The Convergence of Prescribed Combinatorial Ricci Flows for Total Geodesic Curvatures in Spherical Background Geometry

Guangming Hu, Ziping Lei,Yu Sun and Puchun Zhou

Abstract

In this paper, we study the existence and rigidity of (degenerated) circle pattern metric with prescribed total geodesic curvatures in spherical background geometry. To find the (degenerated) circle pattern metric with prescribed total geodesic curvatures, we define some prescribed combinatorial Ricci flows and study the convergence of flows for (degenerated) circle pattern metrics. We solve the prescribed total geodesic curvature problem and provide two methods to find the degenerated circle pattern metric with prescribed total geodesic curvatures. As far as we know, this is the first degenerated result for total geodesic curvatures in spherical background geometry.

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

1 Introduction

1.1 Background

In differential geometry, Hamilton [5] introduced the Ricci flow defined by some equations $\frac{dg_{ij}}{dt}=-2Kg_{ij}$ , where $g_{ij}$ is the Riemannian metric and $K$ is the Gaussian curvature. This is an important tool to solve the Poincaré conjecture [7, 8, 9] and Thurston’s geometrization conjecture [1]. Moreover, the Ricci flow is used to prove the uniformization theorem [2].

In discrete geometry, Chow and Luo [3] constructed the combinatorial Ricci flow which is an analogy of the Ricci flow in smooth case. Given a closed surface $S$ with a triangulation $\mathscr{T}=(V,E,F)$ and a weight $\Phi:E\to[0,\frac{\pi}{2}]$ , then we can define a map $\mathbf{r}=(r_{1},\cdots,r_{|V|}):V\to(0,+\infty)$ , the $\mathbf{r}$ is called a metric on the weighted triangulation $(\mathscr{T},\Phi)$ . Using the metric $\mathbf{r}$ , for any edge $e_{ij}\in E$ joining $v_{i}$ and $v_{j}$ , we can define the length $l_{ij}$ of $e_{ij}$ in spherical (Euclidean, hyperbolic) background geometry, i.e.

l_{ij}=\arccos(\cos r_{i}\cos r_{j}-\sin r_{i}\sin r_{j}\cos(\Phi(e_{ij})))\leavevmode\nobreak\ (\mathbb{S}^{2}),

l_{ij}=\sqrt{r_{i}^{2}+r_{j}^{2}+2r_{i}r_{j}\cos(\Phi(e_{ij}))}\leavevmode\nobreak\ (\mathbb{R}^{2})

and

l_{ij}=\operatorname{arccosh}(\cosh r_{i}\cosh r_{j}+\sinh r_{i}\sinh r_{j}\cos(\Phi(e_{ij})))\leavevmode\nobreak\ (\mathbb{H}^{2}).

Then there exists a local spherical (Euclidean, hyperbolic) metric on each triangle $\triangle_{ijk}\in F$ whose vertices are $v_{i}$ , $v_{j}$ and $v_{k}$ . Gluing these spherical (Euclidean, hyperbolic) triangles along their sides together, then we obtain a spherical (Euclidean, hyperbolic) conical metric with singularities at the vertices on the closed surface $S$ . Denote $\vartheta_{i}^{jk}$ as the angle at the vertex $v_{i}$ in the spherical (Euclidean, hyperbolic) triangle $\triangle_{ijk}\in F$ . The combinatorial Gaussian curvature $K_{i}$ at the vertex $v_{i}$ is defined as

K_{i}=2\pi-\sum_{\triangle_{ijk}\in F}\vartheta_{i}^{jk},

where the sum is taken over all triangles with the vertex $v_{i}$ in $F$ . Then Chow and Luo [3] defined the combinatorial Ricci flow in spherical (Euclidean, hyperbolic) background geometry, i.e.

(1.1)

\frac{dr_{i}}{dt}=-K_{i}\sin r_{i}\leavevmode\nobreak\ (\mathbb{S}^{2}),

(1.2)

\frac{dr_{i}}{dt}=-K_{i}r_{i}\leavevmode\nobreak\ (\mathbb{R}^{2})

and

(1.3)

\frac{dr_{i}}{dt}=-K_{i}\sinh r_{i}\leavevmode\nobreak\ (\mathbb{H}^{2}).

Chow and Luo [3] studied the combinatorial Ricci flow and obtained the long time existence of the solution to the combinatorial Ricci flow. Moreover, the solution to the combinatorial Ricci flow will converge exponentially fast to the metric with constant curvature in some cases. They obtained the following two theorems:

Theorem 1.1.

Suppose $(\mathscr{T},\Phi)$ is a weighted triangulation of a closed connected surface $S$ . Given any initial metric based on the weighted triangulation, the solution to the combinatorial Ricci flow (1.2) in the Euclidean background geometry with the given initial value exists for all time and converges if and only if for any proper subset $I\subset V$ ,

(1.4)

2\pi|I|\chi(S)/|V|>-\sum_{(e,v)\in Lk(I)}(\pi-\Phi(e))+2\pi\chi(F_{I}),

where $F_{I}$ is the subcomplex of $\mathscr{T}$ consisting of all simplex whose vertices are contained in $I$ , $(e,v)\in\operatorname{Lk}(I)$ is a triangle in $F$ with a vertex $v$ in $(e,v)$ belongs to $I$ and neither of the endpoints of the edge $e$ in $(e,v)$ opposite $v$ belongs to $I$ and $\operatorname{Lk}(I)$ is the set of all such triangles.

Furthermore, if the solution converges, then it converges exponentially fast to the metric with constant curvature.

Theorem 1.2.

Suppose $(\mathscr{T},\Phi)$ is a weighted triangulation of a closed connected surface $S$ of negative Euler characteristic. Given any initial metric, the solution to (1.3) in the hyperbolic background geometry with the given initial value exists for all time and converges if and only if the following two conditions hold.

1.

For any three edges $e_{1},e_{2},e_{3}$ forming a null homotopic loop in $S$ , if $\sum_{i=1}^{3}\Phi\left(e_{i}\right)\geq\pi$ , then these three edges form the boundary of a triangle in $F$ .
2.

For any four edges $e_{1},e_{2},e_{3},e_{4}$ forming a null homotopic loop in $S$ , if $\sum_{i=1}^{4}\Phi\left(e_{i}\right)\geq$ $2\pi$ , then these $e_{i}$ ’s form the boundary of the union of two adjacent triangles in $F$ .

Furthermore, if it converges, then it converges exponentially fast to a hyperbolic metric on $S$ so that all vertex angles are $2\pi$ .

Besides, Chow and Luo [3] posed some questions and one of them is whether the limit $\lim_{t\to+\infty}r_{i}(t)$ of the solution $r_{i}(t)$ to combinatorial Ricci flow always exists in the extended set $[0,+\infty]$ when the combinatorial conditions (1.4), 1 and 2 are not valid.

Takatsu [10] studied the question and solved it, based on an infinitesimal description of degenerated circle pattern metrics. She obtained the following theorem:

Theorem 1.3.

Let $(\mathscr{T},\Phi)$ be a weighted triangulation of a surface $S$ of nonpositive Euler characteristic such that $\phi(I)\leq 0$ holds for any subset $I\subset V$ and $Z_{T}:=\{z\in V\mid$ there exists a proper subset $Z$ of $V$ such that $z\in Z$ and $\phi(Z)=0\}$ is nonempty, where

\phi(I):=-\sum_{(e,v)\in Lk(I)}(\pi-\Phi(e))+2\pi\chi\left(F_{I}\right).

Then for any metric $\mathbf{r}$ on $(\mathscr{T},\Phi)$ , the solution $\{\mathbf{r}(t)\}_{t\geq 0}$ to combinatorial Ricci flow with initial data $\mathbf{r}$ does not converge on $\mathbb{R}_{>0}^{|V|}$ at infinity. However,

\lim_{t\rightarrow+\infty}K_{i}(\mathbf{r}(t))=0

holds for any vertex $v_{i}$ .

On the one hand, for $\chi(S)=0$ , the solution $\{\mathbf{r}(t)\}_{t\geq 0}$ to combinatorial Ricci flow does not converge on $\mathbb{R}_{\geq 0}^{|V|}$ at infinity. However, if we fix an arbitrary $v\in V\backslash Z_{T}$ , then the limit

\rho_{u}:=\lim_{t\rightarrow+\infty}\frac{r_{u}(t)}{r_{v}(t)}

exists for any $u\in V$ , where $Z_{T}=\left\{z\in V\mid\rho_{z}=0\right\}$ holds and $\left(\rho_{u}\right)_{u\in V\backslash Z_{T}}$ is a unique circle pattern metric with normalization $\rho_{v}=1$ on a certain weighted triangulation with vertices $V\backslash Z_{T}$ .

On the other hand, for $\chi(S)<0$ , the solution $\{\mathbf{r}(t)\}_{t\geq 0}$ to combinatorial Ricci flow converges on $\mathbb{R}_{\geq 0}^{|V|}$ at infinity, where we have $Z_{T}=\left\{z\in V\mid\lim_{t\rightarrow\infty}r_{z}(t)=0\right\}$ holds and the limit of $\left(r_{v}(t)\right)_{v\in V\backslash Z_{T}}$ at infinity is a unique circle pattern metric on a certain weighted triangulation with vertices $V\backslash Z_{T}$ .

Recently, Nie [6] defined a new combinatorial scalar curvature, i.e. the total geodesic curvature and constructed a convex functional with total geodesic curvature. He obtained the rigidity of circle patterns in spherical background geometry. Followed his work, the last author of this paper and his collaborators [4] defined a combinatorial curvature flow which is an analogy of the prescribed combinatorial Ricci flow. They obtained an algorithm to find the desired ideal circle pattern.

Motivated by [3, 10], we obtain some results for total geodesic curvature in spherical background geometry. In this paper, we will define the prescribed combinatorial Ricci flow for total geodesic curvature in spherical background geometry and study the convergence of the solution to prescribed combinatorial Ricci flow.

1.2 Main results

1.2.1 Spherical conical metrics on surfaces

Given a closed topological surface $S$ and a cellular decomposition $\Sigma=(V,E,F)$ of $S$ . Denote $V,E$ and $F$ as set of $0$ -cells, $1$ -cells and $2$ -cells, respectively. By $|E|$ we denote the number of 1-cells and by $|F|$ we denote the number of 2-cells. We can define a function $\mathbf{r}$ on $V$ and a function $\Phi$ on $E$ , i.e. $\mathbf{r}=(r_{1},\cdots,r_{|F|}):V\to(0,\frac{\pi}{2}]$ and $\Phi:E\to(0,\frac{\pi}{2})$ . Besides, the function $\Phi$ is called a weight.

For any 2-cell $f\in F$ , we choose a auxiliary point $p_{f}\in f$ and add an edge between each vertex on $\partial f$ and $p$ . Then we obtain a triangulation of $S$ as shown in Figure 1 .

Refer to caption — Figure 1: The cellular decomposition and triangulation of $S$

For any 1-cell $e\in E$ , there exists a quadrilateral $Q(e)$ , where the vertices of $Q(e)$ are the end points of $e$ and auxiliary points of the 2-cells on the two sides of $e$ . Denote the two auxiliary points as $p,{p}^{\prime}$ and $f(p),f({p}^{\prime})\in F$ denotes the 2-cells containing the points $p,{p}^{\prime}$ , respectively.

Given the weight $\Phi\in(0,\frac{\pi}{2})^{|E|}$ , for any radii $\mathbf{r}\in(0,\frac{\pi}{2}]^{|F|}$ , by Lemma 3.1, there exists a local spherical metric on the quadrilateral $Q(e)$ such that $Q(e)$ is the corresponding spherical quadrilateral of two spherical disks, where the radii of the disks are $r_{f(p)}$ and $r_{f({p}^{\prime})}$ and their intersection angle is $\Phi(e)$ as shown in Figure 2.

Gluing these spherical quadrilaterals along their sides together, then we obtain a spherical conical metric on the closed topological surface $S$ . Denote the spherical conical metric as $\mu$ , we obtain a spherical conical surface $(S,\mu)$ .

By definition, these spherical disks form a circle pattern $\mathcal{P}$ on $(S,\mu)$ which realizes the weight $\Phi$ . By $(\mu,\mathcal{P})$ we denote the spherical conical metric $\mu$ with circle pattern $\mathcal{P}$ . Besides, if the radii $\mathbf{r}\in(0,\frac{\pi}{2})^{|F|}$ , then the spherical conical metric $\mu$ is called a circle pattern metric. If there exists some radii equal to $\frac{\pi}{2}$ , then the spherical conical metric $\mu$ is called a degenerated circle pattern metric.

Remark 1.4.

Given a closed topological surface $S$ with a cellular decomposition $\Sigma$ , by the above argument, for the weight $\Phi\in(0,\frac{\pi}{2})^{|E|}$ and the radii $\mathbf{r}\in(0,\frac{\pi}{2}]^{|E|}$ , there exists the corresponding circle pattern $\mathcal{P}$ which realizes the weight $\Phi$ and the radii $\mathbf{r}$ .

For a disk $D\subset\mathbb{S}^{2}$ , suppose that the radius of $D$ is $r\in(0,\frac{\pi}{2}]$ . Then the $\partial D$ has constant geodesic curvature $k(\partial D)=\cot r$ . Hence we can define a map $\iota$ , i.e.

\begin{array}[]{cccc}\iota:(0,\frac{\pi}{2}]^{|F|}&\longrightarrow&\mathbb{R}^{|F|}_{\geq 0}\\ \mathbf{r}=(r_{1},\cdots,r_{|F|})^{T}&\longmapsto&k=(\cot r_{1},\cdots,\cot r_{|F|})^{T}\\ \end{array}

The map $\iota$ is bijective.

Besides, for simplicity, in this paper, if the geodesic curvatures $k\in\mathbb{R}_{\geq 0}^{|F|}$ , then $k$ is also called a spherical conical metric. If the geodesic curvatures $k\in\mathbb{R}_{>0}^{|F|}$ , then $k$ is also called a circle pattern metric. If there exists some geodesic curvatures equal to $0$ , then $k$ is also called a degenerated circle pattern metric.

For any prescribed total geodesic curvature $\hat{L}$ , we want to know whether there exists some (degenerated) circle pattern metrics with total geodesic curvature $\hat{L}$ . In other words, we consider the following problem:

Prescribed Total Geodesic Curvature Problem: Is there a (degenerated) circle pattern metric with the prescribed total geodesic curvature $\hat{L}$ ? If it exists, how to find it?

Our first results solve the first part of prescribed total geodesic curvature problem:

Theorem 1.5.

