CIRCLE PACKINGS ON SURFACES WITH PROJECTIVE STRUCTURES AND UNIFORMIZATION. Sadayoshi Kojima, Shigeru Mizushima, and Ser Peow Tan

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1 CIRCLE PACKINGS ON SURFACES WITH PROJECTIVE STRUCTURES AND UNIFORMIZATION Sadayoshi Kojima, Shigeru Mizushima, and Ser Peow Tan Dedicated to Professor Yukio Matsumoto on his 60th birthday Let Σ g be a closed orientable surface of genus g 2 and τ a graph on Σ g with one vertex which lifts to a triangulation of the universal cover. We have shown that the cross ratio parameter space C τ associated with τ, which can be identified with the set of all pairs of a projective structure and a circle packing on it with nerve isotopic to τ, is homeomorphic to R 6g 6, and moreover that the forgetting map of C τ to the space of projective structures is injective. In this paper, we show that the composition of the forgetting map with the uniformization from C τ to the Teichmüller space T g is proper. 1. Introduction In [8], we initiated the study of circle packings on Riemann surfaces with complex projective structures. Since the topic is rather new, here we briefly recall the setting. A projective structure on a surface is, by definition, a geometric structure modeled on the pair of the Riemann sphere Ĉ and the projective linear group PGL 2 (C) acting on Ĉ by projective transformations. Hence it is in particular a complex structure, but finer than the complex structure up to conformal equivalence. We thus would like to call a surface with a projective structure a projective Riemann surface for short throughout this paper. The key observation is that circles/disks are fundamental objects without referring to metrics in one-dimensional complex projective geometry, since a projective transformation sends a circle on Ĉ to a circle on Ĉ. This is despite the fact that PGL 2 (C) does not preserve any metric on the Riemann sphere. Thus, circles on a projective Riemann surface are not metric circles in the usual sense, rather, they are homotopically trivial closed curves which develop to circles in Ĉ via the developing map. A circle in the present paper is a circle in this projective sense, and not a circle with respect to the hyperbolic metric conformally equivalent to the projective structure. Suppose we are given a closed orientable surface Σ g of genus g 2 without any auxiliary structure, and a graph τ on Σ g which lifts to an honest 1

2 2 S. KOJIMA, S. MIZUSHIMA, AND S. P. TAN triangulation of the universal cover Σ g. We are interested in the moduli space of all pairs (S, P) consisting of a projective Riemann surface S with a reference homeomorphism h : Σ g S and a circle packing P on S whose nerve is isotopic to h(τ). In [8], this moduli space is shown to be identified with, what we call, the cross ratio parameter space C τ. Let P g be the space of all projective Riemann surfaces homeomorphic to Σ g up to marked projective equivalence. In other words, it is the space of all marked projective structures on Σ g. To each pair (S, P) C τ, assign its first component and we obtain the forgetting map, f : C τ P g. One of main results in [8] is that when τ has exactly one vertex, C τ is homeomorphic to the Euclidean space of dimension 6g 6 and f is injective. The injectivity implies that each projective Riemann surface admits at most one circle packing dominated by τ. Namely, the packings are in fact rigid. On the other hand, assigning the underlying complex structure to each projective Riemann surface, we obtain the uniformization map u : P g T g, of P g to the Teichmüller space T g, the space of all complex structures on Σ g up to marked conformal equivalence. By taking the Schwarzian derivative of the developing map, a projective structure can be identified with a holomorphic quadratic differential over the underlying Riemann surface, so the uniformization map is a complex vector bundle of rank 3g 3 over T g. For the genus one case, when τ has one vertex, it was shown by Mizushima in [10], with slightly different language, that the composition of the forgetting map with the uniformization map is a homeomorphism. In [8], we conjectured that this holds in general, regardless of the genus g ( 1) and the graph τ. The purpose of this paper is to take the first step towards the conjecture by proving the following properness theorem for τ with one vertex: Theorem 1.1. Let τ be a graph on Σ g (g 2) with one vertex which lifts to an honest triangulation of Σ g and C τ the cross ratio parameter space associated with τ. Then the composition u f : C τ T g of the forgetting map with the uniformization map is proper. To complete the proof of the conjecture for such graphs τ, since C τ in this case was shown to be homeomorphic to R 6g 6, it suffices to show that the map is locally injective. This sort of question for the grafting map based on Tanigawa s properness theorem in [13] was settled by Scannell and Wolf in [12]. See also Faltings [3] and McMullen [11] for earlier proofs of special cases. However, it is not clear if the proofs in the above cited papers can be extended to our setting.

