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Convergence of combinatorial Ricci flows to degenerate circle patterns

We investigate the combinatorial Ricci flow on a surface of nonpositive Euler characteristic when the necessary and sufficient condition for the convergence of the combinatorial Ricci flow is not valid. This observation addresses one of questions raised by B. Chow and F. Luo.


Introduction
On a connected, oriented, closed surface S, although its triangulation T = (V, E, F ) is a topological structure, a metric of constant curvature zero on a weighted triangulation (T, Θ) uniquely determines a pair of a geometric structure of S and a configuration of circles on S realizing the data (T, Θ) (see [5]). Here V, E, F respectively denote the set of all vertices, edges and triangles of T , and we assume that the triangulation T lifts to a triangulation in the universal cover of S. A weighted triangulation (T, Θ) is a triangulation equipped with a weight Θ : E → [0, π/2]. We call r = (r v ) v∈V ∈ R V >0 a metric on (T, Θ) and define the length ℓ(e; r) of an edge e with endpoints v, u with respect to r by ℓ(e; r) := the distance between the centers of two circles on the universal cover of S with radii r v , r u intersecting at angle Θ(e) , which determines the angle at each vertex in triangles of (T, Θ). 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, Θ) having v as one of the vertices. We call K v (r) := 2π − a v (r) the curvature at the vertex v with respect to r. A metric is called a circle pattern metric provided its curvature is identically zero at each vertex. A circle pattern metric on (T, Θ), which connects the topology with the geometry of S, does not always exist. However, a criterion for (T, Θ) to have a circle pattern metric is known, which depends on whether the Euler characteristic χ(S) of S is positive or nonpositive. The case of χ(S) ≤ 0 was first obtained by Thurston [5]. Chow-Luo [2] gave another proof by using the combinatorial Ricci flow, which is a family of metrics {r(t)} t∈[0,T ) that satisfies (π − Θ(e f v(f ) )) + 2πχ(τ U ) < 0, where τ U is the CW-subcomplex of T , consisting of all cells whose vertices are contained in U, and f ∈ 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 f v(f ) in f opposite to v(f ) belongs to U. (III) Given any metric r on (T, Θ), the combinatorial Ricci flow with initial data r exists for all time and converges on R V >0 at infinity. We remark that, instead of (II), the following condition (II ′ ) is used in [5,Theorem 13.7.1] and [2,Theorem 1.2].: (II ′ ) (T, Θ) satisfies the following two conditions.: (i) If e 1 , e 2 , e 3 ∈ E form a null-homotopic loop and if 3 i=1 Θ(e i ) ≥ π, then these three edges form the boundary of a triangle of T . (ii) If e 1 , e 2 , e 3 , e 4 ∈ E form a null-homotopic loop and if 4 i=1 Θ(e i ) = 2π, then these four edges form the boundary of the union of two adjacent triangles of T . The condition (II) implies (II ′ ), and they are equivalent to each other if χ(S) < 0. Although it is stated in [2,Proposition 1.3] that (II) follows from (II ′ ) even if χ(S) = 0, there is a counterexample (see Remark 2.5). Note that Thurston [5, §13.7] treated a cell-division of not only triangles but also quadrangles by adding following condition: (IV) Θ(e) = π/2 if e is an edge of a quadrilateral. Such a weighted quadrangle can be eliminated by subdivision into two weighted triangles by a diagonal of weight 0.
Chow-Luo [2, §7] raised some questions about the combinatorial Ricci flow, one of which is to investigate the combinatorial Ricci flow when (II) is not valid. We address it when the maximum of φ(U) over all nonempty proper subsets U of V is zero. Theorem 1.2. Let (T, Θ) be a weighted triangulation of a surface S of nonpositive Euler characteristic such that φ(U) ≤ 0 holds for any U ⊂ V and Z T := {z ∈ V | there exists a proper subset Z of V such that z ∈ Z and φ(Z) = 0} is nonempty. Then for any metric r on (T, Θ), the combinatorial Ricci flow {r(t)} t≥0 with initial data r does not converge on R V >0 at infinity, however lim t→∞ K v (r(t)) = 0 holds for any vertex v. On the one hand, for χ(S) = 0, {r(t)} t≥0 does not converge on R V ≥0 at infinity. However if we fix an arbitrary v ∈ V \ Z T , then the limit exists for any u ∈ V , where Z T = {z ∈ V | ρ z = 0} holds and (ρ u ) u∈V \Z T is a unique circle patten metric with normalization ρ v = 1 on a certain weighted triangulation with vertices V \ Z T .
