# Rigidity of the Delaunay triangulations of the plane

Song Dai  Center for Applied Mathematics, Tianjin University, Tianjin, 300072,
P.R. China<EMAIL_ADDRESS>and  Tianqi Wu  Department of Mathematics,
Clark University, 950 Main St, Worcester, MA 01610, USA<EMAIL_ADDRESS>
###### Abstract.

We proved a rigidity result for Delaunay triangulations of the plane under
Luo’s discrete conformal change, extending previous results on hexagonal
triangulations. Our result is a discrete analogue of the conformal rigidity of
the plane. We followed Zhengxu He’s analytical approach in his work on the
rigidity of disk patterns, and developed a discrete Schwarz lemma and a
discrete Liouville theorem. The main tools include conformal modulus, discrete
extremal length, and maximum principles in discrete conformal geometry.

###### Contents

  1. 1 Introduction
    1. 1.1 Notations and conventions
    2. 1.2 Acknowledgement
  2. 2 Preparations for the Proof
    1. 2.1 Hyperbolic discrete conformality
    2. 2.2 Discrete Laplacian on graphs
    3. 2.3 Maximum principles for discrete curvature
    4. 2.4 Modulus of annuli
    5. 2.5 Recurrence of electrical networks
  3. 3 Proof of the Main Theorem
    1. 3.1 Estimates on uniformly nondegenerate triangulations
    2. 3.2 A discrete Schwarz lemma
    3. 3.3 Proof of the boundedness of the conformal factor
    4. 3.4 A discrete Liouville theorem
    5. 3.5 Proof of that the conformal factor is constant

##  1\. Introduction

A fundamental property in conformal geometry is that a conformal embedding of
the plane $\mathbb{R}^{2}$ to itself must be a similar transformation. In this
paper we discretize the plane by geodesic triangulations and prove a similar
rigidity result under the notion of discrete conformal change introduced by
Luo [Luo04].

Let $T=(V,E,F)$ be a topological triangulation of a surface with or without
boundary, where $V$ is the set of vertices, $E$ is the set of edges and $F$ is
the set of faces. Denote $|T|$ as the underlying space of the complex $T$. A
_PL (piecewise linear) metric_ on $T$ is a function,
$l:E\rightarrow\mathbb{R}_{+}$ such that every triangle $\triangle ijk\in F$
could form a Euclidean triangle under the length $l$. Luo [Luo04] introduced
the following notion of _discrete conformality_.

######  Definition 1.1 ([Luo04]).

Two PL metrics $l,l^{\prime}$ on $T=(V,E,F)$ are discretely conformal if there
exists a function $u:V\rightarrow\mathbb{R}$ such that for any edge $ij\in E$,

$l^{\prime}_{ij}=e^{\frac{1}{2}(u_{i}+u_{j})}l_{ij}.$

In this case, $u$ is called a discrete conformal factor, and we denote
$l^{\prime}=u*l$.

Given a PL metric $l$ on $T$, let $\theta^{i}_{jk}$ denote the inner angle at
the vertex $i$ in the Euclidean triangle $\triangle ijk$ under the metric $l$.
The PL metric $l$ is called

  1. (a)

_uniformly nondegenerate_ if there exists a constant $\epsilon>0$ such that
$\theta^{i}_{jk}\geq\epsilon$ for all $\triangle ijk$ in $T$, and

  2. (b)

_Delaunay_ if $\theta^{k_{1}}_{ij}+\theta^{k_{2}}_{ij}\leq\pi$ for any pair of
adjacent triangles $\triangle ijk_{1}$ and $\triangle ijk_{2}$ in $T$, and

  3. (c)

_uniformly Delaunay_ if there exists a constant $\epsilon>0$ such that
$\theta^{k_{1}}_{ij}+\theta^{k_{2}}_{ij}\leq\pi-\epsilon$ for any pair of
adjacent triangles $\triangle ijk_{1}$ and $\triangle ijk_{2}$ in $T$.

######  Remark 1.2.

The Delaunay condition is equivalent to that for every pair of adjacent
triangles $\triangle ijk_{1},\triangle ijk_{2}\in F$, if the Euclidean
quadrilateral $(ik_{1}jk_{2})$ is isometrically embedded in $\mathbb{C}$, then
$k_{2}\notin\emph{int}(D_{ijk_{1}})$, where $\emph{int}(D_{ijk_{1}})$ is the
interior of the circumscribed disk of $\triangle ijk_{1}$.

A map $\phi:|T|\rightarrow\mathbb{C}$ is called a _geodesic embedding_ if for
every $ij\in E$, $\phi$ maps $ij$ to a segment connecting $\phi(i)$ and
$\phi(j)$, and $\phi$ maps $|T|$ homeomorphically to its image. If further
$\phi$ is surjective, we call $\phi$ is a _geodesic homeomorphism_ or a
_geodesic triangulation_. It is clear that a geodesic embedding $\phi$ gives a
PL metric $l(\phi)$, or $l$ for short, by using the Euclidean distance. A
geodesic embedding $\phi$ is called _(uniformly) Delaunay_ if $l(\phi)$ is
(uniformly) Delaunay. The main result of the paper is the following.

######  Theorem 1.3.

Suppose $\phi:|T|\rightarrow\mathbb{C}$ is a geodesic homeomorphism and
$\phi^{\prime}:|T|\rightarrow\mathbb{C}$ is a geodesic embedding with the
induced PL metric $l,l^{\prime}$ respectively, such that

(a) $l,l^{\prime}$ are both uniformly nondegenerate,

(b) $l$ is uniformly Delaunay and $l^{\prime}$ is Delaunay,

(c) $l$ is discretely conformal to $l^{\prime}$, i.e., $l^{\prime}=u*l$ for
some $u\in\mathbb{R}^{V}$.
Then $l$ and $l^{\prime}$ differ by a constant scaling, i.e., $u$ is constant
on $V$.

Wu-Gu-Sun [WGS15] first proved Theorem 1.3 for the special case where
$\phi(T)$ is a regular hexagonal triangulation and $\phi^{\prime}(T)$
satisfies the uniformly acute condition, i.e. all the inner angles are no more
than $\frac{\pi}{2}-\epsilon$ for some constant $\epsilon>0$. Luo-Sun-Wu
[LSW22] and Dai-Ge-Ma [DGM22] generalized Wu-Gu-Sun’s result by allowing
$l^{\prime}$ to be only Delaunay rather than uniformly acute. All these works
essentially rely on the lattice structure of the regular hexagonal
triangulation, and apparently cannot be generalized to triangulations without
translational invariance.

######  Remark 1.4.

In [BPS15], Bobenko-Pinkall-Springborn observed the relation between Luo’s
discrete conformality and the hyperbolic polyhedra in $\mathbb{H}^{3}$. In
fact the rigidity problem in this paper corresponds to the Cauchy rigidity of
certain hyperbolic polyhedra and the Delaunay condition corresponds to the
convexity of the hyperbolic polyhedra.

To prove Theorem 1.3, we follow the approach developed by Zhengxu He in his
state-of-the-art work on the rigidity of disk patterns [He99]. Theorem 1.3
immediately follows from the following two propositions.

######  Proposition 1.5.

Under the conditions of Theorem 1.3, the discrete conformal factor $u$ is
bounded on $V$. Furthermore, the condition (b) could be relaxed to that both
$l,l^{\prime}$ are just Delaunay.

######  Proposition 1.6.

Under the conditions of Theorem 1.3, if the discrete conformal factor $u$ is
bounded on $V$, then it is constant on $V$.

The proof of Proposition 1.5 relies on a discrete Schwarz lemma, and
estimating conformal moduli for annuli. The proof of Proposition 1.6 is by
constructing a discretely conformal geometric flow from $l$ toward
$l^{\prime}$, keeping the surface flat. The derivative of the conformal factor
in this flow is known to be discrete harmonic, and thus constant by a discrete
Liouville theorem.

###  1.1. Notations and conventions

Given $0<r<r^{\prime}$, denote $D_{r}=\\{z\in\mathbb{C}:|z|<r\\}$ and
$A_{r,r^{\prime}}=\\{z\in\mathbb{C}:r<|z|<r^{\prime}\\}$. We also denote
$D=D_{1}$ as the unit open disk. Given a subset $X$ of $\mathbb{C}$, $X^{c}$
denotes the complement $\mathbb{C}\backslash X$ and $\partial X$ denotes the
boundary of $X$ in $\mathbb{C}$ and $\text{int}(X)$ denotes the interior of
$X$ in $\mathbb{C}$ and

$\text{diam}(X)=\sup\\{|z-z^{\prime}|:z,z^{\prime}\in X\\},$

denotes the diameter of $X$. Given two subsets $X,Y$ of $\mathbb{C}$, the
distance between $X,Y$ is denoted by

$d(X,Y)=\inf\\{|z-z^{\prime}|:z\in X,z^{\prime}\in Y\\}.$

Given $i\in V$, denote $\deg(i)$ as the number of neighbors of $i$ and $N_{i}$
as set of neighbors of $i$, i.e., $N_{i}=\\{j\in V:ij\in E\\}$. Furthermore,
we denote $R_{i}$ as the union of the triangles in $T$ containing $i$. $R_{i}$
is always viewed as the underlying space of the subcomplex generated by the
triangles containing $i$. Such $R_{i}$ is called a _1-ring neighborhood_ of
$i$ if $R_{i}$ is homeomorphic to a closed disk with vertex $i$ mapped to the
center of the disk. Given a subset $V_{0}$ of $V$, a vertex $i\in
V_{0}\subseteq V$ is called an _inner point_ of $V_{0}$, if $N_{i}\subseteq
V_{0}$ and $R_{i}$ is a 1-ring neighborhood of $i$. We denote
$\text{int}(V_{0})$ as the set of inner points of $V_{0}$, and $\partial
V_{0}=V_{0}-\text{int}(V_{0})$. In particular, $\partial V$ is the set of
vertices of $T$ that are on the boundary of the surface $|T|$.

