial u_{k}}\\\ \end{array}\right)$

is symmetric with $\\{t(1,1,1)^{T}|t\in\mathbb{R}\\}$ contained in its kernel.
We further have the following result on the rank of the Jacobian matrix
$\Lambda^{E}_{ijk}$.

######  Lemma 2.11.

Suppose $(M,\mathcal{T},\varepsilon,\eta)$ is a weighted triangulated surface
with the weights $\varepsilon:V\rightarrow\\{0,1\\}$ and
$\eta:E\rightarrow\mathbb{R}$ satisfying the structure conditions (1.6) and
(1.7). $\\{ijk\\}\in F$ is a topological triangle in the triangulated surface.
Then the rank of the Jacobian matrix
$\Lambda^{E}_{ijk}=\frac{\partial(\theta_{i},\theta_{j},\theta_{k})}{\partial(u_{i},u_{j},u_{k})}$
is $2$ for any nondegenerate Euclidean discrete conformal structure on the
triangle $\\{ijk\\}$.

Proof. By the chain rules, we have

$\displaystyle\frac{\partial(\theta_{i},\theta_{j},\theta_{k})}{\partial(u_{i},u_{j},u_{k})}=\frac{\partial(\theta_{i},\theta_{j},\theta_{k})}{\partial(l_{jk},l_{ik},l_{ij})}\cdot\frac{\partial(l_{jk},l_{ik},l_{ij})}{\partial(u_{i},u_{j},u_{k})}.$
(2.35)

By the derivative cosine law ([7], Lemma A1), we have

$\displaystyle\frac{\partial(\theta_{i},\theta_{j},\theta_{k})}{\partial(l_{jk},l_{ik},l_{ij})}=\frac{1}{A}\left(\begin{array}[]{ccc}l_{jk}&&\\\
&l_{ik}&\\\ &&l_{ij}\\\
\end{array}\right)\left(\begin{array}[]{ccc}1&-\cos\theta_{k}&-\cos\theta_{j}\\\
-\cos\theta_{k}&1&-\cos\theta_{i}\\\ -\cos\theta_{j}&-\cos\theta_{i}&1\\\
\end{array}\right),$

which has rank $2$ and kernel $\\{t(l_{jk},l_{ik},l_{ij})|t\in\mathbb{R}\\}$
for $(l_{jk},l_{ik},l_{ij})$ satisfying the triangle inequalities.

Note that $d_{ij}=\frac{\partial l_{ij}}{\partial
u_{i}}=\frac{\varepsilon_{i}r_{i}^{2}+\eta_{ij}r_{i}r_{j}}{l_{ij}}$. By direct
calculations,

$\displaystyle\frac{\partial(l_{jk},l_{ik},l_{ij})}{\partial(u_{i},u_{j},u_{k})}=$
$\displaystyle\left(\begin{array}[]{ccc}0&d_{jk}&d_{kj}\\\ d_{ik}&0&d_{ki}\\\
d_{ij}&d_{ji}&0\\\ \end{array}\right)$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}l_{jk}^{-1}&&\\\ &l_{ik}^{-1}&\\\
&&l_{ij}^{-1}\\\ \end{array}\right)$
$\displaystyle\cdot\left(\begin{array}[]{ccc}0&\varepsilon_{j}r_{j}+\eta_{jk}r_{k}&\varepsilon_{k}r_{k}+\eta_{jk}r_{j}\\\
\varepsilon_{i}r_{i}+\eta_{ik}r_{k}&0&\varepsilon_{k}r_{k}+\eta_{ik}r_{i}\\\
\varepsilon_{i}r_{i}+\eta_{ij}r_{j}&\varepsilon_{j}r_{j}+\eta_{ij}r_{i}&0\\\
\end{array}\right)\left(\begin{array}[]{ccc}r_{i}&&\\\ &r_{j}&\\\ &&r_{k}\\\
\end{array}\right).$

This implies

$\displaystyle\det\frac{\partial(l_{jk},l_{ik},l_{ij})}{\partial(u_{i},u_{j},u_{k})}=$
$\displaystyle\frac{r_{i}r_{j}r_{k}}{l_{ij}l_{ik}l_{jk}}[2(\varepsilon_{i}\varepsilon_{j}\varepsilon_{k}+\eta_{ij}\eta_{ik}\eta_{jk})r_{i}r_{j}r_{k}+r_{i}\gamma_{i}(\varepsilon_{j}r_{j}^{2}+\varepsilon_{k}r_{k}^{2})$
(2.36)
$\displaystyle+r_{j}\gamma_{j}(\varepsilon_{i}r_{i}^{2}+\varepsilon_{k}r_{k}^{2})+r_{k}\gamma_{k}(\varepsilon_{i}r_{i}^{2}+\varepsilon_{j}r_{j}^{2})]$
$\displaystyle\geq$
$\displaystyle\frac{2r_{i}^{2}r_{j}^{2}r_{k}^{2}}{l_{ij}l_{ik}l_{jk}}[\varepsilon_{i}\varepsilon_{j}\varepsilon_{k}+\eta_{ij}\eta_{ik}\eta_{jk}+\gamma_{i}\varepsilon_{j}\varepsilon_{k}+\gamma_{j}\varepsilon_{i}\varepsilon_{k}+\gamma_{k}\varepsilon_{i}\varepsilon_{j}]$
$\displaystyle=$
$\displaystyle\frac{2r_{i}^{2}r_{j}^{2}r_{k}^{2}}{l_{ij}l_{ik}l_{jk}}(\varepsilon_{i}\varepsilon_{j}+\eta_{ij})(\varepsilon_{i}\varepsilon_{k}+\eta_{ik})(\varepsilon_{j}\varepsilon_{k}+\eta_{jk})$
$\displaystyle>$ $\displaystyle 0,$

where the structure condition (1.7) is used in the second line and the
structure condition (1.6) is used in the last line. (2.36) implies that
$\frac{\partial(l_{jk},l_{ik},l_{ij})}{\partial(u_{i},u_{j},u_{k})}$ is
nonsingular.

Combining (2.35), (2.36) and the fact that the rank of
$\frac{\partial(\theta_{i},\theta_{j},\theta_{k})}{\partial(l_{jk},l_{ik},l_{ij})}$
is $2$ for $(l_{jk},l_{ik},l_{ij})$ satisfying the triangle inequalities, we
have the rank of the Jacobian matrix
$\Lambda^{E}_{ijk}=\frac{\partial(\theta_{i},\theta_{j},\theta_{k})}{\partial(u_{i},u_{j},u_{k})}$
is $2$ for any nondegenerate Euclidean discrete conformal structure on the
triangle $\\{ijk\\}$. $\square$

As a direct consequence of Lemma 2.11, we have the following negative semi-
definiteness for the Jacobian matrix $\Lambda^{E}_{ijk}$.

######  Theorem 2.12.

Suppose $(M,\mathcal{T},\varepsilon,\eta)$ is a weighted triangulated surface
with the weights $\varepsilon:V\rightarrow\\{0,1\\}$ and
$\eta:E\rightarrow\mathbb{R}$ satisfying the structure conditions (1.6) and
(1.7). $\\{ijk\\}\in F$ is a topological triangle in the triangulated surface.
Then the Jacobian matrix
$\Lambda^{E}_{ijk}=\frac{\partial(\theta_{i},\theta_{j},\theta_{k})}{\partial(u_{i},u_{j},u_{k})}$
is negative semi-definite with rank $2$ and kernel
$\\{t(1,1,1)^{T}|t\in\mathbb{R}\\}$ for any nondegenerate Euclidean discrete
conformal structure on the triangle $\\{ijk\\}$.

Proof. By Lemma 2.11, the Jacobian matrix $\Lambda^{E}_{ijk}$ has two nonzero
eigenvalues and one zero eigenvalue. By the continuity of eigenvalues of
$\Lambda^{E}_{ijk}=\frac{\partial(\theta_{i},\theta_{j},\theta_{k})}{\partial(u_{i},u_{j},u_{k})}$
as functions of
$(r_{i},r_{j},r_{k},\eta_{ij},\eta_{ik},\eta_{jk})\in\Omega_{ijk}^{E}$ and the
connectivity of parameterized admissible space $\Omega_{ijk}^{E}$ in Corollary
2.9, to prove
$\frac{\partial(\theta_{i},\theta_{j},\theta_{k})}{\partial(u_{i},u_{j},u_{k})}$
is negative semi-definite, we just need to prove
$\frac{\partial(\theta_{i},\theta_{j},\theta_{k})}{\partial(u_{i},u_{j},u_{k})}$
is negative semi-definite with rank $2$ at some point in $\Omega_{ijk}^{E}$.
By Lemma 2.8, $h_{i}>0,h_{j}>0,h_{k}>0$ at
$(r_{i},r_{j},r_{k},\eta_{ij},\eta_{ik},\eta_{jk})=(1,1,1,1,1,1)\in\Omega_{ijk}^{E}$,
which implies the $\frac{\partial\theta_{i}}{\partial u_{j}}$,
$\frac{\partial\theta_{i}}{\partial u_{k}}$,
$\frac{\partial\theta_{j}}{\partial u_{k}}$ are positive by (2.17) and (2.34).
Then by the following well-known result from linear algebra,
$-\Lambda^{E}_{ijk}$ is positive semi-definite with rank $2$ and null space
$\\{t(1,1,1)^{T}|t\in\mathbb{R}\\}$ at
$(r_{i},r_{j},r_{k},\eta_{ij},\eta_{ik},\eta_{jk})=(1,1,1,1,1,1)\in\Omega_{ijk}^{E}$.

######  Lemma 2.13.

Suppose $A=[a_{ij}]_{n\times n}$ is a symmetric matrix.

(a)

If $a_{ii}>\sum_{j\neq i}|a_{ij}|$ for all indices $i$, then $A$ is positive
definite.

(b)

If $a_{ii}>0$ and $a_{ij}<0$ for all $i\neq j$ so that
$\sum_{i=1}^{n}a_{ij}=0$ for all $j$, then $A$ is positive semi-definite so
that its kernel is 1-dimensional.