Given a closed topological surface $S$ with a cellular decomposition $\Sigma=(V,E,F)$ and the weight $\Phi\in(0,\frac{\pi}{2})^{|E|}$ . For the prescribed total geodesic curvature $\hat{L}=(\hat{L}_{1},\cdots,\hat{L}_{|F|})^{T}$ on the face set $F$ , there exists a circle pattern metric on $S$ with total geodesic curvature $\hat{L}$ if and only if the prescribed total geodesic curvature $\hat{L}\in\mathcal{L}_{1}$ , where

\mathcal{L}_{1}=\{(L_{1},\cdots,L_{|F|})^{T}\in\mathbb{R}^{|F|}_{>0}\leavevmode\nobreak\ |\leavevmode\nobreak\ \sum_{f\in F^{\prime}}L_{f}<2\sum_{e\in E_{F^{\prime}}}\Phi(e),\forall F^{\prime}\subset F\}.

Moreover, the circle pattern metric is unique up to isometry if it exists.

Remark 1.6.

Theorem 1.5 is first proved in [6] and we provide a concrete proof in this paper.

Theorem 1.7.

Given a closed topological surface $S$ with a cellular decomposition $\Sigma=(V,E,F)$ and the weight $\Phi\in(0,\frac{\pi}{2})^{|E|}$ . For the prescribed total geodesic curvature $\hat{L}=(\hat{L}_{1},\cdots,\hat{L}_{|F|})^{T}$ on the face set $F$ , there exists a degenerated circle pattern metric on $S$ with total geodesic curvature $\hat{L}$ if and only if the prescribed total geodesic curvature $\hat{L}\in\bar{\partial}\mathcal{L}$ , where

\bar{\partial}\mathcal{L}=\bigcup_{1\leq m\leq|F|-1,1\leq i_{1}<\cdots<i_{m}\leq|F|}\mathcal{L}_{i_{1}\cdots i_{m}}\cup\{0\},

\mathcal{L}_{i_{1}\cdots i_{m}}=\left\{\begin{array}[]{l|l}(0,\cdots,L_{i_{1}},\cdots,0,\cdots,L_{i_{m}},0,\cdots,0)^{T}\in\mathbb{R}^{|F|}&\begin{array}[]{l}\sum_{f\in F_{i_{1}\cdots i_{m}}^{\prime}}L_{f}<2\sum_{e\in E_{F_{i_{1}\cdots i_{m}}^{\prime}}}\Phi(e),\\ \forall F_{i_{1}\cdots i_{m}}^{\prime}\subset F_{i_{1}\cdots i_{m}};L_{i_{j}}>0,1\leq j\leq m\end{array}\end{array}\right\}.

Moreover, the degenerated circle pattern metric is unique up to isometry if it exists.

Theorem 1.8.

Given a closed topological surface $S$ with a cellular decomposition $\Sigma=(V,E,F)$ and the weight $\Phi\in(0,\frac{\pi}{2})^{|E|}$ . For the prescribed total geodesic curvature $\hat{L}=(\hat{L}_{1},\cdots,\hat{L}_{|F|})^{T}$ on the face set $F$ , there exists a spherical conical metric on $S$ with total geodesic curvature $\hat{L}$ if and only if the prescribed total geodesic curvature $\hat{L}\in\mathcal{L}$ , where $\mathcal{L}=\mathcal{L}_{1}\cup\bar{\partial}\mathcal{L}$ . Moreover, the spherical conical metric is unique up to isometry if it exists.

1.2.2 Prescribed combinatorial Ricci flows for total geodesic curvatures

Given $\hat{L}=(\hat{L}_{1},\cdots,\hat{L}_{|F|})^{T}\in\mathcal{L}_{1}$ , then we can define the prescribed combinatorial Ricci flows as follows, i.e.

(1.5)

\frac{dk_{i}}{dt}=-(L_{i}-\hat{L}_{i})k_{i},\leavevmode\nobreak\ \leavevmode\nobreak\ \forall i\in F.

For any $k\in\mathbb{R}_{i_{1}\cdots i_{m}}^{|F|}$ , where $1\leq m\leq|F|-1$ , $1\leq i_{1}<\cdots<i_{m}\leq|F|$ and

\mathbb{R}^{|F|}_{i_{1}\cdots i_{m}}=\{(0,\cdots,k_{i_{1}},0,\cdots,k_{i_{m}},\cdots,0)\in\mathbb{R}^{|F|}\leavevmode\nobreak\ |\leavevmode\nobreak\ k_{i_{1}},\cdots,k_{i_{m}}>0\},

then $\tilde{k}=(k_{i_{1}},\cdots,k_{i_{m}})\in\mathbb{R}_{>0}^{m}$ . Given $\hat{L}\in\mathcal{L}_{i_{1}\cdots i_{m}}$ , we construct the prescribed combinatorial Ricci flows as follows, i.e.

(1.6)

\frac{d\tilde{k}_{i}}{dt}=-(L_{i}(\tilde{k})-\hat{L}_{i})\tilde{k}_{i},\leavevmode\nobreak\ \leavevmode\nobreak\ \forall i\in F_{i_{1}\cdots i_{m}},

where $F_{i_{1}\cdots i_{m}}=\{i_{1},\cdots,i_{m}\}$ .

Given the prescribed total geodesic curvature $\hat{L}\in\mathcal{L}_{i_{1}\cdots i_{m}}$ , we can define the prescribed combinatorial Ricci flows as follows, i.e.

(1.7)

\frac{dk_{i}}{dt}=-(L_{i}-\hat{L}_{i})k_{i},\leavevmode\nobreak\ \leavevmode\nobreak\ \forall i\in F_{i_{1}\cdots i_{m}},\leavevmode\nobreak\ \frac{dk_{i}}{dt}=-L_{i}k_{i},\leavevmode\nobreak\ \leavevmode\nobreak\ \forall i\in F\setminus F_{i_{1}\cdots i_{m}}.

The following results solve the second part of prescribed total geodesic curvature problem:

Theorem 1.9.

Given a closed topological surface $S$ with a cellular decomposition $\Sigma=(V,E,F)$ , the weight $\Phi\in(0,\frac{\pi}{2})^{|E|}$ and the prescribed total geodesic curvature $\hat{L}=(\hat{L}_{1},\cdots,\hat{L}_{|F|})^{T}\in\mathcal{L}_{1}$ on the face set $F$ . For any initial geodesic curvature $k(0)\in\mathbb{R}_{>0}^{|F|}$ , the solution of the prescribed combinatorial Ricci flows (1.5) converges to the unique circle pattern metric with the total geodesic curvature $\hat{L}$ up to isometry.

Remark 1.10.

Theorem 1.9 is first proved in [4] and we provide a new proof in this paper.

Theorem 1.11.

Given a closed topological surface $S$ with a cellular decomposition $\Sigma=(V,E,F)$ , the weight $\Phi\in(0,\frac{\pi}{2})^{|E|}$ and the prescribed total geodesic curvature $\hat{L}\in\mathcal{L}_{i_{1}\cdots i_{m}}(1\leq m\leq|F|-1$ , $1\leq i_{1}<\cdots<i_{m}\leq|F|)$ on the face set $F$ . For any initial geodesic curvature $\tilde{k}(0)\in\mathbb{R}_{>0}^{m}$ on the face set $F_{i_{1}\cdots i_{m}}$ and 0 on the face set $F\setminus F_{i_{1}\cdots i_{m}}$ , the solution of the prescribed combinatorial Ricci flows (1.6) converges to the unique degenerated circle pattern metric with the total geodesic curvature $\tilde{L}=(\hat{L}_{i_{1}},\cdots,\hat{L}_{i_{m}})^{T}$ on $F_{i_{1}\cdots i_{m}}$ and 0 on $F\setminus F_{i_{1}\cdots i_{m}}$ up to isometry. Moreover, if the solution converges to the degenerated circle pattern metric $\hat{k}$ , then $\hat{k}\in\mathbb{R}^{|F|}_{i_{1}\cdots i_{m}}$ .

Theorem 1.12.

Given a closed topological surface $S$ with a cellular decomposition $\Sigma=(V,E,F)$ , the weight $\Phi\in(0,\frac{\pi}{2})^{|E|}$ and the prescribed total geodesic curvature $\hat{L}\in\mathcal{L}_{i_{1}\cdots i_{m}}(1\leq m\leq|F|-1$ , $1\leq i_{1}<\cdots<i_{m}\leq|F|)$ on the face set $F$ . For any initial geodesic curvature $k(0)\in\mathbb{R}_{>0}^{|F|}$ , the solution of the prescribed combinatorial Ricci flows (1.7) converges to the unique degenerated circle pattern metric with the total geodesic curvature $\hat{L}$ up to isometry. Moreover, if the solution converges to the degenerated circle pattern metric $\hat{k}$ , then $\hat{k}\in\mathbb{R}^{|F|}_{i_{1}\cdots i_{m}}$ .

Remark 1.13.

By Theorem 1.11 and Theorem 1.12, for finding the degenerated circle pattern metric with the total geodesic curvature $\hat{L}\in\mathcal{L}_{i_{1}\cdots i_{m}}\leavevmode\nobreak\ (1\leq m\leq|F|-1,1\leq i_{1}<\cdots<i_{m}\leq|F|)$ , we have two methods. The first one is by constructing prescribed combinatorial Ricci flows (1.6) and the second one is by constructing prescribed combinatorial Ricci flows (1.7). Then the solutions of the two flows will converge to the same degenerated circle pattern metric with the total geodesic curvature $\hat{L}$ in different dimensions (see figure 3).

Organization This paper is organized as follows. In section 2, we define the circle patterns on spherical surfaces. In section 3, we study the moduli space of spherical conical metrics. In section 4, we construct the potential functions on bigons and define the moduli space of spherical bigons with given angle. In section 5, we prove the existence and rigidity of circle pattern metrics for prescribed total geodesic curvatures. In section 6, we prove the existence and rigidity of degenerated circle pattern metrics for prescribed total geodesic curvatures. In section 7, we prove the existence and rigidity of spherical conical metrics for prescribed total geodesic curvatures. In section 8, we study the convergence of prescribed combinatorial Ricci flows for circle pattern metrics. In section 9, we study the convergence of prescribed combinatorial Ricci flows for degenerated circle pattern metrics.

2 Circle patterns on spherical surfaces

Let $S$ be a topological surface, then $S$ is called a spherical surface if every point $p$ in $S$ has a neighborhood $U$ such that there is a isometry from $U$ onto an open subset of $\mathbb{S}_{\alpha}^{2}$ , where $\mathbb{S}_{\alpha}^{2}$ is a unit sphere with a cone angle $\alpha>0$ . We also often call the spherical surface $S$ a surface with spherical conical metric.

By $D_{\alpha}(o,r)$ we denote the open disk of radius $r$ in $\mathbb{S}_{\alpha}^{2}$ centered at the cone point $o\in\mathbb{S}_{\alpha}^{2}$ . In this paper, we only consider the disk $D_{\alpha}(o,r)$ with radius $r\in(0,\frac{\pi}{2}]$ . By a bigon in a spherical surface $S$ we denote an open set isometric to the intersection of two disks of radii $r_{1},r_{2}\in(0,\frac{\pi}{2}]$ in $\mathbb{S}_{\alpha}^{2}$ not containing each other. Besides, a bigon does not contain any cone point. The angle of a bigon is the interior angle $\phi$ formed by its two sides.

Using similar analysis in proof of Proposition 2 in [6], we have the following lemma:

Lemma 2.1.

Given some disks $D_{i}=D_{\alpha_{i}}\left(o,r_{i}\right)\leavevmode\nobreak\ (1\leq i\leq n)$ as above with $\alpha_{i}>0,r_{i}\in(0,\frac{\pi}{2}]$ . Let $\eta$ be a local isometry from the disjoint union $D_{1}\sqcup\cdots\sqcup D_{n}$ to a closed spherical surface $\Gamma$ such that

1.

the set $\Gamma\setminus\eta(D_{1}\sqcup\cdots\sqcup D_{n})$ only has finitely many points;
2.

$\eta$ is at most 2 to 1;
3.

the $\eta\left(D_{i}\right)\leavevmode\nobreak\ (1\leq i\leq n)$ do not contain each other.

By $\mathscr{B}$ we denote the set $\mathscr{B}:=\{p\in\Gamma\mid\eta^{-1}(p)\leavevmode\nobreak\ \text{has two points}\}$ . Then $\eta^{-1}(\mathscr{B})$ is a disjoint union of bigons which fills up the boundary of the $D_{i}\leavevmode\nobreak\ (1\leq i\leq n)$ .

For more details, we refer [6] to the readers. Then we can define the circle patterns on spherical surfaces in this paper.

Definition 2.2.

A local isometry $\eta:D_{1}\sqcup\cdots\sqcup D_{n}\rightarrow\Gamma$ as in Lemma 2.1 is called a circle pattern on $\Gamma$ and each connected component of the set $\mathscr{B}\subset\Gamma$ is called a bigon of the circle pattern. For simplicity, we also call $D_{i}\leavevmode\nobreak\ (1\leq i\leq n)$ an immersed disk in $\Gamma$ and call the set of immersed disks $\mathcal{P}=\left\{D_{1},\cdots,D_{n}\right\}$ a circle pattern (so that each bigon is either an intersection component of two such disks $D_{i},D_{j}$ or a "self-intersection component" of a single disk $D_{i}$ ).

Let $(G_{\mathcal{P}},\Phi)$ be the weighted graph on $\Gamma$ with the circle pattern $\mathcal{P}$ which satisfies

1.

the vertex set $V_{\mathcal{P}}$ is the complement of $D_{1}\cup\cdots\cup D_{n}$ in $\Gamma$ : $V_{\mathcal{P}}=\Gamma\backslash D_{1}\cup\cdots\cup D_{n}$ ;
2.

the edges in edge set $E_{\mathcal{P}}$ are 1 to 1 correspondence with the bigons: given a bigon $B$ , then the corresponding edge $e\in E_{\mathcal{P}}$ is a curve in $B$ joining the two endpoints of the bigon $B$ ;
3.

each edge $e\in E_{\mathcal{P}}$ is weighted by the angle $\Phi(e)\in(0,\pi)$ of the corresponding bigon;
4.

each face $f\in F_{\mathcal{P}}$ corresponds to a disk $D_{i}\in\mathcal{P}$ : $f$ is contained in $\bar{D}_{i}$ and each vertex on $\partial f$ is in $\partial D_{i}$ .

Remark 2.3.

Given a closed spherical surface $S$ with a circle pattern $\mathcal{P}$ , then the weighted graph $(G_{\mathcal{P}},\Phi)$ gives a cellular decomposition $\Sigma$ of the surface $S$ (see figure 4).

3 The moduli space of spherical conical metrics

Lemma 3.1.

For any $r_{1},r_{2}\in(0,\frac{\pi}{2}]$ and any $\phi\in(0,\frac{\pi}{2})$ , there exists a bigon of angle $\phi$ which is formed by the disk $D_{i}\subset\mathbb{S}^{2}$ , where the $D_{i}$ has radius $r_{i}$ , $i=1,2$ . Moreover, the bigon is unique up to isometry.

Proof.