Given $l\in\mathbb{R}^{E}$ and $u\in\mathbb{R}^{V}$, if $u*l$ is a PL metric
then

  1. (a)

$\theta^{i}_{jk}(u)=\theta^{i}_{jk}(u,l)$ denotes the inner angle of
$\triangle ijk$ at $i$ under $u*l$, and

  2. (b)

$K_{i}(u)=K_{i}(u,l)$ denotes the discrete curvature at $i$ for
$i\in\text{int}(V)$

$K_{i}(u)=2\pi-\sum_{jk:\triangle ijk\in F}\theta^{i}_{jk}(u).$

###  1.2. Acknowledgement

The authors would like to thank Huabin Ge for the encouragement. The first
author is supported by NSF of China (No.11871283, No.11971244 and
No.12071338). The second author is supported by NSF 1760471.

##  2\. Preparations for the Proof

###  2.1. Hyperbolic discrete conformality

The notion of the hyperbolic discrete conformality was first introduced by
Bobenko-Pinkall-Springborn in [BPS15]. A _piecewise hyperbolic (PH) metric_ on
$T$ is represented by a function $l^{h}:E\rightarrow\mathbb{R}_{+}$ such that
every $\triangle ijk\in F$ could form a hyperbolic triangle under the length
$l^{h}$. Let $l^{h},l^{h\prime}$ be two PH metrics on $T$. We say $l^{h}$ is
_hyperbolic discretely conformal_ to $l^{h\prime}$ if there exists a function
$u^{h}:V\rightarrow\mathbb{R}$ such that for any edge $ij\in E$

$\sinh\frac{l^{h}_{ij}}{2}=e^{\frac{1}{2}(u^{h}_{i}+u^{h}_{j})}\sinh\frac{l^{h\prime}_{ij}}{2}.$

In this case, $u$ is called a _hyperbolic discrete conformal factor_ and we
denote $l^{h\prime}=u^{h}*^{h}l^{h}$.

Denote $\mathbb{D}$ as the hyperbolic space, represented in the Poincaré disk
model. A map $\phi^{h}:|T|\rightarrow\mathbb{D}$ is called a _hyperbolic
geodesic embedding_ if for every $ij\in E$, $\phi^{h}$ maps $ij$ to a
hyperbolic geodesic segment connecting $\phi^{h}(i)$ and $\phi^{h}(j)$, and
$\phi^{h}$ maps $|T|$ homeomorphically to its image. It is clear that a
hyperbolic geodesic embedding $\phi^{h}$ gives a PH metric $l^{h}(\phi^{h})$,
or $l^{h}$ for short, by using the hyperbolic distance. A hyperbolic geodesic
embedding $\phi^{h}$ is called _Delaunay_ if for any pair of adjacent
triangles $\triangle ijk_{1},\triangle ijk_{2}$ in $T$,
$\phi^{h}(k_{2})\notin\text{int}(D_{ijk_{1}})$ where $D_{ijk_{1}}$ is the
circumscribed disk of $\phi^{h}(\triangle ijk_{1})$. Notice that in the
Poincaré disk model, a hyperbolic disk in $\mathbb{D}$ is also a Euclidean
disk in $\mathbb{C}$.

Given $i\in\text{int}(V)$ and a Euclidean geodesic embedding
$\phi:R_{i}\rightarrow D$, we say that $\phi$ _induces_ the hyperbolic
geodesic embedding $\phi^{h}:R_{i}\rightarrow\mathbb{D}$ if
$\phi(j)=\phi^{h}(j)$ for all $j\in\\{i\\}\cup N_{i}$. Such $\phi^{h}$ exists
if $l(\phi)$ is non-degenerate and small, and $\phi(R_{i})$ is away from
$\partial D$.

######  Lemma 2.1.

Let $\phi:R_{i}\rightarrow D$ be a (Euclidean) geodesic embedding where all
the inner angles in are at least $\epsilon>0$. Suppose

(2.1) $l_{ij}\leq(1-|\phi(i)|^{2})\sin\epsilon,\quad\text{ for every }j\in
N_{i}.$

Then there exists a hyperbolic geodesic embedding
$\phi^{h}:R_{i}\rightarrow\mathbb{D}$ such that $\phi^{h}$ coincides with
$\phi$ on the set of vertices, i.e., $\phi(j)=\phi^{h}(j)$ for any $j\in
N_{i}\cup\\{i\\}$.

###### Proof.

Denote $m=\deg(i)$ and $z_{0}=\phi(i)$. Let $z_{1},...,z_{m}$ be the points in
$\phi(N_{i})$ such that $z_{1}-z_{0},...,z_{m}-z_{0}$ are counterclockwise.
Let $\exp_{z_{0}}$ be the exponential map with respect to the hyperbolic
metric at $z_{0}$. Identifying $T_{z_{0}}\mathbb{D}$ with $\mathbb{C}$ by
translating $z_{0}$ to the origin, denote
$v(z)=\exp_{z_{0}}^{-1}z\in\mathbb{C}$ for any $z\in\mathbb{D}$. Then we only
need to show the following claims.

(2.2) $\arg(\frac{v(z_{k+1})}{v(z_{k})})\in(0,\pi),\quad k=1,\cdots,m,$

and

(2.3)
$\sum\limits_{k=1}^{m}\arg\big{(}\frac{v(z_{k+1})}{v(z_{k})}\big{)}=2\pi,$

where $z_{m+1}=z_{1}$.

We first show the claim (2.2). Fix $k\in\\{1,\cdots,m\\}$, denote

$P=\\{z\in\mathbb{C}:\arg(\frac{z-z_{0}}{z_{k}-z_{0}})\in(0,\pi)\\},$

and

$P_{h}=\\{z\in\mathbb{D}:\arg(\frac{v(z)}{v(z_{k})})\in(0,\pi)\\}.$

See Figure 1 for illustrations.

(a) Case 1

(b) Case 2

Figure 1.

Since $\phi$ is a geodesic embedding and $z_{1}-z_{0},...,z_{m}-z_{0}$ are
counterclockwise, we have $z_{k+1}\in P$. We need to show $z_{k+1}\in P_{h}$.
Let $\gamma_{h}$ be the entire geodesic connecting $z_{0}$ and $z_{k}$ with
respect to the hyperbolic metric. If $\gamma_{h}$ is a straight line, then
$z_{k+1}\in P_{h}=P$ and we are done. Otherwise, $\gamma_{h}$ is a Euclidean
circular arc and orthogonal to the boundary of the unit disk $D$. We denote
$z_{*}$ and $R$ as the Euclidean center and the radius, respectively, or this
circular arc. Then

$R^{2}+1=|z_{*}|^{2}\leq(|z_{0}|+R)^{2}.$

So

$1-|z_{0}|^{2}\leq 2R|z_{0}|<2R.$

Then by Equation (2.1)

$\sin\frac{\angle
z_{0}z_{*}z_{k}}{2}=\frac{|z_{k}-z_{0}|}{2R}\leq\frac{(1-|z_{0}|^{2})\sin\epsilon}{2R}<\sin\epsilon.$

So

$\angle z_{0}z_{k+1}z_{k}\geq\epsilon>\frac{1}{2}\angle z_{0}z_{*}z_{k},$

and

$\angle z_{0}z_{k+1}z_{k}\leq\pi-\epsilon<\pi-\frac{1}{2}\angle
z_{0}z_{*}z_{k}.$

Then from the knowledge of the plane geometry, we see $z_{k}\in P_{h}$. We
finish the proof of claim (2.2).