One can refer to [7] for a proof of Lemma 2.13. Therefore, $\Lambda^{E}_{ijk}$
is negative semi-definite with rank $2$ and kernel
$\\{t(1,1,1)^{T}|t\in\mathbb{R}\\}$ for any point
$(r_{i},r_{j},r_{k},\eta_{ij},\eta_{ik},\eta_{jk})\in\Omega_{ijk}^{E}$.
$\square$

As a direct corollary of Theorem 2.12, we have the following result on the
Jacobian matrix
$\Lambda^{E}=\frac{\partial(K_{1},\cdots,K_{N})}{\partial(u_{1},\cdots,u_{N})}$.

######  Corollary 2.14.

Suppose $(M,\mathcal{T},\varepsilon,\eta)$ is a weighted triangulated surface
with the weights $\varepsilon:V\rightarrow\\{0,1\\}$ and
$\eta:E\rightarrow\mathbb{R}$ satisfying the structure conditions (1.6) and
(1.7). Then the Jacobian matrix
$\Lambda^{E}=\frac{\partial(K_{1},\cdots,K_{N})}{\partial(u_{1},\cdots,u_{N})}$
is symmetric and positive semi-definite with rank $N-1$ and kernel
$\\{t\mathbf{1}\in\mathbb{R}^{N}|t\in\mathbb{R}\\}$ for all nondegenerate
Euclidean discrete conformal structures on $(M,\mathcal{T},\varepsilon,\eta)$.

Proof. This follows from Theorem 2.12 and the fact that
$\Lambda^{E}=-\sum_{\\{ijk\\}\in F}\Lambda^{E}_{ijk}$, where
$\Lambda^{E}_{ijk}$ is extended by zeros to an $N\times N$ matrix so that
$\Lambda^{E}_{ijk}$ acts on a vector $(v_{1},\cdots,v_{N})$ only on the
coordinates corresponding to vertices $v_{i}$, $v_{j}$ and $v_{k}$ in the
triangle $\\{ijk\\}$. $\square$

######  Remark 14.

Under an additional condition that the signed distance of geometric center to
the edges are all positive for any triangle $\\{ijk\\}\in F$, Glickenstein
[40] and Glickenstein-Thomas [41] obtained the positive semi-definiteness of
the Jacobian matrix
$\Lambda^{E}=\frac{\partial(K_{1},\cdots,K_{N})}{\partial(u_{1},\cdots,u_{N})}$.
Corollary 2.14 generalizes Glickenstein-Thomas’s result in that it allows some
of the signed distance to be negative. For example, in the case that
$\varepsilon\equiv 1$ and $\eta\equiv 2$, which satisfies the structures
conditions (1.6) and (1.7), if $r:V\rightarrow(0,+\infty)$ is a map with
$r\equiv 1$ except $r_{i}=1/5$ for some vertex $i\in V$, then $r$ is a
nondegenerate Euclidean discrete conformal structure on
$(M,\mathcal{T},\varepsilon,\eta)$ with $\Lambda^{E}$ positive semi-definite
by Corollary 2.14. However, in this case we have $h_{i}<0,h_{j}>0,h_{k}>0$ for
a triangle $\\{ijk\\}$ at $i$, which implies
$h_{jk,i}<0,h_{ik,j}>0,h_{ij,k}>0$ by (2.17).

###  2.3 Rigidity of Euclidean discrete conformal structures

Theorem 2.7 and Lemma 2.10 imply the following function

$\displaystyle\mathcal{E}_{ijk}(u_{i},u_{j},u_{k})=\int_{(\overline{u}_{i},\overline{u}_{j},\overline{u}_{k})}^{(u_{i},u_{j},u_{k})}\theta_{i}du_{i}+\theta_{j}du_{j}+\theta_{k}du_{k}$
(2.37)

is a well-defined smooth function on $\Omega^{E}_{ijk}(\eta)$ with
$\nabla_{u_{i}}\mathcal{E}_{ijk}=\theta_{i}$ and
$\mathcal{E}_{ijk}(u_{i}+t,u_{j}+t,u_{k}+t)=\mathcal{E}_{ijk}(u_{i},u_{j},u_{k})+t\pi$,
which is called the Ricci energy function for the triangle $\\{ijk\\}$. By
Theorem 2.12, $\mathcal{E}_{ijk}(u_{i},u_{j},u_{k})$ is a locally concave
function defined on $\Omega^{E}_{ijk}(\eta)$. Set

$\displaystyle\mathcal{E}(u_{1},\cdots,u_{N})=2\pi\sum_{i\in
V}u_{i}-\sum_{\\{ijk\\}\in F}\mathcal{E}_{ijk}(u_{i},u_{j},u_{k})$ (2.38)

to be the Ricci energy function defined on the admissible space $\Omega^{E}$
of nondegenerate Euclidean discrete conformal structures for
$(M,\mathcal{T},\varepsilon,\eta)$. Then $\mathcal{E}$ is a locally convex
function defined on $\Omega^{E}$ with
$\mathcal{E}(u_{1}+t,\cdots,u_{N}+t)=\mathcal{E}(u_{1},\cdots,u_{N})+2t\pi\chi(M)$
and $\nabla_{u_{i}}\mathcal{E}=K_{i}$ by Corollary 2.14, from which the local
rigidity of Euclidean discrete conformal structures follows by the following
well-known result from analysis.

######  Lemma 2.15.

If $W:\Omega\rightarrow\mathbb{R}$ is a $C^{2}$-smooth strictly convex
function defined on a convex domain $\Omega\subseteq\mathbb{R}^{n}$, then its
gradient $\nabla W:\Omega\rightarrow\mathbb{R}^{n}$ is injective.

To prove the global rigidity of Euclidean discrete conformal structures, we
need to extend the inner angles of a triangle $\\{ijk\\}\in F$ defined for
nondegenerate Euclidean discrete conformal structures to be a globally defined
function for $(r_{i},r_{j},r_{k})\in\mathbb{R}^{3}_{>0}$.

######  Lemma 2.16.

Suppose $(M,\mathcal{T},\varepsilon,\eta)$ is a weighted triangulated surface
with the weights $\varepsilon:V\rightarrow\\{0,1\\}$ and
$\eta:E\rightarrow\mathbb{R}$ satisfying the structure conditions (1.6) and
(1.7). $\\{ijk\\}\in F$ is a topological triangle in the triangulated surface.
Then the inner angles $\theta_{i},\theta_{j},\theta_{k}$ of the triangle
$\\{ijk\\}$ defined for nondegenerate Euclidean discrete conformal structures
could be extended by constants to be continuous functions
$\widetilde{\theta}_{i},\widetilde{\theta}_{j},\widetilde{\theta}_{k}$ defined
for $(r_{i},r_{j},r_{k})\in\mathbb{R}^{3}_{>0}$.

Proof. By Theorem 2.7,
$\Omega_{ijk}^{E}(\eta)=\mathbb{R}^{3}_{>0}\setminus\sqcup_{\alpha\in\Lambda}V_{\alpha}$,
where
$\Lambda=\\{q\in\\{i,j,k\\}|A_{q}=\eta_{st}^{2}-\varepsilon_{s}\varepsilon_{t}>0,\\{q,s,t\\}=\\{i,j,k\\}\\}$
and $V_{\alpha}$ is a closed region in $\mathbb{R}^{3}_{>0}$ bounded by the
analytical function in (2.23) defined on $\mathbb{R}^{2}_{>0}$. If
$\Lambda=\varnothing$, then $\Omega_{ijk}^{E}(\eta)=\mathbb{R}^{3}_{>0}$ and
$\theta_{i},\theta_{j},\theta_{k}$ is defined for all
$(r_{i},r_{j},r_{k})\in\mathbb{R}^{3}_{>0}$.

Without loss of generality, suppose $\Lambda\neq\varnothing$ and $V_{i}$ is a
connected component of $\mathbb{R}^{3}_{>0}\setminus\Omega_{ijk}^{E}(\eta)$.
If $(r_{i},r_{j},r_{k})\in\Omega_{ijk}^{E}(\eta)$ tends to a point
$(\overline{r}_{i},\overline{r}_{j},\overline{r}_{k})$ in the boundary
$\partial V_{i}$ of $V_{i}$ in $\mathbb{R}^{3}_{>0}$, we have

$\displaystyle
4l_{ij}^{2}l_{ik}^{2}\sin^{2}\theta_{i}=(l_{ij}+l_{ik}+l_{jk})(l_{ij}+l_{ik}-l_{jk})(l_{ij}-l_{ik}+l_{jk})(-l_{ij}+l_{ik}+l_{jk})\rightarrow
0.$

Note that for any $r_{i},r_{j}>0$,
$\varepsilon_{i}r_{i}^{2}+\varepsilon_{j}r_{j}^{2}+2\eta_{ij}r_{i}r_{j}\geq
2(\varepsilon_{i}\varepsilon_{j}+\eta_{ij})r_{i}r_{j}>0$ by the structure
condition (1.6), we have $l_{ij},l_{ik}$ tend to positive constants and
$\sin\theta_{i}$ tends to zero as
$(r_{i},r_{j},r_{k})\rightarrow(\overline{r}_{i},\overline{r}_{j},\overline{r}_{k})$,
which implies $\theta_{i}$ tends to $0$ or $\pi$. Similarly, we have
$\theta_{j},\theta_{k}$ tends to $0$ or $\pi$.

Note that at $(\overline{r}_{i},\overline{r}_{j},\overline{r}_{k})\in\partial
V_{i}$, we have $h_{i}<0$, $h_{j}>0$ and $h_{k}>0$ by Remark 11. By the
continuity of $h_{i},h_{j},h_{k}$ of
$(r_{i},r_{j},r_{k})\in\mathbb{R}^{3}_{>0}$, there exists some neighborhood
$U$ of $(\overline{r}_{i},\overline{r}_{j},\overline{r}_{k})$ in
$\mathbb{R}^{3}_{>0}$ such that $h_{i}<0$, $h_{j}>0$, $h_{k}>0$ for
$(r_{i},r_{j},r_{k})\in\Omega^{E}_{ijk}(\eta)\cap U$. Combining $h_{k}>0$,
(2.17) and (2.34), we have

$\displaystyle\frac{\partial\theta_{i}}{\partial
u_{j}}=\frac{r_{i}^{2}r_{j}^{2}r_{k}^{2}}{Al^{2}_{ij}}\kappa_{k}h_{k}>0$

for $(r_{i},r_{j},r_{k})\in\Omega^{E}_{ijk}(\eta)\cap U$. Similarly, we have
$\frac{\partial\theta_{i}}{\partial u_{k}}>0$ for
$(r_{i},r_{j},r_{k})\in\Omega^{E}_{ijk}(\eta)\cap U$. By Lemma 2.10, we have
$\frac{\partial\theta_{i}}{\partial u_{i}}=-\frac{\partial\theta_{i}}{\partial
u_{j}}-\frac{\partial\theta_{i}}{\partial u_{k}}<0$ for
$(r_{i},r_{j},r_{k})\in\Omega^{E}_{ijk}(\eta)\cap U$. By the form of $V_{i}$,
i.e.