3 CIRCLE PACKINGS AND UNIFORMIZATION 3 The rest of this paper is organized as follows. In 2, we recall the definition of the cross ratio parameter space from [8] and show that it is properly embedded in the euclidean space. In 3, following the exposition in [6], we briefly review the Thurston coordinates of P g in terms of hyperbolic structures and measured laminations, and show that the projected image of f(c τ ) to the space of measured laminations on Σ g is bounded. The results up to that section are valid for any graph τ. In 4, under the assumption that τ has exactly one vertex, we show that the holonomy map from f(c τ ) to the algebro-geometric quotient of the space of representations of π 1 (Σ g ) in PGL 2 (C) up to conjugacy is proper, and deduce from this that the projection of f(c τ ) to the space of the hyperbolic structures in the Thurston coordinates is also proper. Finally, in 5, we complete the proof of the theorem using Tanigawa s inequality in [13]. 2. The Cross Ratio Parameter Space We first recall briefly how the cross ratio parameter space was defined in [8]. Let Σ g be a closed orientable surface of genus g 2 and τ a graph on Σ g which lifts to an honest triangulation of Σ g. Suppose that S is a projective Riemann surface which lies in f(c τ ) with a reference homeomorphism h : Σ g S and P is a circle packing on S with nerve isotopic to h(τ). In [8], we defined a cross ratio parameter for the pair (S, P), which is a function c : E τ R, where E τ denotes the set of edges of τ. To each edge e of τ is associated a configuration of four circles in the developed image surrounding a preimage ẽ of e, see Figure 1(a). The real value c(e) assigned to the edge e is obtained by taking the imaginary part of the cross ratio (p 14, p 23, p 12, p 13 ) = (p 14 p 12 )(p 23 p 13 ) (p 14 p 13 )(p 23 p 12 ) of the four contact points (p 14, p 23, p 12, p 13 ) of the configuration chosen as in Figure 1(b) with orientation convention (for the definition of the cross ratio of four ordered points, see [1]). It is the modulus of the rectangle obtained by normalizing the configuration by moving the contact point p 13 to, see Figure 2. From this figure, it is clear that the cross ratio is purely imaginary. Moreover, unless the imaginary part is positive, neighboring triangular interstices overlap and circles cannot correspond to a correct packing. Because the developing map is a local homeomorphism, the cross ratio parameter must satisfy certain conditions, best expressed in terms of the associated matrices. If e is an edge of τ with a real value x = c(e), we associate to e the matrix A = ( x ), where x is positive real. The associated

4 4 S. KOJIMA, S. MIZUSHIMA, AND S. P. TAN C 3 ẽ C 2 p 23 p 12 p 13 p 34 p 14 C 4 (a) (b) C 1 Figure 1 C 1 C 3 p 13 = p 14 = x 1 p 34 C 4 C 2 p 12 = 0 p 23 = 1 Figure 2 matrix A represents a transformation which sends the left triangular interstice to the right one in the normalized picture of Figure 1, obtained by sending p 12, p 13 and p 23 to, 0 and 1 respectively. Now, if v is a vertex of τ with valence m, we read off the edges e 1,, e m incident to v in a clockwise direction to obtain a sequence of real values x 1 = c(e 1 ),..., x m = c(e m ) associated to v. Let W j = A 1 A 2 A j = ( aj b j c j d j ), j = 1,...,m

5 where A i is the matrix v of τ, we have (1) and (2) CIRCLE PACKINGS AND UNIFORMIZATION 5 ( ) 0 1 associated to e 1 x i. Then, for each vertex i W v = A 1 A 2 A m = ( ), { aj, c j < 0, b j, d j > 0 for 1 j m 1 except for a 1 = d m 1 = 0. Roughly, the first condition means that a fixed triangular interstice is mapped back to the original position by the composition of the transformations represented by the associated matrices, and this ensures the consistency of the chain of circles surrounding the circle corresponding to v, see Figure 3. The second condition eliminates the case where the chain surrounds the central circle more than once. We note that it does not matter which edge we start out with in the above. C m C 1 C 2 C C 3 C 4 Figure 3 Conversely, if an assignment to the edges of τ, c : E τ R, satisfies conditions (1) and (2) for each vertex, the map is a cross ratio parameter of some packing P on a projective Riemann surface S f(c τ ), where both S and P are determined by c (main lemma of [8]). Hence, the set C τ = {c : E τ R c satisfies (1) and (2) for each vertex}, called the cross ratio parameter space, can be identified with the space of pairs (S, P) of a projective Riemann surface S and a circle packing P on S, and c parameterizes the space of such pairs. By the definition, we see that C τ is a semi-algebraic set in R Eτ defined by equations coming from (1) and inequalities coming from (2) for each vertex