Next we show the claim (2.3). We have

(2.4)
$\arg\big{(}\frac{v(z_{k+1})}{v(z_{k})}\big{)}+\arg\big{(}\frac{v(z_{k})}{z_{k}-z_{0}}\big{)}=\arg\big{(}\frac{z_{k+1}-z_{0}}{z_{k}-z_{0}}\big{)}+\arg\big{(}\frac{v(z_{k+1})}{z_{k+1}-z_{0}}\big{)}+2n\pi,$

for some integer $n$. From the knowledge of plane geometry, we have that

$\arg\big{(}\frac{v(z_{k})}{z_{k}-z_{0}}\big{)}\in(-\frac{\pi}{2},\frac{\pi}{2}).$

Then together with the claim (2.2), both the right hand side and the left hand
side of Equation (2.4) lies in $(-\frac{\pi}{2},\frac{3\pi}{2})$, so $n=0$.
Then we obtain

$\sum\limits_{k=1}^{m}\arg\big{(}\frac{v(z_{k+1})}{v(z_{k})}\big{)}=\sum\limits_{k=1}^{m}\arg\big{(}\frac{z_{k+1}-z_{0}}{z_{k}-z_{0}}\big{)}=2\pi.$

We finish the proof. ∎

######  Remark 2.2.

One can show that such hyperbolic geodesic embedding $\phi^{h}$ exists, if the
Euclidean geodesic embedding $\phi$ maps each triangle in $R_{i}$ to an acute
triangle.

The discrete conformality and the hyperbolic discrete conformality are related
as follows.

######  Lemma 2.3.

Let $\phi,\phi^{\prime}:|T|\rightarrow D$ be two geodesic embeddings with the
induced PL metrics $l,l^{\prime}$ respectively. Suppose both
$\phi,\phi^{\prime}$ induce hyperbolic geodesic embeddings
$\phi^{h},\phi^{h\prime}:|T|\rightarrow\mathbb{D}$ with the induced hyperbolic
PH metrics $l^{h},l^{h\prime}$ respectively. Then $l^{\prime}=u*l$ if and only
if $l^{h\prime}=u^{h}*^{h}l^{h}$ where $u$ and $u^{h}$ are related by

$u^{h}_{i}=u_{i}+\ln\frac{1-|z_{i}|^{2}}{1-|z_{i}^{\prime}|^{2}},$

with $z_{i}=\phi(i)$, $z_{i}^{\prime}=\phi^{\prime}(i)$.

###### Proof.

Denote $d_{h}$ as the hyperbolic distance function on $\mathbb{D}$. Then the
lemma follows from the formula

$\sinh\frac{d_{h}(z_{1},z_{2})}{2}=\frac{|z_{1}-z_{2}|}{\sqrt{(1-|z_{1}|^{2})(1-|z_{2}|^{2})}},$

where $z_{1},z_{2}\in\mathbb{D}$. See [And05]. ∎

######  Remark 2.4.

In the smooth setting, Lemma 2.3 is interpreted as follows. Let $\Omega$ be a
domain in $D$ and $f$ be a smooth map from $\Omega$ to $D$, $w=f(z)$. Denote
$g_{0}=|dz|^{2}$ as the Euclidean metric on $D$ and
$g_{-1}=\frac{4}{(1-|z|^{2})^{2}}|dz|^{2}$ as the hyperbolic metric on $D$.
Suppose $f$ is conformal with respect to $g_{0}$, i.e.
$f^{*}g_{0}=e^{2u}g_{0}$ for some smooth function $u=u(z)$. Then $f$ is also
conformal with respect to $g_{-1}$. In fact

$\displaystyle\quad
f^{*}g_{-1}=f^{*}(\frac{4}{(1-|w|^{2})^{2}}|dw|^{2})=\frac{4}{(1-|f(z)|^{2})^{2}}e^{2u}|dz|^{2}$
$\displaystyle=\big{(}\frac{1-|z|^{2}}{1-|f(z)|^{2}}\big{)}^{2}e^{2u}g_{-1}=e^{2(u+\ln\frac{1-|z|^{2}}{1-|f(z)|^{2}})}g_{-1}.$

###  2.2. Discrete Laplacian on graphs

Let $G=(V,E)$ be a connected simple graph, and $\mu:E\rightarrow[0,+\infty)$
be a function on the set of edges. We call $G_{\mu}=(V,E,\mu)$ a _weighted
graph_ , or an _electrical network_. The _discrete Laplacian operator_
$\Delta_{\mu}:\mathbb{R}^{V}\rightarrow\mathbb{R}^{V}$ is defined as

$(\Delta_{\mu}u)_{i}=\sum\limits_{j:ij\in E}\mu_{ij}(u_{j}-u_{i}).$

We say that $u$ is _harmonic_ at $i\in V$, if $(\Delta_{\mu}u)_{i}=0$. In this
case we have the average property

$u_{i}=\sum\limits_{j:ij\in E}\frac{\mu_{ij}}{\sum\limits_{k:ik\in
E}\mu_{ik}}u_{j}$

if $\mu_{jk}>0$ for all $jk\in E$. So we have the maximum principle.

######  Lemma 2.5 (Maximum principle for discrete harmonic functions).

If $\mu_{ij}>0$ for all $ij\in E$ and $V_{0}$ is a finite proper subset of $V$
and $u$ is harmonic at every point in $V_{0}$, then $u$ achieves its maximum
and minimum on $V-V_{0}$.

We also have the following well-posedness result of the discrete Laplace
equation with Dirichlet boundary condition.

######  Lemma 2.6.

Suppose $\mu_{ij}>0$ for all $ij\in E$ and $V_{0}$ is a finite proper subset
of $V$ and $f$ is a given funciton on $V-V_{0}$. Then the following equation
of $u\in\mathbb{R}^{V}$

$\Delta_{\mu}u=0\text{ in $V_{0}$},\qquad u=f\text{ on $(V-V_{0})$}$

has a unique solution. Furthermore, the map $(\mu,f)\mapsto u$ is smooth.

Lemma 2.6 is well-known. Solving $u\in\mathbb{R}^{V}$ here is indeed solving a
diagonal dominant linear system.

Let $T=(V,E,F)$ be a triangulation and $l$ be a PL metric. Recall that
$K_{i}(u)=2\pi-\sum\limits_{jk:\triangle ijk\in F}\theta^{i}_{jk}$ is the
curvature at $i\in V$ under the metric $u*l$. The operator $K$, $u\mapsto
K(u)$, is smooth with respect to $u$. From the direct calculation or [Luo04],
one have

(2.5) $dK_{i}=-\sum\limits_{j:ij\in E}\mu_{ij}(du_{j}-du_{i}),$

where

(2.6)
$\mu_{ij}=\mu_{ij}(u)=\frac{1}{2}(\cot\theta^{k_{1}}_{ij}(u)+\cot\theta^{k_{2}}_{ij}(u))$

for adjacent triangles $\triangle ijk_{1},\triangle ijk_{2}\in F$. It is not
difficult to check that if $u*l$ is uniformly Delaunay then
$\mu_{ij}\geq\epsilon$ for some constant $\epsilon=\epsilon(T,l,u)>0$. From
Equation (2.5), we see that the linearization of $-K$ is in fact a discrete
Laplacian operator with respect to $G_{\mu}=(V,E,\mu)$.

###  2.3. Maximum principles for discrete curvature

Maximum principle plays a very important role in partial differential
equations and geometric analysis. For the discrete conformal geometry, the
curvature $K$, which is clearly nonlinear as an operator on the set of
conformal factors, also satisfies the (strong) maximum principle.

The following lemma is a corollary of Theorem 3.1 in [LSW22]. In [DGM22] there
is another proof of the maximum principle for a special case. Let $T=(V,E,F)$
be a triangulated surface.

######  Lemma 2.7.

Suppose $i\in V$ and $R_{i}$ is a $1$-ring neighborhood. Let
$\phi,\phi^{\prime}:R_{i}\rightarrow\mathbb{C}$ be two Delaunay geodesic
embeddings with induced PL metric $l,l^{\prime}$ respectively. If
$l^{\prime}=u*l$, then

$u_{i}\leq\max\limits_{j\in N_{i}}u_{j},\qquad u_{i}\geq\min\limits_{j\in
N_{i}}u_{j}.$

Furthermore, if $u_{i}=\max\limits_{j\in N_{i}}u_{j}$ or
$u_{i}=\min\limits_{j\in N_{i}}u_{j}$, then $u$ is a constant on $\\{i\\}\cup
N_{i}$.

As a direct corollary, we have the following maximum principle.

######  Lemma 2.8.

Let $T=(V,E,F)$ be a triangulation of a compact surface with boundary. Let
$\phi,\phi^{\prime}:|T|\rightarrow\mathbb{C}$ be two Delaunay geodesic
embeddings with induced PL metric $l,l^{\prime}$ respectively. Suppose $\phi$
and $\phi^{\prime}$ are discretely conformal, $l^{\prime}=u*l$. Then

$\max\limits_{j\in V}u_{j}=\max\limits_{j\in\partial
V}u_{j},\qquad\min\limits_{j\in V}u_{j}=\min\limits_{j\in\partial V}u_{j}.$

Furthermore, if $\max\limits_{j\in\emph{int}(V)}u_{j}=\max\limits_{j\in
V}u_{j}$ or $\min\limits_{j\in\emph{int}(V)}u_{j}=\min\limits_{j\in V}u_{j}$,
then $u$ is a constant on $V$.