$\displaystyle
V_{i}=\\{(r_{i},r_{j},r_{k})\in\mathbb{R}^{3}_{>0}|\kappa_{i}\geq\frac{-B_{i}+\sqrt{\Delta_{i}}}{2A_{i}}\\}=\\{(r_{i},r_{j},r_{k})\in\mathbb{R}^{3}_{>0}|r_{i}\leq\frac{2A_{i}}{-B_{i}+\sqrt{\Delta_{i}}}\\},$

we have $\theta_{i}\rightarrow\pi$ as
$(r_{i},r_{j},r_{k})\rightarrow(\overline{r}_{i},\overline{r}_{j},\overline{r}_{k})$,
which implies $\theta_{j}\rightarrow 0$, $\theta_{k}\rightarrow 0$ as
$(r_{i},r_{j},r_{k})\rightarrow(\overline{r}_{i},\overline{r}_{j},\overline{r}_{k})$
by $\theta_{i}+\theta_{j}+\theta_{k}=\pi$. The same arguments apply to the
other components of $\mathbb{R}^{3}_{>0}\setminus\Omega_{ijk}^{E}(\eta)$.
Therefore, we can extend the inner angle functions
$\theta_{i},\theta_{j},\theta_{k}$ defined on $\Omega^{E}_{ijk}(\eta)$ to be
continuous functions defined on $\mathbb{R}^{3}_{>0}$ by setting

$\displaystyle\widetilde{\theta}_{i}(r_{i},r_{j},r_{k})=\left\\{\begin{array}[]{ll}\theta_{i},&\hbox{if
$(r_{i},r_{j},r_{k})\in\Omega^{E}_{ijk}(\eta)$;}\\\ \pi,&\hbox{if
$(r_{i},r_{j},r_{k})\in V_{i}$;}\\\ 0,&\hbox{otherwis.}\end{array}\right.$
(2.39)

$\square$

By Lemma 2.16, we can further extend the combinatorial curvature function $K$
defined on $\Omega^{E}$ to be defined for $r\in\mathbb{R}^{N}_{>0}$ by

$\displaystyle\widetilde{K}_{i}=2\pi-\sum_{\\{ijk\\}\in
F}\widetilde{\theta}_{i},$ (2.40)

where $\widetilde{\theta}_{i}$ is the extension of $\theta_{i}$ defined in
Lemma 2.16 by (2.39). The extended combinatorial curvature $\widetilde{K}_{i}$
still satisfies the discrete Gauss-Bonnet formula
$\sum_{i=1}^{N}\widetilde{K}_{i}=2\pi\chi(M)$. The vector
$r\in\mathbb{R}^{N}_{>0}$ is called a generalized Euclidean discrete conformal
structure for the weighted triangulated surface
$(M,\mathcal{T},\varepsilon,\eta)$. Sometimes, we also call a vector
$u=(u_{1},\cdots,u_{N})=(\ln r_{1},\cdots,\ln r_{N})\in\mathbb{R}^{N}$ as a
generalized Euclidean discrete conformal structure for the weighted
triangulated surface $(M,\mathcal{T},\varepsilon,\eta)$.

Recall the following definition of closed continuous $1$-form and extension of
locally convex function of Luo [60], which is a generalization of Bobenko-
Pinkall-Spingborn’s extension introduced in [3].

######  Definition 5 ([60], Definition 2.3).

A differential 1-form $w=\sum_{i=1}^{n}a_{i}(x)dx^{i}$ in an open set
$U\subset\mathbb{R}^{n}$ is said to be continuous if each $a_{i}(x)$ is
continuous on $U$. A continuous differential 1-form $w$ is called closed if
$\int_{\partial\tau}w=0$ for each triangle $\tau\subset U$.

######  Theorem 2.17 ([60], Corollary 2.6).

Suppose $X\subset\mathbb{R}^{n}$ is an open convex set and $A\subset X$ is an
open subset of $X$ bounded by a real analytic codimension-1 submanifold in
$X$. If $w=\sum_{i=1}^{n}a_{i}(x)dx_{i}$ is a continuous closed 1-form on $A$
so that $F(x)=\int_{a}^{x}w$ is locally convex on $A$ and each $a_{i}$ can be
extended continuous to $X$ by constant functions to a function
$\widetilde{a}_{i}$ on $X$, then
$\widetilde{F}(x)=\int_{a}^{x}\sum_{i=1}^{n}\widetilde{a}_{i}(x)dx_{i}$ is a
$C^{1}$-smooth convex function on $X$ extending $F$.

By Lemma 2.16 and Theorem 2.17, the locally concave function
$\mathcal{E}_{ijk}$ defined by (2.37) for nondegenerate Euclidean discrete
conformal structures for a triangle $\\{ijk\\}\in F$ could be extended to be a
$C^{1}$ smooth concave function

$\displaystyle\widetilde{\mathcal{E}}_{ijk}(u_{i},u_{j},u_{k})=\int_{(\overline{u}_{i},\overline{u}_{j},\overline{u}_{k})}^{(u_{i},u_{j},u_{k})}\widetilde{\theta}_{i}du_{i}+\widetilde{\theta}_{j}du_{j}+\widetilde{\theta}_{k}du_{k}$
(2.41)

defined on $\mathbb{R}^{3}$ with
$\nabla_{u_{i}}\widetilde{\mathcal{E}}_{ijk}=\widetilde{\theta}_{i}$. As a
result, the locally convex function $\mathcal{E}$ defined by (2.38) for
nondegenerate Euclidean discrete conformal structures on a weighted
triangulated surface could be extended to be a $C^{1}$ smooth convex function

$\displaystyle\widetilde{\mathcal{E}}(u_{1},\cdots,u_{N})=2\pi\sum_{i\in
V}u_{i}-\sum_{\\{ijk\\}\in F}\widetilde{\mathcal{E}}_{ijk}(u_{i},u_{j},u_{k})$
(2.42)

defined on $\mathbb{R}^{N}$ with
$\nabla_{u_{i}}\widetilde{\mathcal{E}}=\widetilde{K}_{i}=2\pi-\sum\widetilde{\theta}_{i}$.

Using the extended Ricci energy function $\widetilde{\mathcal{E}}$, we can
prove the following rigidity for generalized Euclidean discrete conformal
structures on polyhedral surfaces, which is a generalization of Theorem 1.1
(a).

######  Theorem 2.18.

Suppose $(M,\mathcal{T},\varepsilon,\eta)$ is a weighted triangulated surface
with the weights $\varepsilon:V\rightarrow\\{0,1\\}$ and
$\eta:E\rightarrow\mathbb{R}$ satisfying the structure conditions (1.6) and
(1.7). If there exists a nondegenrate Euclidean discrete conformal structure
$r_{A}\in\Omega^{E}$ and a generalized Euclidean discrete conformal structure
$r_{B}\in\mathbb{R}^{N}_{>0}$ such that $K(r_{A})=\widetilde{K}(r_{B})$. Then
$r_{A}=cr_{B}$ for some positive constant $c\in\mathbb{R}$.

Proof. Set

$\displaystyle
f(t)=\widetilde{\mathcal{E}}((1-t)u_{A}+tu_{B})=2\pi\sum_{i=1}^{N}[(1-t)u_{A,i}+tu_{B,i}]+\sum_{\\{ijk\\}\in
F}f_{ijk}(t),$

where $f_{ijk}(t)=-\widetilde{\mathcal{E}}_{ijk}((1-t)u_{A}+tu_{B})$. Then
$f(t)$ is a $C^{1}$ smooth convex function for $t\in[0,1]$ with
$f^{\prime}(0)=f^{\prime}(1)$, which implies that
$f^{\prime}(t)=f^{\prime}(0)$ for any $t\in[0,1]$. Note that the admissible
space $\Omega^{E}$ of nondegenerate Euclidean discrete conformal structures is
an open subset of $\mathbb{R}^{N}$, there exists $\epsilon>0$ such that
$(1-t)u_{A}+tu_{B}$ is nondegenerate for $t\in[0,\epsilon]$. Note that $f(t)$
is smooth for $t\in[0,\epsilon]$, we have

$\displaystyle
f^{\prime\prime}(t)=(u_{B}-u_{A})\Lambda^{E}(u_{B}-u_{A})^{T}=0,\ \forall
t\in[0,\epsilon],$

which implies $u_{B}-u_{A}=\lambda(1,\cdots,1)$ for some constant
$\lambda\in\mathbb{R}$ by Corollary 2.14. Therefore, $r_{B}=cr_{A}$ with
$c=e^{\lambda}$. $\square$

##  3 Hyperbolic discrete conformal structures

###  3.1 Admissible space of hyperbolic discrete conformal structures for a
triangle

In this subsection, we investigate the admissible space
$\Omega^{H}_{ijk}(\eta)$ of hyperbolic discrete conformal structure for a
topological triangle $\\{ijk\\}\in F$ with the weights
$\varepsilon:V\rightarrow\\{0,1\\}$ and $\eta:E\rightarrow\mathbb{R}$
satisfying the structure conditions (1.6) and (1.7). To simplify the
notations, set

$\displaystyle
S_{i}=e^{f_{i}},C_{i}=\sqrt{1+\varepsilon_{i}e^{2f_{i}}},\kappa_{i}=\frac{C_{i}}{S_{i}}.$
(3.1)

Then

$\displaystyle C_{i}^{2}-\varepsilon_{i}S_{i}^{2}=1$ (3.2)

and the edge length $l_{ij}$ is determined by

$\displaystyle\cosh l_{ij}=C_{i}C_{j}+\eta_{ij}S_{i}S_{j}.$ (3.3)

Note that (1.4) defines a positive number by the structure condition (1.6) and
the inequality $(1+a^{2})(1+b^{2})\geq(1+ab)^{2}$. Parallelling to Lemma 2.1
for the Euclidean background geometry, we have the following result on the
triangle inequalities in the hyperbolic background geometry.

