A discrete uniformization theorem for decorated piecewise Euclidean metrics on surfaces

Xu Xu, Chao Zheng School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, P.R.China [email protected] School of Mathematics and Statistics, Wuhan University, Wuhan 430072, P.R. China [email protected]

Abstract.

In this paper, we introduce a new discretization of the Gaussian curvature on surfaces, which is defined as the quotient of the angle defect and the area of some dual cell of a weighted triangulation at the conic singularity. A discrete uniformization theorem for this discrete Gaussian curvature is established on surfaces with non-positive Euler number. The main tools are Bobenko-Lutz’s discrete conformal theory for decorated piecewise Euclidean metrics on surfaces and variational principles with constraints.

Key words and phrases:

Discrete uniformization theorem; Decorated piecewise Euclidean metrics; Discrete Gaussian curvature; Variational principle

MSC (2020): 52C26

1. Introduction

Bobenko-Lutz [2] recently introduced the decorated piecewise Euclidean metrics on surfaces. Suppose $S$ is a connected closed surface and $V$ is a finite non-empty subset of $S$ , we call $(S,V)$ a marked surface. A piecewise Euclidean metric (PE metric) $dist_{S}$ on the marked surface $(S,V)$ is a flat cone metric with the conic singularities contained in $V$ . A decoration on a PE surface $(S,V,dist_{S})$ is a choice of circle of radius $r_{i}\geq 0$ at each point $i\in V$ . These circles in the decoration are called vertex-circles. We denote a decorated PE surface by $(S,V,dist_{S},r)$ and call the pair $(dist_{S},r)$ a decorated PE metric. In this paper, we focus on the case that $r_{i}>0$ for all $i\in V$ and each pair of vertex-circles is separated.

For a PE surface $(S,V,dist_{S})$ , denote $\theta_{i}$ as the cone angle at $i\in V$ . The angle defect

(1)

W:V\rightarrow(-\infty,2\pi),\quad W_{i}=2\pi-\theta_{i},

is used to describe the conic singularities of the PE metric. Let $\mathcal{T}={(V,E,F)}$ be a triangulation of $(S,V)$ , where $V,E,F$ are the sets of vertices, edges and faces respectively. The triangulation $\mathcal{T}$ for a PE surface $(S,V,dist_{S})$ is a geodesic triangulation if the edges are geodesics in the PE metric $dist_{S}$ . We use one index to denote a vertex (such as $i$ ), two indices to denote an edge (such as $\{ij\}$ ) and three indices to denote a face (such as $\{ijk\}$ ) in the triangulation $\mathcal{T}$ . For any decorated geodesic triangle $\{ijk\}\in F$ , there is a unique circle $C_{ijk}$ simultaneously orthogonal to the three vertex-circles at the vertices $i,j,k$ [10]. We call this circle $C_{ijk}$ as the face-circle of the decorated geodesic triangle $\{ijk\}$ and denote its center by $c_{ijk}$ and radius by $r_{ijk}$ . The center $c_{ijk}$ of the face-circle $C_{ijk}$ of the decorated geodesic triangle $\{ijk\}$ is the geometric center introduced by Glickenstein [9] and Glickenstein-Thomas [11] for general discrete conformal structures on surfaces. Denote $\alpha_{ij}^{k}$ as the interior intersection angle of the face-circle $C_{ijk}$ and the edge $\{ij\}$ . Please refer to Figure 1 (left) for the angle $\alpha_{ij}^{k}$ . The edge $\{ij\}$ , shared by two adjacent decorated triangles $\{ijk\}$ and $\{ijl\}$ , is called weighted Delaunay if

(2)

\alpha_{ij}^{k}+\alpha_{ij}^{l}\leq\pi.

The triangulation $\mathcal{T}$ is called weighted Delaunay in the decorated PE metric $(dist_{S},r)$ if every edge in the triangulation is weighted Delaunay. Connecting the center $c_{ijk}$ with the vertices $i,j,k$ by geodesics produces a cellular decomposition of the decorated triangle $\{ijk\}$ . Denote $A_{i}^{jk}$ as the sum of the signed area of the two triangles adjacent to $i$ in the cellular decomposition of the decorated triangle $\{ijk\}$ . The area of the triangle with the vertices $i$ , $j$ , $c_{ijk}$ is positive if it is on the same side of the edge $\{ij\}$ as the decorated triangle $\{ijk\}$ , otherwise it is negative (or zero if $c_{ijk}$ lies in $\{ij\}$ ). Please refer to the shaded domain in Figure 1 (left) for $A_{i}^{jk}$ . Gluing these cells of all decorated triangles isometrically along edges in pairs leads a cellular decomposition of the decorated PE surface $(S,V,dist_{S},r)$ . Set

A_{i}=\sum_{\{ijk\}\in F}A_{i}^{jk}.

Please refer to Figure 1 (right) for $A_{i}$ .

\begin{overpic}[scale={0.4}]{figure_0.pdf} \put(15.0,15.0){$i$} \put(80.5,15.0){$j$} \put(48.5,74.0){$k$} \put(10.0,26.0){$r_{i}$} \put(46.0,35.0){$c_{ijk}$} \put(30.0,55.0){$\alpha_{ik}^{j}$} \put(20.0,30.0){$\alpha_{ik}^{j}$} \put(27.0,12.5){$\alpha_{ij}^{k}$} \put(63.0,12.5){$\alpha_{ij}^{k}$} \end{overpic}

\begin{overpic}[scale={0.6}]{figure_3.pdf} \put(48.0,42.0){$i$} \put(92.0,44.0){$j$} \put(74.0,90.0){$k$} \put(74.0,57.0){$c_{ijk}$} \end{overpic}

Figure 1. Domain of the signed area

A_{i}^{jk}

in a decorated triangle

\{ijk\}

(left) and domain of the area

A_{i}

in a decorated PE surface (right)

Definition 1.1.

Suppose $(S,V,dist_{S},r)$ is a decorated PE surface and $\mathcal{T}$ is a weighted Delaunay triangulation of $(S,V,dist_{S},r)$ . The discrete Gaussian curvature $K_{i}$ at the vertex $i\in V$ is the quotient of the angle defect $W_{i}$ and the area $A_{i}$ of the dual cell at the vertex $i\in V$ , i.e.,

(3)

K_{i}=\frac{W_{i}}{A_{i}}.

Remark 1.2.

In the literature, the discrete curvature is usually defined by the angle defect $W$ in (1). However, the angle defect $W$ is scaling invariant and does not approximate the smooth Gaussian curvature pointwisely on smooth surfaces as the triangulations of the surface become finer and finer. This is supported by the discussions in [3, 7]. For the discrete Gaussian curvature $K$ in (3), it scales by a factor $\frac{1}{u^{2}}$ upon a global rescaling of the decorated PE metric by a factor $u$ . This property is paralleling to that of the smooth Gaussian curvature on surfaces. On the other hand, the definition of the discrete Gaussian curvature $K_{i}$ coincides with the original definition of the Gaussian curvature on smooth surfaces. This implies that the discrete Gaussian curvature $K_{i}$ is a good candidate as a discretization of the smooth Gaussian curvature on surfaces.

Remark 1.3.

According to Definition 1.1, the discrete Gaussian curvature $K_{i}$ defined by (3) seems to depend on the choice of weighted Delaunay triangulations of the decorated PE surface $(S,V,dist_{S},r)$ . We will show that $K_{i}$ is an intrinsic geometric invariant of the decorated PE surface $(S,V,dist_{S},r)$ in the sense that it is independent of the weighted Delaunay triangulations of $(S,V,dist_{S},r)$ . Note that the angle defect $W_{i}$ defined by (1) is an intrinsic geometric invariant of a decorated PE surface, we just need to prove that $A_{i}$ is independent of the choice of weighted Delaunay triangulations. This is true by Lemma 2.8.

Remark 1.4.

The weighted Delaunay triangulation is a natural generalization of the classical Delaunay triangulation. When the weighted Delaunay triangulation is reduced to the classical Delaunay triangulation, i.e. $r_{i}=0$ for all $i\in V$ , the area $A_{i}$ is exactly twice the area of the Voronoi cell at the vertex $i$ . Thus the area $A_{i}$ is a generalization of the area of the Voronoi cell at the vertex $i$ . As a result, the discrete Gaussian curvature in Definition 1.1 generalizes Kouřimská’s definition of discrete Gaussian curvature in [17, 16].

The discrete Yamabe problem for a decorated PE metric $(dist_{S},r)$ on $(S,V)$ asks if there exists a discrete conformal equivalent decorated PE metric on $(S,V)$ with constant discrete Gaussian curvature. The following discrete uniformization theorem solves this problem affirmatively for the discrete Gaussian curvature $K$ in Definition 1.1.

Theorem 1.5.

For any decorated PE metric $(dist_{S},r)$ on a marked surface $(S,V)$ with Euler number $\chi(S)\leq 0$ , there is a discrete conformal equivalent decorated PE metric with constant discrete Gaussian curvature $K$ .

By the relationships of the discrete Gaussian curvature $K$ and the classical discrete Gaussian curvature $W$ , the case $\chi(S)=0$ in Theorem 1.5 is covered by Bobenko-Lutz’s work [2]. Therefore, we just need to prove the case $\chi(S)<0$ in Theorem 1.5.

Remark 1.6.

The discrete Yamabe problem on surfaces for different types of discrete conformal structures with respect to the classical discrete Gaussian curvature $W$ has been extensively studied in the literature. For Thurston’s circle packings on surfaces, the solution of discrete Yamabe problem gives rise to the famous Koebe-Andreev-Thurston Theorem. See also the work of Beardon-Stephenson [1] for the discrete uniformization theorems for circle packings on surfaces. For the vertex scalings introduced by Luo [18] on surfaces, Gu-Luo-Sun-Wu [13], Gu-Guo-Luo-Sun-Wu [12], Springborn [23] and Izmestiev-Prosanov-Wu [15] give nice answers to this problem in different background geometries. Recently, Bobenko-Lutz [2] established the discrete conformal theory for decorated PE metrics and prove the corresponding discrete uniformization theorem. Since Bobenko-Lutz’s discrete conformal theory of decorated PE metrics also applies to the Euclidean vertex scalings and thus generalizes Gu-Luo-Sun-Wu’s result [13] and Springborn’s result [23], Theorem 1.5 also generalizes Kouřimská’s results in [17, 16]. It should be mentioned that Kouřimská [17, 16] constructed counterexamples to the uniqueness of PE metrics with constant discrete Gaussian curvatures. We conjecture that the decorated PE metric with constant discrete Gaussian curvature $K$ in Theorem 1.5 is not unique.