First consider the case $r_{1}\leq r_{2}$ . Let $D_{i}\subset\mathbb{S}^{2}$ be the disk which has radius $r_{i}$ , $i=1,2$ . We suppose that the distance between the centers of $D_{1}$ and $D_{2}$ is $r_{3}$ , the intersection angle of $D_{1}$ and $D_{2}$ is $\phi$ . By the law of cosines in spherical geometry, we have the following identity, i.e.

\cos r_{3}=\cos r_{1}\cos r_{2}+\sin r_{1}\sin r_{2}\cos(\pi-\phi)=\cos r_{1}\cos r_{2}-\sin r_{1}\sin r_{2}\cos\phi.

For $r_{1},r_{2}\in(0,\frac{\pi}{2}]$ , $\phi\in(0,\frac{\pi}{2})$ , we have

\cos(r_{1}+r_{2})<\cos r_{3}=\cos r_{1}\cos r_{2}-\sin r_{1}\sin r_{2}\cos\phi<\cos(r_{1}+r_{2}).

Since $0\leq r_{1}+r_{2},r_{2}-r_{1}\leq\pi$ , $0<r_{3}<r_{1}+r_{2}\leq\pi$ , we know that the $r_{3}$ is unique and $r_{3}\in(r_{2}-r_{1},r_{1}+r_{2})$ . The $D_{1}\cap D_{2}$ is the bigon of angle $\phi$ which is formed by the disks $D_{1},D_{2}\subset\mathbb{S}^{2}$ as shown in Figure 5. Since $r_{3}$ is unique, the bigon is unique up to isometry.

For the case $r_{1}>r_{2}$ , it is similar to the analysis above. This completes the proof. ∎

Remark 3.2.

For the disks $D_{1},D_{2}\subset\mathbb{S}^{2}$ with radii $r_{1},r_{2}\in(0,\frac{\pi}{2}]$ and the intersection angle $\phi\in(0,\frac{\pi}{2})$ , $\bar{D}_{1}$ do not contain the center of $D_{2}$ and $\bar{D}_{2}$ do not contain the center of $D_{1}$ .

Given a cellular decomposition $\Sigma$ with the weight $\Phi\in(0,\frac{\pi}{2})^{|E|}$ on the closed topological surface $S$ , then we can define the moduli space $\mathcal{T}$ of all spherical conical metrics with circle patterns on $S$ which realize the weighted graph $(\Sigma,\Phi)$ , i.e.

\mathcal{T}:=\left\{\begin{array}[]{l|l}(\mu,\mathcal{P})&\begin{array}[]{l}\mu\text{ is a spherical conical metric on }S;\\ \mathcal{P}\text{ is a circle pattern on }(S,\mu)\text{ such that }G_{\mathcal{P}}\\ \text{ is isotopic to}\leavevmode\nobreak\ \Sigma\leavevmode\nobreak\ \text{and realizes the weight}\leavevmode\nobreak\ \Phi\end{array}\end{array}\right\}/\sim,

where $(\mu_{1},\mathcal{P}_{1})\sim(\mu_{2},\mathcal{P}_{2})$ if and only if there exists an isometry $f:S\to S$ such that $f^{*}\mu_{2}=\mu_{1},f^{*}\mathcal{P}_{2}=\mathcal{P}_{1}$ and $f$ is isotopic to the identity map. Then we can define a map $\kappa$ from $(0,\frac{\pi}{2}]^{|F|}$ to $\mathcal{T}$ . By Lemma 3.1, the map $\kappa$ is well-defined. Besides, we have that $(0,\frac{\pi}{2}]^{|F|}\cong\mathcal{T}$ , i.e.

Proposition 3.3.

Given the weighted graph $(\Sigma,\Phi)$ on $S$ , then the map $\kappa$

\begin{array}[]{cccc}\kappa:(0,\frac{\pi}{2}]^{|F|}&\longrightarrow&\mathcal{T}\\ \mathbf{r}=(r_{1},\cdots,r_{|F|})^{T}&\longmapsto&[(\mu,\mathcal{P})]\\ \end{array}

is bijective.

Proof.

If $\kappa(\mathbf{r}_{1})=\kappa(\mathbf{r}_{2})$ , let $\kappa(\mathbf{r}_{i})=[(\mu_{i},\mathcal{P}_{i})],i=1,2$ . Then we know that $(\mu_{1},\mathcal{P}_{1})\sim(\mu_{2},\mathcal{P}_{2})$ . By definition, the radii of the corresponding disks in circle patterns $\mathcal{P}_{1}$ and $\mathcal{P}_{2}$ are equal, i.e. $\mathbf{r}_{1}=\mathbf{r}_{2}$ . The map $\kappa$ is injective.

For any $[(\mu,\mathcal{P})]\in\mathcal{T}$ , there exists a corresponding radii $\mathbf{r}\in(0,\frac{\pi}{2}]^{|F|}$ . Given the weight $\phi$ , by Lemma 3.1, the local spherical metric generated by $\mathbf{r}$ is isometric to the corresponding local metric of $(\mu,\mathcal{P})$ . Hence the spherical metric generated by $\mathbf{r}$ is isometric to the metric of $(\mu,\mathcal{P})$ , i.e. $\kappa(\mathbf{r})=[(\mu,\mathcal{P})]$ . The map $\kappa$ is surjective. ∎

By Proposition 3.3, we know that $(0,\frac{\pi}{2}]^{|F|}\cong\mathcal{T}\cong\mathbb{R}^{|F|}_{\geq 0}$ .

4 Potential functions on bigons

Given $\phi\in(0,\frac{\pi}{2})$ , we consider two disks $D_{1},D_{2}\subset\mathbb{S}^{2}$ . Suppose that the radius of disk $D_{i}$ is $r_{i}\in(0,\frac{\pi}{2}]$ , $i=1,2$ . Besides, let the intersection angle of $D_{1}$ and $D_{2}$ be $\phi$ . By $B$ we denote the bigon which is formed by the intersection of disks $D_{1}$ and $D_{2}$ . Let $\ell_{i}$ denote the length of the sides of bigon $B$ , $i=1,2$ . By $k_{i}$ we denote the geodesic curvature of $\partial D_{i}$ , i.e. $k_{i}=\cot r_{i}$ , $i=1,2$ . By $L_{i}$ we denote the total geodesic curvature of the sides of bigon $B$ , i.e. $L_{i}=\ell_{i}k_{i}$ , $i=1,2$ (see Figure 6).

Lemma 4.1.

Given $\phi\in(0,\frac{\pi}{2})$ , then the 1-form $\omega_{\phi}=\ell_{1}dk_{1}+\ell_{2}dk_{2}$ is closed on $\mathbb{R}^{2}_{>0}$ , $\mathbb{R}_{>0}\times\{k_{2}=0\}$ and $\{k_{1}=0\}\times\mathbb{R}_{>0}$ .

Proof.

By Lemma 8 in [6], the 1-form $\omega_{\phi}$ is closed on $\mathbb{R}^{2}_{>0}$ . On $\mathbb{R}_{>0}\times\{k_{2}=0\}$ , $\omega_{\phi}=\ell_{1}dk_{1}$ . Since $\ell_{1}$ is a smooth function with respect to $k_{1}$ , $d\omega_{\phi}=0$ , $\omega_{\phi}$ is closed. Similarly, we know that $\omega_{\phi}$ is also closed on $\{k_{1}=0\}\times\mathbb{R}_{>0}$ . ∎

Using change of variables $K_{i}=\ln k_{i}$ , $i=1,2$ , by Lemma 4.1 the 1-form $\omega_{\phi}=L_{1}dK_{1}+L_{2}dK_{2}$ is closed on $\mathbb{R}^{2}$ , $\omega_{\phi}=L_{1}dK_{1}$ is closed on $\mathbb{R}\times\{-\infty\}$ and $\omega_{\phi}=L_{2}dK_{2}$ is closed on $\{-\infty\}\times\mathbb{R}$ . On $\mathbb{R}^{2}$ , we can define a potential function

\Lambda_{\phi}(K_{1},K_{2}):=\int_{0}^{\left(K_{1},K_{2}\right)}\omega_{\phi}.

On $\mathbb{R}\times\{-\infty\}$ , we define a potential function

\Lambda_{\phi}(K_{1}):=\int_{0}^{K_{1}}\omega_{\phi}.

On $\{-\infty\}\times\mathbb{R}$ , we define a potential function

\Lambda_{\phi}(K_{2}):=\int_{0}^{K_{2}}\omega_{\phi}.

These three functions are well-defined. Besides, we have that $\nabla\Lambda_{\phi}(K_{1},K_{2})=(L_{1},L_{2})^{T}$ , $\Lambda^{{}^{\prime}}_{\phi}(K_{1})=L_{1}$ and $\Lambda^{{}^{\prime}}_{\phi}(K_{2})=L_{2}$ .

Lemma 4.2.

We have that $\frac{\partial L_{1}}{\partial K_{1}}>0$ on $\mathbb{R}\times\{-\infty\}$ , $\frac{\partial L_{2}}{\partial K_{2}}>0$ on $\{-\infty\}\times\mathbb{R}$ and the matrix

\left(\begin{array}[]{ll}\frac{\partial L_{1}}{\partial K_{1}}&\frac{\partial L_{1}}{\partial K_{2}}\\ \frac{\partial L_{2}}{\partial K_{1}}&\frac{\partial L_{2}}{\partial K_{2}}\end{array}\right)

is positive definite on $\mathbb{R}^{2}$ .

Proof.

By Lemma 9 in [6], we know that on $\mathbb{R}^{2}$ , the matrix

\left(\begin{array}[]{ll}\frac{\partial L_{1}}{\partial K_{1}}&\frac{\partial L_{1}}{\partial K_{2}}\\ \frac{\partial L_{2}}{\partial K_{1}}&\frac{\partial L_{2}}{\partial K_{2}}\end{array}\right)

is positive definite.

On $\mathbb{R}\times\{-\infty\}$ , since $k_{2}=0$ , $L_{2}=0$ . By Gauss-Bonnet formula, we have

\text{Area}(B)=2\phi-L_{1}.

By Lemma 9 in [6], we know that $\frac{\partial\text{Area}(B)}{\partial K_{1}}<0$ , then $\frac{\partial L_{1}}{\partial K_{1}}>0$ . Similarly, we know that $\frac{\partial L_{2}}{\partial K_{2}}>0$ on $\{-\infty\}\times\mathbb{R}$ . ∎

By Lemma 4.2, $\Lambda_{\phi}(K_{1},K_{2})$ is a strictly convex function on $\mathbb{R}^{2}$ , $\Lambda_{\phi}(K_{1})$ is a strictly convex function on $\mathbb{R}\times\{-\infty\}$ and $\Lambda_{\phi}(K_{2})$ is a strictly convex function on $\{-\infty\}\times\mathbb{R}$ .

We can define the moduli space $\mathcal{B}_{\phi}$ of spherical bigons with given angle $\phi\in(0,\frac{\pi}{2})$ , i.e.

\mathcal{B}_{\phi}:=\left\{\begin{array}[]{l|l}B&\begin{array}[]{l}B\text{ is a spherical bigon which is formed by}\\ \text{the intersection of two disks}\leavevmode\nobreak\ D_{1},D_{2}\subset\mathbb{S}^{2}\leavevmode\nobreak\ \text{with}\\ \text{ given intersection angle}\leavevmode\nobreak\ \phi\in(0,\frac{\pi}{2}),\leavevmode\nobreak\ \text{where}\\ \text{the radii}\leavevmode\nobreak\ r_{1},r_{2}\leavevmode\nobreak\ \text{ of the two disks are in }\leavevmode\nobreak\ (0,\frac{\pi}{2}]\\ \end{array}\end{array}\right\}/\sim,

where $B_{1}\sim B_{2}$ if and only if $B_{1}$ and $B_{2}$ are isomorphic.

Then we can construct a map $\gamma$ from $\mathcal{B}_{\phi}$ to $(0,\frac{\pi}{2}]^{2}$ , i.e.

\begin{aligned} \gamma:\mathcal{B}_{\phi}&\longrightarrow(0,\frac{\pi}{2}]^{2}\\ {[B]}&\longmapsto\left(r_{1},r_{2}\right)\end{aligned},

The map $\gamma$ is well-defined. Besides, by Lemma 3.1, $\gamma$ is a bijective map, $\mathcal{B}_{\phi}\cong(0,\frac{\pi}{2}]^{2}$ .

Then we define some subspaces of the moduli space $\mathcal{B}_{\phi}$ , i.e.

\mathcal{B}^{0}_{\phi}:=\left\{\begin{array}[]{l|l}B&\begin{array}[]{l}B\text{ is a spherical bigon which is formed by}\\ \text{the intersection of two disks}\leavevmode\nobreak\ D_{1},D_{2}\subset\mathbb{S}^{2}\leavevmode\nobreak\ \text{with}\\ \text{ given intersection angle}\leavevmode\nobreak\ \phi\in(0,\frac{\pi}{2}),\leavevmode\nobreak\ \text{where}\\ \text{the radii}\leavevmode\nobreak\ r_{1},r_{2}\leavevmode\nobreak\ \text{ of the two disks are}\leavevmode\nobreak\ \frac{\pi}{2}\\ \end{array}\end{array}\right\}/\sim,

\mathcal{B}^{1}_{\phi}:=\left\{\begin{array}[]{l|l}B&\begin{array}[]{l}B\text{ is a spherical bigon which is formed by}\\ \text{the intersection of two disks }\leavevmode\nobreak\ D_{1},D_{2}\subset\mathbb{S}^{2}\leavevmode\nobreak\ \text{with}\\ \text{ given intersection angle}\leavevmode\nobreak\ \phi\in(0,\frac{\pi}{2}),\leavevmode\nobreak\ \text{where}\\ \text{the radii}\leavevmode\nobreak\ r_{1},r_{2}\leavevmode\nobreak\ \text{ of the two disks satisfy}\\ r_{1}\in(0,\frac{\pi}{2}),r_{2}=\frac{\pi}{2}\\ \end{array}\end{array}\right\}/\sim,

\mathcal{B}^{2}_{\phi}:=\left\{\begin{array}[]{l|l}B&\begin{array}[]{l}B\text{ is a spherical bigon which is formed by}\\ \text{the intersection of two disks }\leavevmode\nobreak\ D_{1},D_{2}\subset\mathbb{S}^{2}\leavevmode\nobreak\ \text{with}\\ \text{ given intersection angle}\leavevmode\nobreak\ \phi\in(0,\frac{\pi}{2}),\leavevmode\nobreak\ \text{where}\\ \text{the radii}\leavevmode\nobreak\ r_{1},r_{2}\leavevmode\nobreak\ \text{ of the two disks satisfy}\\ r_{1}=\frac{\pi}{2},r_{2}\in(0,\frac{\pi}{2})\\ \end{array}\end{array}\right\}/\sim,

\mathcal{B}^{3}_{\phi}:=\left\{\begin{array}[]{l|l}B&\begin{array}[]{l}B\text{ is a spherical bigon which is formed by}\\ \text{the intersection of two disks }\leavevmode\nobreak\ D_{1},D_{2}\subset\mathbb{S}^{2}\leavevmode\nobreak\ \text{with}\\ \text{ given intersection angle}\leavevmode\nobreak\ \phi\in(0,\frac{\pi}{2}),\leavevmode\nobreak\ \text{where}\\ \text{the radii}\leavevmode\nobreak\ r_{1},r_{2}\leavevmode\nobreak\ \text{ of the two disks are in }\leavevmode\nobreak\ (0,\frac{\pi}{2})\\ \end{array}\end{array}\right\}/\sim,

It is easy to know that the subspace $\mathcal{B}^{0}_{\phi},\mathcal{B}^{1}_{\phi},\mathcal{B}^{2}_{\phi}$ and $\mathcal{B}^{3}_{\phi}$ are mutually disjoint. Besides, we have that $\mathcal{B}_{\phi}=\mathcal{B}^{0}_{\phi}\sqcup\mathcal{B}^{1}_{\phi}\sqcup\mathcal{B}^{2}_{\phi}\sqcup\mathcal{B}^{3}_{\phi}.$ Similarly, we have that $\mathcal{B}^{0}_{\phi}\cong\{(\frac{\pi}{2},\frac{\pi}{2})\},\mathcal{B}^{1}_{\phi}\cong(0,\frac{\pi}{2})\times\{\frac{\pi}{2}\},\mathcal{B}^{2}_{\phi}\cong\{\frac{\pi}{2}\}\times(0,\frac{\pi}{2})$ and $\mathcal{B}^{3}_{\phi}\cong(0,\frac{\pi}{2})^{2}$ .