######  Lemma 3.1.

Suppose $(M,\mathcal{T},\varepsilon,\eta)$ is a weighted triangulated surface
with the weights $\varepsilon:V\rightarrow\\{0,1\\}$ and
$\eta:E\rightarrow\mathbb{R}$. $\\{ijk\\}\in F$ is a topological triangle in
the triangulation. Then the positive edge lengths $l_{ij},l_{ik},l_{jk}$
defined by (1.4) satisfy the triangle inequalities if and only if $Q^{H}>0$,
where

$\displaystyle Q^{H}=$
$\displaystyle(\varepsilon_{j}\varepsilon_{k}-\eta_{jk}^{2})\kappa_{i}^{2}+(\varepsilon_{i}\varepsilon_{k}-\eta_{ik}^{2})\kappa_{j}^{2}+(\varepsilon_{i}\varepsilon_{j}-\eta_{ij}^{2})\kappa_{k}^{2}+2\gamma_{i}\kappa_{j}\kappa_{k}+2\gamma_{j}\kappa_{i}\kappa_{k}$
(3.4)
$\displaystyle+2\gamma_{k}\kappa_{i}\kappa_{j}+\varepsilon_{i}\eta_{jk}^{2}+\varepsilon_{j}\eta_{ik}^{2}+\varepsilon_{k}\eta_{ij}^{2}+2\eta_{ij}\eta_{ik}\eta_{jk}-\varepsilon_{i}\varepsilon_{j}\varepsilon_{k}.$

Proof. The positive edge lengths $l_{ij},l_{ik},l_{jk}$ defined by (1.4) for
the topological triangle $\\{ijk\\}$ satisfy the triangle inequalities if and
only if

$\displaystyle 0<$ $\displaystyle
4\sinh\frac{l_{ij}+l_{ik}+l_{jk}}{2}\sinh\frac{l_{ij}+l_{ik}-l_{jk}}{2}\sinh\frac{l_{ij}-l_{ik}+l_{jk}}{2}\sinh\frac{-l_{ij}+l_{ik}+l_{jk}}{2}$
(3.5) $\displaystyle=$ $\displaystyle(\cosh(l_{ik}+l_{jk})-\cosh l_{ij})(\cosh
l_{ij}-\cosh(l_{jk}-l_{ik}))$ $\displaystyle=$ $\displaystyle 1+2\cosh
l_{ij}\cosh l_{ik}\cosh
l_{jk}-\cosh^{2}l_{ij}-\cosh^{2}l_{ik}-\cosh^{2}l_{jk}.$

Submitting (3.3) into (3.5), by direct calculations, we have

$\displaystyle 0<$ $\displaystyle
4\sinh\frac{l_{ij}+l_{ik}+l_{jk}}{2}\sinh\frac{l_{ij}+l_{ik}-l_{jk}}{2}\sinh\frac{l_{ij}-l_{ik}+l_{jk}}{2}\sinh\frac{-l_{ij}+l_{ik}+l_{jk}}{2}$
(3.6) $\displaystyle=$
$\displaystyle(1+2C_{i}^{2}C_{j}^{2}C_{k}^{2}-C_{i}^{2}C_{j}^{2}-C_{i}^{2}C_{k}^{2}-C_{j}^{2}C_{k}^{2})$
$\displaystyle+2\gamma_{i}C_{j}C_{k}S_{i}^{2}S_{j}S_{k}+2\gamma_{j}C_{i}C_{k}S_{i}S_{j}^{2}S_{k}+2\gamma_{k}C_{i}C_{j}S_{i}S_{j}S_{k}^{2}$
$\displaystyle+2\eta_{ij}\eta_{ik}\eta_{jk}S_{i}^{2}S_{j}^{2}S_{k}^{2}-\eta_{ij}^{2}S_{i}^{2}S_{j}^{2}-\eta_{ik}^{2}S_{i}^{2}S_{k}^{2}-\eta_{jk}^{2}S_{j}^{2}S_{k}^{2}$
$\displaystyle=$
$\displaystyle(\varepsilon_{i}\varepsilon_{j}-\eta_{ij}^{2})S_{i}^{2}S_{j}^{2}+(\varepsilon_{i}\varepsilon_{k}-\eta_{ik}^{2})S_{i}^{2}S_{k}^{2}+(\varepsilon_{j}\varepsilon_{k}-\eta_{jk}^{2})S_{j}^{2}S_{k}^{2}+2\gamma_{i}C_{j}C_{k}S_{i}^{2}S_{j}S_{k}$
$\displaystyle+2\gamma_{j}C_{i}C_{k}S_{i}S_{j}^{2}S_{k}+2\gamma_{k}C_{i}C_{j}S_{i}S_{j}S_{k}^{2}+2(\varepsilon_{i}\varepsilon_{j}\varepsilon_{k}+\eta_{ij}\eta_{ik}\eta_{jk})S_{i}^{2}S_{j}^{2}S_{k}^{2}$
$\displaystyle=$ $\displaystyle
S_{i}^{2}S_{j}^{2}S_{k}^{2}[(\varepsilon_{j}\varepsilon_{k}-\eta_{jk}^{2})\kappa_{i}^{2}+(\varepsilon_{i}\varepsilon_{k}-\eta_{ik}^{2})\kappa_{j}^{2}+(\varepsilon_{i}\varepsilon_{j}-\eta_{ij}^{2})\kappa_{k}^{2}$
$\displaystyle+2\gamma_{i}\kappa_{j}\kappa_{k}+2\gamma_{j}\kappa_{i}\kappa_{k}+2\gamma_{k}\kappa_{i}\kappa_{j}+\varepsilon_{i}\eta_{jk}^{2}+\varepsilon_{j}\eta_{ik}^{2}+\varepsilon_{k}\eta_{ij}^{2}+2\eta_{ij}\eta_{ik}\eta_{jk}-\varepsilon_{i}\varepsilon_{j}\varepsilon_{k}]$
$\displaystyle=$ $\displaystyle S_{i}^{2}S_{j}^{2}S_{k}^{2}Q^{H},$

where (3.2) is used in the second and third equality. This completes the
proof. $\square$

Comparing Lemma 2.1 with Lemma 3.1, we find that $Q^{H}$ defined for
hyperbolic background geometry in Lemma 3.1 formally has one term symmetric in
$i,j,k$ more than $Q^{E}$ defined for Euclidean background geometry in Lemma
2.1. Set $h_{i},h_{j},h_{k}$ as that in (2.9), then

$\displaystyle Q^{H}=\kappa_{i}h_{i}+\kappa_{j}h_{j}+\kappa_{k}h_{k}+G,$ (3.7)

where $G$ is defined by (2.21) in Corollary 2.6.

######  Lemma 3.2.

Suppose $(M,\mathcal{T},\varepsilon,\eta)$ is a weighted triangulated surface
with the weights $\varepsilon:V\rightarrow\\{0,1\\}$ and
$\eta:E\rightarrow\mathbb{R}$ satisfying the structure conditions (1.6) and
(1.7). $\\{ijk\\}\in F$ is a topological triangle in the triangulated surface.
If we further have $\varepsilon_{j}\varepsilon_{k}-\eta_{jk}^{2}\geq 0$,
$\varepsilon_{i}\varepsilon_{k}-\eta_{ik}^{2}\geq 0$ and
$\varepsilon_{i}\varepsilon_{j}-\eta_{ij}^{2}\geq 0$, then the admissible
space $\Omega^{H}_{ijk}(\eta)$ of hyperbolic discrete conformal structures
$(f_{i},f_{j},f_{k})$ for the triangle $\\{ijk\\}$ is $\mathbb{R}^{3}$ and
then simply connected.

Proof. By Lemma 3.1, we just need to prove $Q^{H}>0$ for any
$(f_{i},f_{j},f_{k})\in\mathbb{R}^{3}$. If one of
$\varepsilon_{i},\varepsilon_{j},\varepsilon_{k}$ is zero, say
$\varepsilon_{i}=0$, we have $\eta_{ij}>0$ and $\eta_{ik}>0$ by the structure
condition (1.6), which implies
$G=2\eta_{ij}\eta_{ik}\eta_{jk}+\varepsilon_{j}\eta_{ik}^{2}+\varepsilon_{k}\eta_{ij}^{2}=(\varepsilon_{j}\eta_{ik}-\varepsilon_{k}\eta_{ij})^{2}+2\eta_{ij}\eta_{ik}(\eta_{jk}+\varepsilon_{j}\varepsilon_{k})>0$
by the structure condition (1.6). By the structure condition (1.7) and the
condition $\varepsilon_{j}\varepsilon_{k}-\eta_{jk}^{2}\geq 0$,
$\varepsilon_{i}\varepsilon_{k}-\eta_{ik}^{2}\geq 0$,
$\varepsilon_{i}\varepsilon_{j}-\eta_{ij}^{2}\geq 0$, we have $h_{i}\geq
0,h_{j}\geq 0,h_{k}\geq 0$, which implies that $Q^{H}\geq G>0$ by (3.7). If
$\varepsilon_{i}=\varepsilon_{j}=\varepsilon_{k}=1$, we have
$\kappa_{i}>1,\kappa_{j}>1,\kappa_{j}>1$ by (3.1) in this case. Combining the
structure condition (1.7) and the assumption
$\varepsilon_{j}\varepsilon_{k}-\eta_{jk}^{2}\geq 0$,
$\varepsilon_{i}\varepsilon_{k}-\eta_{ik}^{2}\geq 0$,
$\varepsilon_{i}\varepsilon_{j}-\eta_{ij}^{2}\geq 0$, we have

$\displaystyle Q^{H}\geq$ $\displaystyle
1-\eta_{jk}^{2}+1-\eta_{ik}^{2}+1-\eta_{ij}^{2}+2\gamma_{i}+2\gamma_{j}+2\gamma_{k}+\eta_{jk}^{2}+\eta_{ik}^{2}+\eta_{ij}^{2}+2\eta_{ij}\eta_{ik}\eta_{jk}-1$
$\displaystyle=$ $\displaystyle 2(\eta_{ij}+1)(\eta_{ik}+1)(\eta_{jk}+1)$
$\displaystyle>$ $\displaystyle 0,$

where the structure condition (1.6) with
$\varepsilon_{i}=\varepsilon_{j}=\varepsilon_{k}=1$ is used in the last
inequality. Therefore, the admissible space $\Omega^{H}_{ijk}(\eta)$ of
hyperbolic discrete conformal structures for the triangle $\\{ijk\\}$ is
$\mathbb{R}^{3}$. $\square$

By Lemma 3.2, we just need to study the admissible space $\Omega^{H}_{ijk}(H)$
in the case that one of $\varepsilon_{j}\varepsilon_{k}-\eta_{jk}^{2}$,
$\varepsilon_{i}\varepsilon_{k}-\eta_{ik}^{2}$,
$\varepsilon_{i}\varepsilon_{j}-\eta_{ij}^{2}$ is negative. Following the case
for Euclidean background geometry, we will give a precise description of the
admissible space $\Omega^{H}_{ijk}(\eta)$ by solving the space of degenerate
hyperbolic discrete conformal structures for the triangle $\\{ijk\\}$.