The main tools for the proof of Theorem 1.5 are Bobenko-Lutz’s discrete conformal theory for decorated PE metrics on surfaces [2] and variational principles with constraints. The main ideas of the paper come from reading of Bobenko-Lutz [2] and Kouřimská [17, 16].

The paper is organized as follows. In Section 2, we briefly recall Bobenko-Lutz’s discrete conformal theory for decorated PE metrics on surfaces. Then we show that $A_{i}$ is independent of the choice of weighted Delaunay triangulations, i.e., Lemma 2.8. We also give some notations and a variational characterization of the area $A_{i}^{jk}$ . In this section, we also extend the energy function $\mathcal{E}$ and the area function $A_{tot}$ . In Section 3, we translate Theorem 1.5 into an optimization problem with constraints, i.e., Lemma 3.2. Using the classical result from calculus, i.e., Theorem 3.3, we translate Lemma 3.2 into Theorem 3.4. By analysing the limit behaviour of sequences of discrete conformal factors, we get an asymptotic expression of the function $\mathcal{E}$ , i.e., Lemma 3.12. In the end, we prove Theorem 3.4.

Acknowledgements
The first author thanks Professor Feng Luo for his invitation to the workshop “Discrete and Computational Geometry, Shape Analysis, and Applications” taking place at Rutgers University, New Brunswick from May 19th to May 21st, 2023. The first author also thanks Carl O. R. Lutz for helpful communications during the workshop.

2. Preliminaries on decorated PE surfaces

2.1. Discrete conformal equivalence and Bobenko-Lutz’s discrete conformal theory

In this subsection, we briefly recall Bobenko-Lutz’s discrete conformal theory for decorated PE metrics on surfaces. Please refer to Bobenko-Lutz’s original work [2] for more details on this. The PE metric $dist_{S}$ on a PE surface with a geodesic triangulation defines a length map $l:E\rightarrow\mathbb{R}_{>0}$ such that $l_{ij},l_{ik},l_{jk}$ satisfy the triangle inequalities for any triangle $\{ijk\}\in F$ . Conversely, given a function $l:E\rightarrow\mathbb{R}_{>0}$ satisfying the triangle inequalities for any face $\{ijk\}\in F$ , one can construct a PE metric on a triangulated surface by isometrically gluing Euclidean triangles along edges in pairs. Therefore, we use $l:E\rightarrow\mathbb{R}_{>0}$ to denote a PE metric and use $(l,r)$ to denote a decorated PE metric on a triangulated surface $(S,V,\mathcal{T})$ .

Definition 2.1 ([2], Proposition 2.2).

Let $\mathcal{T}$ be a triangulation of a marked surface $(S,V)$ . Two decorated PE metrics $(l,r)$ and $(\widetilde{l},\widetilde{r})$ on $(S,V,\mathcal{T})$ are discrete conformal equivalent if and only if there exists a discrete conformal factor $u\in\mathbb{R}^{V}$ such that

(4)

\widetilde{r}_{i}=e^{u_{i}}r_{i},

(5)

\widetilde{l}_{ij}^{2}=(e^{2u_{i}}-e^{u_{i}+u_{j}})r^{2}_{i}+(e^{2u_{j}}-e^{u_{i}+u_{j}})r^{2}_{j}+e^{u_{i}+u_{j}}l_{ij}^{2}

for all $\{ij\}\in E$ .

Remark 2.2.

Note that the inversive distance

(6)

I_{ij}=\frac{l^{2}_{ij}-r^{2}_{i}-r^{2}_{j}}{2r_{i}r_{j}}

between two vertex-circles is invariant under Möbius transformations [6]. Combining (4) and (5) gives $I=\widetilde{I}$ . Since each pair of vertex-circles is required to be separated, we have $I>1$ . Therefore, Definition 2.1 can be regarded as a special case of the inversive distance circle packings introduced by Bowers-Stephenson [4]. One can refer to [5, 14, 19, 24, 25] for more properties of the inversive distance circle packings on triangulated surfaces.

In general, the existence of decorated PE metrics with constant discrete Gaussian curvatures on triangulated surfaces can not be guaranteed if the triangulation is fixed. In the following, we work with a generalization of the discrete conformal equivalence in Definition 2.1, introduced by Bobenko-Lutz [2], which allows the triangulation of the marked surface to be changed under the weighted Delaunay condition.

Definition 2.3 ([2], Definition 4.11).

Two decorated PE metrics $(dist_{S},r)$ and $(\widetilde{dist}_{S},\widetilde{r})$ on the marked surface $(S,V)$ are discrete conformal equivalent if there is a sequence of triangulated decorated PE surfaces $(\mathcal{T}^{0},l^{0},r^{0}),...,(\mathcal{T}^{N},l^{N},r^{N})$ such that

(i):: the decorated PE metric of $(\mathcal{T}^{0},l^{0},r^{0})$ is $(dist_{S},r)$ and the decorated PE metric of $(\mathcal{T}^{N},l^{N},r^{N})$ is $(\widetilde{dist}_{S},\widetilde{r})$ ,
(ii):: each $\mathcal{T}^{n}$ is a weighted Delaunay triangulation of the decorated PE surface $(\mathcal{T}^{n},l^{n},r^{n})$ ,
(iii):: if $\mathcal{T}^{n}=\mathcal{T}^{n+1}$ , then there is a discrete conformal factor $u\in\mathbb{R}^{V}$ such that $(\mathcal{T}^{n},l^{n},r^{n})$ and $(\mathcal{T}^{n+1},l^{n+1},r^{n+1})$ are related by (4) and (5),
(iv):: if $\mathcal{T}^{n}\neq\mathcal{T}^{n+1}$ , then $\mathcal{T}^{n}$ and $\mathcal{T}^{n+1}$ are two different weighted Delaunay triangulations of the same decorated PE surface.

Definition 2.3 defines an equivalence relationship for decorated PE metrics on a marked surface. The equivalence class of a decorated PE metric $(dist_{S},r)$ on $(S,V)$ is also called as the discrete conformal class of $(dist_{S},r)$ and denoted by $\mathcal{D}(dist_{S},r)$ .

Lemma 2.4 ([2]).

The discrete conformal class $\mathcal{D}(dist_{S},r)$ of a decorated PE metric $(dist_{S},r)$ on the marked surface $(S,V)$ is parameterized by $\mathbb{R}^{V}=\{u:V\rightarrow\mathbb{R}\}$ .

For simplicity, for any $(\widetilde{dist}_{S},\widetilde{r})\in\mathcal{D}(dist_{S},r)$ , we denote it by $(dist_{S}(u),r(u))$ for some $u\in\mathbb{R}^{V}$ . Set

\mathcal{C}_{\mathcal{T}}(dist_{S},r)=\{u\in\mathbb{R}^{V}|\ \mathcal{T}\ \text{is a weighted Delaunay triangulation of}\ (S,V,dist_{S}(u),r(u))\}.

For any decorated PE surface, there exists a unique complete hyperbolic surface $\Sigma_{g}$ , i.e., the hyperbolic surface induced by any triangular refinement of its unique weighted Delaunay tessellation. It is homeomorphic to $S\backslash V$ and called as the fundamental discrete conformal invariant of the decorated PE metric $(dist_{S},r)$ . The decoration of $\Sigma_{g}$ is denoted by $\omega:=e^{h}$ and here the height $h$ is related to $u$ by $dh_{i}=-du_{i}$ . The canonical weighted Delaunay tessellation $\mathcal{T}$ of $\Sigma_{g}$ is denoted by $\mathcal{T}_{\Sigma_{g}}^{\omega}$ . Bobenko-Lutz [2] defined the following set

\mathcal{D}_{\mathcal{T}}(\Sigma_{g})=\{\omega\in\mathbb{R}_{>0}^{V}|\mathcal{T}\ \text{refines}\ \mathcal{T}_{\Sigma_{g}}^{\omega}\}

and proved the following proposition.

Proposition 2.5 ([2], Proposition 4.3).

Given a complete hyperbolic surface with ends $\Sigma_{g}$ .

(1):: Each $\mathcal{D}_{\mathcal{T}_{n}}(\Sigma_{g})$ is either empty or the intersection of $\mathbb{R}^{V}_{>0}$ with a closed polyhedral cone.
(2):: There is only a finite number of geodesic tessellations $\mathcal{T}_{1},...,\mathcal{T}_{N}$ of $\Sigma_{g}$ such that $\mathcal{D}_{\mathcal{T}_{n}}(\Sigma_{g})$ $(n=1,...,N)$ is non-empty. In particular, $\mathbb{R}^{V}_{>0}=\bigcup_{n=1}^{N}\mathcal{D}_{\mathcal{T}_{n}}(\Sigma_{g})$ .

Let $P$ be the polyhedral cusp corresponding to the triangulated surface $(S,V,\mathcal{T})$ with fundamental discrete conformal invariant $\Sigma_{g}$ . The polyhedral cusp is convex if and only if $\mathcal{T}$ is a weighted Delaunay triangulation. The set of all heights $h$ of convex polyhedral cusps over the triangulated hyperbolic surface $(\Sigma_{g},\mathcal{T})$ is denoted by $\mathcal{P}_{\mathcal{T}}(\Sigma_{g})\subseteq\mathbb{R}^{V}$ .

Proposition 2.6 ([2], Proposition 4.9).

Given a decorated PE metric $(dist_{S},r)$ on the marked surface $(S,V)$ . Then $\mathcal{C}_{\mathcal{T}}(dist_{S},r)$ , $\mathcal{P}_{\mathcal{T}}(\Sigma_{g})$ and $\mathcal{D}_{\mathcal{T}}(\Sigma_{g})$ are homeomorphic.

Combining Proposition 2.5 and Proposition 2.6 gives the following result.

Lemma 2.7 ([2]).

The set

J=\{\mathcal{T}|\mathcal{C}_{\mathcal{T}}(dist_{S},r)\ \text{has non-empty interior in}\ \mathbb{R}^{V}\}

is a finite set, $\mathbb{R}^{V}=\cup_{\mathcal{T}_{i}\in J}\mathcal{C}_{\mathcal{T}_{i}}(dist_{S},r)$ , and each $\mathcal{C}_{\mathcal{T}_{i}}(dist_{S},r)$ is homeomorphic to a polyhedral cone (with its apex removed) and its interior is homeomorphic to $\mathbb{R}^{V}$ .

2.2. A decorated triangle

Denote $r_{ij}$ as half of the distance of the two intersection points of the face-circle $C_{ijk}$ and the edge $\{ij\}$ . Denote $h_{ij}^{k}$ as the signed distance of the center $c_{ijk}$ to the edge $\{ij\}$ , which is defined to be positive if the center is on the same side of the line determined by $\{ij\}$ as the triangle $\{ijk\}$ and negative otherwise (or zero if the center is on the line). Note that $h_{ij}^{k}$ is symmetric in the indices $i$ and $j$ . By Figure 3, we have

(7)

h_{ij}^{k}=r_{ij}\cot\alpha_{ij}^{k}.