For the hyperbolic setting, we have the following modified version of the
maximum principle.

######  Lemma 2.9.

Suppose $i\in V$ and $R_{i}$ is a $1$-ring neighborhood. Let
$\phi^{h},\phi^{h\prime}:R_{i}\rightarrow\mathbb{D}$ be two Delaunay
hyperbolic geodesic embeddings with induced PH metric $l^{h},l^{h\prime}$
respectively. Suppose $\phi^{h}$ and $\phi^{h\prime}$ are hyperbolic
discretely conformal, $l^{h\prime}=u^{h}*^{h}l^{h}$. Then $u_{i}^{h}<0$
implies that

$u^{h}_{i}>\min\limits_{j\in N_{i}}u^{h}_{j}.$

###### Proof.

Since the hyperbolic discrete conformality and conformal factor are invariant
under the hyperbolic isometric group, we may assume $\phi^{h}(i)=0$,
$\phi^{h\prime}(i)=0$. Then $\phi^{h},\phi^{h\prime}$ are induced by Euclidean
geodesic embeddings $\phi,\phi^{\prime}$ with PL metric $l,l^{\prime}$
respectively. Notice that the hyperbolic isometric group preserves circles, so
$l,l^{\prime}$ are also Delaunay. From Lemma 2.3, since
$l^{h\prime}=u^{h}*^{h}l^{h}$, we have $l$ and $l^{\prime}$ are also
discretely conformal. Specifically $l^{\prime}=u*l$ where

$u_{j}=u^{h}_{j}-\ln\frac{1-|z_{j}|^{2}}{1-|z_{j}^{\prime}|^{2}}.$

In particular $u_{i}=u^{h}_{i}$. From the maximum principle Lemma 2.7, there
exists $j_{0}\in N_{i}$ such that $u_{j_{0}}\leq u_{i}$. Suppose
$u^{h}_{i}<0$. Then

$|z^{\prime}_{j_{0}}|=l^{\prime}_{ij_{0}}=e^{\frac{1}{2}(u_{i}+u_{j_{0}})}l_{ij_{0}}<l_{ij_{0}}=|z_{j_{0}}|.$

Therefore

$u_{j_{0}}^{h}=u_{j_{0}}+\ln\frac{1-|z_{j_{0}}|^{2}}{1-|z_{j_{0}}^{\prime}|^{2}}\leq
u_{i}+\ln\frac{1-|z_{j_{0}}|^{2}}{1-|z_{j_{0}}^{\prime}|^{2}}<u_{i}=u_{i}^{h}.$

So

$u^{h}_{i}>\min\limits_{j\in N_{i}}u^{h}_{j}.$

∎

As a direct corollary, we have

######  Lemma 2.10.

Let $T=(V,E,F)$ be a triangulation of a compact surface with boundary. Let
$\phi^{h},\phi^{h\prime}:|T|\rightarrow\mathbb{D}$ be two Delaunay hyperbolic
geodesic embeddings with induced PH metric $l^{h},l^{h\prime}$ respectively.
Suppose $\phi^{h}$ and $\phi^{h\prime}$ are hyperbolic discretely conformal,
$l^{h\prime}=u^{h}*^{h}l^{h}$. Then $u^{h}_{i}<0$ implies that

$u^{h}_{i}>\min\limits_{j\in\partial V}u^{h}_{j}.$

######  Remark 2.11.

In the smooth setting, the maximum principles above are interpreted as
follows. Let $\Omega$ be a domain in $\mathbb{C}$. Let $f$ be a conformal map
from $\Omega$ to $\mathbb{C}$ with respect to the Euclidean metric
$g_{0}=|dz|^{2}$. Suppose $f^{*}g_{0}=e^{2u}g_{0}$. From the curvature
formula, for $g=g(z)|dz|^{2}$,

$K_{g}=-\frac{2}{g(z)}\partial_{z}\partial_{\bar{z}}\ln g(z),$

we have $\triangle u=0$. So $u$ satisfies the maximum principle as in Lemma
2.8.

On the other hand, let $\Omega$ be a domain in $D$. Let $f$ be a conformal map
from $\Omega$ to $D$ with respect to the hyperbolic metric
$g_{-1}=\frac{4}{(1-|z|^{2})^{2}}|dz|^{2}$. Suppose
$f^{*}g_{-1}=e^{2u^{h}}g_{-1}$. From the curvature formula, we have

$\triangle_{g_{-1}}u^{h}=e^{2u^{h}}-1,$

where $\triangle_{g_{-1}}=(1-|z|^{2})^{2}\partial_{z}\partial_{\bar{z}}$. So
$u^{h}$ satisfies the maximum principle as in Lemma 2.10.

###  2.4. Modulus of annuli

We briefly review the notion of conformal modulus. The definitions and
properties discussed here are mostly well-known. One may refer [Ahl10] and
[LV73] for more comprehensive introductions.

A _closed annulus_ is a subset of $\mathbb{C}$ that is homeomorphic to
$\\{z\in\mathbb{C}:1\leq|z|\leq 2\\}$. An _(open) annulus_ is the interior of
a closed annulus. Given an annulus $A$, denote $\Gamma=\Gamma(A)$ as the set
of smooth simple closed curves in $A$ separating the two boundary components
of $A$. A real-valued Borel measurable function $f$ on $A$ is called
_admissible_ if $\int_{\gamma}fds\geq 1$ for all $\gamma\in\Gamma$. Here $ds$
denotes the element of arc length. The _(conformal) modulus_ of $A$ is defined
as

$\text{Mod}(A)=\inf\\{\|f\|^{2}_{2}:f\text{ is admissible}\\},$

where $\|f\|^{2}_{2}$ denotes the integral of $(f(z))^{2}$ against the 2-dim
Lebesgue measure on $A$. From the definition it is straightforward to verify
that $\text{Mod}(A)$ is conformally invariant. Furthermore, if $f:A\rightarrow
A^{\prime}$ is a K-quasiconformal homeomorphism between two annuli, then

$\frac{1}{K}\cdot\text{Mod}(A)\leq\text{Mod}(A^{\prime})\leq{K}\cdot\text{Mod}(A).$

Given $0<r<r^{\prime}$, denote $A_{r,r^{\prime}}$ as the annulus
$\\{z\in\mathbb{C}:r<|z|<r^{\prime}\\}$. It is well-known that

$\text{Mod}(A_{r,r^{\prime}})=\frac{1}{2\pi}\ln\frac{r^{\prime}}{r}.$

Intuitively, the modulus measures the relative thickness of an annulus. If an
annulus $A$ in $\mathbb{C}\backslash\\{0\\}$ contains $A_{r,r^{\prime}}$, then
it is “thicker” than $A_{r,r^{\prime}}$ and we have the monotonicity

$\text{Mod}(A)\geq\text{Mod}(A_{r,r^{\prime}})=\frac{1}{2\pi}\ln\frac{r^{\prime}}{r}.$

On the other hand, we have that

######  Lemma 2.12.

Suppose $A\subseteq\mathbb{C}\backslash\\{0\\}$ is an annulus separating $0$
from the infinity. If $\text{Mod}(A)\geq 100$, then $A\supseteq A_{r,2r}$ for
some $r>0$.

###### Proof.

Deonte $B$ as the bounded component of $\mathbb{C}-A$, and $r=\max\\{|z|:z\in
B\\}$ and $r^{\prime}=\min\\{|z|:z\in\mathbb{(}B\cup A)^{c}\\}$. If
$r^{\prime}\geq 2r$ we are done. So we may assume $r^{\prime}<2r$.

Then $D_{2r}\cap\gamma\neq\emptyset$ for all $\gamma\in\Gamma(A)$. Let $f$ be
the function on $A$ such that $f(z)=\frac{1}{r}$ on $A\cap D_{3r}$ and
$f(z)=0$ on $A\backslash D_{3r}$. If $\gamma\in\Gamma$ and $\gamma\subseteq
D_{3r}$,

$\int_{\gamma}fds=s(\gamma)\cdot\frac{1}{r}\geq
2\cdot\text{diam}(B)\cdot\frac{1}{r}\geq 2r\cdot\frac{1}{r}>1.$

If $\gamma\in\Gamma$ and $\gamma\not\subseteq D_{3r}$, then $\gamma$ is a
connected curve connecting $D_{2r}$ and $D_{3r}^{c}$ and

$\int_{\gamma}fds\geq
d(D_{2r},D_{3r}^{c})\cdot\frac{1}{r}=r\cdot\frac{1}{r}=1.$

So $f$ is admissible and

$\text{Mod}(A)\leq\int_{A}f^{2}=\frac{1}{r^{2}}\cdot\text{Area}(A\cap
D_{3r})\leq\frac{\pi(3r)^{2}}{r^{2}}=9\pi<100.$

This contradicts with our assumption. ∎

######  Remark 2.13.

To some extend, Lemma 2.12 is a consequence of Teichmüller’s result on
extremal annuli (see Theorem 4-7 in [Ahl10]). The constant $100$ is chosen for
convenience and should not be optimal.