Parallelling to the Euclidean case, we have the following result on the signs
of $h_{i},h_{j},h_{k}$ for degenerate hyperbolic discrete conformal structures
for a triangle $\\{ijk\\}$.

######  Lemma 3.3.

Suppose $(M,\mathcal{T},\varepsilon,\eta)$ is a weighted triangulated surface
with the weights $\varepsilon:V\rightarrow\\{0,1\\}$ and
$\eta:E\rightarrow\mathbb{R}$ satisfying the structure conditions (1.6) and
(1.7). $\\{ijk\\}\in F$ is a topological triangle in the triangulated surface.
If $(f_{i},f_{j},f_{k})$ is a degenerate hyperbolic discrete conformal
structure for the triangle $\\{ijk\\}$, then one of $h_{i},h_{j},h_{k}$ is
negative and the others are positive.

Proof. By Lemma 3.1, if $(f_{i},f_{j},f_{k})$ is a degenerate hyperbolic
discrete conformal structure for the triangle $\\{ijk\\}$, then
$Q^{H}=\kappa_{i}h_{i}+\kappa_{j}h_{j}+\kappa_{k}h_{k}+G\leq 0$. By Lemma 3.2,
at least one of $\varepsilon_{j}\varepsilon_{k}-\eta_{jk}^{2}$,
$\varepsilon_{i}\varepsilon_{k}-\eta_{ik}^{2}$,
$\varepsilon_{i}\varepsilon_{j}-\eta_{ij}^{2}$ is negative. Without loss of
generality, assume $\varepsilon_{j}\varepsilon_{k}-\eta_{jk}^{2}<0$. By
Corollary 2.6, we have $G>0$ under the condition
$\varepsilon_{j}\varepsilon_{k}-\eta_{jk}^{2}<0$, which implies
$Q^{E}=\kappa_{i}h_{i}+\kappa_{j}h_{j}+\kappa_{k}h_{k}\leq-G<0$ by $Q^{H}\leq
0$ and then at least one of $h_{i},h_{j},h_{k}$ is negative. Following the
arguments for Lemma 2.2, there is no subset $\\{s,t\\}\subseteq\\{i,j,k\\}$
such that $h_{s}\leq 0$ and $h_{t}\leq 0$. Following the arguments for
Corollary 2.3, one of $h_{i},h_{j},h_{k}$ is negative and the others are
positive. As the proof is parallelling to that for Lemma 2.2 and Corollary 2.3
in the Euclidean background geometry, we omit the details here. $\square$

Suppose $(f_{i},f_{j},f_{k})\in\mathbb{R}^{3}$ is a degenerate hyperbolic
discrete conformal structure for the triangle $\\{ijk\\}\in F$, then one of
$h_{i},h_{j},h_{k}$ is negative by Lemma 3.3. Without loss of generality,
assume $h_{i}<0$, then
$(\eta_{jk}^{2}-\varepsilon_{j}\varepsilon_{k})\kappa_{i}>\gamma_{j}\kappa_{k}+\gamma_{k}\kappa_{j}\geq
0$ by the structure condition (1.7). As $(f_{i},f_{j},f_{k})$ is a degenerate
hyperbolic discrete conformal structure, we have $Q^{H}\leq 0$ by Lemma 3.1,
which is equivalent to

$\displaystyle A_{i}\kappa_{i}^{2}+B_{i}\kappa_{i}+C_{i}\geq 0,$ (3.8)

where

$\displaystyle A_{i}=$
$\displaystyle\eta_{jk}^{2}-\varepsilon_{j}\varepsilon_{k}>0,$ (3.9)
$\displaystyle B_{i}=$
$\displaystyle-2\gamma_{j}\kappa_{k}-2\gamma_{k}\kappa_{j}\leq 0,$
$\displaystyle C_{i}=$
$\displaystyle(\eta_{ik}^{2}-\varepsilon_{i}\varepsilon_{k})\kappa_{j}^{2}+(\eta_{ij}^{2}-\varepsilon_{i}\varepsilon_{j})\kappa_{k}^{2}-2\gamma_{i}\kappa_{j}\kappa_{k}-G.$

Parallelling to the Euclidean case, we have the following result for the
discriminant of (3.8) in the hyperbolic background geometry.

######  Lemma 3.4.

Suppose $(M,\mathcal{T},\varepsilon,\eta)$ is a weighted triangulated surface
with the weights $\varepsilon:V\rightarrow\\{0,1\\}$ and
$\eta:E\rightarrow\mathbb{R}$ satisfying the structure conditions (1.6).
$\\{ijk\\}\in F$ is a topological triangle in the triangulated surface. If
$A_{i}=\eta_{jk}^{2}-\varepsilon_{j}\varepsilon_{k}>0$, then the discriminant
$\Delta_{i}=B_{i}^{2}-4A_{i}C_{i}$ for (3.8) is positive, where
$A_{i},B_{i},C_{i}$ are defined by (3.9).

Proof. By Corollary 2.6, we have $G>0$ under the assumption
$A_{i}=\eta_{jk}^{2}-\varepsilon_{j}\varepsilon_{k}>0$. Then the proof is
reduced to the case in Lemma 2.5, which has been completed. $\square$

######  Remark 15.

One can also take $Q^{H}$ as a quadratic function of $\kappa_{j}$ or
$\kappa_{k}$ and define $\Delta_{j}$, $\Delta_{k}$ similarly. By symmetry,
under the same conditions as that in Lemma 3.4, if
$A_{j}=\eta_{ik}^{2}-\varepsilon_{i}\varepsilon_{k}>0$, then $\Delta_{j}>0$
and if $A_{k}=\eta_{ij}^{2}-\varepsilon_{i}\varepsilon_{j}>0$, then
$\Delta_{k}>0$.

Parallelling to Theorem 2.7, we have the following characterization for the
admissible space $\Omega^{H}_{ijk}(\eta)$ of hyperbolic discrete conformal
structures for the triangle $\\{ijk\\}\in F$.

######  Theorem 3.5.

Suppose $(M,\mathcal{T},\varepsilon,\eta)$ is a weighted triangulated surface
with the weights $\varepsilon:V\rightarrow\\{0,1\\}$ and
$\eta:E\rightarrow\mathbb{R}$ satisfying the structure conditions (1.6) and
(1.7). $\\{ijk\\}\in F$ is a topological triangle in the triangulated surface.
Then the admissible space $\Omega_{ijk}^{H}(\eta)\subseteq\mathbb{R}^{3}$ of
hyperbolic discrete conformal structures for the triangle $\\{ijk\\}\in F$ is
nonempty and simply connected with analytical boundary components.
Furthermore,

$\displaystyle\Omega^{H}_{ijk}(\eta)=\mathbb{R}^{3}\setminus\sqcup_{\alpha\in\Lambda}V_{\alpha},$

where
$\Lambda=\\{q\in\\{i,j,k\\}|A_{q}=\eta_{st}^{2}-\varepsilon_{s}\varepsilon_{t}>0,\\{q,s,t\\}=\\{i,j,k\\}\\}$,
$\sqcup_{\alpha\in\Lambda}V_{\alpha}$ is a disjoint union of $V_{\alpha}$ with

$\displaystyle V_{i}=$
$\displaystyle\left\\{(f_{i},f_{j},f_{k})\in\mathbb{R}^{3}|\kappa_{i}\geq\frac{-B_{i}+\sqrt{\Delta_{i}}}{2A_{i}}\right\\}$
(3.10) $\displaystyle=$
$\displaystyle\left\\{(f_{i},f_{j},f_{k})\in\mathbb{R}^{3}|f_{i}\leq-\frac{1}{2}\ln\left[\left(\frac{-B_{i}+\sqrt{\Delta_{i}}}{2A_{i}}\right)^{2}-\varepsilon_{i}\right]\right\\}$

being a closed region in $\mathbb{R}^{3}$ bounded by an analytical function
defined on $\mathbb{R}^{2}$ and $V_{j},V_{k}$ defined similarly.

The proof of Theorem 3.5 is parallelling to that of Theorem 2.7, so we omit
the details here.

######  Remark 16.

If $(f_{i},f_{j},f_{k})\in V_{i}$ is a degenerate hyperbolic discrete
conformal structure for the triangle $\\{ijk\\}$, then $h_{i}\leq 0$ by
$-2h_{i}=2A_{i}\kappa_{i}+B_{i}$ and (3.10), which further implies
$h_{i}<0,h_{j}>0,h_{k}>0$ by Lemma 3.3.

Parallelling to the Euclidean case, we can introduce the following
parameterized hyperbolic admissible space

$\displaystyle\Omega_{ijk}^{H}=\left\\{(f_{i},f_{j},f_{k},\eta_{ij},\eta_{ik},\eta_{jk})\in\mathbb{R}^{6}|\eta\
\text{satisfies}\ (\ref{structure condition 1}),(\ref{structure condition 2})\
\text{and }(f_{i},f_{j},f_{k})\in\Omega_{ijk}^{H}(\eta)\right\\}.$

As a direct corollary of Theorem 3.5, we have the following result for the
parameterized hyperbolic admissible space $\Omega_{ijk}^{H}$ parallelling to
Corollary 2.9 for the parameterized Euclidean admissible space
$\Omega_{ijk}^{E}$.