Since $r_{ij}>0$ and $\alpha_{ij}^{k}\in(0,\pi)$ , if $h_{ij}^{k}<0$ , then $\alpha_{ij}^{k}\in(\frac{\pi}{2},\pi)$ . The equality (7) implies that (2) is equivalent to

(8)

h_{ij}^{k}+h_{ij}^{l}\geq 0

for any adjacent triangles $\{ijk\}$ and $\{ijl\}$ sharing a common edge $\{ij\}$ . Therefore, the equality (8) also characterizes a weighted Delaunay triangulation $\mathcal{T}$ for a decorated PE metric $(l,r)$ on $(S,V)$ . Due to this fact, the equality (8) is usually used to define the weighted Delaunay triangulations of decorated PE surfaces. See [5, 8] and others for example. Then $A_{i}^{jk}$ can be written as

(9)

A_{i}^{jk}=\frac{1}{2}l_{ij}h_{ij}^{k}+\frac{1}{2}l_{ki}h_{ki}^{j}.

Since $h_{ij}^{k},h_{ki}^{j}$ are the signed distances, thus $A_{i}^{jk}$ is an algebraic sum of the area of triangles, i.e. a signed area.

Lemma 2.8.

The area $A_{i}$ is independent of the choice of weighted Delaunay triangulations of a decorated PE surface.

Proof.

Suppose a decorated quadrilateral $\{ijlk\}$ is in a face of the weighted Delaunay tessellation of a decorated PE surface, then there exist two weighted Delaunay triangulations $\mathcal{T}_{1}$ and $\mathcal{T}_{2}$ of the decorated PE surface with an edge $\{jk\}$ in $\mathcal{T}_{1}$ flipped to another edge $\{il\}$ in $\mathcal{T}_{2}$ . Please refer to Figure 2. We just need to prove the signed area $A^{jk}_{i}$ in $\mathcal{T}_{1}$ is equal to the signed area $A_{i}^{kl}+A_{i}^{jl}$ in $\mathcal{T}_{2}$ . In $\mathcal{T}_{1}$ , the signed area at the vertex $i$ in $\{ijlk\}$ is $A_{i}^{jk}=\frac{1}{2}l_{ki}h_{ki}^{j}+\frac{1}{2}l_{ij}h_{ij}^{k}$ . In $\mathcal{T}_{2}$ , the signed area at the vertex $i$ in $\{ijlk\}$ is

	$\displaystyle A_{i}^{kl}+A_{i}^{jl}$	$\displaystyle=\frac{1}{2}l_{ki}h_{ki}^{l}+\frac{1}{2}l_{il}h_{il}^{k}+\frac{1}{2}l_{ij}h_{ij}^{l}+\frac{1}{2}l_{il}h_{il}^{j}$
		$\displaystyle=\frac{1}{2}l_{ki}h_{ki}^{l}+\frac{1}{2}l_{ij}h_{ij}^{l}+\frac{1}{2}l_{il}(h_{il}^{k}+h_{il}^{j}).$

Since $\mathcal{T}_{1}$ and $\mathcal{T}_{2}$ are two weighted Delaunay triangulations of the same decorated PE metric on $(S,V)$ , then $h_{il}^{k}+h_{il}^{j}=0$ by (8). One can also refer to [2] (Proposition 3.4) for this. Moreover, $h_{ki}^{l}=h_{ki}^{j}$ and $h_{ij}^{l}=h_{ij}^{k}$ . Then $A_{i}^{kl}+A_{i}^{jl}=A_{i}^{jk}$ .

\begin{overpic}[scale={0.6}]{figure_2.pdf} \put(4.0,20.0){$i$} \put(41.5,20.0){$l$} \put(23.0,41.0){$k$} \put(17.0,2.0){$j$} \put(3.5,24.5){$r_{i}$} \put(14.0,17.0){$h_{ij}^{k}$} \put(16.5,27.0){$h_{ki}^{j}$} \put(23.5,19.0){$c_{ijk}$} \par\put(56.5,20.0){$i$} \put(94.0,20.0){$l$} \put(75.0,41.0){$k$} \put(69.0,2.0){$j$} \put(56.0,24.5){$r_{i}$} \put(76.0,19.0){$c_{ijk}$} \put(66.5,17.0){$h_{ij}^{l}$} \put(69.0,27.0){$h_{ki}^{l}$} \put(76.0,21.0){$h_{il}^{j}$} \end{overpic}

Figure 2. Weighted Delaunay triangulation

\mathcal{T}_{1}

(left) and weighted Delaunay triangulation

\mathcal{T}_{2}

(right).

Q.E.D.

Denote $c_{ij}$ as the center of the edge $\{ij\}$ , which is obtained by projecting the center $c_{ijk}$ to the line determined by $\{ij\}$ . Denote $d_{ij}$ as the signed distance of $c_{ij}$ to the vertex $i$ , which is positive if $c_{ij}$ is on the same side of $i$ as $j$ along the line determined by $\{ij\}$ and negative otherwise (or zero if $c_{ij}$ is the same as $i$ ). In general, $d_{ij}\neq d_{ji}$ . Since the face-circle $C_{ijk}$ is orthogonal to the vertex-circle at the vertex $j$ , we have

(10)

r_{ijk}^{2}+r_{j}^{2}=d_{jk}^{2}+(h^{i}_{jk})^{2}=d_{ji}^{2}+(h^{k}_{ij})^{2}.

Please refer to Figure 3 for this. Moreover, we have the following explicit expressions of $d_{ij}$ and $h_{ij}^{k}$ due to Glickenstein [9], i.e.,

(11)

d_{ij}=\frac{r_{i}^{2}+r_{i}r_{j}I_{ij}}{l_{ij}},

and

(12)

h_{ij}^{k}=\frac{d_{ik}-d_{ij}\cos\theta_{jk}^{i}}{\sin\theta_{jk}^{i}},

where $\theta^{i}_{jk}$ is the inner angle of the triangle $\{ijk\}$ at the vertex $i$ . The equality (11) implies that $d_{ij}\in\mathbb{R}$ is independent of the existence of the center $c_{ijk}$ . Since each pair of vertex-circles is required to be separated, then $I>1$ . This implies

d_{rs}>0,\ \ \forall\{r,s\}\subseteq\{i,j,k\}.

\begin{overpic}[scale={0.4}]{figure_1.pdf} \put(15.0,15.0){$i$} \put(80.5,15.0){$j$} \put(48.5,74.0){$k$} \put(30.0,55.0){$\alpha_{ik}^{j}$} \put(20.0,30.0){$\alpha_{ik}^{j}$} \put(37.5,27.0){$\alpha_{ij}^{k}$} \put(27.0,12.5){$\alpha_{ij}^{k}$} \put(63.0,12.5){$\alpha_{ij}^{k}$} \put(10.0,25.0){$r_{i}$} \put(46.0,25.0){$h_{ij}^{k}$} \put(36.0,40.0){$h_{ki}^{j}$} \put(53.0,43.5){$h_{jk}^{i}$} \put(31.0,21.0){$d_{ij}$} \put(45.0,36.0){$c_{ijk}$} \put(44.0,16.0){$c_{ij}$} \put(34.0,16.0){$r_{ij}$} \put(26.0,42.0){$c_{ik}$} \put(66.0,46.0){$c_{jk}$} \end{overpic}

Figure 3. Dates for a decorated triangle

\{ijk\}\in F

The following lemma gives some useful formulas.

Lemma 2.9 ([14, 24, 25]).

Let $\{ijk\}$ be a decorated triangle with the edge lengths $l_{ij},l_{jk},l_{ki}$ defined by (5). If the decorated triangle $\{ijk\}$ is non-degenerate, then

(13)

\frac{\partial\theta_{jk}^{i}}{\partial u_{j}}=\frac{\partial\theta_{ki}^{j}}{\partial u_{i}}=\frac{h_{ij}^{k}}{l_{ij}},\ \ \ \frac{\partial\theta_{jk}^{i}}{\partial u_{i}}=-\frac{\partial\theta_{jk}^{i}}{\partial u_{j}}-\frac{\partial\theta_{jk}^{i}}{\partial u_{k}},

where

(14)

h_{ij}^{k}=\frac{r_{i}^{2}r_{j}^{2}r_{k}^{2}}{2A_{ijk}l_{ij}}[\kappa_{k}^{2}(1-I_{k}^{2})+\kappa_{j}\kappa_{k}\gamma_{i}+\kappa_{i}\kappa_{k}\gamma_{j}]=\frac{r_{i}^{2}r_{j}^{2}r_{k}^{2}}{2A_{ijk}l_{ij}}\kappa_{k}h_{k}

with $A_{ijk}=\frac{1}{2}l_{ij}l_{jk}\sin\theta_{ki}^{j}$ , $\gamma_{i}=I_{jk}+I_{ij}I_{ki}$ , $\kappa_{i}:=r_{i}^{-1}$ and

(15)

\displaystyle h_{i}=\kappa_{i}(1-I_{jk}^{2})+\kappa_{j}\gamma_{k}+\kappa_{k}\gamma_{j}.

As a direct application of Lemma 2.9, we have the following result.

Lemma 2.10.

The area $A_{ijk}(u)$ of each decorated triangle $\{ijk\}\in F$ is an analytic function with

(16)

\frac{\partial A_{ijk}}{\partial u_{i}}=A_{i}^{jk}.

Proof.

By (14), we have

h^{k}_{ij}=\frac{d_{ik}-d_{ij}\cos\theta_{jk}^{i}}{\sin\theta_{jk}^{i}},\quad h_{ki}^{j}=\frac{d_{ij}-d_{ik}\cos\theta_{jk}^{i}}{\sin\theta_{jk}^{i}}.

Direct calculations give

(17)

h_{ki}^{j}=d_{ij}\sin\theta_{jk}^{i}-h^{k}_{ij}\cos\theta_{jk}^{i}.

Combining (5), (6) and (11), it is easy to check that

(18)

\frac{\partial l_{ij}}{\partial u_{i}}=d_{ij}.