Lemma 4.3.

We have that $\mathbb{R}\times\{-\infty\}\cong\Delta_{1},\{-\infty\}\times\mathbb{R}\cong\Delta_{2}$ , where $\Delta_{1}=\{(x,0)\leavevmode\nobreak\ |\leavevmode\nobreak\ 0<x<2\phi\},\Delta_{2}=\{(0,y)\leavevmode\nobreak\ |\leavevmode\nobreak\ 0<y<2\phi\}$ .

Proof.

On $\mathbb{R}\times\{-\infty\}$ , by Lemma 4.2, it is easy to know that $\Lambda^{{}^{\prime}}_{\phi}(K_{1})$ is a embedding map. For the map $\Lambda^{{}^{\prime}}_{\phi}(K_{1})$ , we have

	$\displaystyle\Lambda^{{}^{\prime}}_{\phi}(K_{1}):\mathbb{R}$	$\displaystyle\longrightarrow\mathbb{R}_{>0}$
	$\displaystyle K_{1}$	$\displaystyle\longmapsto L_{1}(K_{1}).$

On $\mathbb{R}\times\{-\infty\}$ , $L_{2}=0$ . By Gauss-Bonnet formula, we have that $\text{Area}(B)=2\phi-L_{1}>0$ , i.e. $0<L_{1}<2\phi$ . Hence the image set of $\Lambda^{{}^{\prime}}_{\phi}(K_{1})$ is contained in $\tilde{\Delta}_{1}$ , where $\tilde{\Delta}_{1}=\{x\leavevmode\nobreak\ |\leavevmode\nobreak\ 0<x<2\phi\}$ . Then we will show that $0$ and $2\phi$ are the limit points of the map $\Lambda^{{}^{\prime}}_{\phi}(K_{1})$ .

We can choose a sequence $\{n\}_{n\in\mathbb{N}_{+}}$ , then $k_{1}(n)=e^{n}\to+\infty\leavevmode\nobreak\ (n\to+\infty),r_{1}(n)=\operatorname{arccot}k_{1}(n)\to 0\leavevmode\nobreak\ (n\to+\infty)$ . Hence we have that $\text{Area}(B(n))\to 0\leavevmode\nobreak\ (n\to+\infty)$ . On $\mathbb{R}\times\{-\infty\}$ , $L_{2}(n)=0$ . By Gauss-Bonnet formula, we have that $\text{Area}(B(n))=2\phi-L_{1}(n)\to 0\leavevmode\nobreak\ (n\to+\infty)$ , i.e. $L_{1}(n)\to 2\phi\leavevmode\nobreak\ (n\to+\infty)$ .

We choose another sequence $\{-n\}_{n\in\mathbb{N}_{+}}$ , then $k_{1}(-n)=e^{-n}$ . Since $\ell_{1}(-n)\leq\pi$ , then $0\leq L_{1}(-n)=\ell_{1}(-n)k_{1}(-n)\leq\pi e^{-n}\to 0\leavevmode\nobreak\ (n\to+\infty)$ . Hence we have that $L_{1}(-n)\to 0\leavevmode\nobreak\ (n\to+\infty)$ .

Since $0$ and $2\phi$ are the limit points of the map $\Lambda^{{}^{\prime}}_{\phi}(K_{1})$ and $\Lambda^{{}^{\prime}}_{\phi}(K_{1})$ is a embedding map, it is easy to know that $\Lambda^{{}^{\prime}}_{\phi}(K_{1})$ is homeomorphism from $\mathbb{R}$ to $\tilde{\Delta}_{1}$ , i.e. $\mathbb{R}\times\{-\infty\}\cong\Delta_{1}$ .

Using the above same argument, we know that $\{-\infty\}\times\mathbb{R}\cong\Delta_{2}$ . ∎

By Lemma 4.3, we have that $\mathcal{B}^{1}_{\phi}\cong(0,\frac{\pi}{2})\times\{\frac{\pi}{2}\}\cong\mathbb{R}\times\{-\infty\}\cong\Delta_{1}$ and $\mathcal{B}^{2}_{\phi}\cong\{\frac{\pi}{2}\}\times(0,\frac{\pi}{2})\cong\{-\infty\}\times\mathbb{R}\cong\Delta_{2}$ . By Lemma 11 in [6], $\mathcal{B}^{3}_{\phi}\cong(0,\frac{\pi}{2})^{2}\cong\mathbb{R}^{2}\cong\Delta_{3}$ , where $\Delta_{3}=\{(x,y)\in\mathbb{R}^{2}_{>0}\leavevmode\nobreak\ |\leavevmode\nobreak\ x+y<2\phi\}$ . Besides, we have $\mathcal{B}^{0}_{\phi}\cong\{(\frac{\pi}{2},\frac{\pi}{2})\}\cong\{(0,0)\}$ .

5 Existence and rigidity of circle pattern metrics for prescribed total geodesic curvatures

Given a closed topological surface $S$ and a cellular decomposition $\Sigma=(V,E,F)$ of $S$ . For simplicity, we can suppose that $F=\{1,\cdots,|F|\}$ . By $L_{f,e}$ we denote the total geodesic curvature of an arc $C_{f,e}$ , where the arc $C_{f,e}$ is the side of the bigon corresponding to 2-cell $f\in F$ and 1-cell $e\in E$ . By $B_{e}$ we denote the bigon corresponding to $e$ (see Figure 7).

Then we can define the total geodesic curvature $L_{f}$ of a 2-cell $f$ , i.e. $L_{f}:=\sum_{e\in E_{f}}L_{f,e}$ , where $E_{f}$ is the set of 1-cells contained in $\partial f$ .

Given the weight $\Phi\in(0,\frac{\pi}{2})^{|E|}$ , for $\mathbf{r}=(r_{1},\cdots,r_{|F|})\in(0,\frac{\pi}{2})^{|F|}$ , using change of variables $K_{i}=\ln\cot r_{i}$ , $i=1,\cdots,|F|$ , we consider the new variable $K=(K_{1},\cdots,K_{|F|})\in\mathbb{R}^{|F|}$ .

We can define a potential function

\Lambda(K):=\sum_{e\in E}\Lambda_{\Phi(e)}(K_{f_{1}(e)},K_{f_{2}(e)})

on $\mathbb{R}^{|F|}$ , where $f_{1}(e)$ and $f_{2}(e)$ are the 2-cells on two sides of 1-cell $e$ . Besides, we have

(5.1)

\displaystyle\frac{\partial\Lambda(K)}{\partial K_{i}}=\sum_{e\in E_{i}}\frac{\partial\Lambda_{\Phi(e)}(K_{f_{1}(e)},K_{f_{2}(e)})}{\partial K_{i}}=\sum_{e\in E_{i}}L_{i,e}=L_{i}>0,\leavevmode\nobreak\ 1\leq i\leq|F|.

Hence the Hessian of $\Lambda$ is a Jacobi matrix, i.e.

\text{Hess}\leavevmode\nobreak\ \Lambda=M=\begin{pmatrix}\frac{\partial L_{1}}{\partial K_{1}}&\cdots&\frac{\partial L_{1}}{\partial K_{|F|}}\\ \vdots&\ddots&\vdots\\ \frac{\partial L_{|F|}}{\partial K_{1}}&\cdots&\frac{\partial L_{|F|}}{\partial K_{|F|}}\\ \end{pmatrix}.

Proposition 5.1.

The Jacobi matrix $M$ is positive definite.

Proof.

By Lemma 9 in [6], we have that

\frac{\partial L_{i}}{\partial K_{i}}=\frac{\partial(\sum_{e\in E_{i}}L_{i,e})}{\partial K_{i}}=\sum_{e\in E_{i}}\frac{\partial L_{i,e}}{\partial K_{i}}>0,\leavevmode\nobreak\ i=1,\cdots,|F|,

and

\frac{\partial L_{i}}{\partial K_{j}}=\frac{\partial(\sum_{e\in E_{i}}L_{i,e})}{\partial K_{j}}=\sum_{e\in E_{i}}\frac{\partial L_{i,e}}{\partial K_{j}}\leq 0,\leavevmode\nobreak\ 1\leq i\neq j\leq|F|.

Hence we obtain

\left|\frac{\partial L_{i}}{\partial K_{i}}\right|-\sum_{j\neq i}\left|\frac{\partial L_{j}}{\partial K_{i}}\right|=\frac{\partial L_{i}}{\partial K_{i}}+\sum_{j\neq i}\frac{\partial L_{j}}{\partial K_{i}}=\frac{\partial(\sum_{f\in F}L_{f})}{\partial K_{i}}.

Besides, we have

\sum_{f\in F}L_{f}=\sum_{f\in F}\sum_{e\in E_{f}}L_{f,e}=\sum_{e\in E}\sum_{f\in F_{\{}e\}}L_{f,e},

where $F_{\{e\}}$ is the set of 2-cells on two sides of 1-cell $e$ . Then by Gauss-Bonnet formula, we have

\sum_{f\in F_{\{}e\}}L_{f,e}=2\Phi(e)-\text{Area}(B_{e}).

Hence we obtain

\sum_{f\in F}L_{f}=\sum_{e\in E}2\Phi(e)-\text{Area}(B_{e}).

By Lemma 9 in [6], we have

\frac{\partial(\sum_{f\in F}L_{f})}{\partial K_{i}}=-\sum_{e\in E}\frac{\partial(\text{Area}(B_{e}))}{\partial K_{i}}>0,

Then we obtain

\left|\frac{\partial L_{i}}{\partial K_{i}}\right|-\sum_{j\neq i}\left|\frac{\partial L_{j}}{\partial K_{i}}\right|=\frac{\partial(\sum_{f\in F}L_{f})}{\partial K_{i}}>0.

Hence $M$ is a strictly diagonally dominant matrix with positive diagonal entries, i.e. $M$ is positive definite. ∎

Corollary 5.2.

The potential function $\Lambda$ is strictly convex on $\mathbb{R}^{|F|}$ .

Proof.

By Proposition 5.1, $\text{Hess}\leavevmode\nobreak\ \Lambda$ is positive definite, then $\Lambda$ is strictly convex on $\mathbb{R}^{|F|}$ . ∎

For a subset $F^{\prime}\subset F$ , by $E_{F^{\prime}}$ we denote the set

E_{F^{\prime}}=\{e\in E\leavevmode\nobreak\ |\leavevmode\nobreak\ \exists f\in F^{\prime},s.t.\leavevmode\nobreak\ e\in E_{f}\}.

Then we have the following theorem, i.e.

Theorem 5.3.

$\nabla\Lambda$ is a homeomorphism from $\mathbb{R}^{|F|}$ to $\mathcal{L}_{1}$ , where

\mathcal{L}_{1}=\{(L_{1},\cdots,L_{|F|})^{T}\in\mathbb{R}^{|F|}_{>0}\leavevmode\nobreak\ |\leavevmode\nobreak\ \sum_{f\in F^{\prime}}L_{f}<2\sum_{e\in E_{F^{\prime}}}\Phi(e),\forall F^{\prime}\subset F\}.

Proof.

By (5.1), we have

	$\displaystyle\nabla\Lambda:\mathbb{R}^{\|F\|}$	$\displaystyle\longrightarrow\mathbb{R}^{\|F\|}_{>0}$
	$\displaystyle K=(K_{1},\cdots,K_{\|F\|})^{T}$	$\displaystyle\mapsto L=(L_{1},\cdots,L_{\|F\|})^{T}.$

For any subset $F^{\prime}\subset F$ , we obtain

\sum_{f\in F^{\prime}}L_{f}=\sum_{f\in F^{\prime}}\sum_{e\in E_{f}}L_{f,e}=\sum_{e\in E_{F^{\prime}}}\sum_{f\in F_{\{e\}}\cap F^{\prime}}L_{f,e}.

Then by Gauss-Bonnet formula, we have

\text{Area}(B_{e})=2\Phi(e)-\sum_{f\in F_{\{e\}}}L_{f,e}>0,

i.e. $\sum_{f\in F_{\{e\}}}L_{f,e}<2\Phi(e)$ . Hence we obtain

\sum_{f\in F^{\prime}}L_{f}=\sum_{e\in E_{F^{\prime}}}\sum_{f\in F_{\{e\}}\cap F^{\prime}}L_{f,e}\leq\sum_{e\in E_{F^{\prime}}}\sum_{f\in F_{\{e\}}}L_{f,e}<2\sum_{e\in E_{F^{\prime}}}\Phi(e),

hence $(L_{1},\cdots,L_{|F|})^{T}\in\mathcal{L}_{1}$ , $\nabla\Lambda$ is a map from $\mathbb{R}^{|F|}$ to $\mathcal{L}_{1}$ .

By Proposition 5.1 and Corollary 5.2, $\nabla\Lambda$ is a embedding map. Hence we only need to show that the image set of $\nabla\Lambda$ is $\mathcal{L}_{1}$ . Now we need to analysis the boundary of its image.

Choose a point $a=(a_{1},\cdots,a_{|F|})^{T}\in\partial\mathbb{R}^{|F|}$ , we define two subset $W_{1},W_{2}\subset F$ , i.e.

W_{1}=\{i\in F\leavevmode\nobreak\ |\leavevmode\nobreak\ a_{i}=+\infty\},\leavevmode\nobreak\ W_{2}=\{i\in F\leavevmode\nobreak\ |\leavevmode\nobreak\ a_{i}=-\infty\},

we know that $W_{1}\neq\emptyset$ or $W_{2}\neq\emptyset$ .

