1 Main results

There are our main results.

Definition 1 Weighted triangulation

A weighted triangulation \((T, \Theta )\) is a triangulation equipped with a weight \(\Theta : E → [0, \pi /2]\).

Definition 2 Circle packing Metric, piecewise linear length, Cone angle

We 1call \(r=\left(r_{v}\right)_{v \in V} \in \mathbb {R}_{{\gt}0}^{V}\) a metric on ( \(\left. T, \Theta \right)\) and define the length \(\ell (e ; r)\) of an edge \(e\) with endpoints \(v, u\) with respect to \(r\) by

\[ \ell (e ; r):=\binom {\text{ the distance between the centers of two circles on the universal }}{\text{ cover of } S \text{ with radii } r_{v}, r_{u} \text{ intersecting at angle } \Theta (e)}, \]

Which determines the angle at each vertex in triangles of \((T, \Theta )\). The cone angle \(a_{v}(r)\) at the vertex \(v\) with respect to \(r\) is the sum of each angle at \(v\) in all triangles of \((T, \Theta )\) having \(v\) as one of the vertices.

Definition 3 combinatorial curvature

We call \(K_v (r) := 2π − a_v (r)\) the curvature at the vertex \(v\) with respect to \(r\)

Definition 4 circle pattern metric

A metric is called a circle pattern metric provided its curvature is identically zero at each vertex.

Definition 5 combinatorial Ricci flow

the combinatorial Ricci flow, which is a family of metrics \(\{ r (t)\} _{t \in [0, T)}\) that satisfies

\[ \frac{d}{d t} r_{v}(t)=-K_{v}(r(t)) \sigma _{S}\left(r_{v}(t)\right), \quad \sigma _{S}\left(r_{v}\right):=\left\{ \begin{array}{ll} r_{v} & \text{ if } \chi (S)=0 \\ \sinh r_{v} & \text{ if } \chi (S){\lt}0 \end{array}\right. \]

Theorem 6 Thurston-Chow-Luo theorem

Let \((T, \Theta )\) be a weighted triangulation of a surface \(S\) of nonpositive Euler characteristic. The following conditions (I), (II), and (III) are equivalent to each other.

(I) There exists a unique circle pattern metric up to a scalar multiple if \(\chi (S)=0\)

(II) It holds for any nonempty proper subset \(U\) of \(V\) that

\[ \phi (U):=-\sum _{f \in \operatorname {Lk}(U)}\left(\pi -\Theta \left(e_{v(f)}^{f}\right)\right)+2 \pi \chi \left(\tau _{U}\right){\lt}0 \]

Where \(\tau _{U}\) is the \(C W\) -subcomplex of \(T\), consisting of all cells whose vertices are contained in \(U\), and \(f \in \operatorname {Lk}(U)\) is an element in \(F\) such that one vertex \(v (f)\) in \(f\) belongs to \(U\) and neither of the endpoints of the edge \(e_{v (f)}^{f}\) in \(f\) opposite \(v (f)\) belongs to \(U\).

(III) Given any metric \(r\) on \((T, \Theta )\), the combinatorial Ricci flow with initial data \(r\) exists for all time and converges on \(\mathbb {R}_{{\gt}0}^{V}\) at infinity.

Definition 7 phi def

For any nonempty proper subset \(U\) of \(V\) that

\[ \phi (U):=-\sum _{f \in \operatorname {Lk}(U)}\left(\pi -\Theta \left(e_{v(f)}^{f}\right)\right)+2 \pi \chi \left(\tau _{U}\right){\lt}0 \]

Theorem 8 Main Theorem

Let \((T, \Theta )\) be a weighted triangulation of a surface \(S\) of nonpositive Euler characteristic such that \(\phi (U) \leq 0\) holds for any \(U \subset V\) 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. Then for any metric \(r\) on \((T, \Theta )\), the combinatorial Ricci flow \(\{ r (t)\} _{t \geq 0}\) with initial data \(r\) does not converge on \(\mathbb {R}_{{\gt}0}^{V}\) at infinity. However

\[ \lim _{t \rightarrow \infty } K_{v}(r(t))=0 \]

Holds for any vertex \(v\).

On the one hand, for \(\chi (S)=0,\{ r (t)\} _{t \geq 0}\) 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 \(\chi (S){\lt}0,\{ r (t)\} _{t \geq 0}\) 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 \(V \backslash Z_{T}\).