Differentiating $A_{ijk}=\frac{1}{2}l_{ij}l_{jk}\sin\theta_{ki}^{j}$ with respect to $u_{i}$ gives

	$\displaystyle\frac{\partial A_{ijk}}{\partial u_{i}}$	$\displaystyle=\frac{1}{2}\frac{\partial l_{ij}}{\partial u_{i}}l_{jk}\sin\theta_{ki}^{j}+\frac{1}{2}l_{ij}l_{jk}\cos\theta_{ki}^{j}\frac{\partial\theta_{ki}^{j}}{\partial u_{i}}$
		$\displaystyle=\frac{1}{2}d_{ij}l_{jk}\sin\theta_{ki}^{j}+\frac{1}{2}l_{ij}l_{jk}\cos\theta_{ki}^{j}\frac{h^{k}_{ij}}{l_{ij}}$
		$\displaystyle=\frac{1}{2}d_{ij}l_{ki}\sin\theta_{jk}^{i}+\frac{1}{2}l_{jk}\cos\theta_{ki}^{j}h^{k}_{ij}$
		$\displaystyle=\frac{1}{2}d_{ij}l_{ki}\sin\theta_{jk}^{i}+\frac{1}{2}(l_{ij}-l_{ki}\cos\theta_{jk}^{i})h^{k}_{ij}$
		$\displaystyle=\frac{1}{2}l_{ki}(d_{ij}\sin\theta_{jk}^{i}-h^{k}_{ij}\cos\theta_{jk}^{i})+\frac{1}{2}l_{ij}h^{k}_{ij}$
		$\displaystyle=\frac{1}{2}l_{ki}h_{ki}^{j}+\frac{1}{2}l_{ij}h_{ij}^{k}$
		$\displaystyle=A_{i}^{jk},$

where the second equality uses (18) and (13), the third equality uses the sine laws and the penultimate line uses (17). Q.E.D.

Remark 2.11.

One can refer to Glickenstein [9] for a nice geometric explanation of the result in Lemma 2.10.

2.3. The extended energy function and the extended area function

There exists a geometric relationship between the decorated triangle $\{ijk\}$ and the geometry of hyperbolic polyhedra in $3$ -dimensional hyperbolic space. Specially, there is a generalized hyperbolic tetrahedra in $\mathbb{H}^{3}$ with one ideal vertex and three hyper-ideal vertices corresponding to a decorated triangle $\{ijk\}$ . Please refer to [2] for more details on this fact. Springborn [22] found the following explicit formula for the truncated volume $\mathrm{Vol}(ijk)$ of this generalized hyperbolic tetrahedra

(19)	$\displaystyle 2\mathrm{Vol}(ijk)=$	$\displaystyle\mathbb{L}(\theta_{jk}^{i})+\mathbb{L}(\theta_{ki}^{j})+\mathbb{L}(\theta_{ij}^{k})$
		$\displaystyle+\mathbb{L}(\frac{\pi+\alpha_{ki}^{j}+\alpha_{ij}^{k}-\theta_{jk}^{i}}{2})+\mathbb{L}(\frac{\pi+\alpha_{ki}^{j}-\alpha_{ij}^{k}-\theta_{jk}^{i}}{2})$
		$\displaystyle+\mathbb{L}(\frac{\pi-\alpha_{ki}^{j}+\alpha_{ij}^{k}-\theta_{jk}^{i}}{2})+\mathbb{L}(\frac{\pi-\alpha_{ki}^{j}-\alpha_{ij}^{k}-\theta_{jk}^{i}}{2})$
		$\displaystyle+\mathbb{L}(\frac{\pi+\alpha_{jk}^{i}+\alpha_{ij}^{k}-\theta_{ki}^{j}}{2})+\mathbb{L}(\frac{\pi+\alpha_{jk}^{i}-\alpha_{ij}^{k}-\theta_{ki}^{j}}{2})$
		$\displaystyle+\mathbb{L}(\frac{\pi-\alpha_{jk}^{i}+\alpha_{ij}^{k}-\theta_{ki}^{j}}{2})+\mathbb{L}(\frac{\pi-\alpha_{jk}^{i}-\alpha_{ij}^{k}-\theta_{ki}^{j}}{2})$
		$\displaystyle+\mathbb{L}(\frac{\pi+\alpha_{jk}^{i}+\alpha_{ki}^{j}-\theta_{ij}^{k}}{2})+\mathbb{L}(\frac{\pi+\alpha_{jk}^{i}-\alpha_{ki}^{j}-\theta_{ij}^{k}}{2})$
		$\displaystyle+\mathbb{L}(\frac{\pi-\alpha_{jk}^{i}+\alpha_{ki}^{j}-\theta_{ij}^{k}}{2})+\mathbb{L}(\frac{\pi-\alpha_{jk}^{i}-\alpha_{ki}^{j}-\theta_{ij}^{k}}{2}),$

where

(20)

\mathbb{L}(x)=-\int_{0}^{x}\log|2\sin(t)|dt

is Milnor’s Lobachevsky function. Milnor’s Lobachevsky function is bounded, odd, $\pi$ -periodic and smooth except at integer multiples of $\pi$ . Please refer to [20, 21] for more information on Milnor’s Lobachevsky function $\mathbb{L}(x)$ .

Set

(21)		$\displaystyle F_{ijk}(u_{i},u_{j},u_{k})=$	$\displaystyle-2\mathrm{Vol}(ijk)+\theta_{jk}^{i}u_{i}+\theta_{ki}^{j}u_{j}+\theta_{ij}^{k}u_{k}$
(21)			$\displaystyle+(\frac{\pi}{2}-\alpha_{ij}^{k})\lambda_{ij}+(\frac{\pi}{2}-\alpha_{ki}^{j})\lambda_{ki}+(\frac{\pi}{2}-\alpha_{jk}^{i})\lambda_{jk},$

where $\cosh\lambda_{ij}=I_{ij}$ . Then $\nabla F_{ijk}=(\theta_{jk}^{i},\theta_{ki}^{j},\theta_{ij}^{k})$ and

(22)

F_{ijk}((u_{i},u_{j},u_{k})+c(1,1,1))=F_{ijk}(u_{i},u_{j},u_{k})+c\pi

for $c\in\mathbb{R}$ . Furthermore, on a decorated PE surface $(S,V,l,r)$ with a weighted Delaunay triangulation $\mathcal{T}$ , Bobenko-Lutz [2] defined the following function

(23)

\mathcal{H}_{\mathcal{T}}(u)=\sum_{\{ijk\}\in F}F_{ijk}(u_{i},u_{j},u_{k})=-2\mathrm{Vol}(P_{h})+\sum_{i\in V}\theta_{i}u_{i}+\sum_{\{ij\}\in E_{\mathcal{T}}}(\pi-\alpha_{ij})\lambda_{ij},

where $P_{h}$ is the convex polyhedral cusp defined by the heights $h\in\mathbb{R}^{V}$ , $\theta_{i}=\sum_{\{ijk\}\in F_{\mathcal{T}}}\theta^{i}_{jk}$ and $\alpha_{ij}=\alpha_{ij}^{k}+\alpha_{ij}^{l}$ . Note that the function $\mathcal{H}_{\mathcal{T}}(u)$ defined by (23) differs from its original definition in [2] (Equation 4-9) by some constant. By (22), for $c\in\mathbb{R}$ , we have

\mathcal{H}_{\mathcal{T}}(u+c\mathbf{1})=\mathcal{H}_{\mathcal{T}}(u)+c|F|\pi.

Using the function $\mathcal{H}_{\mathcal{T}}$ , we define the following energy function

\mathcal{E}_{\mathcal{T}}(u)=-\mathcal{H}_{\mathcal{T}}(u)+2\pi\sum_{i\in V}u_{i},

which is well-defined on $\mathcal{C}_{\mathcal{T}}(dist_{S},r)$ with $\nabla_{u_{i}}\mathcal{E}_{\mathcal{T}}=2\pi-\sum_{\{ijk\}\in F_{\mathcal{T}}}\theta^{i}_{jk}=W_{i}$ . Moreover, for $c\in\mathbb{R}$ , we have

(24)	$\displaystyle\mathcal{E}_{\mathcal{T}}(u+c\mathbf{1})=$	$\displaystyle-\mathcal{H}_{\mathcal{T}}(u+c\mathbf{1})+2\pi\sum_{i\in V}(u_{i}+c)$
	$\displaystyle=$	$\displaystyle-\mathcal{H}_{\mathcal{T}}(u)-c\|F\|\pi+2\pi\sum_{i\in V}u_{i}+2c\|V\|\pi$
	$\displaystyle=$	$\displaystyle\mathcal{E}_{\mathcal{T}}(u)+2c\pi\chi(S),$

where $2|V|-|F|=2\chi(S)$ is used in the last line.

Theorem 2.12 ([2], Proposition 4.13).

For a discrete conformal factor $u\in\mathbb{R}^{V}$ , let $\mathcal{T}$ be a weighted Delaunay triangulation of the decorated PE surface $(S,V,dist_{S}(u),r(u))$ . The map

(25)		$\displaystyle\mathcal{H}:\ \mathbb{R}^{V}$	$\displaystyle\rightarrow\mathbb{R},$
(25)		$\displaystyle u$	$\displaystyle\mapsto\mathcal{H}_{\mathcal{T}}(u)$

is well-defined, concave, and twice continuously differentiable over $\mathbb{R}^{V}$ .

Therefore, the function $\mathcal{E}_{\mathcal{T}}(u)$ defined on $\mathcal{C}_{\mathcal{T}}(dist_{S},r)$ can be extended to be

(26)

\mathcal{E}(u)=-\mathcal{H}(u)+2\pi\sum_{i\in V}u_{i}=-\sum_{\{ijk\}\in F}F_{ijk}(u_{i},u_{j},u_{k})+2\pi\sum_{i\in V}u_{i}

defined on $\mathbb{R}^{V}$ .

Definition 2.13.

Suppose $(S,V,\mathcal{T})$ is a triangulated surface with a decorated PE metric $(l,r)$ . The area function $A^{\mathcal{T}}_{tot}$ on $(S,V,\mathcal{T})$ is defined to be

A^{\mathcal{T}}_{tot}:\ \mathcal{C}_{\mathcal{T}}(dist_{S},r)\rightarrow\mathbb{R},\\ \quad\ A^{\mathcal{T}}_{tot}(u)=\sum_{\{ijk\}\in F}A_{ijk}(u).

By Lemma 2.10, we have the following result.

Corollary 2.14.

The function $A^{\mathcal{T}}_{tot}$ is an analytic function with

(27)

\frac{\partial A^{\mathcal{T}}_{tot}}{\partial u_{i}}=A_{i}.

Lemma 2.8 and Corollary 2.14 imply the following result, which shows the function $A^{\mathcal{T}}_{tot}$ defined on $\mathcal{C}_{\mathcal{T}}(dist_{S},r)$ can be extended.

Theorem 2.15.