We choose a sequence $\{a^{n}\}_{n=1}^{+\infty}\subset\mathbb{R}^{|F|}$ such that $\lim_{n\to+\infty}a^{n}=a$ , where $a^{n}=(a_{1}^{n},\cdots,a_{|F|}^{n})^{T}$ . Then we have

\nabla\Lambda(a^{n})=L(a^{n})\overset{\Delta}{=}L^{n},\leavevmode\nobreak\ k_{i}^{n}\overset{\Delta}{=}\exp(a_{i}^{n}),\leavevmode\nobreak\ \forall n\geq 1.

Now we need to show that $\{L^{n}\}_{n=1}^{+\infty}$ converges to the boundary of $\mathcal{L}_{1}$ .

If $W_{1}\neq\emptyset$ , we know that for each $i\in W_{1}$ , $k_{i}^{n}=\exp(a_{i}^{n})\to+\infty\leavevmode\nobreak\ (n\to+\infty)$ , and $r_{i}^{n}=\operatorname{arccot}k_{i}^{n}\to 0\leavevmode\nobreak\ (n\to+\infty)$ . Besides, note that

(5.2)

\displaystyle\lim_{n\to+\infty}\sum_{f\in W_{1}}L_{f}^{n}=\lim_{n\to+\infty}\sum_{f\in W_{1}}\sum_{e\in E_{f}}L_{f,e}^{n}=\lim_{n\to+\infty}\sum_{e\in E_{W_{1}}}\sum_{f\in W_{1}\cap F_{\{e\}}}L_{f,e}^{n}.

Now we need to show

(5.3)

\displaystyle\lim_{n\to+\infty}\sum_{f\in W_{1}\cap F_{\{e\}}}L_{f,e}^{n}=2\Phi(e).

Since $W_{1}\neq\emptyset$ and $e\in E_{W_{1}}$ , the set $W_{1}\cap F_{\{e\}}$ has 1 or 2 elements.

If $W_{1}\cap F_{\{e\}}$ has 1 elements, we can suppose that $W_{1}\cap F_{\{e\}}=\{h\}$ and $F_{\{e\}}=\{h,j\}$ . By Gauss-Bonnet formula, we have

(5.4)

\displaystyle L_{h,e}^{n}+L_{j,e}^{n}=2\Phi(e)-\text{Area}(B_{e}^{n}).

Since $j\notin W_{1}$ , we know that $a_{j}^{n},k_{j}^{n}\nrightarrow+\infty,r_{j}^{n}\nrightarrow 0\leavevmode\nobreak\ (n\to+\infty)$ . Since $h\in W_{1}$ , we know that $r_{h}^{n}\to 0\leavevmode\nobreak\ (n\to+\infty)$ . Then we have that $\ell_{j,e}^{n},\text{Area}(B_{e}^{n})\to 0\leavevmode\nobreak\ (n\to+\infty)$ . Hence we obtain $L_{j,e}^{n}=\ell_{j,e}^{n}k_{j}^{n}\to 0\leavevmode\nobreak\ (n\to+\infty)$ .

Using (5.4), we have

\lim_{n\to+\infty}L_{h,e}^{n}=2\Phi(e)-\lim_{n\to+\infty}\text{Area}(B_{e}^{n})=2\Phi(e),

i.e. $\lim_{n\to+\infty}\sum_{f\in W_{1}\cap F_{\{e\}}}L_{f,e}^{n}=2\Phi(e)$ .

If $W_{1}\cap F_{\{e\}}$ has 2 elements, we can suppose that $W_{1}\cap F_{\{e\}}=\{m,s\}$ , then $F_{\{e\}}=\{m,s\}$ . By Gauss-Bonnet formula, we have

(5.5)

\displaystyle L_{m,e}^{n}+L_{s,e}^{n}=2\Phi(e)-\text{Area}(B_{e}^{n}).

Since $m,s\in W_{1}$ , we know that $k_{m}^{n},k_{s}^{n}\to+\infty\leavevmode\nobreak\ (n\to+\infty)$ , $r_{m}^{n},r_{s}^{n}\to 0\leavevmode\nobreak\ (n\to+\infty)$ . Then we have $\text{Area}(B_{e}^{n})\to 0\leavevmode\nobreak\ (n\to+\infty)$ . Using (5.5), we obtain

\lim_{n\to+\infty}L_{m,e}^{n}+L_{s,e}^{n}=2\Phi(e)-\lim_{n\to+\infty}\text{Area}(B_{e}^{n})=2\Phi(e),

i.e. $\lim_{n\to+\infty}\sum_{f\in W_{1}\cap F_{\{e\}}}L_{f,e}^{n}=2\Phi(e)$ .

Hence if $W_{1}\neq\emptyset$ , by (5.2) and (5.3), we have

\lim_{n\to+\infty}\sum_{f\in W_{1}}L_{f}^{n}=\lim_{n\to+\infty}\sum_{e\in E_{W_{1}}}\sum_{f\in W_{1}\cap F_{\{e\}}}L_{f,e}^{n}=\sum_{e\in E_{W_{1}}}\lim_{n\to+\infty}\sum_{f\in W_{1}\cap F_{\{e\}}}L_{f,e}^{n}=2\sum_{e\in E_{W_{1}}}\Phi(e),

i.e. $\{L^{n}\}_{n=1}^{+\infty}$ converges to the boundary of $\mathcal{L}_{1}$ .

If $W_{2}\neq\emptyset$ , then for each $i\in W_{2}$ , $k_{i}^{n}=\exp(a_{i}^{n})\to 0\leavevmode\nobreak\ (n\to+\infty)$ . For any 1-cell $e\in E_{i}$ , since $0\leq\ell_{i,e}^{n}\leq 2\pi$ and $k_{i}^{n}\to 0\leavevmode\nobreak\ (n\to+\infty)$ , then $\lim_{n\to+\infty}L_{i,e}^{n}=\lim_{n\to+\infty}\ell_{i,e}^{n}k_{i}^{n}=0$ . Hence we obtain

\lim_{n\to+\infty}L_{i}^{n}=\lim_{n\to+\infty}\sum_{e\in E_{i}}L_{i,e}^{n}=0,

i.e. $\{L^{n}\}_{n=1}^{+\infty}$ converges to the boundary of $\mathcal{L}_{1}$ .

By Brouwer’s Theorem on the Invariance of Domain and the above analysis, we know that the image set of $\nabla\Lambda$ is $\mathcal{L}_{1}$ . Hence $\nabla\Lambda$ is a homeomorphism from $\mathbb{R}^{|F|}$ to $\mathcal{L}_{1}$ . ∎

We can construct a map $\varsigma$ from $\mathbb{R}^{|F|}_{>0}$ to $\mathbb{R}^{|F|}$ , i.e.

\begin{array}[]{cccc}\varsigma:\mathbb{R}^{|F|}_{>0}&\longrightarrow&\mathbb{R}^{|F|}\\ k=(k_{1},\cdots,k_{|F|})^{T}&\longmapsto&K=(\ln k_{1},\cdots,\ln k_{|F|})^{T},\\ \end{array}

the map $\varsigma$ is a homeomorphism. By Theorem 5.3, we know that the map $\mathcal{E}_{1}:=\nabla\Lambda\circ\varsigma$ , i.e.

\begin{array}[]{cccc}\mathcal{E}_{1}:\mathbb{R}^{|F|}_{>0}&\longrightarrow&\mathcal{L}_{1}\\ k&\longmapsto&\mathcal{E}_{1}(k)=\nabla\Lambda(K)=L(K)\\ \end{array}

is a homeomorphism from $\mathbb{R}^{|F|}_{>0}$ to $\mathcal{L}_{1}$ . Then we completed the proof of Theorem 1.5.

6 Existence and rigidity of degenerated circle pattern metrics for prescribed total geodesic curvatures

By $\mathbb{R}^{|F|}_{i_{1}\cdots i_{m}}$ we denote the set

\mathbb{R}^{|F|}_{i_{1}\cdots i_{m}}=\{(0,\cdots,k_{i_{1}},0,\cdots,k_{i_{m}},\cdots,0)\in\mathbb{R}^{|F|}\leavevmode\nobreak\ |\leavevmode\nobreak\ k_{i_{1}},\cdots,k_{i_{m}}>0\}.

By $\partial_{i}\mathbb{R}^{|F|}_{\geq 0}$ we denote the set

\partial_{i}\mathbb{R}^{|F|}_{\geq 0}=\{(k_{1},\cdots,k_{|F|})\in\mathbb{R}^{|F|}\leavevmode\nobreak\ |\leavevmode\nobreak\ i\leavevmode\nobreak\ \text{components}>0,\leavevmode\nobreak\ |F|-i\leavevmode\nobreak\ \text{components}=0\}.

Then we have

\partial_{1}\mathbb{R}^{|F|}_{\geq 0}=\bigcup_{1\leq i\leq|F|}\{(0,\cdots,k_{i},\cdots,0)\in\mathbb{R}^{|F|}\leavevmode\nobreak\ |\leavevmode\nobreak\ k_{i}>0\}=\bigcup_{1\leq i\leq|F|}\mathbb{R}^{|F|}_{i},

\partial_{2}\mathbb{R}^{|F|}_{\geq 0}=\bigcup_{1\leq i<j\leq|F|}\{(0,\cdots,k_{i},\cdots,k_{j},\cdots,0)\in\mathbb{R}^{|F|}\leavevmode\nobreak\ |\leavevmode\nobreak\ k_{i},k_{j}>0\}=\bigcup_{1\leq i<j\leq|F|}\mathbb{R}^{|F|}_{ij},

\cdots

\partial_{|F|-1}\mathbb{R}^{|F|}_{\geq 0}=\bigcup_{1\leq i_{1}<\cdots<i_{|F|-1}\leq|F|}\{(k_{i},\cdots,0,\cdots,k_{i_{|F|-1}})\in\mathbb{R}^{|F|}\leavevmode\nobreak\ |\leavevmode\nobreak\ k_{i_{1}},\cdots,k_{i_{|F|-1}}>0\}=\bigcup_{1\leq i_{1}<\cdots<i_{|F|-1}\leq|F|}\mathbb{R}^{|F|}_{i_{1}\cdots i_{|F|-1}}.

For $1\leq m\leq|F|-1$ , we have that

\partial_{m}\mathbb{R}^{|F|}_{\geq 0}=\bigcup_{1\leq i_{1}<\cdots<i_{m}\leq|F|}\mathbb{R}^{|F|}_{i_{1}\cdots i_{m}}.

By $\bar{\partial}\mathbb{R}^{|F|}_{\geq 0}$ we denote the set

\bar{\partial}\mathbb{R}^{|F|}_{\geq 0}=\bigcup_{m=1}^{|F|-1}\partial_{m}\mathbb{R}^{|F|}_{\geq 0}\cup\{0\}.

Then we consider the set $\mathbb{R}^{|F|}_{1\cdots m}$ , i.e.

\mathbb{R}^{|F|}_{1\cdots m}=\{(k_{1},\cdots,k_{m},0,\cdots,0)\in\mathbb{R}^{|F|}\leavevmode\nobreak\ |\leavevmode\nobreak\ k_{1},\cdots,k_{m}>0\}.

Given the weight $\Phi\in(0,\frac{\pi}{2})^{|E|}$ , for $\mathbf{r}=(r_{1},r_{2},\cdots,r_{m},\frac{\pi}{2},\cdots,\frac{\pi}{2})\in(0,\frac{\pi}{2})^{m}\times\{\frac{\pi}{2}\}^{|F|-m}$ , using change of variables $K_{i}=\ln\cot r_{i}$ , $i=1,\cdots,m$ , we consider the new variable $K=(K_{1},\cdots,K_{m})\in\mathbb{R}^{m}$ .

We can define a potential function

\Lambda_{1}(K):=\sum_{e\in E,f_{1}(e),f_{2}(e)\in F_{m}}\Lambda_{\Phi(e)}(K_{f_{1}(e)},K_{f_{2}(e)})+\sum_{e\in E,f_{1}(e)\in F_{m},f_{2}(e)\notin F_{m}}\Lambda_{\Phi(e)}(K_{f_{1}(e)})

on $\mathbb{R}^{m}$ , where $F_{m}=\{1,2,\cdots,m\}$ , $f_{1}(e)$ and $f_{2}(e)$ are the 2-cells on two sides of 1-cell $e$ . Besides, note that at least one of these two sums exists.

For $1\leq i\leq m$ , we have

	$\displaystyle\frac{\partial\Lambda_{1}(K)}{\partial K_{i}}$	$\displaystyle=\sum_{e\in E_{i},f_{1}(e),f_{2}(e)\in F_{m}}\frac{\partial\Lambda_{\Phi(e)}(K_{f_{1}(e)},K_{f_{2}(e)})}{\partial K_{i}}+\sum_{e\in E_{i},f_{1}(e)\in F_{m},f_{2}(e)\notin F_{m}}\frac{\partial\Lambda_{\Phi(e)}(K_{f_{1}(e)})}{\partial K_{i}}$
		$\displaystyle=\sum_{e\in E_{i},f_{1}(e),f_{2}(e)\in F_{m}}L_{i,e}(K_{f_{1}(e)},K_{f_{2}(e)})+\sum_{e\in E_{i},f_{1}(e)\in F_{m},f_{2}(e)\notin F_{m}}L_{i,e}(K_{i}).$

Since $i\in F_{m}$ , then for any 1-cell $e\in E_{i}$ , we know that $f_{1}(e),f_{2}(e)\in F_{m}$ or $f_{1}(e)\in F_{m},f_{2}(e)\notin F_{m}$ . Hence we have

(6.1)

\displaystyle L_{i}=\sum_{e\in E_{i}}L_{i,e}=\sum_{e\in E_{i},f_{1}(e),f_{2}(e)\in F_{m}}L_{i,e}(K_{f_{1}(e)},K_{f_{2}(e)})+\sum_{e\in E_{i},f_{1}(e)\in F_{m},f_{2}(e)\notin F_{m}}L_{i,e}(K_{i})=\frac{\partial\Lambda_{1}(K)}{\partial K_{i}}.

Then the Hessian of $\Lambda_{1}$ is a Jacobi matrix, i.e.

\text{Hess}\leavevmode\nobreak\ \Lambda_{1}=M_{1}=\begin{pmatrix}\frac{\partial L_{1}}{\partial K_{1}}&\cdots&\frac{\partial L_{1}}{\partial K_{m}}\\ \vdots&\ddots&\vdots\\ \frac{\partial L_{m}}{\partial K_{1}}&\cdots&\frac{\partial L_{m}}{\partial K_{m}}\\ \end{pmatrix}.

Proposition 6.1.

The Jacobi matrix $M_{1}$ is positive definite.

Proof.

For any $1\leq i\leq m$ , by (6.1), we have

\frac{\partial L_{i}}{\partial K_{i}}=\sum_{e\in E_{i},f_{1}(e),f_{2}(e)\in F_{m}}\frac{\partial L_{i,e}(K_{f_{1}(e)},K_{f_{2}(e)})}{\partial K_{i}}+\sum_{e\in E_{i},f_{1}(e)\in F_{m},f_{2}(e)\notin F_{m}}\frac{\partial L_{i,e}(K_{i})}{\partial K_{i}}.