On the other hand, for χ(S) < 0, {r(t)} t≥0 converges on R V ≥0 at infinity, where we have Z T = {z ∈ V | lim t→∞ r z (t) = 0} holds and the limit of (r v (t)) v∈V \Z T at infinity is a unique circle patten metric on a certain weighted triangulation with vertices V \ Z T .
It should be mentioned that weighted triangulations on a torus not satisfying (II ′ -i) are previously studied by Yamada [6], where a part of Theorem 1.2 was stated and partially proved for |V | ≤ 3 by analyzing individual cases. We present a completely different proof for the general cases, based on an infinitesimal description of degenerate circle patten metrics.
The organization of this paper is as follows. Section 2 is devoted to recalling the relevant facts and terminology from weighted triangulations, circle pattern metrics and the combinatorial Ricci flow. In Section 3, we give the infinitesimal description of degenerate circle patten metrics and prove Theorem 1.2.
Acknowledgment. The author would like to thank Professor Hiroshi Matsuzoe for bringing this topic to her attention and stimulating discussions. She also expresses her thanks to Mr. Masahiro Yamada for seminars and discussions.

Preliminaries
Throughout this paper, let S be a connected, oriented, closed surface of nonpositive Euler characteristic. Most of the results in this section hold true for a general surface if modified appropriately. See [5], [2], [3] [4] and references therein for details.

Weighted Triangulation.
A triangulation T = (V, E, F ) of S is always assumed to lift to a triangulation in the universal cover of S, where V, E, F respectively denote the set of all vertices (0-cells), edges (1-cells) and triangles (2-cells) of T . In particular, there is no null-homotopic loop formed by at most two edges. Although a 2-cell f ∈ F is an open set in S, if its boundary ∂f contains a vertex v ∈ V and an edge e ∈ E, then we write v ∈ f and e ⊂ f instead of v ∈ ∂f and e ⊂ ∂f , respectively. Similarly for U ⊂ V , we write f ∩ U and e ∩ U instead of ∂f ∩ U and ∂e ∩ U, respectively. Let us fix a weighted triangulation (T, Θ) of S, where Θ : E → [0, π/2] is a weight. For f ∈ Lk(U), we denote by v(f ) a unique element in U such that v(f ) ∈ f . Definition 2.3. For U ⊂ V , let τ U be the CW-subcomplex of T , consisting of all cells whose vertices are contained in U. We say that U is connected if τ U is connected. Set However the converse is not true in general. For j = 1, 2, 3, define where the right-hand side is additive with respect to connected components of τ U . Θ(e f v(f ) ) > π, |Lk(U)| = 3.
If φ(U) = 0, then τ U is contractible and we have Remark 2.5. If e 1 , e 2 , e 3 ∈ E form a null-homotopic loop but not the boundary of a triangle of T , then there exits U ⊂ V such that τ U is contractible and Thus the contrapositive of the implication from (II) to (II ′ -i) and similarly to (II ′ -ii) holds.
Conversely, if there exists a nonempty proper connected subset U of V such that φ(U) ≥ 0,  However, for a weighted triangulation of a torus on which any triangle has three distinct vertices, (II ′ ) implies (II) as well as the case of χ(S) < 0. Definition 2.6. For k = 3, 4, define such that e, e 1 , e 2 form a null-homotopic loop. Remark 2.8. For Z ∈ Z 4 , there are two diagonal edges in f Z , neither of which belongs to E.
and if the weight of e is 0, then, at each endpoint of e, the angle in f Z is same as the sum of the angles in f 1 Z , f 2 Z . Thus f Z can be divided into two weighted triangles as well as a weighted quadrangle satisfying (IV).

Circle Pattern Metrics.