For a discrete conformal factor $u\in\mathbb{R}^{V}$ , let $\mathcal{T}$ be a weighted Delaunay triangulation of the decorated PE surface $(S,V,dist_{S}(u),r(u))$ . The map

(28)		$\displaystyle A_{tot}:\ \mathbb{R}^{V}$	$\displaystyle\rightarrow\mathbb{R},$
(28)		$\displaystyle u$	$\displaystyle\mapsto A^{\mathcal{T}}_{tot}(u)$

is well-defined and once differentiable.

Proof.

By Corollary 2.14, the function $A_{tot}$ is once differentiable in the interior of any $\mathcal{C}_{\mathcal{T}}(dist_{S},r)$ . At the boundary of $\mathcal{C}_{\mathcal{T}}(dist_{S},r)$ , the weighted triangulations induce the same weighted Delaunay tessellation. The conclusion follows from Lemma 2.8. Q.E.D.

3. The proof of Theorem 1.5

3.1. Variational principles with constraints

In this subsection, we translate Theorem 1.5 into an optimization problem with inequality constraints by variational principles, which involves the function $\mathcal{E}$ defined by (26).

Proposition 3.1.

The set

\mathcal{A}=\{u\in\mathbb{R}^{V}|A_{tot}(u)\leq 1\}.

is an unbounded closed subset of $\mathbb{R}^{V}$ .

Proof.

By Theorem 2.15, the set $\mathcal{A}$ is a closed subset of $\mathbb{R}^{V}$ . Since $A_{ijk}((u_{i},u_{j},u_{k})+(c,c,c))=e^{2c}A_{ijk}(u_{i},u_{j},u_{k})$ , thus $A_{tot}(u+c\mathbf{1})=e^{2c}A_{tot}(u)$ . Then $A_{tot}(u+c\mathbf{1})=e^{2c}A_{tot}(u)\leq 1$ is equivalent to $c\leq-\frac{1}{2}\log A_{tot}(u)$ . This implies that the ray $\{u+c\mathbf{1}|c\leq-\frac{1}{2}\log A_{tot}(u)\}$ stays in the set $\mathcal{A}$ . Hence the set $\mathcal{A}$ is unbounded. Q.E.D.

According to Proposition 3.1, we have following result.

Lemma 3.2.

If $\chi(S)<0$ and the function $\mathcal{E}(u)$ attains a minimum in the set $\mathcal{A}$ , then the minimum value point of $\mathcal{E}(u)$ lies at the boundary of $\mathcal{A}$ , i.e.,

\partial\mathcal{A}=\{u\in\mathbb{R}^{V}|A_{tot}(u)=1\}.

Furthermore, there exists a decorated PE metric with constant discrete Gaussian curvature $K$ in the discrete conformal class.

Proof.

Suppose the function $\mathcal{E}(u)$ attains a minimum at $u\in\mathcal{A}$ . Taking $c_{0}=-\frac{1}{2}\log A_{tot}(u)$ , then $c_{0}\geq 0$ by $A_{tot}(u)\leq 1$ . By the proof of Proposition 3.1, $u+c_{0}\mathbf{1}\in\mathcal{A}$ . Hence, by the additive property of the function $\mathcal{E}$ in (24), we have

\mathcal{E}(u)\leq\mathcal{E}(u+c_{0}\mathbf{1})=\mathcal{E}(u)+2c_{0}\pi\chi(S).

This implies $c_{0}\leq 0$ by $\chi(S)<0$ . Then $c_{0}=0$ and $A_{tot}(u)=1$ . Therefore, the minimum value point of $\mathcal{E}(u)$ lies in the set $\partial\mathcal{A}=\{u\in\mathbb{R}^{V}|A_{tot}(u)=1\}$ . The conclusion follows from the following claim.
Claim : Up to scaling, the decorated PE metrics with constant discrete Gaussian curvature $K$ in the discrete conformal class are in one-to-one correspondence with the critical points of the function $\mathcal{E}(u)$ under the constraint $A_{tot}(u)=1$ .

We use the method of Lagrange multipliers to prove this claim. Set

G(u,\mu)=\mathcal{E}(u)-\mu(A_{tot}(u)-1),

where $\mu\in\mathbb{R}$ is a Lagrange multiplier. If $u$ is a critical point of the function $\mathcal{E}$ under the constraint $A_{tot}(u)=1$ , then by (27) and the fact $\nabla_{u_{i}}\mathcal{E}=W_{i}$ , we have

0=\frac{\partial G(u,\mu)}{\partial u_{i}}=\frac{\partial\mathcal{E}(u)}{\partial u_{i}}-\mu\frac{\partial A_{tot}(u)}{\partial u_{i}}=W_{i}-\mu A_{i}.

This implies

W_{i}=\mu A_{i}.

Since the anger defect $W$ defined by (1) satisfies the following discrete Gauss-Bonnet formula

\sum_{i\in V}W_{i}=2\pi\chi(S),

we have

2\pi\chi(S)=\sum_{i\in V}W_{i}=\mu\sum_{i\in V}A_{i}=\mu A_{tot}=\mu.

under the constraint $A_{tot}(u)=1$ . Therefore, the discrete Gaussian curvature

K_{i}=\frac{W_{i}}{A_{i}}=2\pi\chi(S)

for any $i\in V$ . Q.E.D.

3.2. Reduction to Theorem 3.4

By Lemma 3.2, we just need to prove that the function $\mathcal{E}(u)$ attains the minimum in the set $\mathcal{A}$ . Recall the following classical result from calculus.

Theorem 3.3.

Let $A\subseteq\mathbb{R}^{m}$ be a closed set and $f:A\rightarrow\mathbb{R}$ be a continuous function. If every unbounded sequence $\{u_{n}\}_{n\in\mathbb{N}}$ in $A$ has a subsequence $\{u_{n_{k}}\}_{k\in\mathbb{N}}$ such that $\lim_{k\rightarrow+\infty}f(u_{n_{k}})=+\infty$ , then $f$ attains a minimum in $A$ .

One can refer to [16] (Section 4.1) for a proof of Theorem 3.3. The majority of the conditions in Theorem 3.3 is satisfied, including the set $\mathcal{A}$ is a closed subset of $\mathbb{R}^{V}$ by Proposition 3.1 and the function $\mathcal{E}$ is continuous by Theorem 2.12. To prove Theorem 1.5, we just need to prove the following theorem.

Theorem 3.4.

If $\chi(S)<0$ and $\{u_{n}\}_{n\in\mathbb{N}}$ is an unbounded sequence in $\mathcal{A}$ , then there exists a subsequence $\{u_{n_{k}}\}_{k\in\mathbb{N}}$ of $\{u_{n}\}_{n\in\mathbb{N}}$ such that $\lim_{k\rightarrow+\infty}\mathcal{E}(u_{n_{k}})=+\infty$ .

3.3. Behaviour of sequences of discrete conformal factors

Let $\{u_{n}\}_{n\in\mathbb{N}}$ be an unbounded sequence in $\mathbb{R}^{V}$ . Denote its coordinate sequence at $j\in V$ by $\{u_{j,n}\}_{n\in\mathbb{N}}$ . Motivated by [17], we call the sequence $\{u_{n}\}_{n\in\mathbb{N}}$ with the following properties as a “good” sequence.

(1):: It lies in one cell $\mathcal{C}_{\mathcal{T}}(dist_{S},r)$ of $\mathbb{R}^{V}$ ;
(2):: There exists a vertex $i^{*}\in V$ such that $u_{i^{*},n}\leq u_{j,n}$ for all $j\in V$ and $n\in\mathbb{N}$ ;
(3):: Each coordinate sequence $\{u_{j,n}\}_{n\in\mathbb{N}}$ either converges, diverges properly to $+\infty$ , or diverges properly to $-\infty$ ;
(4):: For any $j\in V$ , the sequence $\{u_{j,n}-u_{i^{*},n}\}_{n\in\mathbb{N}}$ either converges or diverges properly to $+\infty$ .

By Lemma 2.7, it is obvious that every sequence of discrete conformal factors in $\mathbb{R}^{V}$ possesses a “good” subsequence. Hence, the “good” sequence could be chosen without loss of generality.

In the following arguments, we use the following notations

(29)

l^{n}_{ij}=\sqrt{r^{2}_{i,n}+r^{2}_{j,n}+2I_{ij}r_{i,n}r_{j,n}},

(30)

r_{i,n}=e^{u_{i,n}}r_{i},

(31)

(l^{n}_{ij})^{2}=(e^{2u_{i,n}}-e^{u_{i,n}+u_{j,n}})r^{2}_{i}+(e^{2u_{j,n}}-e^{u_{i,n}+u_{j,n}})r^{2}_{j}+e^{{u_{i,n}+u_{j,n}}}l_{ij}^{2}.

For a decorated triangle $\{ijk\}\in F$ in $(S,V,\mathcal{T})$ , set

(32)

\mathcal{C}_{ijk}=\{(u_{i},u_{j},u_{k})\in\mathbb{R}^{3}|u\in\mathcal{C}_{\mathcal{T}}(dist_{S},r)\}.

Let $(u_{i,n},u_{j,n},u_{k,n})_{n\in\mathbb{N}}$ be a coordinate sequence in $\mathcal{C}_{ijk}$ . Then the edge lengths $l^{n}_{ij},l^{n}_{jk},l^{n}_{ki}$ satisfy the triangle inequalities for all $n\in\mathbb{N}$ .

Lemma 3.5.

There exists no sequence in $\mathcal{C}_{ijk}$ such that as $n\rightarrow+\infty$ ,

u_{r,n}\rightarrow+\infty,\quad u_{s,n}\rightarrow+\infty,\quad u_{t,n}\leq C,

where $\{r,s,t\}=\{i,j,k\}$ and $C$ is a constant.

Proof.

Without loss of generality, we assume $\lim u_{i,n}=+\infty$ , $\lim u_{j,n}=+\infty$ and the sequence $u_{k,n}\leq C_{1}$ . The equality (30) implies $\lim r_{i,n}=+\infty$ , $\lim r_{j,n}=+\infty$ and the sequence $r_{k,n}\leq C_{2}$ . Here $C_{1},C_{2}$ are constants. By (29), we have

	$\displaystyle(l^{n}_{jk}+l^{n}_{ki})^{2}=$	$\displaystyle r^{2}_{i,n}+r^{2}_{j,n}+2r^{2}_{k,n}+2I_{jk}r_{j,n}r_{k,n}+2I_{ki}r_{k,n}r_{i,n}$
		$\displaystyle+2\sqrt{(r^{2}_{j,n}+r^{2}_{k,n}+2I_{jk}r_{j,n}r_{k,n})(r^{2}_{k,n}+r^{2}_{i,n}+2I_{ki}r_{k,n}r_{i,n})}.$

Note that $I_{ij}>1$ , then

\lim\frac{r^{2}_{k,n}+I_{jk}r_{j,n}r_{k,n}+I_{ki}r_{k,n}r_{i,n}+\sqrt{(r^{2}_{j,n}+r^{2}_{k,n}+2I_{jk}r_{j,n}r_{k,n})(r^{2}_{k,n}+r^{2}_{i,n}+2I_{ki}r_{k,n}r_{i,n})}}{I_{ij}r_{i,n}r_{j,n}}<1.