By Lemma 9 in [6] and Lemma 4.2, we have $\frac{\partial L_{i,e}(K_{f_{1}(e)},K_{f_{2}(e)})}{\partial K_{i}}>0,\frac{\partial L_{i,e}(K_{i})}{\partial K_{i}}>0$ . Besides, at least one of these two sums exists, then we obtain that $\frac{\partial L_{i}}{\partial K_{i}}>0$ for $1\leq i\leq m$ .

For $1\leq i\neq j\leq m$ , by (6.1), we have

\frac{\partial L_{i}}{\partial K_{j}}=\sum_{e\in E_{i},f_{1}(e),f_{2}(e)\in F_{m}}\frac{\partial L_{i,e}(K_{f_{1}(e)},K_{f_{2}(e)})}{\partial K_{j}}.

For any 1-cell $e\in E_{i}$ such that $f_{1}(e),f_{2}(e)\in F_{m}$ , we know that $i\in F_{\{e\}}$ . If $j\in F_{\{e\}}$ , by Lemma 9 in [6], then we have

\frac{\partial L_{i,e}(K_{f_{1}(e)},K_{f_{2}(e)})}{\partial K_{j}}=\frac{\partial L_{i,e}(K_{i},K_{j})}{\partial K_{j}}<0.

If $j\notin F_{\{e\}}$ , then we have

\frac{\partial L_{i,e}(K_{f_{1}(e)},K_{f_{2}(e)})}{\partial K_{j}}=0.

Then we obtain $\frac{\partial L_{i}}{\partial K_{j}}\leq 0$ for $1\leq i\neq j\leq m$ .

Hence for $1\leq i\leq m$ , we have

	$\displaystyle\left\|\frac{\partial L_{i}}{\partial K_{i}}\right\|-\sum_{1\leq j\neq i\leq m}\left\|\frac{\partial L_{j}}{\partial K_{i}}\right\|$	$\displaystyle=\frac{\partial(\sum_{f\in F_{m}}L_{f})}{\partial K_{i}}=\frac{\partial(\sum_{f\in F_{m}}\sum_{e\in E_{f}}L_{f,e})}{\partial K_{i}}=\frac{\partial(\sum_{e\in E_{F_{m}}}\sum_{f\in F_{m}\cap F_{\{e\}}}L_{f,e})}{\partial K_{i}}$
		$\displaystyle=\sum_{e\in E_{F_{m}}}\frac{\partial(\sum_{f\in F_{m}\cap F_{\{e\}}}L_{f,e})}{\partial K_{i}}=\sum_{e\in E_{i}}\frac{\partial(\sum_{f\in F_{m}\cap F_{\{e\}}}L_{f,e})}{\partial K_{i}}.$

For 1-cell $e\in E_{i}$ , we can suppose that $F_{\{e\}}=\{i,h\}$ . Then $F_{m}\cap F_{\{e\}}=\{i\}$ or $\{i,h\}$ .

If $F_{m}\cap F_{\{e\}}=\{i,h\}$ , by Gauss-Bonnet formula, we have

\sum_{f\in F_{m}\cap F_{\{e\}}}L_{f,e}=L_{i,e}+L_{h,e}=2\Phi(e)-\text{Area}(B_{e}).

By Lemma 9 in [6], we have

\frac{\partial(\sum_{f\in F_{m}\cap F_{\{e\}}}L_{f,e})}{\partial K_{i}}=-\frac{\partial\text{Area}(B_{e})}{\partial K_{i}}>0.

If $F_{m}\cap F_{\{e\}}=\{i\}$ , then by Lemma 4.2, we have

\frac{\partial(\sum_{f\in F_{m}\cap F_{\{e\}}}L_{f,e})}{\partial K_{i}}=\frac{\partial L_{i,e}(K_{i})}{\partial K_{i}}>0.

Then we obtain

\left|\frac{\partial L_{i}}{\partial K_{i}}\right|-\sum_{1\leq j\neq i\leq m}\left|\frac{\partial L_{j}}{\partial K_{i}}\right|=\sum_{e\in E_{i}}\frac{\partial(\sum_{f\in F_{m}\cap F_{\{e\}}}L_{f,e})}{\partial K_{i}}>0.

Hence $M_{1}$ is a strictly diagonally dominant matrix with positive diagonal entries, i.e. $M_{1}$ is positive definite. ∎

Corollary 6.2.

The potential function $\Lambda_{1}$ is strictly convex on $\mathbb{R}^{m}$ .

Proof.

By Proposition 6.1, $\text{Hess}\leavevmode\nobreak\ \Lambda_{1}$ is positive definite, then $\Lambda_{1}$ is strictly convex on $\mathbb{R}^{m}$ . ∎

Theorem 6.3.

$\nabla\Lambda_{1}$ is a homeomorphism from $\mathbb{R}^{m}$ to $\tilde{\mathcal{L}}_{1\cdots m}$ , where

\tilde{\mathcal{L}}_{1\cdots m}=\{(L_{1},\cdots,L_{m})^{T}\in\mathbb{R}^{m}_{>0}\leavevmode\nobreak\ |\leavevmode\nobreak\ \sum_{f\in F_{m}^{\prime}}L_{f}<2\sum_{e\in E_{F_{m}^{\prime}}}\Phi(e),\forall F_{m}^{\prime}\subset F_{m}\}.

Proof.

By (6.1), we have

	$\displaystyle\nabla\Lambda_{1}:\mathbb{R}^{m}$	$\displaystyle\longrightarrow\mathbb{R}^{m}_{>0}$
	$\displaystyle K=(K_{1},\cdots,K_{m})^{T}$	$\displaystyle\mapsto L=(L_{1},\cdots,L_{m})^{T}.$

For any subset $F_{m}^{\prime}\subset F_{m}$ , we obtain

\sum_{f\in F_{m}^{\prime}}L_{f}=\sum_{f\in F_{m}^{\prime}}\sum_{e\in E_{f}}L_{f,e}=\sum_{e\in E_{F_{m}^{\prime}}}\sum_{f\in F_{\{e\}}\cap F_{m}^{\prime}}L_{f,e}.

Then by Gauss-Bonnet formula, we have

\text{Area}(B_{e})=2\Phi(e)-\sum_{f\in F_{\{e\}}}L_{f,e}>0,

i.e. $\sum_{f\in F_{\{e\}}}L_{f,e}<2\Phi(e)$ . Hence we obtain

\sum_{f\in F_{m}^{\prime}}L_{f}=\sum_{e\in E_{F_{m}^{\prime}}}\sum_{f\in F_{\{e\}}\cap F_{m}^{\prime}}L_{f,e}\leq\sum_{e\in E_{F_{m}^{\prime}}}\sum_{f\in F_{\{e\}}}L_{f,e}<2\sum_{e\in E_{F_{m}^{\prime}}}\Phi(e),

hence $(L_{1},\cdots,L_{m})^{T}\in\tilde{\mathcal{L}}_{1\cdots m}$ , $\nabla\Lambda_{1}$ is a map from $\mathbb{R}^{m}$ to $\tilde{\mathcal{L}}_{1\cdots m}$ .

By Proposition 6.1 and Corollary 6.2, $\nabla\Lambda_{1}$ is a embedding map. Using the method in the proof of Theorem 5.3, we have a similar result that the image set of $\nabla\Lambda_{1}$ is $\tilde{\mathcal{L}}_{1\cdots m}$ . Hence $\nabla\Lambda_{1}$ is a homeomorphism from $\mathbb{R}^{m}$ to $\tilde{\mathcal{L}}_{1\cdots m}$ . ∎

We can construct a map $\varsigma_{1}$ from $\mathbb{R}^{m}_{>0}$ to $\mathbb{R}^{m}$ , i.e.

\begin{array}[]{cccc}\varsigma_{1}:\mathbb{R}^{m}_{>0}&\longrightarrow&\mathbb{R}^{m}\\ k=(k_{1},\cdots,k_{m})^{T}&\longmapsto&K=(\ln k_{1},\cdots,\ln k_{m})^{T},\\ \end{array}

the map $\varsigma_{1}$ is a homeomorphism. By Theorem 6.3, we know that the map $\tilde{\mathcal{E}}:=\nabla\Lambda_{1}\circ\varsigma_{1}$ , i.e.

\begin{array}[]{cccc}\tilde{\mathcal{E}}:\mathbb{R}^{m}_{>0}&\longrightarrow&\tilde{\mathcal{L}}_{1\cdots m}\\ k&\longmapsto&\tilde{\mathcal{E}}(k)=\nabla\Lambda_{1}(K)=L(K)\\ \end{array}

is a homeomorphism from $\mathbb{R}^{m}_{>0}$ to $\tilde{\mathcal{L}}_{1\cdots m}$ . In other words, there exists a homeomorphism from $\mathbb{R}^{|F|}_{1\cdots m}$ to $\mathcal{L}_{1\cdots m}$ , where

\mathcal{L}_{1\cdots m}=\left\{\begin{array}[]{l|l}(L_{1},\cdots,L_{m},0,\cdots,0)^{T}\in\mathbb{R}^{m}_{>0}\times\{0\}^{|F|-m}&\begin{array}[]{l}\sum_{f\in F_{m}^{\prime}}L_{f}<2\sum_{e\in E_{F_{m}^{\prime}}}\Phi(e),\forall F_{m}^{\prime}\subset F_{m}\end{array}\end{array}\right\}.

Similarly, for $1\leq m\leq|F|-1$ , $1\leq i_{1}<\cdots<i_{m}\leq|F|$ , by $F_{i_{1}\cdots i_{m}}$ we denote the set $F_{i_{1}\cdots i_{m}}=\{i_{1},\cdots,i_{m}\}$ . Then there exists a homeomorphism from $\mathbb{R}^{|F|}_{i_{1}\cdots i_{m}}$ to $\mathcal{L}_{i_{1}\cdots i_{m}}$ , where

\mathcal{L}_{i_{1}\cdots i_{m}}=\left\{\begin{array}[]{l|l}(0,\cdots,L_{i_{1}},\cdots,0,\cdots,L_{i_{m}},0,\cdots,0)^{T}\in\mathbb{R}^{|F|}&\begin{array}[]{l}\sum_{f\in F_{i_{1}\cdots i_{m}}^{\prime}}L_{f}<2\sum_{e\in E_{F_{i_{1}\cdots i_{m}}^{\prime}}}\Phi(e),\\ \forall F_{i_{1}\cdots i_{m}}^{\prime}\subset F_{i_{1}\cdots i_{m}};L_{i_{j}}>0,1\leq j\leq m\end{array}\end{array}\right\}.

Besides, we define the set $\tilde{\mathcal{L}}_{i_{1}\cdots i_{m}}$ , i.e.

\tilde{\mathcal{L}}_{i_{1}\cdots i_{m}}=\left\{\begin{array}[]{l|l}(L_{i_{1}},\cdots,L_{i_{m}})^{T}\in\mathbb{R}_{>0}^{m}&\begin{array}[]{l}\sum_{f\in F_{i_{1}\cdots i_{m}}^{\prime}}L_{f}<2\sum_{e\in E_{F_{i_{1}\cdots i_{m}}^{\prime}}}\Phi(e),\leavevmode\nobreak\ \forall F_{i_{1}\cdots i_{m}}^{\prime}\subset F_{i_{1}\cdots i_{m}}\end{array}\end{array}\right\}.

By $\bar{\partial}\mathcal{L}$ we denote the set

\bar{\partial}\mathcal{L}=\bigcup_{1\leq m\leq|F|-1,1\leq i_{1}<\cdots<i_{m}\leq|F|}\mathcal{L}_{i_{1}\cdots i_{m}}\cup\{0\}.

Then by above argument, we completed the proof of Theorem 1.7.

7 Existence and rigidity of spherical conical metrics for prescribed total geodesic curvatures

By $\mathcal{E}_{i_{1}\cdots i_{m}}$ we denote the homeomorphism from $\mathbb{R}^{|F|}_{i_{1}\cdots i_{m}}$ to $\mathcal{L}_{i_{1}\cdots i_{m}}$ and by $\mathcal{L}$ we denote the set $\mathcal{L}=\mathcal{L}_{1}\cup\bar{\partial}\mathcal{L}$ . Then we can define a map $\mathcal{E}$ from $\mathbb{R}^{|F|}_{\geq 0}$ to $\mathcal{L}$ , i.e.

(7.4)

\displaystyle\mathcal{E}(k):=\left\{\begin{array}[]{lc}\mathcal{E}_{1}(k),&k\in\mathbb{R}^{|F|}_{>0},\\ \mathcal{E}_{i_{1}\cdots i_{m}}(k),&k\in\mathbb{R}^{|F|}_{i_{1}\cdots i_{m}},1\leq m\leq|F|-1,1\leq i_{1}<\cdots<i_{m}\leq|F|,\\ 0,&k=0.\end{array}\right.

Theorem 7.1.

$\mathcal{E}$ is a homeomorphism from $\mathbb{R}^{|F|}_{\geq 0}$ to $\mathcal{L}$ such that the interior of $\mathbb{R}^{|F|}_{\geq 0}$ maps to the interior of $\mathcal{L}$ and $\bar{\partial}\mathbb{R}^{|F|}_{\geq 0}$ maps to $\bar{\partial}\mathcal{L}$ .

Proof.

Since $\mathcal{E}_{1}$ , $\mathcal{E}_{i_{1}\cdots i_{m}}(1\leq m\leq|F|-1,1\leq i_{1}<\cdots<i_{m}\leq|F|)$ are homeomorphisms, by (7.4), then $\mathcal{E}$ is a bijective map from $\mathbb{R}^{|F|}_{\geq 0}$ to $\mathcal{L}$ . We only need to show that $\mathcal{E}$ and $\mathcal{E}^{-1}$ are continuous from interior to boundary.

We choose a sequence $\{k^{n}\}_{n=1}^{+\infty}\subset\mathbb{R}^{|F|}_{>0}$ such that $\lim_{n\to+\infty}k^{n}=0$ , where $k^{n}=(k_{1}^{n},\cdots,k_{|F|}^{n})^{T}$ . Then we have $\mathcal{E}_{1}(k^{n})=(L_{1}(k^{n}),\cdots,L_{|F|}(k^{n}))^{T}$ , where $L_{i}(k^{n})=\sum_{e\in E_{i}}\ell_{i,e}^{n}k_{i}^{n}$ , $1\leq i\leq|F|$ . Since $0\leq\ell_{i,e}^{n}\leq\pi,k_{i}^{n}\to 0\leavevmode\nobreak\ (n\to+\infty)$ , then we have $\ell_{i,e}^{n}k_{i}^{n}\to 0\leavevmode\nobreak\ (n\to+\infty)$ , $L_{i}(k^{n})=\sum_{e\in E_{i}}\ell_{i,e}^{n}k_{i}^{n}\to 0\leavevmode\nobreak\ (n\to+\infty)$ , $1\leq i\leq|F|$ . Hence we obtain $\mathcal{E}_{1}(k^{n})\to 0\leavevmode\nobreak\ (n\to+\infty)$ .