###  2.5. Recurrence of electrical networks

Discrete harmonic functions are closely related to the theory of electrical
networks. Let $G_{\mu}=(V,E,\mu)$ be a weighted graph, i.e. a graph $G=(V,E)$
with a function $\mu:E\rightarrow[0,\infty)$. A weighted graph $G_{\mu}$ could
be viewed as an electrical network, where $\mu_{ij}$ denotes the conductance
of the edge $ij$. Consider a function $u:V\rightarrow\mathbb{R}$, which
denotes the electric potentials at the vertices. Then $u$ is harmonic at $i$
if and only if the outward electric flux at $i$ is $0$. The theory of
electrical networks is closely related to discrete extremal length, originally
introduced by Duffin [Duf62]. Here we briefly review the theory of discrete
extremal length, adapted to our setting. All the definitions and properties
here are well-known and one may read [He99] for references.

Let $V_{1},V_{2}$ be two nonempty disjoint subsets of $V$. Denote
$\Gamma(V_{1},V_{2})$ as the set of the paths joining $V_{1}$ and $V_{2}$. An
_edge metric_ is a function $m:V\rightarrow[0,\infty)$. An edge metric $m$ is
called $\Gamma(V_{1},V_{2})$-admissible if $\sum\limits_{e\in\gamma}m(e)\geq
1$ for every $\gamma\in\Gamma(V_{1},V_{2})$. The _conductance_ of
$\Gamma(V_{1},V_{2})$ is defined as

$\text{Cond}(V_{1},V_{2})=\inf\\{\sum\limits_{e\in E}\mu(e)m^{2}(e):~{}m\text{
is $\Gamma(V_{1},V_{2})$-admissible.}\\}.$

The _resistance_ of $\Gamma(V_{1},V_{2})$ is defined as

$\text{Res}(V_{1},V_{2})=\frac{1}{\text{Cond}(V_{1},V_{2})}.$

An electrical network $G_{\mu}$ is called _connected_ if
$G^{\prime}_{\mu}=(V,E\setminus\\{e\in E:\mu(e)=0\\})$ is connected. Let
$V_{0}$ be a nonempty finite subset of vertices in $G_{\mu}$. Then a connected
electric network $G_{\mu}$ is called _recurrent_ if
$\text{Res}(V_{0},\infty)=\infty$, and _transient_ otherwise. The recurrency
of $G_{\mu}$ is independent of the choice of $V_{0}$. The following lemma is
well-known and shows that the recurrency implies a discrete Liouville
property. See Lemma 5.5 in [He99] for a proof.

######  Lemma 2.14.

Suppose $G_{\mu}$ is recurrent and $u$ is a bounded function on $V$, if $u$ is
harmonic on every point of $V$, then $u$ is constant.

To check that an electrical network $G_{\mu}$ is recurrent, one may study the
vertex extremal length of the graph $G$ as follows.

Let $G=(V,E)$ be a graph. A _vertex metric_ is a function
$\eta:V\rightarrow[0,\infty)$. A vertex metric $\eta$ is called
$\Gamma(V_{1},V_{2})$-admissible if $\sum\limits_{v\in\gamma}\eta(v)\geq 1$
for every $\gamma\in\Gamma(V_{1},V_{2})$. The _vertex modulus_ of
$\Gamma(V_{1},V_{2})$ is defined as

$\text{Mod}(V_{1},V_{2})=\inf\\{\sum\limits_{v\in V}\eta^{2}(v):~{}\eta\text{
is $\Gamma(V_{1},V_{2})$-admissible.}\\}.$

The _vertex extremal length_ of $\Gamma(V_{1},V_{2})$ is defined as

$\text{VEL}(V_{1},V_{2})=\frac{1}{\text{Mod}(V_{1},V_{2})}.$

Let $V_{0}$ be a nonempty finite subset of vertices in a connected graph $G$.
Then $G$ is called _VEL -parabolic_ if $\text{VEL}(V_{0},\infty)=\infty$, and
_VEL -hyperbolic_ otherwise. The definition is independent of the choice of
$V_{0}$.

The relation between the VEL-parabolicity and the recurrence is as follows.

######  Lemma 2.15 (Lemma 5.4 in [He99]).

Let $C>0$ be a constant. Suppose that for each vertex $v$, we have
$\sum\limits_{v\in e}\mu(e)\leq C$. Then for any mutually disjoint, nonempty
subsets $V_{1}$ and $V_{2}$ of $V$, we have

$\text{VEL}(V_{1},V_{2})\leq 2C\cdot\text{Res}(V_{1},V_{2}).$

In particular, if $G$ is VEL-parabolic and $G_{\mu}$ is connected, then
$G_{\mu}$ is recurrent.

##  3\. Proof of the Main Theorem

We will prove our main Theorem 1.3 by proving Propositions 1.5 and 1.6. We
will first derive needed estimates on uniformly nondegenerate triangulations
in Section 3.1. A discrete Schwarz lemma is developed in Section 3.2, for the
proof of Proposition 1.5 in Section 3.3. A discrete Liouville theorem is
developed in Section 3.4, for the proof of Proposition 1.6 in Section 3.5.

###  3.1. Estimates on uniformly nondegenerate triangulations

######  Lemma 3.1.

Suppose $\phi:|T|\rightarrow\mathbb{C}$ is a geodesic embedding and all the
inner angles in $l=l(\phi)$ are at least $\epsilon$. Then we have the
following.

(a) For all $i\in V$, $\deg(i)\leq\frac{2\pi}{\epsilon}$.

(b) For all $\triangle ijk\in F$,

$\sin\epsilon\leq\frac{l_{ij}}{l_{ik}}\leq\frac{1}{\sin\epsilon}.$

(c) For all $\triangle ijk$,

$\frac{\sin^{2}\epsilon}{2}l_{ij}^{2}\leq\emph{Area}(\phi(\triangle
ijk))\leq\frac{1}{2\sin\epsilon}l_{ij}^{2}.$

(d) There exists a constant $\delta=\delta(\epsilon)>0$ such that for all
$\triangle ijk\in F$ with $i,j,k\in\emph{int}(V)$,

$d(U_{ijk}^{c},\phi(\triangle ijk))\geq\delta\cdot\emph{diam}(\phi(\triangle
ijk)),$

where

$U_{ijk}=\emph{int}(\phi(R_{i}))\cup\emph{int}(\phi(R_{j}))\cup\emph{int}(\phi(R_{k})).$

(e) Suppose $a\in V$ and $\phi(a)=0$. Assume $r>0$ is such that

$\phi(R_{a})\subseteq D_{r}\subseteq\phi(|T|).$

Denote $V_{1}=\\{i\in V:\phi(i)\in D_{r}\\}$ and $T_{1}$ as the subcomplex
generated by $V_{1}$. Then there exists a constant $C=C(\epsilon)>0$ such that
$D_{r/C}\subseteq\phi(|T_{1}|).$

###### Proof.

(a) It is from

$2\pi\geq\sum\limits_{jk:\triangle ijk\in
F}\theta^{i}_{jk}\geq\sum\limits_{jk:\triangle ijk\in
F}\epsilon=\deg(i)\cdot\epsilon.$

(b) This is by the sine law.

(c) This estimate is straightforward from the area formula

$\text{Area}(\phi(\triangle
ijk))=\frac{1}{2}l_{ij}l_{ik}\sin\theta^{i}_{jk}=\frac{1}{2}l_{ij}^{2}\frac{\sin\theta^{j}_{ik}}{\sin\theta^{k}_{ij}}\sin\theta^{i}_{jk}.$

(d) We may normalize $\text{diam}(\phi(\triangle ijk))=1$ and $\phi(i)=0$. By
part (a), there are finitely many possible combinatorial structures of the
natural triangulation of $R_{i}\cup R_{j}\cup R_{k}$. Fixing a combinatorial
structure, $d(U_{ijk}^{c},\phi(\triangle ijk))$ is positive and continuously
determined by $\phi(a)$’s for $a\in N_{i}\cup N_{j}\cup N_{k}$. By the
compactness $d(U_{ijk}^{c},\phi(\triangle ijk))$ has a positive lower bound
$\delta=\delta(\epsilon)$.