Therefore, there exists $n\in\mathbb{N}$ such that $(l^{n})^{2}_{ij}=r^{2}_{i,n}+r^{2}_{j,n}+2I_{ij}r_{i,n}r_{j,n}>(l^{n}_{jk}+l^{n}_{ki})^{2}$ , i.e., $l^{n}_{ij}>l^{n}_{jk}+l^{n}_{ki}$ . This contradicts the triangle inequality $l^{n}_{ij}<l^{n}_{jk}+l^{n}_{ki}$ . Q.E.D.

Combining Lemma 3.5 and the connectivity of the triangulation $\mathcal{T}$ , we have the following result.

Corollary 3.6.

For a discrete conformal factor $u\in\mathbb{R}^{V}$ , let $\mathcal{T}$ be a weighted Delaunay triangulation of the decorated PE surface $(S,V,dist_{S}(u),r(u))$ . For any decorated triangle $\{ijk\}\in F$ in $\mathcal{T}$ , at least two of the three sequences $(u_{i,n}-u_{i^{*},n})_{n\in\mathbb{N}}$ , $(u_{j,n}-u_{i^{*},n})_{n\in\mathbb{N}}$ , $(u_{k,n}-u_{i^{*},n})_{n\in\mathbb{N}}$ converge.

To characterize the function $F_{ijk}(u_{i},u_{j},u_{k})$ in (21), we need the following lemmas.

Lemma 3.7.

Assume that the sequence $(u_{i,n})_{n\in\mathbb{N}}$ diverges properly to $+\infty$ and the sequences $(u_{j,n})_{n\in\mathbb{N}}$ and $(u_{k,n})_{n\in\mathbb{N}}$ converge. Then the sequence $(\theta^{i,n}_{jk})_{n\in\mathbb{N}}$ converges to zero. Furthermore, if the sequences $(\theta^{j,n}_{ki})_{n\in\mathbb{N}}$ and $(\theta^{k,n}_{ij})_{n\in\mathbb{N}}$ converge to non-zero constants, then

(1):: the sequences $(h_{jk}^{i,n})_{n\in\mathbb{N}}$ , $(h_{ki}^{j,n})_{n\in\mathbb{N}}$ and $(h_{ij}^{k,n})_{n\in\mathbb{N}}$ converge;
(2):: the sequences $(\alpha^{i,n}_{jk})_{n\in\mathbb{N}}$ , $(\alpha^{j,n}_{ki})_{n\in\mathbb{N}}$ and $(\alpha^{k,n}_{ij})_{n\in\mathbb{N}}$ converge.

Proof.

By the assumption, we have $\lim r_{i,n}=+\infty$ , $\lim r_{j,n}=c_{1}$ and $\lim r_{k,n}=c_{2}$ , where $c_{1},c_{2}$ are positive constants. The equality (29) implies

(33)

\lim\frac{l_{ij}^{n}}{r_{i,n}}=1,\ \lim\frac{l_{ki}^{n}}{r_{i,n}}=1,\ \lim l_{jk}^{n}=c_{3},

where $c_{3}$ is a positive constant. By the cosine law, we have

\lim\cos\theta^{i,n}_{jk}=\lim\frac{-(l_{jk}^{n})^{2}+(l_{ij}^{n})^{2}+(l_{ki}^{n})^{2}}{2l_{ij}^{n}l_{ki}^{n}}=1.

This implies $\lim\theta^{i,n}_{jk}=0$ .

Suppose the sequences $(\theta^{j,n}_{ki})_{n\in\mathbb{N}}$ and $(\theta^{k,n}_{ij})_{n\in\mathbb{N}}$ converge to non-zero constants. Then

(34)

\lim\frac{A_{ijk}^{n}}{r_{i,n}}=\lim\frac{l^{n}_{ij}l^{n}_{jk}\sin\theta^{j,n}_{ki}}{2r_{i,n}}=c_{4}

for some constant $c_{4}>0$ .

(1): Since $\kappa_{i}=\frac{1}{r_{i}}$ , then $\lim\kappa_{i,n}=0$ , $\lim\kappa_{j,n}=\frac{1}{c_{1}}$ and $\lim\kappa_{k,n}=\frac{1}{c_{2}}$ . By (15), we have

	$\displaystyle\lim h_{i,n}=$	$\displaystyle\lim(\kappa_{i,n}(1-I_{jk}^{2})+\kappa_{j,n}\gamma_{k}+\kappa_{k,n}\gamma_{j})=c_{5}>0,$
	$\displaystyle\lim h_{j,n}=$	$\displaystyle\lim(\kappa_{j,n}(1-I_{ki}^{2})+\kappa_{i,n}\gamma_{k}+\kappa_{k,n}\gamma_{i})=c_{6},$
	$\displaystyle\lim h_{k,n}=$	$\displaystyle\lim(\kappa_{k,n}(1-I_{ij}^{2})+\kappa_{i,n}\gamma_{j}+\kappa_{j,n}\gamma_{i})=c_{7},$

where $c_{5},c_{6},c_{7}$ are constants. Note that $c_{6},c_{7}$ may be non-positive. The equalities (14) and (34) imply

	$\displaystyle\lim h_{jk}^{i,n}=$	$\displaystyle\lim\frac{r_{i,n}^{2}r_{j,n}^{2}r_{k,n}^{2}}{2A^{n}_{ijk}l^{n}_{jk}}\kappa_{i,n}h_{i,n}=\frac{c_{1}^{2}c^{2}_{2}c_{5}}{2c_{3}c_{4}}>0,$
	$\displaystyle\lim h_{ki}^{j,n}=$	$\displaystyle\lim\frac{r_{i,n}^{2}r_{j,n}^{2}r_{k,n}^{2}}{2A^{n}_{ijk}l^{n}_{ki}}\kappa_{j,n}h_{j,n}=\frac{c_{1}c^{2}_{2}c_{6}}{2c_{4}},$
	$\displaystyle\lim h_{ij}^{k,n}=$	$\displaystyle\lim\frac{r_{i,n}^{2}r_{j,n}^{2}r_{k,n}^{2}}{2A^{n}_{ijk}l^{n}_{ij}}\kappa_{k,n}h_{k,n}=\frac{c_{1}^{2}c_{2}c_{7}}{2c_{4}}.$

Hence the sequences $(h_{jk}^{i,n})_{n\in\mathbb{N}}$ , $(h_{ki}^{j,n})_{n\in\mathbb{N}}$ and $(h_{ij}^{k,n})_{n\in\mathbb{N}}$ converge.

(2): The equality (11) implies

(35)

\lim d^{n}_{jk}=\lim\frac{r_{j,n}^{2}+r_{j,n}r_{k,n}I_{jk}}{l^{n}_{jk}}=\frac{c_{1}^{2}+c_{1}c_{2}I_{jk}}{c_{3}}>0.

By (10), we have

\lim(r^{n}_{ijk})^{2}=\lim[(d^{n}_{jk})^{2}+(h^{i,n}_{jk})^{2}-r_{j,n}^{2}]=c_{8}.

where $c_{8}$ is a constant. Note that $h^{i}_{jk}=r_{ijk}\cos\alpha_{jk}^{i}$ . Hence,

	$\displaystyle\lim\cos\alpha_{jk}^{i,n}=$	$\displaystyle\lim\frac{h_{jk}^{i,n}}{r^{n}_{ijk}}=\frac{c_{1}^{2}c^{2}_{2}c_{5}}{2c_{3}c_{4}\sqrt{c_{8}}}>0,$
	$\displaystyle\lim\cos\alpha_{ki}^{j,n}=$	$\displaystyle\lim\frac{h_{ki}^{j,n}}{r^{n}_{ijk}}=\frac{c_{1}c^{2}_{2}c_{6}}{2c_{4}\sqrt{c_{8}}},$
	$\displaystyle\lim\cos\alpha_{ij}^{k,n}=$	$\displaystyle\lim\frac{h_{ij}^{k,n}}{r^{n}_{ijk}}=\frac{c_{1}^{2}c_{2}c_{7}}{2c_{4}\sqrt{c_{8}}}.$

Then the sequences $(\alpha^{i,n}_{jk})_{n\in\mathbb{N}}$ , $(\alpha^{j,n}_{ki})_{n\in\mathbb{N}}$ and $(\alpha^{k,n}_{ij})_{n\in\mathbb{N}}$ converge. Q.E.D.

Lemma 3.8.

\lim h^{i,n}_{jk}=+\infty,\ \lim h^{j,n}_{ki}=+\infty,\ \lim h^{k,n}_{ij}=-\infty.

Proof.

Lemma 3.7 shows that $\lim\theta^{i,n}_{jk}=0$ , thus $\lim(\theta^{j,n}_{ki}+\theta^{k,n}_{ij})=\pi$ . Since $\lim\theta^{j,n}_{ki}=0$ , then $\lim\theta^{k,n}_{ij}=\pi$ . Then

(36)

\lim\frac{A_{ijk}^{n}}{r_{i,n}}=\lim\frac{l^{n}_{ij}l^{n}_{jk}\sin\theta^{j,n}_{ki}}{r_{i,n}}=0.