For a weighted triangulation (T, Θ) of S, we call r ∈ R V >0 a metric on (T, Θ). We define the length ℓ(e; r) of an edge e with endpoints v, u with respect to r by ℓ(e; r) := which is the distance between the centers of two circles on the universal cover of S with radii r v , r u intersecting at angle Θ(e). Needless to say, if χ(S) = 0, then S is a torus and its universal cover is the Euclidean plane E 2 . On the other hand, the universal cover of a surface of negative Euler characteristic is the hyperbolic plane H 2 . Let us first recall the existence of a unique circle pattern in the universal cover of S and properties of the triangle formed by the centers of the three circles.  [3]) For any three nonobtuse angles Θ 1 , Θ 2 , Θ 3 ∈ [0, π/2] and any three radii r 1 , r 2 , r 3 > 0, there is a configuration of three circles C 1 , C 2 , C 3 in both E 2 and H 2 , unique up to isometry, having radii r i and meeting in angles Θ i . Moreover, for the triangle formed by the centers of the three circles C 1 , C 2 , C 3 , if we regard the angle θ i at the center of C i in the triangle as a function of r 1 , r 2 , r 3 , then we have Although three edges of any triangle of T differ from each other, vertices of a triangle may happen to coincide. When one vertex v of f ∈ F is different from the remaining two vertices of f , we denote by e f v the edge opposite to v in f , and by θ f v (r) the angle at v in f with respect to r. In this case, we have where e, e ′ are the edges in f adjacent to v. Let a v (r) be the cone angle at the vertex v with respect to r, which is the sum of each angle at v in all triangles of (T, Θ) having v as one of the vertices. We define by K v (r) := 2π−a v (r) the curvature at the vertex v with respect to r. For χ(S) = 0, we easily check the invariance of the curvature under uniform scalings of the metric. Moreover, the total curvature is always zero due to the Descartes-Gauss-Bonnet Theorem.
>0 . In addition, for a weighted triangulation of a closed ball D in E 2 and its metric r, if we define the cone angle a v (r) at the vertex v as well as the case of χ(S) = 0 and define the curvature by if v is on the boundary of D, In general, the curvature is bounded below by the function φ.
Furthermore, it holds for any nonempty proper subset A curvature uniquely determines a metric, up to a scalar multiple if χ(S) = 0. This follows from the fact that the curvature can be regarded as the gradient of a convex function. To see this, we define an injective map X S : then ψ is strictly convex when restricted to X S (P ) and K is injective when restricted to P .
On the other hand, if χ(S) < 0, then ψ is strictly convex, consequently K is injective.
where ψ is given in Proposition 2.12. Thus the combinatorial flow can be regarded as the gradient flow of a convex function and the combinatorial Ricci flow exists for all time (see also [2,Proposition 3.4]). Moreover, for χ(S) = 0, the product v∈V r v (t) is constant in t ≥ 0 since we deduce from Proposition 2.10 that As mentioned in Theorem 1.

2])
Let h : R n → R be a smooth convex function and ξ : [0, ∞) → R n a gradient flow of h. It holds for any τ > 0 and ξ * ∈ R n that where the inequality follows from the convexity of h. Hence |∇h(ξ(t))| 2 is nonincreasing, which leads to Since the convexity of h asserts Combining this with the above inequality implies We apply the convexity of h again to have Substituting this into the above inequality and dividing with 2/τ 2 yields the proposition.

Infinitesimal description of degenerate circle patten metric
Throughout this section, we fix a weighted triangulation (T, Θ) of S such that φ(U) ≤ 0 holds for any U ⊂ V and Z T = {z ∈ Z |Z ∈ Z} = ∅. Then the sets defined by and Θ 0 := Θ| E 0 form a weighted cell-division (T 0 , Θ 0 ) of S into triangles and quadrangles satisfying (II) and (IV). For n ≥ 1, let us inductively define Z n−1 , Z n , V n by For each Z ∈ Z ∪ {V }, define where we set V V := ∅. Then the correspondence from {V Z } Z∈Z∪{V } to {U ∈ C(V n )} 0≤n≤N is bijective. It turns out that V ′ = Z T and V V = V 0 .
3.1. Key lemmas. Fix an arbitrary Z ∈ Z. Moreover for each U ∈ C(Z ′ ) ∩ Z 4 , we fix a choice of a diagonal edge e U in f U , and divide f U into two triangles f 1 U , f 2 U by e U . We define by Θ Z (e) = Θ(e) for e ∈ E, and 0 otherwise. For j = 1, 2, 3 and v ∈ V Z , puť there is a continuous map c from a closed ball D in R 2 to the closure of f Z so that the restriction of c to the interior of D is a homeomorphism to f Z . The preimages of In order to give an infinitesimal description of degenerate circle patten metrics, let us define the curvature-like function K Z : R V >0 → R V Z as follows. Since H 2 is infinitesimally identified with R 2 , regardless of the Euler characteristic of S, we define the length L(e; r) of e ∈ E Z with endpoints u, v ∈ V Z with respect to r ∈ R V >0 by L(e; r) := r 2 v + r 2 u + 2r v r u cos Θ(e) as well as χ(S) = 0. Similarly, the angle θ f,3 v (r) at v in f ∈ F 3 Z with respect to r is defined by where e, e ′ are the edges in f adjacent to v. Moreover, given f ∈ F 2 Z with vertices v, u ∈ V Z and w ∈ V Z , define the angle θ f,2 v (r) at v in f with respect to r by θ f,2 v (r) := arccos .