######  Corollary 3.6.

Suppose $(M,\mathcal{T},\varepsilon,\eta)$ is a weighted triangulated surface
with the weights $\varepsilon:V\rightarrow\\{0,1\\}$ and
$\eta:E\rightarrow\mathbb{R}$ satisfying the structure conditions (1.6) and
(1.7). $\\{ijk\\}\in F$ is a topological triangle in the triangulated surface.
Then the parameterized hyperbolic admissible space $\Omega_{ijk}^{H}$ is
connected.

The proof for Corollary 3.6 is the same as that for Corollary 2.9, so we omit
the details of the proof here. Parallelling to the Euclidean case, the
parameterized admissible space $\Omega_{ijk}^{H}$ contains some points with
good properties.

######  Lemma 3.7.

The point $p=(f_{i},f_{j},f_{k},\eta_{ij},\eta_{ik},\eta_{jk})=(0,0,0,1,1,1)$
is contained in $\Omega_{ijk}^{H}$. Furthermore,
$h_{i}(p)>0,h_{j}(p)>0,h_{k}(p)>0$.

Proof. It is straight forward to check that
$(\eta_{ij},\eta_{ik},\eta_{jk})=(1,1,1)$ satisfies the structure conditions
(1.6) and (1.7). By direct calculations, we have

$\displaystyle h_{i}(p)=$
$\displaystyle(\varepsilon_{j}\varepsilon_{k}-1)\sqrt{1+\varepsilon_{i}}+(1+\varepsilon_{j})\sqrt{1+\varepsilon_{k}}+(1+\varepsilon_{k})\sqrt{1+\varepsilon_{j}}$
$\displaystyle\geq$
$\displaystyle\sqrt{1+\varepsilon_{j}}+\sqrt{1+\varepsilon_{k}}-\sqrt{1+\varepsilon_{i}}$
$\displaystyle\geq$ $\displaystyle 2-\sqrt{2}$ $\displaystyle>$ $\displaystyle
0$

by $\varepsilon_{i},\varepsilon_{j},\varepsilon_{k}\in\\{0,1\\}$. Similarly,
we have $h_{j}(p)>0,h_{k}(p)>0$. Furthermore,
$G=2+\varepsilon_{i}+\varepsilon_{j}+\varepsilon_{k}-\varepsilon_{i}\varepsilon_{j}\varepsilon_{k}>0$
in this case by $\varepsilon:V\rightarrow\\{0,1\\}$. Therefore,

$Q^{H}=\kappa_{i}h_{i}+\kappa_{j}h_{j}+\kappa_{k}h_{k}+G>0$

at $p$, which implies $p\in\Omega_{ijk}^{H}$ by Lemma 3.1. $\square$

###  3.2 Negative definiteness of the Jacobian matrix in the hyperbolic case

Suppose $\\{ijk\\}$ is a nondegenerate hyperbolic triangle with edge lengths
given by (1.4) and $\theta_{i},\theta_{j},\theta_{k}$ are the inner angles at
the vertices $i,j,k$ in the triangle respectively. Set $u_{i}$ to be the
following function of $f_{i}$

$\displaystyle
u_{i}=\left\\{\begin{array}[]{ll}f_{i},&\hbox{$\varepsilon_{i}=0$;}\\\
\frac{1}{2}\ln\left(\frac{\sqrt{1+e^{2f_{i}}}-1}{\sqrt{1+e^{2f_{i}}}+1}\right),&\hbox{$\varepsilon_{i}=1$.}\end{array}\right.$
(3.11)

Then

$\displaystyle\frac{\partial f_{i}}{\partial
u_{i}}=\sqrt{1+\varepsilon_{i}e^{2f_{i}}}=C_{i}.$ (3.12)

######  Lemma 3.8 ([41, 96]).

Suppose $(M,\mathcal{T},\varepsilon,\eta)$ is a weighted triangulated surface
with the weights $\varepsilon:V\rightarrow\\{0,1\\}$ and
$\eta:E\rightarrow\mathbb{R}$. Suppose $(f_{i},f_{j},f_{k})\in\mathbb{R}^{3}$
is a nondegenerate hyperbolic discrete conformal structure for the topological
triangle $\\{ijk\\}\in F$. Then

$\displaystyle\frac{\partial\theta_{i}}{\partial
u_{j}}=\frac{\partial\theta_{j}}{\partial
u_{i}}=\frac{S_{i}^{2}S_{j}^{2}S_{k}}{A\sinh^{2}l_{ij}}[(\varepsilon_{i}\varepsilon_{j}-\eta_{ij}^{2})\kappa_{k}+\gamma_{i}\kappa_{j}+\gamma_{j}\kappa_{i}]=\frac{S_{i}^{2}S_{j}^{2}S_{k}}{A\sinh^{2}l_{ij}}h_{k},$
(3.13)

where $A=\sinh l_{ij}\sinh l_{ik}\sin\theta_{i}$ and $u_{i}$ is defined by
(3.11).

Proof. By the chain rules,

$\displaystyle\frac{\partial\theta_{i}}{\partial
u_{j}}=\frac{\partial\theta_{i}}{\partial l_{jk}}\frac{\partial
l_{jk}}{\partial u_{j}}+\frac{\partial\theta_{i}}{\partial
l_{ik}}\frac{\partial l_{ik}}{\partial
u_{j}}+\frac{\partial\theta_{i}}{\partial l_{ij}}\frac{\partial
l_{ij}}{\partial u_{j}}.$ (3.14)

The derivative cosine law ([7], Lemma A1) for hyperbolic triangles gives

$\displaystyle\frac{\partial\theta_{i}}{\partial l_{jk}}=\frac{\sinh
l_{jk}}{A},\ \frac{\partial\theta_{i}}{\partial l_{ij}}=\frac{-\sinh
l_{jk}\cos\theta_{j}}{A},$ (3.15)

where $A=\sinh l_{ij}\sinh l_{ik}\sin\theta_{i}$. By (1.4) and (3.12), we have

$\displaystyle\frac{\partial l_{jk}}{\partial u_{j}}=\frac{1}{\sinh
l_{jk}}(\varepsilon_{j}S_{j}^{2}C_{k}+\eta_{jk}S_{j}S_{k}C_{j}),\
\frac{\partial l_{ik}}{\partial u_{j}}=0,\ \frac{\partial l_{ij}}{\partial
u_{j}}=\frac{1}{\sinh
l_{ij}}(\varepsilon_{j}S_{j}^{2}C_{i}+\eta_{ij}S_{i}S_{j}C_{j}).$ (3.16)

Submitting (3.15) and (3.16) into (3.14), by direct calculations, we have

$\displaystyle\frac{\partial\theta_{i}}{\partial u_{j}}=$
$\displaystyle\frac{1}{A}(\varepsilon_{j}S_{j}^{2}C_{k}+\eta_{jk}S_{j}S_{k}C_{j})+\frac{-\sinh
l_{jk}\cos\theta_{j}}{A}\frac{1}{\sinh
l_{ij}}(\varepsilon_{j}S_{j}^{2}C_{i}+\eta_{ij}S_{i}S_{j}C_{j})$ (3.17)
$\displaystyle=$
$\displaystyle\frac{1}{A\sinh^{2}l_{ij}}[(\cosh^{2}l_{ij}-1)(\varepsilon_{j}S_{j}^{2}C_{k}+\eta_{jk}S_{j}S_{k}C_{j})$
$\displaystyle+(\cosh l_{ik}-\cosh l_{ij}\cosh
l_{jk})(\varepsilon_{j}S_{j}^{2}C_{i}+\eta_{ij}S_{i}S_{j}C_{j})]$
$\displaystyle=$
$\displaystyle\frac{S_{i}^{2}S_{j}^{2}S_{k}}{A\sinh^{2}l_{ij}}[(\varepsilon_{i}\varepsilon_{j}-\eta_{ij}^{2})\kappa_{k}+\gamma_{i}\kappa_{j}+\gamma_{j}\kappa_{i}]$
$\displaystyle=$
$\displaystyle\frac{S_{i}^{2}S_{j}^{2}S_{k}}{A\sinh^{2}l_{ij}}h_{k},$

where the hyperbolic cosine law is used in the second equality and the
definition (1.4) for hyperbolic length is used in the third equality. Note
that (3.17) is symmetric in the indices $i$ and $j$, this implies
$\frac{\partial\theta_{i}}{\partial u_{j}}=\frac{\partial\theta_{j}}{\partial
u_{i}}$. $\square$

######  Remark 17.

The result in Lemma 3.8 was previously obtained by Glickenstein-Thomas [41]
and Zhang-Guo-Zeng-Luo-Yau-Gu [96]. Here we give a proof by direct
calculations for completeness. By (3.13) and Remark 16, if
$(\eta_{ij},\eta_{ik},\eta_{jk})$ defined on the triangle $\\{ijk\\}$
satisfies the structure conditions (1.6) and (1.7) and
$(f_{i},f_{j},f_{k})\in\Omega_{ijk}^{H}(\eta)$ tends to a point
$(\overline{f}_{i},\overline{f}_{j},\overline{f}_{k})\in\partial V_{i}$ with
$V_{i}\neq\emptyset$, then $\frac{\partial\theta_{i}}{\partial
u_{j}}\rightarrow+\infty$, $\frac{\partial\theta_{i}}{\partial
u_{k}}\rightarrow+\infty$. Recall the following formula obtained by
Glickenstein-Thomas ([41], Proposition 9)

$\frac{\partial A_{ijk}}{\partial u_{i}}=\frac{\partial\theta_{j}}{\partial
u_{i}}(\cosh l_{ij}-1)+\frac{\partial\theta_{k}}{\partial u_{i}}(\cosh
l_{ik}-1)$ (3.18)

for the area $A_{ijk}$ of the hyperbolic triangle $\\{ijk\\}$, we have

$-\frac{\partial\theta_{i}}{\partial u_{i}}=\frac{\partial A_{ijk}}{\partial
u_{i}}+\frac{\partial\theta_{j}}{\partial
u_{i}}+\frac{\partial\theta_{k}}{\partial
u_{i}}=\frac{\partial\theta_{j}}{\partial u_{i}}\cosh
l_{ij}+\frac{\partial\theta_{k}}{\partial u_{i}}\cosh l_{ik}$ (3.19)

by the area formula for the hyperbolic triangle $\\{ijk\\}$. (3.19) implies
$\frac{\partial\theta_{i}}{\partial u_{i}}\rightarrow-\infty$ as
$(f_{i},f_{j},f_{k})\rightarrow(\overline{f}_{i},\overline{f}_{j},\overline{f}_{k})\in\partial
V_{i}$.