By the proof of Lemma 3.7, we have

\lim h_{jk}^{i,n}=\lim\frac{r_{i,n}^{2}r_{j,n}^{2}r_{k,n}^{2}}{2A^{n}_{ijk}l^{n}_{jk}}\kappa_{i,n}h_{i,n}=\lim\frac{r_{i,n}^{2}c_{1}^{2}c_{2}^{2}}{2A^{n}_{ijk}c_{3}}\cdot\frac{1}{r_{i,n}}c_{5}=+\infty,

where (36) is used and $c_{1},c_{2},c_{3},c_{5}$ are positive constants. Similar to (35), we have

	$\displaystyle\lim d^{n}_{ji}=$	$\displaystyle\lim\frac{r_{j,n}^{2}+r_{i,n}r_{j,n}I_{ij}}{l^{n}_{ij}}=c_{9},$
	$\displaystyle\lim d^{n}_{ki}=$	$\displaystyle\lim\frac{r_{k,n}^{2}+r_{i,n}r_{k,n}I_{ki}}{l^{n}_{ki}}=c_{10}.$

Here $c_{9},c_{10}$ are positive constants. By (10), we have

	$\displaystyle(r^{n}_{ijk})^{2}$	$\displaystyle=(d^{n}_{jk})^{2}+(h^{i,n}_{jk})^{2}-r_{j,n}^{2}$
		$\displaystyle=(d^{n}_{ji})^{2}+(h^{k,n}_{ij})^{2}-r_{j,n}^{2}$
		$\displaystyle=(d^{n}_{ki})^{2}+(h^{j,n}_{ki})^{2}-r_{k,n}^{2}.$

This implies $\lim r^{n}_{ijk}=+\infty$ , $\lim(h^{k,n}_{ij})^{2}=+\infty$ and $\lim(h^{j,n}_{ki})^{2}=+\infty$ . Therefore, we have the following four cases

$(i)$ :: $\lim h^{k,n}_{ij}=+\infty$ and $\lim h^{j,n}_{ki}=+\infty$ ;
$(ii)$ :: $\lim h^{k,n}_{ij}=-\infty$ and $\lim h^{j,n}_{ki}=-\infty$ ;
$(iii)$ :: $\lim h^{k,n}_{ij}=+\infty$ and $\lim h^{j,n}_{ki}=-\infty$ ;
$(iv)$ :: $\lim h^{k,n}_{ij}=-\infty$ and $\lim h^{j,n}_{ki}=+\infty$ .

For the case $(i)$ , since $\lim h_{jk}^{i,n}>0$ , $\lim h^{k,n}_{ij}>0$ and $\lim h^{j,n}_{ki}>0$ . This implies that the center $c_{ijk}$ of the face-circle $C_{ijk}$ lies in the interior of the triangle $\{ijk\}$ by the definition of $h_{jk}^{i},h^{k}_{ij},h^{j}_{ki}$ . However, in this case, $\lim h_{jk}^{i,n},\lim h^{k,n}_{ij},\lim h^{j,n}_{ki}$ are bounded. This is a contradiction.

\begin{overpic}[scale={0.8}]{figure_4.pdf} \put(48.0,11.0){$i$} \put(96.0,-2.0){$j$} \put(82.0,-1.0){$k$} \end{overpic}

Figure 4. Domain of the center

c_{ijk}

in the decorated triangle on surface

Both the cases $(ii)$ and $(iii)$ imply $d_{kj}<0$ . This contradicts with the fact that $d_{rs}>0$ for any $\{r,s\}\subseteq\{i,j,k\}$ . Indeed, the center $c_{ijk}$ lies in the red region in Figure 4 in the case $(ii)$ and lies in the blue region in Figure 4 in the case $(iii)$ . By projecting the center $c_{ijk}$ to the line determined by $\{jk\}$ , we have $d_{kj}<0$ . This completes the proof. Q.E.D.

Remark 3.9.

Similar to the proof of Lemma 3.8, if the sequence $(\theta^{k,n}_{ij})_{n\in\mathbb{N}}$ converges to zero, then $\lim h^{i,n}_{jk}=+\infty,\ \lim h^{j,n}_{ki}=-\infty,\ \lim h^{k,n}_{ij}=+\infty.$

Consider a star-shaped $s$ -sided polygon in the marked surface with boundary vertices $1,\cdots,s$ ordered cyclically ( $v_{s+1}=v_{1}$ ). Please refer to Figure 5. Let $i\in V$ be a vertex such that the sequence $(u_{i,n})_{n\in\mathbb{N}}$ diverges properly to $+\infty$ and the sequences $(u_{j,n})_{n\in\mathbb{N}}$ converge for $j\sim i$ .

\begin{overpic}[scale={0.5}]{figure_5.pdf} \put(74.0,90.0){$j+1$} \put(92.0,44.0){$j$} \put(87.0,14.0){$j-1$} \put(50.0,38.0){$i$} \put(78.0,16.0){$0$} \put(82.0,21.0){$\pi$} \put(85.0,39.0){$0$} \put(85.0,46.0){$\pi$} \put(70.0,82.0){$0$} \put(64.0,86.0){$\pi$} \put(38.0,92.0){$0$} \put(33.0,90.0){$\pi$} \put(35.0,100.0){$s$} \put(6.0,55.0){$0$} \put(5.0,48.0){$\pi$} \put(-4.0,50.0){$1$} \put(18.0,18.0){$0$} \put(23.0,14.0){$\pi$} \put(16.0,8.0){$2$} \put(53.0,4.0){$0$} \put(59.0,5.0){$\pi$} \end{overpic}

Figure 5. A star triangulation of a star-shaped

s

-sided polygon

Lemma 3.10.

The sequences of inner angles at the boundary vertices in the triangles of a star-shaped polygon converge to non-zero constants.

Proof.

As $\lim u_{i,n}=+\infty$ and $\lim u_{j,n}=C$ for $j\sim i$ . By Lemma 3.7, for any $j=1,...,s$ , we have $\lim\theta^{i,n}_{j-1,j}=0$ and hence $\lim(\theta^{j-1,n}_{i,j}+\theta^{j,n}_{i,j-1})=\pi$ . We prove the result by contradiction. Without loss of generality, we assume $\lim\theta^{j-1,n}_{i,j}=\pi$ and $\lim\theta^{j,n}_{i,j-1}=0$ in the triangle $\{i,j-1,j\}$ . Then for $n$ large enough, we have

l^{n}_{i,j-1}<l^{n}_{i,j}.

By Lemma 3.8, we have $\lim h^{j,n}_{i,j-1}=+\infty,\ \lim h^{j-1,n}_{i,j}=-\infty,\ \lim h^{i,n}_{j-1,j}=+\infty$ . Since the edge $\{i,j\}$ is weighted Delaunay, thus by (8), we have

h^{j-1,n}_{i,j}+h^{j+1,n}_{i,j}\geq 0.

This implies $\lim h^{j+1,n}_{i,j}=+\infty$ .

In the triangle $\{i,j,j+1\}$ , suppose the sequences $(\theta^{j,n}_{i,j+1})_{n\in\mathbb{N}}$ and $(\theta^{j+1,n}_{i,j})_{n\in\mathbb{N}}$ converge to non-zero constants. By Lemma 3.7, the sequences $(h_{i,j}^{j+1,n})_{n\in\mathbb{N}}$ and $(h_{i,j+1}^{j,n})_{n\in\mathbb{N}}$ converge. This contradicts $\lim h^{j+1,n}_{i,j}=+\infty$ . Hence the sequences $(\theta^{j,n}_{i,j+1})_{n\in\mathbb{N}}$ or $(\theta^{j+1,n}_{i,j})_{n\in\mathbb{N}}$ converge to zero. By Lemma 3.8 and Remark 3.9, we have $\lim\theta^{j,n}_{i,j+1}=\pi$ , $\lim\theta^{j+1,n}_{i,j}=0$ and $\lim h^{j,n}_{i,j+1}=-\infty$ . Then for $n$ large enough, we have

l^{n}_{i,j}<l^{n}_{i,j+1}.

Please refer to Figure 5. By induction, for $n$ large enough, we have

l^{n}_{i,j-1}<l^{n}_{i,j}<l^{n}_{i,j+1}<l^{n}_{i,j+2}<...<l^{n}_{i,j-1}.

This is a contradiction. Q.E.D.

Combining (21), Lemma 3.7 and Lemma 3.10, we have the following result.

Corollary 3.11.

Proof.

By the definition of $F_{ijk}(u_{i},u_{j},u_{k})$ in (21), we have

	$\displaystyle F_{ijk}(u_{i,n},u_{j,n},u_{k,n})=$	$\displaystyle-2\mathrm{Vol}^{n}(ijk)+\theta_{jk}^{i,n}u_{i,n}+\theta_{ki}^{j,n}u_{j,n}+\theta_{ij}^{k,n}u_{k,n}$
		$\displaystyle+(\frac{\pi}{2}-\alpha_{ij}^{k,n})\lambda_{ij}+(\frac{\pi}{2}-\alpha_{ki}^{j,n})\lambda_{ki}+(\frac{\pi}{2}-\alpha_{jk}^{i,n})\lambda_{jk}.$

Combining Lemma 3.7 and Lemma 3.10 gives that $\lim\theta^{i,n}_{jk}=0$ , the sequences $(\theta^{j,n}_{ki})_{n\in\mathbb{N}}$ and $(\theta^{k,n}_{ij})_{n\in\mathbb{N}}$ converge to non-zero constants and the sequences $(\alpha^{i,n}_{jk})_{n\in\mathbb{N}}$ , $(\alpha^{j,n}_{ki})_{n\in\mathbb{N}}$ and $(\alpha^{k,n}_{ij})_{n\in\mathbb{N}}$ converge. Combining the continuity of Milnor’s Lobachevsky function defined by (20) and the definition of the truncated volume $\mathrm{Vol}(ijk)$ defined by (19), we have that the sequence $(\mathrm{Vol}^{n}(ijk))_{n\in\mathbb{N}}$ converges. Note that $\lambda_{ij}=\mathrm{arccosh}I_{ij}$ keeps invariant. Hence,

\lim F_{ijk}(u_{i,n},u_{j,n},u_{k,n})=\lim\theta_{jk}^{i,n}u_{i,n}+c_{11}

for some constant $c_{11}$ . By (30), we have $u_{i,n}=\log r_{i,n}-\log r_{i}$ . Then

	$\displaystyle\lim\theta_{jk}^{i,n}u_{i,n}=$	$\displaystyle\lim\sin\theta^{i,n}_{jk}(\log r_{i,n}-\log r_{i})$
	$\displaystyle=$	$\displaystyle\lim\frac{2A_{ijk}^{n}}{l_{ij}^{n}l_{ki}^{n}}\log r_{i,n}$
	$\displaystyle=$	$\displaystyle 2c_{4}\lim\frac{\log r_{i,n}}{r_{i,n}}$
	$\displaystyle=$	$\displaystyle 0,$

where the equalities (33) and (34) is used in the second line and $\lim_{x\rightarrow+\infty}\frac{1}{x}\log x=0$ is used in the third line. Therefore, $\lim F_{ijk}(u_{i,n},u_{j,n},u_{k,n})=c_{11}$ . Q.E.D.

The following lemma gives an asymptotic expression of the function $\mathcal{E}$ .

Lemma 3.12.

There exists a convergent sequence $\{D_{n}\}_{n\in\mathbb{N}}$ such that the function $\mathcal{E}$ satisfies

\mathcal{E}(u_{n})=D_{n}+2\pi\left(u_{i^{*},n}\chi(S)+\sum_{j\in V}(u_{j,n}-u_{i^{*},n})\right).

Proof.