(e) Pick $\delta$ as in part (d) and we claim that $C=1+2/\delta$ is a desired
constant. Let us prove by contradiction and assume that there exists $z\in
D_{r/C}\backslash\phi(|T_{1}|)$. Then there exists a triangle $\triangle
ijk\in F$ such that $z\in\triangle ijk$. Then $\triangle ijk$ is not a
triangle in $T_{1}$ and we may assume $i\notin V_{1}$, so $|\phi(i)|\geq r$.
Since $\phi(R_{a})\subseteq D_{r}$, we have $ai\notin E$. So $a\neq i,j,k$,
then $0=\phi(a)\notin U_{ijk}$. Then

$\displaystyle\quad r/C\geq|0-z|\geq d(U_{ijk}^{c},\triangle
ijk)\geq\delta\cdot\text{diam}(\triangle ijk)$
$\displaystyle\geq\delta\cdot|\phi(i)-z|\geq\delta\cdot(r-r/C)=(r/C)\cdot\delta(C-1)=2r/C$

and we get a contradiction. ∎

###  3.2. A discrete Schwarz lemma

Recall that the Schwarz lemma says that any holomorphic map $f:D\rightarrow D$
satisfies that $|f^{\prime}(0)|\leq 1$ if $f(0)=0$. Here we prove a discrete
weaker version of the Schwarz lemma. Let $T$ be a triangulated surface.

######  Proposition 3.2.

Suppose $\phi,\phi^{\prime}$ are geodesic embeddings of $|T|$ into
$\mathbb{C}$ with induced PL metrics $l,l^{\prime}$. Assume both
$l,l^{\prime}$ satisfy the uniformly nondegenerate condition with constant
$\epsilon>0$ and the Delaunay condition. If $r,r^{\prime}>0$ and $T_{0}$ is a
finite subcomplex of $T$ satisfing that

$\phi(|T_{0}|)\subseteq D_{r},\quad
D_{r^{\prime}}\subseteq\phi^{\prime}(|T_{0}|),$

then there exists a constant $M=M(\epsilon)>0$ such that for every $i\in V$
satisfying $\phi^{\prime}(i)\in D_{r^{\prime}/2}$, we have

$u_{i}\geq\log\frac{r^{\prime}}{r}-M.$

###### Proof.

Without loss of generality, we assume $\epsilon\leq\pi/6$. By scaling, we may
assume $r=\frac{1}{4}\sin^{3}\epsilon\leq\frac{1}{4}$ and $r^{\prime}=1$.
Denote

$V_{1}=\\{i\in V:\phi^{\prime}(i)\in D=D_{1}\\}$

and $T_{1}=T(V_{1})$ as the subcomplex generated by $V_{1}$. Then
$\phi,\phi^{\prime}$ map $|T_{1}|$ into $D$. Let $z_{i}=\phi(i)$,
$z_{i}^{\prime}=\phi^{\prime}(i)$. Denote

$u^{h}_{i}=u_{i}+\ln\frac{1-|z_{i}|^{2}}{1-|z_{i}^{\prime}|^{2}}.$

We claim $u_{i}^{h}\geq 0$ for every $i\in V_{1}$. Just let $i$ attain the
minimum of $u^{h}$ in $V_{1}$. Here $\text{int}(V_{1})$ and $\partial V_{1}$
are with respect to $T$, and defined in Section 1.1.
(1) If $i\in\text{int}(V_{1})$ and
$l^{\prime}_{ij}<(1-|z^{\prime}_{i}|^{2})\sin\epsilon$, for every $ij\in E$,
then from Lemma 2.1, $\phi^{\prime}$ induces a hyperbolic geodesic embedding
$\phi^{h\prime}$ from the $1$-ring neighborhood $R_{i}$ of $i_{0}$ into
$\mathbb{D}$. Since $\phi^{\prime}$ is Delaunay, $\phi^{h\prime}$ is also
Delaunay. Since $\phi(|V_{1}|)\subset D_{\sin^{3}\epsilon/4}$, for the same
reason $\phi$ also induces a Delaunay hyperbolic geodesic embedding $\phi^{h}$
from $R_{i}$ into $\mathbb{D}$. Then the hyperbolic maximum principle Lemma
2.9 implies $u_{i}^{h}\geq 0$.
(2) If $i\in\text{int}(V_{1})$ and there exists $j\in V_{1}$, $ij\in E$ such
that $l^{\prime}_{ij}\geq(1-|z^{\prime}_{i}|^{2})\sin\epsilon$, then from
Lemma 3.1 (b),

$e^{(u_{i}-u_{j})/2}=\frac{e^{(u_{i}+u_{k})/2}}{e^{(u_{j}+u_{k})/2}}=\frac{l^{\prime}_{ik}}{l_{ik}}\frac{l_{jk}}{l^{\prime}_{jk}}\geq\sin^{2}\epsilon$

where $\triangle ijk\in F$. So

$\displaystyle\quad
e^{u_{i}^{h}}=e^{u_{i}}\cdot\frac{1-|z_{i}|^{2}}{1-|z_{i}^{\prime}|^{2}}=\frac{l^{\prime}_{ij}}{l_{ij}}\cdot
e^{(u_{i}-u_{j})/2}\cdot\frac{1-|z_{i}|^{2}}{1-|z_{i}^{\prime}|^{2}}$
$\displaystyle\geq\frac{l^{\prime}_{ij}}{l_{ij}}\cdot\sin^{2}\epsilon\cdot\frac{1-|z_{i}|^{2}}{1-|z_{i}^{\prime}|^{2}}\geq\sin^{3}\epsilon\cdot\frac{1-|z_{i}|^{2}}{l_{ij}}\geq\sin^{3}\epsilon\cdot\frac{1/2}{2r}=1.$

(3) If $i\in\partial V_{1}$, then there exists $j\in V$ such that
$\phi^{\prime}(j)\notin D$. Then $l^{\prime}_{ij}\geq 1-|z^{\prime}_{i}|.$
Since we assume $\epsilon\leq\frac{\pi}{6}$, then
$l^{\prime}_{ij}\geq(1-|z^{\prime}_{i}|^{2})\sin\epsilon$. As the estimates
above, we also have $e^{u_{i}^{h}}\geq 1.$

So $u^{h}_{i}\geq 0$ for every $i\in V_{1}$. Then for $i\in V$ satisfying
$\phi^{\prime}(i)\in D_{1/2}$, set $M=-\ln\frac{\sin^{3}\epsilon}{8}$, and we
have

$u_{i}=u_{i}^{h}-\ln\frac{1-|z_{i}|^{2}}{1-|z_{i}^{\prime}|^{2}}\geq-\ln\frac{1-|z_{i}|^{2}}{1-|z_{i}^{\prime}|^{2}}\geq\ln(1-|z_{i}^{\prime}|^{2})\geq\ln\frac{1}{2}=\ln\frac{r^{\prime}}{r}-M.$

∎

###  3.3. Proof of the boundedness of the conformal factor

In this section, we prove that the discrete conformal factor $u$ is bounded,
i.e. Proposition 1.5.

###### Proof of Proposition 1.5.

Without loss of generality, we may assume that $\phi^{\prime}\circ\phi^{-1}$
is linear on each triangle $\phi(\triangle ijk)$. Then
$\phi^{\prime}\circ\phi^{-1}$ is $K$-quasiconformal for some constant
$K=K(\epsilon)>0$. We will prove the boundedness of $u$ by showing that for
every $j_{1},j_{2}\in V$,

$|u_{j_{1}}-u_{j_{2}}|\leq 2M+2\ln C+\ln C^{\prime}-\ln 2,$

where $M=M(\epsilon)$ is the constant given in Proposition 3.2 and
$C=C(\epsilon)$ is the constant given in Lemma 3.1 (e) and
$C^{\prime}=C^{\prime}(\epsilon)=e^{200\pi K}$.

Assume $j_{1},j_{2}\in V$. For convenience, let us assume
$\phi(j_{1})=\phi^{\prime}(j_{1})=0$ by translations. Pick $r>0$ sufficiently
large such that $|\phi(j_{2})|<r/(2C)$ and $\phi(R_{j_{1}})\subseteq D_{r}$.
Let

$V_{1}=\\{i\in V:\phi(i)\in D_{r}\\}\text{ and }V_{2}=\\{i\in V:\phi(i)\in
D_{CC^{\prime}r}\\}.$

Denote $T_{1},T_{2}$ as the subcomplexes generated by $V_{1},V_{2}$
respectively. Then by Lemma 3.1 (e) we have

(3.1) $\\{\phi(j_{1}),\phi(j_{2})\\}\subseteq D_{r/(2C)}\subseteq
D_{r/C}\subseteq\phi(|T_{1}|)\subseteq D_{r}.$

And

(3.2) $D_{C^{\prime}r}\subseteq\phi(|T_{2}|)\subseteq D_{CC^{\prime}r}.$

Recall that $A_{r_{1},r_{2}}=\\{z\in\mathbb{C}:r_{1}<|z|<z_{2}\\}$. Then
$A=A_{r,C^{\prime}r}$ separates $\phi(|T_{1}|)$ and $\phi(|T_{2}|)^{c}$, and
then $A^{\prime}=\phi^{\prime}\circ\phi^{-1}(A)$ separates
$\phi^{\prime}(T_{1})$ and $\phi^{\prime}(T_{2})^{c}$. Furthermore

$\text{Mod}(A^{\prime})\geq\frac{1}{K}\cdot\text{Mod}(A)=\frac{1}{K}\cdot\frac{1}{2\pi}\ln\frac{C^{\prime}r}{r}=100.$