Lemma 3.8 shows that the Jacobian matrix

$\displaystyle\Lambda^{H}_{ijk}:=\frac{\partial(\theta_{i},\theta_{j},\theta_{k})}{\partial(u_{i},u_{j},u_{k})}=\left(\begin{array}[]{ccc}\frac{\partial\theta_{i}}{\partial
u_{i}}&\frac{\partial\theta_{i}}{\partial
u_{j}}&\frac{\partial\theta_{i}}{\partial u_{k}}\\\
\frac{\partial\theta_{j}}{\partial u_{i}}&\frac{\partial\theta_{j}}{\partial
u_{j}}&\frac{\partial\theta_{j}}{\partial u_{k}}\\\
\frac{\partial\theta_{k}}{\partial u_{i}}&\frac{\partial\theta_{k}}{\partial
u_{j}}&\frac{\partial\theta_{k}}{\partial u_{k}}\\\ \end{array}\right)$

is symmetric. We further have the following result on the rank of the Jacobian
matrix $\Lambda^{H}_{ijk}$.

######  Lemma 3.9.

Suppose $(M,\mathcal{T},\varepsilon,\eta)$ is a weighted triangulated surface
with the weights $\varepsilon:V\rightarrow\\{0,1\\}$ and
$\eta:E\rightarrow\mathbb{R}$ satisfying the structure conditions (1.6) and
(1.7). $\\{ijk\\}\in F$ is a topological triangle in the triangulated surface.
Then the rank of the Jacobian matrix
$\Lambda^{H}_{ijk}=\frac{\partial(\theta_{i},\theta_{j},\theta_{k})}{\partial(u_{i},u_{j},u_{k})}$
is $3$ for all nondegenerate hyperbolic discrete conformal structures on the
triangle $\\{ijk\\}$.

Proof. By the chain rules, we have

$\displaystyle\frac{\partial(\theta_{i},\theta_{j},\theta_{k})}{\partial(u_{i},u_{j},u_{k})}=\frac{\partial(\theta_{i},\theta_{j},\theta_{k})}{\partial(l_{jk},l_{ik},l_{ij})}\cdot\frac{\partial(l_{jk},l_{ik},l_{ij})}{\partial(u_{i},u_{j},u_{k})}.$
(3.20)

The derivative cosine law ([7], Lemma A1) gives

$\displaystyle\frac{\partial(\theta_{i},\theta_{j},\theta_{k})}{\partial(l_{jk},l_{ik},l_{ij})}=\frac{1}{A}\left(\begin{array}[]{ccc}\sinh
l_{jk}&&\\\ &\sinh l_{ik}&\\\ &&\sinh l_{ij}\\\
\end{array}\right)\left(\begin{array}[]{ccc}1&-\cos\theta_{k}&-\cos\theta_{j}\\\
-\cos\theta_{k}&1&-\cos\theta_{i}\\\ -\cos\theta_{j}&-\cos\theta_{i}&1\\\
\end{array}\right),$

which implies

$\displaystyle\det\left(\frac{\partial(\theta_{i},\theta_{j},\theta_{k})}{\partial(l_{jk},l_{ik},l_{ij})}\right)$
$\displaystyle=$ $\displaystyle\frac{\sinh l_{ij}\sinh l_{ik}\sinh
l_{jk}}{A^{3}}\det\left(\begin{array}[]{ccc}1&-\cos\theta_{k}&-\cos\theta_{j}\\\
-\cos\theta_{k}&1&-\cos\theta_{i}\\\ -\cos\theta_{j}&-\cos\theta_{i}&1\\\
\end{array}\right)$ $\displaystyle=$ $\displaystyle\frac{-\sinh l_{ij}\sinh
l_{ik}\sinh
l_{jk}}{A^{3}}(-1+\cos\theta_{i}^{2}+\cos\theta_{j}^{2}+\cos\theta_{k}^{2}+2\cos\theta_{i}\cos\theta_{j}\cos\theta_{k})$
$\displaystyle=$ $\displaystyle\frac{-4\sinh l_{ij}\sinh l_{ik}\sinh
l_{jk}}{A^{3}}$
$\displaystyle\cdot\cos\frac{\theta_{i}+\theta_{j}+\theta_{k}}{2}\cos\frac{\theta_{i}+\theta_{j}-\theta_{k}}{2}\cos\frac{\theta_{i}-\theta_{j}+\theta_{k}}{2}\cos\frac{\theta_{i}-\theta_{j}-\theta_{k}}{2}.$

By the area formula for hyperbolic triangles, we have
$\theta_{i}+\theta_{j}+\theta_{k}\in(0,\pi)$, which implies
$\frac{\theta_{i}+\theta_{j}+\theta_{k}}{2},\frac{\theta_{i}+\theta_{j}-\theta_{k}}{2},\frac{\theta_{i}-\theta_{j}+\theta_{k}}{2},\frac{\theta_{i}-\theta_{j}-\theta_{k}}{2}\in(-\frac{\pi}{2},\frac{\pi}{2})$.
Then
$\det\large(\frac{\partial(\theta_{i},\theta_{j},\theta_{k})}{\partial(l_{jk},l_{ik},l_{ij})}\large)<0$.

By (1.4) and (3.12), we have

$\displaystyle\frac{\partial(l_{jk},l_{ik},l_{ij})}{\partial(u_{i},u_{j},u_{k})}$
$\displaystyle=$ $\displaystyle\left(\begin{array}[]{ccc}\frac{1}{\sinh
l_{jk}}&&\\\ &\frac{1}{\sinh l_{ik}}&\\\ &&\frac{1}{\sinh l_{ij}}\\\
\end{array}\right)$
$\displaystyle\cdot\left(\begin{array}[]{ccc}0&\varepsilon_{j}S_{j}^{2}C_{k}+\eta_{jk}S_{j}S_{k}C_{j}&\varepsilon_{k}S_{k}^{2}C_{j}+\eta_{jk}S_{j}S_{k}C_{k}\\\
\varepsilon_{i}S_{i}^{2}C_{k}+\eta_{ik}S_{i}S_{k}C_{i}&0&\varepsilon_{k}S_{k}^{2}C_{i}+\eta_{ik}S_{i}S_{k}C_{k}\\\
\varepsilon_{i}S_{i}^{2}C_{j}+\eta_{ij}S_{i}S_{j}C_{i}&\varepsilon_{j}S_{j}^{2}C_{i}+\eta_{ij}S_{i}S_{j}C_{j}&0\\\
\end{array}\right).$

This implies

$\displaystyle\det\left(\frac{\partial(l_{jk},l_{ik},l_{ij})}{\partial(u_{i},u_{j},u_{k})}\right)$
(3.21) $\displaystyle=$ $\displaystyle\frac{S_{i}S_{j}S_{k}}{\sinh l_{ij}\sinh
l_{ik}\sinh l_{jk}}$
$\displaystyle\cdot[2(\varepsilon_{i}\varepsilon_{j}\varepsilon_{k}+\eta_{ij}\eta_{ik}\eta_{jk})S_{i}S_{j}S_{k}C_{i}C_{j}C_{k}+\gamma_{i}S_{i}C_{i}(\varepsilon_{k}S_{k}^{2}C^{2}_{j}+\varepsilon_{j}S_{j}^{2}C_{k}^{2})$
$\displaystyle+\gamma_{j}S_{j}C_{j}(\varepsilon_{i}S_{i}^{2}C_{k}^{2}+\varepsilon_{k}S_{k}^{2}C_{i}^{2})+\gamma_{k}S_{k}C_{k}(\varepsilon_{i}S_{i}^{2}C_{j}^{2}+\varepsilon_{j}S_{j}^{2}C_{i}^{2})]$
$\displaystyle\geq$
$\displaystyle\frac{2S_{i}^{2}S_{j}^{2}S_{k}^{2}C_{i}C_{j}C_{k}}{\sinh
l_{ij}\sinh l_{ik}\sinh
l_{jk}}[\varepsilon_{i}\varepsilon_{j}\varepsilon_{k}+\eta_{ij}\eta_{ik}\eta_{jk}+\gamma_{i}\varepsilon_{j}\varepsilon_{k}+\gamma_{j}\varepsilon_{i}\varepsilon_{k}+\gamma_{k}\varepsilon_{i}\varepsilon_{j}]$
$\displaystyle=$
$\displaystyle\frac{2S_{i}^{2}S_{j}^{2}S_{k}^{2}C_{i}C_{j}C_{k}}{\sinh
l_{ij}\sinh l_{ik}\sinh
l_{jk}}(\varepsilon_{i}\varepsilon_{j}+\eta_{ij})(\varepsilon_{i}\varepsilon_{k}+\eta_{ik})(\varepsilon_{j}\varepsilon_{k}+\eta_{jk})$
$\displaystyle>$ $\displaystyle 0,$

where the structure condition (1.7) is used in the third line and the
structure condition (1.6) is used in the last line.

Therefore,
$\det(\frac{\partial(\theta_{i},\theta_{j},\theta_{k})}{\partial(u_{i},u_{j},u_{k})})<0$
by (3.20),
$\det\large(\frac{\partial(\theta_{i},\theta_{j},\theta_{k})}{\partial(l_{jk},l_{ik},l_{ij})}\large)<0$
and (3.21), which implies the rank of the Jacobian matrix
$\Lambda^{H}_{ijk}=\frac{\partial(\theta_{i},\theta_{j},\theta_{k})}{\partial(u_{i},u_{j},u_{k})}$
is $3$. $\square$

As a consequence of Lemma 3.8 and Lemma 3.9, we have the following result on
the negative definiteness of the Jacobian matrix
$\Lambda^{H}_{ijk}=\frac{\partial(\theta_{i},\theta_{j},\theta_{k})}{\partial(u_{i},u_{j},u_{k})}$.