By (26), we have

	$\displaystyle\mathcal{E}(u_{n})=$	$\displaystyle-\sum_{\{ijk\}\in F}F_{ijk}(u_{i,n},u_{j,n},u_{k,n})+2\pi\sum_{j\in V}u_{j,n}$
	$\displaystyle=$	$\displaystyle-\sum_{\{ijk\}\in F}F_{ijk}((u_{i,n},u_{j,n},u_{k,n})-u_{i^{},n}(1,1,1))-\pi\|F\|u_{i^{},n}+2\pi\sum_{j\in V}u_{j,n}$
	$\displaystyle=$	$\displaystyle D_{n}-\pi(2\|V\|-2\chi(S))u_{i^{*},n}+2\pi\sum_{j\in V}u_{j,n}$
	$\displaystyle=$	$\displaystyle D_{n}+2\pi\left(u_{i^{},n}\chi(S)+\sum_{j\in V}(u_{j,n}-u_{i^{},n})\right),$

where $D_{n}=-\sum_{\{ijk\}\in F}F_{ijk}((u_{i,n},u_{j,n},u_{k,n})-u_{i^{*},n}(1,1,1))$ , the equation (22) is used in the second line and $2|V|-|F|=2\chi(S)$ is used in the third line. The sequence $\{D_{n}\}_{n\in\mathbb{N}}$ converges by Corollary 3.6 and Corollary 3.11. Q.E.D.

The following lemma gives the influence of the sequence $(u_{n})_{n\in\mathbb{N}}$ on the area $A_{ijk}$ of a decorated triangle $\{ijk\}$ .

Lemma 3.13.

For a discrete conformal factor $u\in\mathbb{R}^{V}$ , let $\mathcal{T}$ be a weighted Delaunay triangulation of the decorated PE surface $(S,V,dist_{S}(u),r(u))$ . Assume the sequences $(u_{j,n}-u_{i^{*},n})_{n\in\mathbb{N}}$ , $(u_{k,n}-u_{i^{*},n})_{n\in\mathbb{N}}$ converge for $\{ijk\}$ in $\mathcal{T}$ with edge lengths $l_{ij}^{n},l_{jk}^{n},l_{ki}^{n}$ .

(a):

If $(u_{i,n}-u_{i^{*},n})_{n\in\mathbb{N}}$ converges, there exists a convergent sequence of real numbers $(C_{n})_{n\in\mathbb{N}}$ such that

(37)

\log A_{ijk}^{n}=C_{n}+2u_{i^{*},n}.

(b):

If $(u_{i,n}-u_{i^{*},n})_{n\in\mathbb{N}}$ diverges to $+\infty$ , there exists a convergent sequence of real numbers $(C_{n})_{n\in\mathbb{N}}$ such that

(38)

\log A_{ijk}^{n}=C_{n}+u_{i,n}+u_{i^{*},n}.

Proof.

Applying (31) to $A_{ijk}=\frac{1}{2}l_{ij}l_{jk}\sin\theta_{ki}^{j}$ gives

	$\displaystyle A^{n}_{ijk}=$	$\displaystyle\frac{1}{2}l_{ij}^{n}l_{jk}^{n}\sin\theta_{ki}^{j,n}$
	$\displaystyle=$	$\displaystyle\frac{1}{2}\sin\theta_{ki}^{j,n}\sqrt{r^{2}_{i}e^{2u_{i,n}}+r^{2}_{j}e^{2u_{j,n}}+(l_{ij}^{2}-r^{2}_{i}-r^{2}_{j})e^{(u_{i,n}+u_{j,n})}}$
		$\displaystyle\times\sqrt{r^{2}_{j}e^{2u_{j,n}}+r^{2}_{k}e^{2u_{k,n}}+(l_{jk}^{2}-r^{2}_{j}-r^{2}_{k})e^{(u_{j,n}+u_{k,n})}}.$

Then

	$\displaystyle\log A^{n}_{ijk}=$	$\displaystyle\log(\frac{1}{2}\sin\theta_{ki}^{j,n})+2u_{i^{*},n}$
		$\displaystyle+\frac{1}{2}\log(r^{2}_{i}e^{2(u_{i,n}-u_{i^{},n})}+r^{2}_{j}e^{2(u_{j,n}-u_{i^{},n})}+(l_{ij}^{2}-r^{2}_{i}-r^{2}_{j})e^{(u_{i,n}-u_{i^{},n})+(u_{j,n}-u_{i^{},n})})$
		$\displaystyle+\frac{1}{2}\log(r^{2}_{j}e^{2(u_{j,n}-u_{i^{},n})}+r^{2}_{k}e^{2(u_{k,n}-u_{i^{},n})}+(l_{jk}^{2}-r^{2}_{j}-r^{2}_{k})e^{(u_{j,n}-u_{i^{},n})+(u_{k,n}-u_{i^{},n})})$
	$\displaystyle=$	$\displaystyle\log(\frac{1}{2}\sin\theta_{ki}^{j,n})+u_{i,n}+u_{i^{*},n}$
		$\displaystyle+\frac{1}{2}\log(r^{2}_{i}+r^{2}_{j}e^{2(u_{j,n}-u_{i^{},n})-2(u_{i,n}-u_{i^{},n})}+(l_{ij}^{2}-r^{2}_{i}-r^{2}_{j})e^{-(u_{i,n}-u_{i^{},n})+(u_{j,n}-u_{i^{},n})})$
		$\displaystyle+\frac{1}{2}\log(r^{2}_{j}e^{2(u_{j,n}-u_{i^{},n})}+r^{2}_{k}e^{2(u_{k,n}-u_{i^{},n})}+(l_{jk}^{2}-r^{2}_{j}-r^{2}_{k})e^{(u_{j,n}-u_{i^{},n})+(u_{k,n}-u_{i^{},n})}).$

If the sequence $(u_{i,n}-u_{i^{*},n})_{n\in\mathbb{N}}$ converges, then $\log A_{ijk}^{n}=C_{n}+2u_{i^{*},n}$ . If the sequence $(u_{i,n}-u_{i^{*},n})_{n\in\mathbb{N}}$ diverges to $+\infty$ , then the sequence $(\theta_{ki}^{j,n})_{n\in\mathbb{N}}$ converges to a non-zero constant in $(0,\pi)$ by Lemma 3.10. This implies $\log A_{ijk}^{n}=C_{n}+u_{i,n}+u_{i^{*},n}$ . In both cases, the sequence $(C_{n})_{n\in\mathbb{N}}$ converges. Q.E.D.

3.4. Proof of Theorem 3.4

Let $\{u_{n}\}_{n\in\mathbb{N}}$ be an unbounded “good” sequence. Suppose $\chi(S)<0$ and $\{u_{n}\}_{n\in\mathbb{N}}$ is an unbounded sequence in $\mathcal{A}$ . Combining $\chi(S)<0$ and Lemma 3.12, we just need to prove that $\lim_{n\rightarrow+\infty}u_{i^{*},n}=-\infty$ . By the definition of “good” sequence, the sequence $\left(\sum_{j\in V}(u_{j,n}-u_{i^{*},n})\right)_{n\in\mathbb{N}}$ converges to a finite number or diverges properly to $+\infty$ .

If $\left(\sum_{j\in V}(u_{j,n}-u_{i^{*},n})\right)_{n\in\mathbb{N}}$ converges to a finite number, then the sequence $(u_{j,n}-u_{i^{*},n})_{n\in\mathbb{N}}$ converges for all $j\in V$ . Since the sequence $\{u_{n}\}_{n\in\mathbb{N}}$ lies in $\mathcal{A}$ , the area $A_{ijk}$ of each triangle is bounded from above. This implies $\{u_{i^{*},n}\}_{n\in\mathbb{N}}$ is bounded from above by (37). Then $\{u_{i^{*},n}\}_{n\in\mathbb{N}}$ converges to a finite number or diverges properly to $-\infty$ . Suppose $\{u_{i^{*},n}\}_{n\in\mathbb{N}}$ converges to a finite number. Since $(u_{j,n}-u_{i^{*},n})_{n\in\mathbb{N}}$ converges for all $j\in V$ , then $\{u_{j,n}\}_{n\in\mathbb{N}}$ are bounded for all $j\in V$ , which implies $\{u_{n}\}_{n\in\mathbb{N}}$ is bounded. This contradicts the assumption that $\{u_{n}\}_{n\in\mathbb{N}}$ is unbounded. Therefore, the sequence $\{u_{i^{*},n}\}_{n\in\mathbb{N}}$ diverges properly to $-\infty$ .

If $\left(\sum_{j\in V}(u_{j,n}-u_{i^{*},n})\right)_{n\in\mathbb{N}}$ diverges properly to $+\infty$ , then there exists at least one vertex $i\in V$ such that the sequence $(u_{i,n}-u_{i^{*},n})_{n\in\mathbb{N}}$ diverges properly to $+\infty$ . By Corollary 3.6, the sequences $(u_{j,n}-u_{i^{*},n})_{n\in\mathbb{N}}$ and $(u_{k,n}-u_{i^{*},n})_{n\in\mathbb{N}}$ converge for $j\sim i$ and $k\sim i$ . Since the area $A_{ijk}$ of each triangle is bounded from above, thus $u_{i,n}+u_{i^{*},n}\leq C$ and $u_{j,n}+u_{i^{*},n}\leq C$ by (38), where $C$ is a constant. Then $(u_{i,n}-u_{i^{*},n})+2u_{i^{*},n}\leq C$ . This implies $\{u_{i^{*},n}\}_{n\in\mathbb{N}}$ diverges properly to $-\infty$ . Q.E.D.

Remark 3.14.

For the case $\chi(S)>0$ , Kouřimská [17, 16] gave the existence of PE metrics with constant discrete Gaussian curvatures. However, we can not get similar results. The main difference is that the edge length defined by (5) involves the square term of discrete conformal factors, such as $e^{2u_{i}}$ , while the edge length defined by the vertex scalings only involves the mixed product of the first order terms, i.e., $e^{u_{i}+u_{j}}$ . Indeed, in this case, we can define the set $\mathcal{A}_{+}=\{u\in\mathbb{R}^{V}|A_{tot}(u)\geq 1\}$ , which is an unbounded closed subset of $\mathbb{R}^{V}$ . Under the conditions that $\chi(S)>0$ and the function $\mathcal{E}(u)$ attains a minimum in the set $\mathcal{A}_{+}$ , Lemma 3.2 still holds. Using Theorem 3.3, we just need to prove Theorem 3.4 under the condition $\chi(S)>0$ . However, we can not get a good asymptotic expression of the area $A_{ijk}$ . The asymptotic expression of the area $A_{ijk}$ in (38) involves $u_{i,n}+u_{i^{*},n}$ , which is not enough for this case.

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