For any point $k\in\mathbb{R}_{i_{1}\cdots i_{m}}^{|F|}$ , $1\leq m\leq|F|-1$ , $1\leq i_{1}<\cdots<i_{m}\leq|F|$ , we choose another sequence $\{k^{n}\}_{n=1}^{+\infty}\subset\mathbb{R}^{|F|}_{>0}$ such that $\lim_{n\to+\infty}k^{n}=k$ . Then we need to show that $\mathcal{E}_{1}(k^{n})\to\mathcal{E}_{i_{1}\cdots i_{m}}(k)\leavevmode\nobreak\ (n\to+\infty)$ . For simplicity, we only show that $\mathcal{E}_{1}(k^{n})\to\mathcal{E}_{1\cdots m}(k)=(L_{1}(k),\cdots,L_{m}(k),0,\cdots,0)\leavevmode\nobreak\ (n\to+\infty)$ , the others use the same method.

We can suppose that $k=(k_{1},\cdots,k_{m},0,\cdots,0)^{T}$ , since $k_{i}^{n}\to 0,\leavevmode\nobreak\ m+1\leq i\leq|F|$ , by the analysis above, then we have $L_{i}(k^{n})=\sum_{e\in E_{i}}\ell_{i,e}^{n}k_{i}^{n}\to 0\leavevmode\nobreak\ (n\to+\infty),\leavevmode\nobreak\ m+1\leq i\leq|F|$ . Since the total geodesic curvature $L_{i}$ is continuous on $\mathbb{R}^{|F|}_{\geq 0}$ , then we have $L_{i}(k^{n})\to L_{i}(k)\leavevmode\nobreak\ (n\to+\infty)$ , $1\leq i\leq m$ , i.e. $\mathcal{E}_{1}(k^{n})\to\mathcal{E}_{1\cdots m}(k)\leavevmode\nobreak\ (n\to+\infty)$ . Hence the map $\mathcal{E}$ is continuous.

For any $L\in\bar{\partial}\mathcal{L}$ , there exists a small enough neighborhood $U$ of $L$ such that $\mathcal{E}^{-1}(U)$ is bounded in $\mathbb{R}^{|F|}_{\geq 0}$ . We choose a sequence $\{L^{n}\}_{n=1}^{+\infty}\subset U$ such that $\lim_{n\to+\infty}L^{n}=L$ . By $\{k^{n}\}_{n=1}^{+\infty}\subset\mathbb{R}_{\geq 0}^{|F|}$ and $k$ we denote the image of $\{L^{n}\}_{n=1}^{+\infty}$ and $L$ under the map $\mathcal{E}^{-1}$ , respectively. Then we only need to show that $k^{n}\to k\leavevmode\nobreak\ (n\to+\infty)$ .

If $k^{n}\nrightarrow k\leavevmode\nobreak\ (n\to+\infty)$ , since $\{k^{n}\}_{n=1}^{+\infty}\subset\mathcal{E}^{-1}(U)$ and $\mathcal{E}^{-1}(U)$ is bounded, then there exists a subsequence of $\{k^{n}\}_{n=1}^{+\infty}$ , which is still denoted by $\{k^{n}\}_{n=1}^{+\infty}$ , such that $k^{n}\to k^{\prime}\leavevmode\nobreak\ (n\to+\infty)$ , where $k^{\prime}\neq k$ . Since $\mathcal{E}$ is continuous, then $\mathcal{E}(k^{n})=L^{n}\to\mathcal{E}(k^{\prime})\leavevmode\nobreak\ (n\to+\infty)$ . Besides, we know that $L^{n}\to L=\mathcal{E}(k)\leavevmode\nobreak\ (n\to+\infty)$ , then we obtain $\mathcal{E}(k)=\mathcal{E}(k^{\prime})$ . This is a contradiction since the map $\mathcal{E}$ is injective. Hence the map $\mathcal{E}^{-1}$ is continuous, $\mathcal{E}$ is a homeomorphism from $\mathbb{R}^{|F|}_{\geq 0}$ to $\mathcal{L}$ . ∎

By above argument, we completed the proof of Theorem 1.8.

8 Convergence of prescribed combinatorial Ricci flows for circle pattern metrics

Given $\hat{L}=(\hat{L}_{1},\cdots,\hat{L}_{|F|})^{T}\in\mathcal{L}_{1}$ , then we can define the prescribed combinatorial Ricci flows as follows, i.e.

(8.1)

\frac{dk_{i}}{dt}=-(L_{i}-\hat{L}_{i})k_{i},\leavevmode\nobreak\ \leavevmode\nobreak\ \forall i\in F.

Using change of variables $K_{i}=\ln k_{i}$ , we can rewrite flows (8.1) as the following equivalent prescribed combinatorial Ricci flows

(8.2)

\frac{dK_{i}}{dt}=-(L_{i}-\hat{L}_{i}),\leavevmode\nobreak\ \leavevmode\nobreak\ \forall i\in F.

Theorem 8.1.

Given a closed topological surface $S$ with a cellular decomposition $\Sigma=(V,E,F)$ , the weight $\Phi\in(0,\frac{\pi}{2})^{|E|}$ and the prescribed total geodesic curvature $\hat{L}=(\hat{L}_{1},\cdots,\hat{L}_{|F|})^{T}\in\mathcal{L}_{1}$ on the face set $F$ . For any initial geodesic curvature $k(0)\in\mathbb{R}_{>0}^{|F|}$ , the solution of the prescribed combinatorial Ricci flows (8.1) exists for all time $t\in[0,+\infty)$ and is unique.

Proof.

We can only consider the equivalent flows (8.2). Since all $-(L_{i}-\hat{L}_{i})$ are smooth functions on $\mathbb{R}_{>0}^{|F|}$ , by Peano’s existence theorem in ODE theory, we know that the solution of flows (8.2) exists on $[0,\epsilon)$ , where $\epsilon>0$ .

By Gauss-Bonnet formula, we have that

\text{Area}(B_{e})=2\Phi(e)-L_{i,e}-L_{j,e}>0,\leavevmode\nobreak\ \leavevmode\nobreak\ j\in F_{\{e\}},

hence we obtain $0<L_{i,e}<2\Phi(e)$ . Since $L_{i}=\sum_{e\in E_{i}}L_{i,e}$ , we have that $0<L_{i}<2\sum_{e\in E_{i}}\Phi(e)$ . Besides, since $\hat{L}\in\mathcal{L}_{1}$ , by definition, we have $0<\hat{L}_{i}<\sum_{f\in F}\hat{L}_{f}<2\sum_{e\in E_{F}}\Phi(e)$ . Then we obtain that

|L_{i}-\hat{L}_{i}|\leq|L_{i}|+|\hat{L}_{i}|\leq 2\sum_{e\in E_{i}}\Phi(e)+2\sum_{e\in E_{F}}\Phi(e)<+\infty.

Hence $|L_{i}-\hat{L}_{i}|$ is uniformly bounded by a constant, which depends only on the weight $\Phi$ and cellular decomposition $\Sigma$ . By the extension theorem of solution in ODE theory, the solution of flows (8.2) exists for all time $t\in[0,+\infty)$ . By existence and uniqueness theorem of solution in ODE theory, the solution of flows (8.2) is unique. ∎

We need the following result in the theory of negative gradient flows.

Lemma 8.2.

([10], Proposition 2.13) Let $h:\mathbb{R}^{n}\rightarrow\mathbb{R}$ be a smooth convex function and let $\xi:[0,+\infty)\rightarrow\mathbb{R}^{n}$ be a negative gradient flow of $h$ . It holds for any $\tau>0$ and $\xi^{*}\in\mathbb{R}^{n}$ that

|\nabla h(\xi(\tau))|^{2}\leq\left|\nabla h\left(\xi^{*}\right)\right|^{2}+\frac{1}{\tau^{2}}\left|\xi^{*}-\xi(0)\right|^{2}.

Then we can prove the Theorem 1.9.

Proof of Theorem 1.9.

We can only consider the equivalent flows (8.2). By Theorem 8.1, we suppose that the solution of flows (8.2) is $K(t)$ , $t\in[0,+\infty)$ . We construct a function $\tilde{\Lambda}=\Lambda-\sum_{f\in F}K_{f}\hat{L}_{f}$ , by Corollary 5.2, the function $\tilde{\Lambda}$ is convex on $\mathbb{R}^{|F|}$ . Besides, we have that $\nabla\tilde{\Lambda}=\nabla\Lambda-\hat{L}=L-\hat{L},$ then we obtain

\frac{dK}{dt}=-\nabla\tilde{\Lambda}(K(t)).

Since $\hat{L}\in\mathcal{L}_{1}$ , by Theorem 5.3, there exists the unique $\hat{K}\in\mathbb{R}^{|F|}$ such that $\nabla\Lambda(\hat{K})=L(\hat{K})=\hat{L}$ . By Lemma 8.2, for any $t>0$ , we have

|\nabla\tilde{\Lambda}(K(t))|^{2}\leq|\nabla\tilde{\Lambda}(\hat{K})|^{2}+\frac{1}{t^{2}}|\hat{K}-K(0)|^{2},

i.e. we obtain

|L(K(t))-L(\hat{K})|^{2}\leq\frac{1}{t^{2}}|\hat{K}-K(0)|^{2}\to 0\leavevmode\nobreak\ (t\to+\infty).

Hence we know that $L(K(t))\to L(\hat{K})=\hat{L}\leavevmode\nobreak\ (t\to+\infty)$ , i.e. $\nabla\Lambda(K(t))\to\nabla\Lambda(\hat{K})\leavevmode\nobreak\ (t\to+\infty)$ . Then by Theorem 5.3, we have that $K(t)\to\hat{K}\leavevmode\nobreak\ (t\to+\infty).$ This completes the proof. ∎

9 Convergence of prescribed combinatorial Ricci flows for degenerated circle pattern metrics

Given the prescribed total geodesic curvature $\hat{L}\in\mathcal{L}_{i_{1}\cdots i_{m}}$ , where $1\leq m\leq|F|-1$ , $1\leq i_{1}<\cdots<i_{m}\leq|F|$ , we can define the prescribed combinatorial Ricci flows as follows, i.e.

(9.1)

\frac{dk_{i}}{dt}=-(L_{i}-\hat{L}_{i})k_{i},\leavevmode\nobreak\ \leavevmode\nobreak\ \forall i\in F_{i_{1}\cdots i_{m}},\leavevmode\nobreak\ \frac{dk_{i}}{dt}=-L_{i}k_{i},\leavevmode\nobreak\ \leavevmode\nobreak\ \forall i\in F\setminus F_{i_{1}\cdots i_{m}},

where $F_{i_{1}\cdots i_{m}}=\{i_{1},\cdots,i_{m}\}$ . Then we study the existence and convergence of solution of the flows (9.1).

Using change of variables $K_{i}=\ln k_{i}$ , we can rewrite flows (9.1) as the following equivalent prescribed combinatorial Ricci flows

(9.2)

\frac{dK_{i}}{dt}=-(L_{i}-\hat{L}_{i}),\leavevmode\nobreak\ \leavevmode\nobreak\ \forall i\in F_{i_{1}\cdots i_{m}},\leavevmode\nobreak\ \frac{dK_{i}}{dt}=-L_{i},\leavevmode\nobreak\ \leavevmode\nobreak\ \forall i\in F\setminus F_{i_{1}\cdots i_{m}}.

By similar argument in the proof of Theorem 8.1, we have the following theorem.

Theorem 9.1.

Given a closed topological surface $S$ with a cellular decomposition $\Sigma=(V,E,F)$ , the weight $\Phi\in(0,\frac{\pi}{2})^{|E|}$ and the prescribed total geodesic curvature $\hat{L}\in\mathcal{L}_{i_{1}\cdots i_{m}}$ on the face set $F$ . For any initial geodesic curvature $k(0)\in\mathbb{R}_{>0}^{|F|}$ , the solution of the prescribed combinatorial Ricci flows (9.1) exists for all time $t\in[0,+\infty)$ and is unique.

For simplicity, we consider the set $F_{m}=\{1,\cdots,m\}$ , where $1\leq m\leq|F|-1$ . Given $\Phi\in(0,\frac{\pi}{2})^{|E|}$ , $\hat{L}=(\hat{L}_{1},\cdots,\hat{L}_{m},0,\cdots,0)^{T}\in\mathcal{L}_{1\cdots m}$ , then we study the following prescribed combinatorial Ricci flows, i.e.

(9.3)

\frac{dk_{i}}{dt}=-(L_{i}-\hat{L}_{i})k_{i},\leavevmode\nobreak\ \leavevmode\nobreak\ 1\leq i\leq m,\leavevmode\nobreak\ \frac{dk_{i}}{dt}=-L_{i}k_{i},\leavevmode\nobreak\ \leavevmode\nobreak\ m+1\leq i\leq|F|.

We can also consider the equivalent prescribed combinatorial Ricci flows, i.e.

(9.4)

\frac{dK_{i}}{dt}=-(L_{i}-\hat{L}_{i}),\leavevmode\nobreak\ \leavevmode\nobreak\ 1\leq i\leq m,\leavevmode\nobreak\ \frac{dK_{i}}{dt}=-L_{i},\leavevmode\nobreak\ \leavevmode\nobreak\ m+1\leq i\leq|F|.

Remark 9.2.

For any $\tilde{K}=(\tilde{K}_{1},\cdots,\tilde{K}_{m})^{T}\in\mathbb{R}^{m}$ , we have that $\nabla\Lambda_{1}(\tilde{K})=(L_{1}(\tilde{K}),\cdots,L_{m}(\tilde{K}))^{T}$ . By (6.1), we know that $L_{i}(\tilde{K})(1\leq i\leq m)$ is actually the total geodesic curvature at the face $i$ when the radii $\mathbf{r}=(\operatorname{arccot}e^{\tilde{K}_{1}},\cdots,\operatorname{arccot}e^{\tilde{K}_{m}},\frac{\pi}{2},\cdots,\frac{\pi}{2})^{T}$ or when the geodesic curvatures $k=(e^{\tilde{K}_{1}},\cdots,e^{\tilde{K}_{m}},0,\cdots,0)^{T}$ or when the $K=(\tilde{K}_{1},\cdots,\tilde{K}_{m},-\infty,\cdots,-\infty)^{T}$ .
If we use geodesic curvatures $\tilde{k}=(\tilde{k}_{1},\cdots,\tilde{k}_{m})^{T}\in\mathbb{R}^{m}_{>0}$ as the variable of $L_{i}$ , then the $L_{i}(\tilde{k})(1\leq i\leq m)$ is actually the total geodesic curvature at the face $i$ when the geodesic curvatures $k=(\tilde{k}_{1},\cdots,\tilde{k}_{m},0,\cdots,0)^{T}$ .