Finally, define the angle which is invariant under uniform scalings of the metric. We first prove that K Z is independent of the choice of diagonal edges. In what follows, the adjacency, connectivity of vertices and the link of a subset of vertices are respectively defined with respect to (T, Θ), not (T D , Θ D ). Proposition 3.1. Fix v ∈ V Z and take U ∈ C(Z ′ ) ∩ Z 4 such that v ∈ f U . Let v 1 , v 2 , u be the vertices of f U other than v such that v 1 , v 2 are adjacent to v. We divide f U into the two triangles f 1 , f 2 (resp. f, f ′ ) by the diagonal edge in f U not intersecting v i (resp. v), where v i ∈ f i for i = 1, 2 (resp. v ∈ f ). Then it holds for any r ∈ R V >0 that where j, j 1 , j 2 ∈ {1, 2, 3} satisfy f ∈F j Z and f i ∈F j i Z for i = 1, 2. Proof. We first observe for i = 1, 2 that In the case of j = 1, we see that θ f,1 v (r) = π. If u ∈ V Z (resp. u ∈ V Z ), then we find j i = 2 (resp. j i = 1) and Assume j = 2 and v 1 ∈ V Z . It turns out that If moreover u ∈ V Z (resp. u ∈ V Z ), then (j 1 , j 2 ) = (3, 2) (resp. (j 1 , j 2 ) = (2, 1)) and Finally, if j = 3, then we find As well as the case of j = 1, 2, regardless of the position of u, we have These with the trigonometric addition formulae conclude the proposition.
Given any v ∈ V Z and U ∈ C(Z ′ ) ∩ Z 4 such that v ∈ f U , we denote by e f U v the diagonal edge in f U not intersecting v. Letf U be the triangle obtained by dividing f U by e f U v and containing v. For j = 1, 2, 3, definē , and it is natural to define θ f U ,j v (r) := θf U ,j v (r). Proposition 3.1 asserts ≥0 at infinity and satisfies the following conditions: • We have Z T = {z ∈ V | lim t→∞ r z (t) = 0} and lim t→∞ r v (t) = r v for any v ∈ V 0 . • For each U ∈ Z, if we fix an arbitrary v ∈ V U , then the limit of {(r u (t)/r v (t)) u∈V U } t≥0 at infinity exists and the limit is parallel to the restriction r| V U of r to V U . • For any U ∈ Z ∪ {V } and v ∈ V U , u ∈ U ′ , we have lim t→∞ r u (t)/r v (t) = 0.
Proof. Let u, w be the vertices of f other than v. We can assume that n(w) ≤ n(u) ≤ n(v) without loss of generality. Note that n(v) = 0. For x = u, w, there exists ρ x ∈ R ≥0 such that We have ρ x > 0 if and only if n(x) = n(v), in which case ρ x r x = r v holds. The condition j = 1 is equivalent to n(w) ≤ n(u) < n(v). We similarly see that j = 2 (resp. j = 3) if and only if n(w) < n(u) = n(v) (resp. n(w) = n(u) = n(v)).

Substituting this into
completes the proof of (3.2). To prove (3.3), let us classify triangles .
Let us setf U as f U itself if U ∈ C(Z ′ ) ∩ Z 3 , and as the triangle obtained by dividing f U by e f U v and containing v if U ∈ C(Z ′ )∩Z 4 . For i = 1, 2, let u i , w i be the vertices of f i ∈ F 1 U (v) other than v such that u i ∈ U and w i ∈ V Z ∪ V Z , hence n(w i ) ≤ n(v) < n(u i ). Then w 1 , w 2 become the vertices off U , and e f i u i = ef U w k for (i, k) = (1, 2), (2, 1). By U ∈ Z, we have ). Arguing similarly for (3.2), we have for (i, k) = (1, 2), (2, 1) that Since the condition j = 1 is equivalent to n(w 1 ), n(w 2 ) < n(v), the trigonometric addition formulae with (3.5) imply (3.3). Similarly, the condition j = 2 (resp. j = 3) is equivalent to min{n(w 1 ), n(w 2 )} < max{n(w 1 ), n(w 2 )} = n(v) (resp. n(w 1 ) = n(w 2 ) = n(v)), and we have the desired results.