Then by Lemma 2.12 there exists $r^{\prime}>0$ such that
$A_{r^{\prime},2r^{\prime}}\subseteq A^{\prime}$. So
$A_{r^{\prime},2r^{\prime}}$ separates $\phi^{\prime}(T_{1})$ and
$\phi^{\prime}(T_{2})^{c}$ and then

(3.3) $\phi^{\prime}(|T_{1}|)\subseteq D_{r^{\prime}}$

and

(3.4) $\\{\phi^{\prime}(j_{1}),\phi^{\prime}(j_{2})\\}\subseteq
D_{r^{\prime}}\subseteq D_{2r^{\prime}}\subseteq\phi^{\prime}(|T_{2}|).$

By Proposition 3.2, Equation (3.2) and Equation (3.4), both
$u_{j_{1}},u_{j_{2}}$ are at least

$\ln\frac{2r^{\prime}}{CC^{\prime}r}-M=\ln\frac{r^{\prime}}{r}+\ln\frac{2}{CC^{\prime}}-M.$

Again by Proposition 3.2, Equation (3.3) and Equation (3.1), both $-u_{j_{1}}$
and $-u_{j_{2}}$ are at least

$\ln\frac{r/C}{r^{\prime}}-M=\ln\frac{r}{r^{\prime}}-\ln C-M.$

So both $u_{j_{1}}$ and $u_{j_{2}}$ are in the interval

$[\ln\frac{r^{\prime}}{r}+\ln\frac{2}{CC^{\prime}}-M,\ln\frac{r^{\prime}}{r}+\ln
C+M],$

and $|u_{j_{1}}-u_{j_{2}}|$ is bounded by the length of this interval

$2M+\ln C-\ln\frac{2}{CC^{\prime}}=2M+2\ln C+\ln C^{\prime}-\ln 2.$

∎

###  3.4. A discrete Liouville theorem

In this section, we prove the following discrete Liouville theorem.

######  Proposition 3.3.

Suppose $\phi:|T|\rightarrow\mathbb{C}$ is a geodesic homeomorphism, and $l$
satisfies the uniformly nondegenerate condition and the Delaunay condition.
Given the weight $\mu\in\mathbb{R}_{\geq 0}^{E}$ defined as in equation (2.6),
then any bounded function $u$ on $V$ is constant if $u$ is harmonic at every
point of $V$.

###### Proof.

By Lemma 2.14, it suffices to show $(V,E,\mu)$ is recurrent. Let us assume $l$
is uniformly nondegerate with constant $\epsilon>0$. Then for all $i\in V$,

$\sum_{j:ij\in
E}\mu_{ij}\leq\deg(i)\cdot\cot\epsilon\leq\frac{2\pi\cot\epsilon}{\epsilon}.$

Then by Lemma 2.15, it suffices to show $(V,E)$ is VEL-parabolic and
$(V,E,\mu)$ is connected.

First we show that if two subsets $V_{1},V_{2}$ are separated by an annulus,
then the vertex extremal length has a lower bound.

######  Lemma 3.4.

Let $V_{1},V_{2}$ be two nonempty subsets of $V$ such that
$\phi(V_{1})\subseteq D_{r_{1}}$, $\phi(V_{2})\subseteq D^{c}_{r_{2}}$.
Suppose for every $i\in V_{1}$, $\phi(|R_{i}|)\subseteq D_{r_{2}}$. Then there
is a constant $C=C(\epsilon)>0$ such that

$\text{VEL}(V_{1},V_{2})\geq C\big{(}1-(\frac{r_{1}}{r_{2}})^{2}\big{)}.$

In particular, if $r_{2}\geq 2r_{1}$, there is a constant $C=C(\epsilon)>0$
such that $\text{VEL}(V_{1},V_{2})\geq C.$

###### Proof.

Denote $d_{M}(i)=\max\limits_{j:ji\in E}d(\phi(i),\phi(j))$. Consider the
vertex metric $\eta$ as follows. For $i\in V$, if $\phi(i)\in D_{r_{2}}^{c}$,
then define $\eta(i)=0$; if $\phi(i)\in D_{r_{2}}$ and there exists $j\in V$,
$ji\in E$ such that $\phi(j)\in D_{r_{2}}$, then define
$\eta(i)=\frac{d_{M}(i)}{r_{2}-r_{1}}$; if $\phi(i)\in D_{r_{2}}$ and there is
no $j\in V$, $ji\in E$ such that $\phi(j)\in D_{r_{2}}$, then define
$\eta(v)=0$.

First we check $\eta$ is $\Gamma(V_{1},V_{2})$-admissible. Let
$\gamma=\\{i_{0},...,i_{k}\\}$ be a path joining $V_{1}$ and $V_{2}$ such that
$i_{0},\cdots,i_{k-1}\in D_{r_{2}}$ and $i_{k}\in D_{r_{2}}^{c}$. Then from
the assumptions

$\displaystyle\sum\limits_{s=0}^{k-1}\eta(i_{s})=\frac{1}{r_{2}-r_{1}}\sum\limits_{s=0}^{k-1}d_{M}(i_{s})$
$\displaystyle\geq\frac{1}{r_{2}-r_{1}}\sum\limits_{s=0}^{k-1}d(\phi(i_{s}),\phi(i_{s+1}))\geq\frac{1}{r_{2}-r_{1}}d(\phi(i_{0}),\phi(i_{k+1})).$

Since $\phi(i_{0})\in V_{1}\subseteq D_{r_{1}}$ and $\phi(i_{k+1})\in
D_{r_{2}}^{c}$, we obtain
$\sum\limits_{i\in\gamma}\eta(i)\geq\frac{1}{r_{2}-r_{1}}(r_{2}-r_{1})=1.$

Next we estimate an upper bound of $\sum\limits_{i\in V}\eta^{2}(i)$. We only
need to consider the vertices where $\eta$ are nonzero. For $i\in V$,
$\eta(i)\neq 0$, since $\phi(i)\in D_{r_{2}}$ and there is $j\in V$ such that
$\phi(j)\in D_{r_{2}}$ and $ji\in E$, we have $d(\phi(i),\phi(j))\leq 2r_{2}$.
Then from Lemma 3.1 (a) and (b), there is a constant $C_{1}=C_{1}(\epsilon)>0$
such that $d_{M}(i)\leq C_{1}r_{2}$. So for every $i\in V$, $\eta(i)\neq 0$,
we have $\phi(R_{i})\subset D_{(1+C_{1})r_{2}}.$ Then from Lemma 3.1 (a) and
(b) and (c), there is a constant $C_{2}=C_{2}(\epsilon)>0$ such that

$\sum\limits_{i\in V}\eta^{2}(i)=\sum\limits_{\eta(i)\neq
0}\frac{d_{M}^{2}(i)}{(r_{2}-r_{1})^{2}}\leq\sum\limits_{\eta(i)\neq
0}\frac{C_{2}\text{Area}(\phi(R_{i}))}{(r_{2}-r_{1})^{2}}.$

Since every triangle is calculated at most three times in $\phi(|R_{i}|)$,
$i\in V$, we obtain

$\sum\limits_{i\in
V}\eta^{2}(i)\leq\frac{3C_{2}\text{Area}(D_{(1+C_{1})r_{2}})}{(r_{2}-r_{1})^{2}}=\frac{1}{C}\frac{r_{2}^{2}}{(r_{2}-r_{1})^{2}}.$

So
$\text{Mod}(V_{1},V_{2})\leq\frac{1}{C}\frac{r_{2}^{2}}{(r_{2}-r_{1})^{2}}.$
We finish the proof. ∎

The vertex extremal length has a property of additivity, which is from Lemma
5.1 in [He99].

######  Lemma 3.5 (Lemma 5.1 in [He99]).

Let $V_{1},V_{2},\cdots,V_{2m}$ be mutually disjoint, nonvoid subsets of
vertices such that for $i_{1}<i_{2}<i_{3}$, $V_{i_{2}}$ separates $V_{i_{1}}$
from $V_{i_{3}}$, i.e. every path joining $V_{i_{1}}$ and $V_{i_{3}}$ must
pass through a vertex in $V_{i_{2}}$. We allow $V_{2m}=\infty.$ Then we have

$\text{VEL}(V_{1},V_{2m})\geq\sum\limits_{i=1}^{m}\text{VEL}(V_{2k-1},V_{2k}).$

Then combining Lemma 3.4 and Lemma 3.5, we can show the following lemma.

######  Lemma 3.6.

$(V,E)$ is VEL-parabolic.