######  Theorem 3.10.

Suppose $(M,\mathcal{T},\varepsilon,\eta)$ is a weighted triangulated surface
with the weights $\varepsilon:V\rightarrow\\{0,1\\}$ and
$\eta:E\rightarrow\mathbb{R}$ satisfying the structure conditions (1.6) and
(1.7). $\\{ijk\\}\in F$ is a topological triangle in the triangulated surface.
Then the Jacobian matrix
$\Lambda^{H}_{ijk}=\frac{\partial(\theta_{i},\theta_{j},\theta_{k})}{\partial(u_{i},u_{j},u_{k})}$
is symmetric and negative definite for all nondegenerate hyperbolic discrete
conformal structures on the triangle $\\{ijk\\}$.

Proof. By Lemma 3.9, all the three eigenvalues of the Jacobian matrix
$\Lambda^{H}_{ijk}$ are nonzero. Taking the Jacobian matrix
$\Lambda^{H}_{ijk}$ as a matrix-valued function of
$(f_{i},f_{j},f_{k},\eta_{ij},\eta_{ik},\eta_{jk})$ in the parameterized
admissible space $\Omega^{H}_{ijk}$. By the continuity of eigenvalues of
$\Lambda^{H}_{ijk}$ and the connectivity of the parameterized admissible space
$\Omega^{H}_{ijk}$ in Corollary 3.6, to prove the negative definiteness of
$\Lambda^{H}_{ijk}=\frac{\partial(\theta_{i},\theta_{j},\theta_{k})}{\partial(u_{i},u_{j},u_{k})}$,
we just need to find a point $p\in\Omega^{H}_{ijk}$ such that the eigenvalues
of $\Lambda^{H}_{ijk}$ at $p$ are negative. Taking
$p=(f_{i},f_{j},f_{k},\eta_{ij},\eta_{ik},\eta_{jk})=(0,0,0,1,1,1)$, then
$p\in\Omega_{ijk}^{H}$ and $h_{i}(p)>0,h_{j}(p)>0,h_{k}(p)>0$ by Lemma 3.7,
which implies that $\frac{\partial\theta_{i}}{\partial
u_{j}}=\frac{\partial\theta_{j}}{\partial u_{i}}>0$ and
$\frac{\partial\theta_{i}}{\partial u_{k}}=\frac{\partial\theta_{k}}{\partial
u_{i}}>0$ at $p$ by Lemma 3.8. By (3.19), we have
$-\frac{\partial\theta_{i}}{\partial u_{i}}>\frac{\partial\theta_{j}}{\partial
u_{i}}+\frac{\partial\theta_{k}}{\partial u_{i}}$ at $p$. Therefore,
$\Lambda^{H}_{ijk}$ is negative definite and has three negative eigenvalues at
$p$ by Lemma 2.13. $\square$

As a direct corollary of Theorem 3.10, we have the following result on the
Jacobian matrix
$\Lambda^{H}=\frac{\partial(K_{1},\cdots,K_{N})}{\partial(u_{1},\cdots,u_{N})}$
for hyperbolic discrete conformal structures.

######  Corollary 3.11.

Suppose $(M,\mathcal{T},\varepsilon,\eta)$ is a weighted triangulated surface
with the weights $\varepsilon:V\rightarrow\\{0,1\\}$ and
$\eta:E\rightarrow\mathbb{R}$ satisfying the structure conditions (1.6) and
(1.7). Then the Jacobian matrix
$\Lambda^{H}=\frac{\partial(K_{1},\cdots,K_{N})}{\partial(u_{1},\cdots,u_{N})}$
for hyperbolic discrete conformal structures is symmetric and positive
definite for all nondegenerate hyperbolic discrete conformal structures on
$(M,\mathcal{T},\varepsilon,\eta)$.

The proof for Corollary 3.11 is the same as that for Corollary 2.14, so we
omit the details of the proof here.

###  3.3 Rigidity of hyperbolic discrete conformal structures

Theorem 3.5 and Lemma 3.8 imply the following Ricci energy function for the
triangle $\\{ijk\\}$

$\displaystyle\mathcal{E}_{ijk}(u_{i},u_{j},u_{k})=\int_{(\overline{u}_{i},\overline{u}_{j},\overline{u}_{k})}^{(u_{i},u_{j},u_{k})}\theta_{i}du_{i}+\theta_{j}du_{j}+\theta_{k}du_{k}$
(3.22)

is a well-defined smooth function on $\Omega^{H}_{ijk}(\eta)$ with
$\nabla_{u_{i}}\mathcal{E}_{ijk}=\theta_{i}$. By Theorem 3.10,
$\mathcal{E}_{ijk}(u_{i},u_{j},u_{k})$ is a locally strictly concave function
defined on $\Omega^{H}_{ijk}(\eta)$ under the structure conditions (1.6) and
(1.7). Set

$\displaystyle\mathcal{E}(u_{1},\cdots,u_{N})=2\pi\sum_{i\in
V}u_{i}-\sum_{\\{ijk\\}\in F}\mathcal{E}_{ijk}(u_{i},u_{j},u_{k})$ (3.23)

to be the Ricci energy function defined on the admissible space $\Omega^{H}$
of nondegenerate hyperbolic discrete conformal structures for
$(M,\mathcal{T},\varepsilon,\eta)$. Then $\mathcal{E}$ is a locally strictly
convex function defined on $\Omega^{H}$ with $\nabla_{u_{i}}\mathcal{E}=K_{i}$
by Corollary 3.11, from which the local rigidity of hyperbolic discrete
conformal structures follows by Lemma 2.15.

To prove the global rigidity of hyperbolic discrete conformal structures, we
need to extend the inner angles in a hyperbolic triangle $\\{ijk\\}\in F$
defined for nondegenerate hyperbolic discrete conformal structures to be a
globally defined function for $(f_{i},f_{j},f_{k})\in\mathbb{R}^{3}$.
Parallelling to Lemma 2.16, we have the following extension for inner angles
of hyperbolic triangles.

######  Lemma 3.12.

Suppose $(M,\mathcal{T},\varepsilon,\eta)$ is a weighted triangulated surface
with the weights $\varepsilon:V\rightarrow\\{0,1\\}$ and
$\eta:E\rightarrow\mathbb{R}$ satisfying the structure conditions (1.6) and
(1.7). $\\{ijk\\}\in F$ is a topological triangle in the triangulated surface.
Then the inner angles $\theta_{i},\theta_{j},\theta_{k}$ of the triangle
$\\{ijk\\}$ defined for nondegenerate hyperbolic discrete conformal structures
could be extended by constants to be continuous functions
$\widetilde{\theta}_{i},\widetilde{\theta}_{j},\widetilde{\theta}_{k}$ defined
for $(f_{i},f_{j},f_{k})\in\mathbb{R}^{3}$.

Proof. By Theorem 3.5,
$\Omega_{ijk}^{H}(\eta)=\mathbb{R}^{3}\setminus\sqcup_{\alpha\in\Lambda}V_{\alpha}$,
where
$\Lambda=\\{q\in\\{i,j,k\\}|A_{q}=\eta_{st}^{2}-\varepsilon_{s}\varepsilon_{t}>0,\\{q,s,t\\}=\\{i,j,k\\}\\}$
and $V_{\alpha}$ is a closed region in $\mathbb{R}^{3}$ bounded by the
analytical function in (3.10) defined on $\mathbb{R}^{2}$. If
$\Lambda=\varnothing$, then $\Omega_{ijk}^{H}(\eta)=\mathbb{R}^{3}$ and
$\theta_{i},\theta_{j},\theta_{k}$ is defined for all
$(f_{i},f_{j},f_{k})\in\mathbb{R}^{3}$.

Without loss of generality, suppose $\Lambda\neq\varnothing$ and $V_{i}$ is a
connected component of $\mathbb{R}^{3}\setminus\Omega_{ijk}^{H}(\eta)$.
Suppose $(f_{i},f_{j},f_{k})\in\Omega_{ijk}^{H}(\eta)$ tends to a point
$(\overline{f}_{i},\overline{f}_{j},\overline{f}_{k})\in\partial V_{i}$ in
$\mathbb{R}^{3}$. By direct calculations, we have

$\displaystyle
4\sinh\frac{l_{ij}+l_{ik}+l_{jk}}{2}\sinh\frac{l_{ij}+l_{ik}-l_{jk}}{2}\sinh\frac{l_{ij}-l_{ik}+l_{jk}}{2}\sinh\frac{-l_{ij}+l_{ik}+l_{jk}}{2}$
(3.24) $\displaystyle=$ $\displaystyle(\cosh(l_{jk}+l_{ik})-\cosh
l_{ij})(\cosh l_{ij}-\cosh(l_{jk}-l_{ik}))$ $\displaystyle=$
$\displaystyle(\cosh^{2}l_{jk}-1)(\cosh l_{ik}^{2}-1)-(\cosh l_{jk}\cosh
l_{ik}-\cosh l_{ij})^{2}$ $\displaystyle=$
$\displaystyle\sinh^{2}l_{jk}\sinh^{2}l_{ik}-\sinh^{2}l_{jk}\sinh^{2}l_{ik}\cos^{2}\theta_{k}$
$\displaystyle=$
$\displaystyle\sinh^{2}l_{jk}\sinh^{2}l_{ik}\sin^{2}\theta_{k}.$

Combining Lemma 3.1, (3.6) and the hyperbolic sine law, (3.24) implies that
$\theta_{i},\theta_{j},\theta_{k}$ tends to $0$ or $\pi$ as
$(f_{i},f_{j},f_{k})\rightarrow(\overline{f}_{i},\overline{f}_{j},\overline{f}_{k})\in\partial
V_{i}$.

Note that for $(\overline{f}_{i},\overline{f}_{j},\overline{f}_{k})\in\partial
V_{i}$, we have $h_{i}<0,h_{j}>0,h_{k}>0$ by Remark 16. By the continuity of
$h_{i},h_{j},h_{k}$ of $(f_{i},f_{j},f_{k})\in\mathbb{R}^{3}$, there exists
some neighborhood $U$ of
$(\overline{f}_{i},\overline{f}_{j},\overline{f}_{k})$ in $\mathbb{R}^{3}$