Proposition 3.4.
Fix v ∈ V 0 and consider a degenerate curve {r(t)} t≥0 of order r ∈ R V >0 .
where we use the notation θf v for the sum of angles at v inf iff contains an edge whose both endpoints are v.
Proof. Suppose that all the vertices of f, f U are different from each other. The relation (3.6) follows from the continuity of the trigonometric and hyperbolic functions. To prove (3.7), we use the same notations in the proof of (3.3). Then (3.4) and (3.5) hold. By V V = ∅, we have n(w i ) = n(v) = 0 < n(u i ) and for (i, k) = (1, 2), (2, 1) that , if χ(S) < 0.
This and the trigonometric addition formulae with (3.5) complete the proof of (3.7). The case that some vertices of f, f U coincide is proved similarly.
Corollary 3.5. Let {r(t)} t≥0 be a degenerate curve of order r. It holds for v ∈ V Z that For the curvature K V on the weighted cell-division (T 0 , Θ 0 ) of S, we have for v ∈ V 0 that (3.9) lim t→∞ K v (r(t)) = K V,v (r).
For U ∈ Z ∪ {V }, since K U (r) depends only on r| V U , we can regard K U : The following corollary immediately follows from the proofs of Propositions 3.3, 3.4. Corollary 3.6. Given U ∈ Z ∪ {V } and v ∈ V U , we assume that a curve {r(t)} t≥0 satisfies for any u ∈ U ′ , w ∈ V U . Moreover for any u ∈ V U , if the limit exists and ρ u > 0, then we have We will show that (T D , Θ D ) and K Z satisfy a series of the following lemmas in a similar way to the proof of the results for (T, Θ) and K mentioned in Section 2.
Lemma 3.7. It holds for any r ∈ R V >0 that v∈V Z K Z,v (r) = 0. Proof. For a degenerate curve {r(t)} t≥0 of order r, Proposition 2.10 and Corollary 3.5 yield where we use the same notations V D , V Z for their preimages under a continuous map c from D to the closure of f Z . In order to analyze the behavior of w∈V Z K D,w (r(t)), define for j = 1, 2, 3 Lemma 3.8. It holds for any nonempty proper subset U of V Z that φ Z (U) < 0.
Proof. Let U be the nonempty subset of V Z of the smallest cardinality such that φ Z (U) ≥ 0. The additivity of the right-hand side of φ Z (U) with respect to connected components of τ Z U leads to the connectivity of τ Z U in T D . By the theory of planar graphs, we have χ(τ Z U ) ≤ 1 with equality if and only if τ Z U is contractible. If Lk Z (U) = ∅, then any two vertices in V D \ U are not joined by e ∈ E D , which contradicts V Z ⊂ V D \ U. Hence Lk Z (U) = ∅. It follows from the fact Θ Z (e) ≤ π/2 that Furthermore, it holds for any nonempty proper subset U of V Z and r Proof. Let U be a subset of V Z , possibly U = ∅, V Z . By an argument similar to τ U , we find , where the inequality follows from Proposition 2.9 for the case of χ(S) = 0. Similarly, for f ∈ F 1 U ∩ F j Z with j = 2, 3 having v ∈ U as one of the vertices, we notice (3.12) lim . This leads to proving the first claim. Moreover, in the case of |V Z | ≥ 2, if U = ∅, V Z , then the connectivity of V Z ensures that at least one of is nonempty, which implies the second claim by (3.11), (3.12).
Remark 3.10. Arguing similarly to Lemma 3.9, we find that the first claim in Proposition 2.11 holds for a sequence in R V >0 converging on [0, ∞] V . We moreover observe from the proof of Lemma 3.9 that, for a sequence {r(n)} n∈N in R V >0 converging on [0, ∞] V as n → ∞, we have lim if and only if it holds for any u ∈ U and v ∈ V \ U adjacent to u that For distinct vertices v, u ∈ V Z of f ∈ F 3 Z , Proposition 2.9 for χ(S) = 0 yields (3.14) Similarly, we find for distinct vertices v, u ∈ V Z of f ∈ F 2 Z that with equality in the inequality if and only if Θ Z (e f v ) = Θ Z (e f u ) = π/2. Thus we conclude ∂x v and the Poincaré lemma together with the regularity of K Z gives the desired ψ Z .