###### Proof.

We construct infinitely many $V_{k}$ by induction starting from a finite
subset $V_{1}$ of $V$. Given a finite subset $V_{k}$ of $V$, choose $r$ large
enough such that $\bigcup\limits_{i\in V_{k}}\phi(R_{i})\subseteq D_{r}$.
Denote $\tilde{V}=\\{i\in V:\phi(i)\in D_{2r}\\}$. Consider
$V^{\prime}=\bigcup\limits_{i\in\tilde{V}}R_{i}$,
$V^{\prime\prime}=\bigcup\limits_{i\in V^{\prime}}R_{i}.$ Choose $r^{\prime}$
such that $\phi(V^{\prime\prime})\subseteq D_{r^{\prime}}$. Since the interior
of $\phi(R_{i})$, $i\in V$ cover the plane, we can choose a finite subset
$V^{\prime}_{k+1}$ such that $\partial
D_{r^{\prime}}\subseteq\bigcup\limits_{i\in
V^{\prime}_{k+1}}\text{int}(\phi(R_{i}))$, and we may assume $\partial
D_{r^{\prime}}\cap\text{int}(\phi(R_{i}))\neq\emptyset$. Then set

$V_{k+1}=\\{i\in V:\text{ there exists }j\in V^{\prime}_{k+1}\text{ such that
}ji\in E\\}.$

We claim $\phi(V_{k+1})\subseteq D^{c}_{2r}$. If not, then there exists $i\in
V^{\prime}_{k+1}$ such that $i\in V^{\prime}$. Then $\phi(R_{i})\subseteq
D_{r^{\prime}}$, which does not intersect with $\partial D_{r^{\prime}}$.
Contradiction.

As construction above, we obtain that for $i\in V_{k}$, $\phi(R_{i})\subseteq
D_{r}$ and $\phi(V_{k+1})\subseteq D^{c}_{2r}$. Since $\partial
D_{r^{\prime}}\subseteq\bigcup\limits_{i\in V^{\prime}_{k+1}}\phi(R_{i})$, it
is clearly $V_{k+1}$ separates $V_{k}$ and $V_{k+2}$. So the conditions in
Lemma 3.4 and Lemma 3.5 hold. ∎

It remains to show that $G_{\mu}=(V,E,\mu)$ is connected, i.e.,
$G^{\prime}_{\mu}$ is connected. We notice that if $\mu_{ij}=0$ then the
points $\phi(i),\phi(k_{1}),\phi(j),\phi(k_{2})$ are co-circle. So to obtain
$G^{\prime}_{\mu}$, we just delete, for finitely many times, the interior
edges of a polygon, whose vertices are co-circle. So $G^{\prime}_{\mu}$ is
connected. ∎

###  3.5. Proof of that the conformal factor is constant

In this section, we show that $u$ is constant, i.e. Proposition 1.6, then
finish the whole proof of main Theorem 1.3.

###### Proof of Proposition 1.6.

In the proof, we will construct a deformation of discrete conformal factors
$u(t)$. To avoid being confused with $u$ and $u(t)$, we use the notation
$\bar{u}$ instead of $u$ in the statement of Proposition 1.6. In other words,
we assume $l^{\prime}=\bar{u}*l$ and want to prove that $\bar{u}$ is bounded.

Let us prove by contradiction and assume that $\bar{u}$ is not constant.
Without loss of generality, we can do a scaling and assume

$\inf_{i\in V}\bar{u}_{i}<0<\sup_{i\in V}\bar{u}_{i}$

and

$-\inf_{i\in V}\bar{u}_{i}=\sup_{i\in V}\bar{u}_{i}=|\bar{u}|_{\infty}.$

By a standard compactness arguement, there exists a constant
$\delta_{0}=\delta_{0}(\epsilon)$, such that for every function
$u:V\rightarrow\mathbb{R}$ satisfying $|u|_{\infty}<2\delta_{0}$, $u*l$
satisfies the triangle inequalities and is uniformly nondegenerate and
uniformly Delaunay.

Denote $\delta=\min\\{\delta_{0},|\bar{u}|_{\infty}\\}$. Pick a sequence of
increasing finite subsets $V_{n}$ of $V$ such that
$\bigcup\limits_{n=1}^{\infty}V_{n}=V$. We use the notation in Section 2.2.
For each $n\in\mathbb{N}$, we will construct a smooth
$\mathbb{R}^{V_{n}}$-valued function $u^{(n)}(t)=[u_{i}^{(n)}(t)]_{i\in
V_{n}}$ on $(-2\delta,2\delta)$ such that

  1. (a)

$u^{(n)}(0)=0$, and

  2. (b)

$\dot{u}_{i}^{(n)}(t)=\bar{u}_{i}/|\bar{u}|_{\infty}$ if $i\in\partial V_{n}$,
and

  3. (c)

if $i\in\text{int}(V_{n})$ then

(3.5) $\sum_{j:ij\in
E}\mu_{ij}(u^{(n)}(t))(\dot{u}_{j}^{(n)}(t)-\dot{u}_{i}^{(n)}(t))=0$

where $\mu_{ij}(u)$ is defined for all $ij\in E(V_{n})$ as in Equation (2.6).

The conditions (b) and (c) give an autonomous ODE system on

$\mathcal{U}_{n}=\\{u\in\mathbb{R}^{V_{n}}:|u|_{\infty}<2\delta\\}.$

Notice that $\mu_{ij}(u)>0$ if $u\in\mathcal{U}_{n}$. Then by Lemma 2.6,
$\dot{u}^{(n)}(t)$ is smoothly determined by $u^{(n)}(t)$ on
$\mathcal{U}_{n}$. Given the initial condition $u^{(n)}(0)=0$, assume the
maximum existence interval for this ODE system on $\mathcal{U}_{n}$ is
$(t_{\min},t_{\max})$ where $t_{\min}\in[-\infty,0)$ and
$t_{\max}\in(0,\infty]$. By the maximum principle Lemma 2.5, for all $i\in
V_{n}$

$|\dot{u}^{(n)}|_{\infty}\leq\max_{j\in\partial
V_{n}}|\dot{u}_{j}^{(n)}|=\max_{j\in\partial
V_{n}}|\bar{u}_{j}|/|\bar{u}|_{\infty}\leq 1.$

So $|u^{(n)}(t)|_{\infty}\leq t\leq t_{\max}$ for all $t\in[0,t_{\max})$. By
the maximality of $t_{\max}$, $t_{\max}\geq 2\delta$ and for a similar reason
$t_{\min}\leq-2\delta$. So $u^{(n)}(t)$ is indeed well-defined on
$(-2\delta,2\delta)$. By Equation (2.5) and Equation (3.5),
$K_{i}(u^{(n)}(t))=0$ for all $i\in\text{int}(V_{n})$. Then by Lemma 2.5, for
all $i\in V_{n}$

$\displaystyle\quad|\bar{u}_{i}-u_{i}^{(n)}(\delta)|\leq\max_{j\in\partial
V_{n}}|\bar{u}_{j}-u_{j}^{(n)}(\delta)|$ (3.6)
$\displaystyle=\max_{j\in\partial
V_{n}}\left(\bar{u}_{j}-\delta\cdot\frac{\bar{u}_{j}}{|\bar{u}|_{\infty}}\right)\leq(1-\frac{\delta}{|\bar{u}|_{\infty}})|\bar{u}|_{\infty}=|\bar{u}|_{\infty}-\delta.$

By picking a subsequence, we may assume that $u^{(n)}_{i}$ converge to
$u_{i}^{*}$ on $[0,\delta]$ uniformly for all $i\in V$. Then
$u^{*}=[u_{i}^{*}]_{i\in V}$ satisfies the following properties.

(a) $u^{*}_{i}(t)$ is $1$-Lipschitz for all $i\in V$. As a consequence, for
all $i\in V$, $u_{i}^{*}(t)$ is differentiable at a.e. $t\in[0,\delta]$.

(b) $u^{*}(t)*l$ is uniformly nondegenerate and uniformly Delaunay for all
$t\in[0,\delta]$.

(c) For all $i\in V$, $K_{i}(u^{*}(t))=0$. As a consequence for a.e.
$t\in[0,\delta]$,

$0=\frac{d}{dt}K_{i}(u^{*}(t))=\sum_{j:ij\in
E}\mu_{ij}(u^{*}(t))(\dot{u}^{*}_{j}(t)-\dot{u}^{*}_{i}(t)),$

for all $i\in V$.

(d) By Proposition 3.3, $\dot{u}^{*}(t)$ is constant on $V$ for a.e.
$t\in[0,\delta]$. As a consequence $u_{i}^{*}(\delta)$ equals to a constant
$c$ independent on $i\in V$.

(f) By Equation (3.5),

$|\bar{u}_{i}-c|=|\bar{u}_{i}-u^{*}_{i}(\delta)|\leq|\bar{u}|_{\infty}-\delta$

for all $i\in V$. As a consequence we get the following contradiction

$2|\bar{u}|_{\infty}=|\sup_{i\in V}\bar{u}_{i}-\inf_{i\in
V}\bar{u}_{i}|\leq|\sup_{i\in V}\bar{u}_{i}-c|+|\inf_{i\in
V}\bar{u}_{i}-c|\leq 2|\bar{u}|_{\infty}-2\delta.$

∎

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