For $\tilde{k}=(\tilde{k}_{1},\cdots,\tilde{k}_{m})^{T}\in\mathbb{R}^{m}_{>0}$ , we construct the prescribed combinatorial Ricci flows as follows, i.e.

(9.5)

\frac{d\tilde{k}_{i}}{dt}=-(L_{i}(\tilde{k})-\hat{L}_{i})\tilde{k}_{i},\leavevmode\nobreak\ \leavevmode\nobreak\ 1\leq i\leq m.

Using change of variables $\tilde{K}_{i}=\ln\tilde{k}_{i}$ , we can rewrite flows (9.5) as the following equivalent prescribed combinatorial Ricci flows

(9.6)

\frac{d\tilde{K}_{i}}{dt}=-(L_{i}(\tilde{K})-\hat{L}_{i}),\leavevmode\nobreak\ \leavevmode\nobreak\ 1\leq i\leq m.

By similar argument in the proof of Theorem 8.1, we have the following theorem.

Theorem 9.3.

Given a closed topological surface $S$ with a cellular decomposition $\Sigma=(V,E,F)$ , the weight $\Phi\in(0,\frac{\pi}{2})^{|E|}$ and the prescribed total geodesic curvature $\hat{L}\in\mathcal{L}_{1\cdots m}$ on the face set $F$ . For any initial geodesic curvature $\tilde{k}(0)\in\mathbb{R}_{>0}^{m}$ on the face set $F_{m}$ and 0 on the face set $F\setminus F_{m}$ , the solution of the prescribed combinatorial Ricci flows (9.5) exists for all time $t\in[0,+\infty)$ and is unique.

Then we study the convergence of solution to the flows (9.5).

Theorem 9.4.

Given a closed topological surface $S$ with a cellular decomposition $\Sigma=(V,E,F)$ , the weight $\Phi\in(0,\frac{\pi}{2})^{|E|}$ and the prescribed total geodesic curvature $\hat{L}=(\hat{L}_{1},\cdots,\hat{L}_{m},0\cdots,0)^{T}\in\mathcal{L}_{1\cdots m}$ on the face set $F$ . For any initial geodesic curvature $\tilde{k}(0)\in\mathbb{R}_{>0}^{m}$ on the face set $F_{m}$ and 0 on the face set $F\setminus F_{m}$ , the solution of the prescribed combinatorial Ricci flows (9.5) converges to the unique degenerated circle pattern metric with the total geodesic curvature $\tilde{L}=(\hat{L}_{1},\cdots,\hat{L}_{m})^{T}$ on $F_{m}$ and 0 on $F\setminus F_{m}$ up to isometry. Moreover, if the solution converges to the degenerated circle pattern metric $\hat{k}$ , then $\hat{k}\in\mathbb{R}^{|F|}_{1\cdots m}$ .

Proof.

We can only consider the equivalent flows (9.6). By Theorem 9.3, we suppose that the solution of flows (9.6) is $\tilde{K}(t)$ , $t\in[0,+\infty)$ . We construct a function $\tilde{\Lambda}_{1}(\tilde{K})=\Lambda_{1}(\tilde{K})-\sum_{f\in F_{m}}\tilde{K}_{f}\hat{L}_{f}$ . By Corollary 6.2, the function $\tilde{\Lambda}_{1}$ is convex on $\mathbb{R}^{m}$ . Besides, we have that

\frac{\partial\tilde{\Lambda}_{1}(\tilde{K})}{\partial\tilde{K}_{i}}=\frac{\partial\Lambda_{1}(\tilde{K})}{\partial\tilde{K}_{i}}-\hat{L}_{i}=L_{i}(\tilde{K})-\hat{L}_{i},\leavevmode\nobreak\ \leavevmode\nobreak\ 1\leq i\leq m,

then we obtain

\frac{d\tilde{K}}{dt}=-\nabla\tilde{\Lambda}_{1}(\tilde{K}(t)).

Since $\tilde{L}\in\tilde{\mathcal{L}}_{1\cdots m}$ , by Theorem 6.3, there exists the unique $\tilde{K}=(\hat{K}_{1},\cdots,\hat{K}_{m})^{T}\in\mathbb{R}^{m}$ such that $\nabla\Lambda_{1}(\tilde{K})=(L_{1}(\tilde{K}),\cdots,L_{m}(\tilde{K}))^{T}=(\hat{L}_{1},\cdots,\hat{L}_{m})^{T}=\tilde{L}$ . By Lemma 8.2, for any $t>0$ , we have

|\nabla\tilde{\Lambda}_{1}(\tilde{K}(t))|^{2}\leq|\nabla\tilde{\Lambda}_{1}(\tilde{K})|^{2}+\frac{1}{t^{2}}|\tilde{K}-\tilde{K}(0)|^{2},

i.e. we obtain

\sum_{i=1}^{m}(L_{i}(\tilde{K}(t))-\hat{L}_{i})^{2}\leq\frac{1}{t^{2}}|\tilde{K}-\tilde{K}(0)|^{2}\to 0\leavevmode\nobreak\ (t\to+\infty).

Hence for $1\leq i\leq m$ , we know that $L_{i}(\tilde{K}(t))\to\hat{L}_{i}=L_{i}(\tilde{K})\leavevmode\nobreak\ (t\to+\infty)$ and $L(\tilde{K}(t))\to\tilde{L}=L(\tilde{K})\leavevmode\nobreak\ (t\to+\infty)$ . Hence we have that $\nabla\Lambda_{1}(\tilde{K}(t))\to\nabla\Lambda_{1}(\tilde{K})\leavevmode\nobreak\ (t\to+\infty)$ . Then by Theorem 6.3, we have that $\tilde{K}(t)\to\tilde{K}\leavevmode\nobreak\ (t\to+\infty)$ . This completes the proof. ∎

Remark 9.5.

By Remark 9.2 and Theorem 9.4, we know that $(\tilde{K}_{1}(t),\cdots,\tilde{K}_{m}(t),-\infty,\cdots,-\infty)^{T}\\ \to(\hat{K}_{1},\cdots,\hat{K}_{m},-\infty,\cdots,-\infty)^{T}\leavevmode\nobreak\ (t\to+\infty)$ and $L_{i}(\tilde{K}_{1}(t),\cdots,\tilde{K}_{m}(t),-\infty,\cdots,-\infty)\to\hat{L}_{i}\leavevmode\nobreak\ (t\to+\infty)$ , $1\leq i\leq m$ . Besides, we have that $\hat{L}_{i}=L_{i}(\hat{K}_{1},\cdots,\hat{K}_{m},-\infty,\cdots,-\infty)$ .

For any $k\in\mathbb{R}_{i_{1}\cdots i_{m}}^{|F|}$ , then $\tilde{k}=(k_{i_{1}},\cdots,k_{i_{m}})\in\mathbb{R}_{>0}^{m}$ . Given $\hat{L}\in\mathcal{L}_{i_{1}\cdots i_{m}}$ , we construct the prescribed combinatorial Ricci flows as follows, i.e.

(9.7)

\frac{d\tilde{k}_{i}}{dt}=-(L_{i}(\tilde{k})-\hat{L}_{i})\tilde{k}_{i},\leavevmode\nobreak\ \leavevmode\nobreak\ \forall i\in F_{i_{1}\cdots i_{m}}.

Using change of variables $\tilde{K}_{i}=\ln\tilde{k}_{i}$ , we can rewrite flows (9.7) as the following equivalent prescribed combinatorial Ricci flows

(9.8)

\frac{d\tilde{K}_{i}}{dt}=-(L_{i}(\tilde{K})-\hat{L}_{i}),\leavevmode\nobreak\ \leavevmode\nobreak\ \forall i\in F_{i_{1}\cdots i_{m}}.

We can use the same techniques above to obtain the long time existence of the solution to prescribed combinatorial Ricci flows (9.7) and Theorem 1.11.

Then we study the convergence of solution to the flows (9.1).

Theorem 9.6.

Given a closed topological surface $S$ with a cellular decomposition $\Sigma=(V,E,F)$ , the weight $\Phi\in(0,\frac{\pi}{2})^{|E|}$ and the prescribed total geodesic curvature $\hat{L}=(\hat{L}_{1},\cdots,\hat{L}_{m},0\cdots,0)^{T}\in\mathcal{L}_{1\cdots m}$ on the face set $F$ . For any initial geodesic curvature $k(0)\in\mathbb{R}_{>0}^{|F|}$ , the solution of the prescribed combinatorial Ricci flows (9.3) converges to the unique degenerated circle pattern metric with the total geodesic curvature $\hat{L}$ up to isometry. Moreover, if the solution converges to the degenerated circle pattern metric $\hat{k}$ , then $\hat{k}\in\mathbb{R}^{|F|}_{1\cdots m}$ .

Proof.

We consider the equivalent flows (9.4). By Theorem 9.1, we suppose that the solution of flows (9.4) is $K(t)$ , $t\in[0,+\infty)$ . We construct a function $\bar{\Lambda}=\Lambda-\sum_{f\in F_{m}}K_{f}\hat{L}_{f}$ . By Corollary 5.2, the function $\bar{\Lambda}$ is convex on $\mathbb{R}^{|F|}$ . Besides, we have that $\nabla\bar{\Lambda}=\nabla\Lambda-\hat{L}=L-\hat{L},$ then we obtain

\frac{dK}{dt}=-\nabla\bar{\Lambda}(K(t)).

By Lemma 8.2, for any $t>0$ and $\bar{K}(t)=(\tilde{K}_{1}(t),\cdots,\tilde{K}_{m}(t),-\ln t,\cdots,\ln t)^{T}\in\mathbb{R}^{|F|}$ , we have

|\nabla\bar{\Lambda}(K(t))|^{2}\leq|\nabla\bar{\Lambda}(\bar{K}(t))|^{2}+\frac{1}{t^{2}}|\bar{K}(t)-K(0)|^{2},

By Theorem 9.4, we obtain that $\tilde{K}_{i}(t)\to\hat{K}_{i}\leavevmode\nobreak\ (t\to+\infty)$ , $1\leq i\leq m$ . Hence we know that the function $\tilde{K}_{i}(t)\leavevmode\nobreak\ (1\leq i\leq m)$ is bounded, then we have

\frac{1}{t^{2}}|\bar{K}(t)-K(0)|^{2}\to 0\leavevmode\nobreak\ (t\to+\infty).

Besides, we have

|\nabla\bar{\Lambda}(\bar{K}(t))|^{2}=\sum_{i=1}^{m}(L_{i}(\bar{K}(t))-\hat{L}_{i})^{2}+\sum_{i=m+1}^{|F|}L_{i}^{2}(\bar{K}(t)).

Since $\bar{K}(t)=(\tilde{K}_{1}(t),\cdots,\tilde{K}_{m}(t),-\ln t,\cdots,\ln t)^{T}\to(\hat{K}_{1},\cdots,\hat{K}_{m},-\infty,\cdots,-\infty)^{T}\leavevmode\nobreak\ (t\to+\infty)$ , by Remark 9.5, we have that $L_{i}(\bar{K}(t))\to L_{i}(\hat{K}_{1},\cdots,\hat{K}_{m},-\infty,\cdots,-\infty)=\hat{L}_{i}\leavevmode\nobreak\ (t\to+\infty)$ , $1\leq i\leq m$ . Besides, the geodesic curvature $k_{i}=e^{-\ln t}=\frac{1}{t}\to 0\leavevmode\nobreak\ (t\to+\infty)$ , $m+1\leq i\leq|F|$ , then $L_{i}(\bar{K}(t))\to 0\leavevmode\nobreak\ (t\to+\infty)$ , $m+1\leq i\leq|F|$ . Hence we have

|\nabla\bar{\Lambda}(\bar{K}(t))|^{2}\to 0\leavevmode\nobreak\ (t\to+\infty).

Then we obtain

|\nabla\bar{\Lambda}(K(t))|^{2}\to 0\leavevmode\nobreak\ (t\to+\infty),

i.e. $L(K(t))\to\hat{L}=(\hat{L}_{1},\cdots,\hat{L}_{m},0\cdots,0)^{T}\leavevmode\nobreak\ (t\to+\infty)$ .

We define some functions $k_{i}(t)=e^{K_{i}(t)}$ , $1\leq i\leq|F|$ and some constants $\hat{k}_{i}=e^{\hat{K}_{i}}$ , $1\leq i\leq m$ . Then it is easy to know that $k(t)=(k_{1}(t),\cdots,k_{|F|}(t))^{T}$ is the solution of the prescribed combinatorial Ricci flows (9.3) and $\hat{k}=(\hat{k}_{1},\cdots,\hat{k}_{m},0,\cdots,0)^{T}\in\mathbb{R}_{1\cdots m}^{|F|}$ .

For simplicity, we can define $L(k):=L(\ln k_{1},\cdots,\ln k_{|F|})$ . Then we have $\hat{L}_{i}=L_{i}(\hat{k}),\leavevmode\nobreak\ 1\leq i\leq m$ and $\hat{L}=L(\hat{k})$ . Hence we know that $L(k(t))\to L(\hat{k})=\hat{L}\leavevmode\nobreak\ (t\to+\infty)$ . Since $\{k(t)\leavevmode\nobreak\ |\leavevmode\nobreak\ t\in[0,+\infty)\}\subset\mathbb{R}_{>0}^{|F|}$ and $\hat{k}\in\mathbb{R}_{1\cdots m}^{|F|}$ , by (7.4), we have that $\mathcal{E}(k(t))\to\mathcal{E}(\hat{k})\leavevmode\nobreak\ (t\to+\infty)$ . By Theorem 7.1, $\mathcal{E}^{-1}$ is continuous, then we have that $k(t)\to\hat{k}\leavevmode\nobreak\ (t\to+\infty)$ . This completes the proof. ∎

We can use the same techniques above to obtain the Theorem 1.12.

10 Acknowledgments

Guangming Hu is supported by NSF of China (No. 12101275). Ziping Lei is supported by NSF of China (No. 12122119). Puchun Zhou is supported by Shanghai Science and Technology Program [Project No. 22JC1400100].

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Guangming Hu, [email protected]
College of Science, Nanjing University of Posts and Telecommunications, Nanjing, 210003, P.R. China.

Ziping Lei, [email protected]
School of Mathematics, Renmin University of China, Beijing, 100872, P.R. China.

Yu Sun, [email protected]
School of mathematics and physics, Nanjing institute of technology, Nanjing, 211100, P.R. China.

Puchun Zhou, [email protected]
School of Mathematical Sciences, Fudan University, Shanghai, 200433, P.R. China

	$\displaystyle\nabla\Lambda:\mathbb{R}^{\|F\|}$	$\displaystyle\longrightarrow\mathbb{R}^{\|F\|}_{>0}$
	$\displaystyle K=(K_{1},\cdots,K_{\|F\|})^{T}$	$\displaystyle\mapsto L=(L_{1},\cdots,L_{\|F\|})^{T}.$