The following fact of linear algebra plays a key role in the proof of the convexity of ψ Z . for any 1 ≤ i, j ≤ n with i = j, then A is semi-positive definite and its kernel is onedimensional with basis (1, . . . , 1). Lemma 3.13. The function ψ Z is convex. Furthermore, K Z is injective when restricted to Proof. For distinct vertices u, v ∈ V Z , we observe from (3.13)-(3.15) and Lemma 3.7 that then Proposition 3.12 leads to the convexity of ψ Z . The injectivity of K Z | P Z follows from the second order Taylor expansion of ψ Z . If there exists v ∈ V Z satisfying (3.16), then |V Z | = 1, consequently K Z | P Z is injective. Indeed for such v, we find F 3 Z (v) = ∅ by (3.14).
≥ 0, which contradicts Lemma 3.8. Arguing similarly, if w, w ′ are joined by some e ∈ E Z , then each of three edges e, e f v , e f ′ v and e, e f u , e f ′ u form a null-homotopic loop and the sum of the weights of the three edges is at least π. However at least one of the three edges do not form the boundary of an element in F D , contradicting Lemma 3.8. Thus w, w ′ are not joined by e ∈ E Z , implying Z ∈ Z 4 and Θ(e) = π/2 for e ∈ {e f v(f ) } f ∈Lk(Z) . Moreover, we have Θ Z (ef v ) = Θ Z (ef ′ v ) = π/2, regardless of the position ofũ,ũ ′ by Z ∈ Z 4 and (3.15). We see thatũ =ũ ′ sinceũ =ũ ′ implies φ Z ({v}) = 0. Ifũ ∈ V Z , then there exists e ∈ {e f v(f ) } f ∈Lk(Z) such that e joins w ′ ,ũ. We find that ef v , e f u , e f ′ u , e form a null-homotopic loop and there exists U V Z such thatũ ′ ∈ U, Lk Z (U) = {ef v , e f u , e f ′ u , e} and φ Z (U) = 0, which never happens by Lemma 3.8. However ifũ ∈ V Z , then by F 3 Z (v) = ∅, there existsf ∈ F 2 Z such that ef w ⊂f . Forŵ ∈f ∩ V Z ,ũ =ũ ′ impliesŵ = w ′ , henceŵ, w are joined by some e ∈ {e f v(f ) } f ∈Lk(Z) . The three edges efũ, efũ, e form a null-homotopic loop and surroundũ, which leads to a contradiction as well asũ ∈ V Z . Thus F 2 Z (v) = ∅, and v is surrounded by elements in F 1 Z , that is, V Z = {v} by the construction of Z. Lemma 3.14. (cf. [5] [2, §6.1], [4, §7]) There exists a unique r ∈ P Z such that K Z (r) = 0.
Proof. Define the hyperplane Q Z of R V Z and the convex polytope Y Z in Q Z respectively by Note that P Z , Y Z are homeomorphic to R |V Z |−1 . Lemmas 3.7, 3.9 yield K Z (P Z ) ⊂ Y Z , and the map K Z : P Z → Y Z is continuously and injectively extended to the map between the one point compactifications of P Z , Y Z . Then by the invariance of domain theorem, the extended map is an open map. Since the one point compactifications of P Z , Y Z are Hausdorff, compact and connected, the extended map is surjective hence K Z (P Z ) = Y Z . By Lemma 3.8, Y Z contains the origin. This with Lemma 3.13 gives a unique r ∈ P Z such that K Z (r) = 0.

Main Theorem.
We prove the following theorem, which implies Theorem 1.2.
Proof of Theorem 1.2. The first claim follows from Theorem 3.18. In the case of χ(S) = 0, v∈V r v (t) is constant in t ≥ 0, which implies that lim t→∞ r v (t) = ∞ for v ∈ V 0 , that is {r(t)} t≥0 does not converges on R V ≥0 at infinity. To prove the rest of the claim, let (T 0 , Θ 0 ) be the weighted cell-devision defined in (3.1). Then a desired weighted triangulation is (T 0 , Θ 0 ) itself if C(Z T ) ∩ Z 4 = ∅, and is obtained by adding diagonal edges of weight 0 to (T 0 , Θ 0 ) if C(Z T ) ∩ Z 4 = ∅.