6 6 S. KOJIMA, S. MIZUSHIMA, AND S. P. TAN v. We first show that the strict inequalities of (2) can be replaced by nonstrict inequalities, namely, for each vertex v and with the same notation as before, consider the set of conditions { aj, c j 0, b j, d j 0 for 1 j m 1 (3) except for a 1 = d m 1 = 0. Lemma 2.1. The conditions (1) and (2) are equivalent to (1) and (3). Proof. It is sufficient to prove that (2) follows from (1) and (3). We have the identity ( ) ( )( ) aj+1 b j+1 aj b = j 0 1 c j+1 d j+1 c j d j 1 x j+1 ( ) bj a = j + b j x (4) j+1 d j c j + d j x j+1 for j = 1,...,m 1. (2) is then shown to follow from (1) and (3) by induction as follows. We first show that a j < 0 and b j 1 > 0 for j = 2,...,m 1. Set the inductive hypotheses to be a j < 0 and b j 1 > 0. Since a 2 = b 1 = 1 < 0, the hypotheses are true for j = 2. Assume that (1) and (3) hold and the hypotheses are true for j = k where 2 k m 2. Then b k 0 by (3). If b k = 0, b k+1 = a k + b k x k+1 = a k < 0 by (4) and the hypotheses, contradicting b k+1 0. Thus b k > 0. Also a k+1 = b k < 0, hence the hypotheses are true for j = k+1. Combining the above induction with b m 1 = a m = 1 < 0 gives the strict inequalities for a j s and b j s in (2). A similar argument is applied for the strict inequalities for the c j s and d j s. This lemma tells us that the condition (2) does not divide a connected component of the algebraic set determined by (1). It just chooses appropriate connected components. With this, we can easily prove Lemma 2.2. The inclusion map of C τ into R Eτ is proper, where E τ is the set of edges of τ. Remark. The above result holds for a general τ with no restriction on the number of vertices. Proof. Let {c n } be a sequence of points in the intersection of a compact set in R Eτ with C τ. Then, there exists a convergent subsequence. Let c denote the limit of the subsequence. Each c n satisfies the conditions (1) and (2). Each entry of the product of the matrices in the conditions (1) and (2) is a polynomial in x i s and thus is continuous. Hence, the limit c satisfies the conditions (1) and (3), and also lies in C τ by lemma 2.1.

7 CIRCLE PACKINGS AND UNIFORMIZATION 7 3. Measured Laminations We recall Thurston s parameterization of P g and then see how f(c τ ) is located with respect to the parameterization. The space we are concerned with is the one which consists of isotopy classes of measured laminations on Σ g, denoted by ML g. A measured lamination is defined to be a closed subset on Σ g locally homeomorphic to a product of a totally disconnected subset of the interval with an interval together with a transverse measure. Although a measured lamination is a topological concept, once we put a hyperbolic metric on Σ g, its support is canonically realized as a disjoint union of simple geodesics which forms a closed subset on the surface. Such a lamination is called a measured geodesic lamination, and hence ML g can be identified with the set of measured geodesic laminations on a hyperbolic surface homeomorphic to Σ g. Then, ML g has a natural topology induced from the weak topology on measures on the set of undirected geodesics (see [14]), and is known to be homeomorphic to the Euclidean space of dimension 6g 6. The detailed account of the theory in terms of the measured foliation, which is the notion equivalent to the measured geodesic lamination (see [9]), can be found in [4, 15]. Thurston has shown that any projective Riemann surface corresponds uniquely to a hyperbolic surface pleated along geodesic laminations with a bending measure. Following [6], we briefly review his parameterization. Start with a projective Riemann surface S which is not a hyperbolic surface. Consider the set of maximal disks in the universal cover S. Each maximal disk is naturally endowed with the hyperbolic metric, the boundary of each disk intersects the ideal boundary of S in two or more points and we can take the convex hull of these ideal boundary points. It can be shown that this gives a stratification of S by ideal polygons, and ideal bigons foliated by parallel lines joining the two ideal vertices of the bigons. The polygonal parts support a canonical hyperbolic metric. Collapsing each bigon foliated by the parallel lines in S to a line and taking the quotient of the resultant by the action of the fundamental group, we obtain a hyperbolic surface H. A hyperbolic surface is in particular a projective Riemann surface and there is a natural section s : T g P g to the uniformization map u : P g T g by assigning a hyperbolic surface to each conformal class of a Riemann surface. Thus assigning H to each S, we obtain a hyperbolization map π : P g s(t g ). Also the stratification defines a geodesic lamination λ on H by taking the union of the collapsed lines. Moreover, using the convex hull of the ideal points of the maximal disk not in the disk but in the 3-dimensional

8 8 S. KOJIMA, S. MIZUSHIMA, AND S. P. TAN hyperbolic space H 3, we can assign a transverse bending measure supported on λ. This defines a pleating map β : P g ML g. Thurston showed that the pair of these maps is in fact a homeomorphism parameterizing P g : = (π, β) : P g s(tg ) ML g. Here we show Lemma 3.1. β(f(c τ )) is bounded in ML g. Remark. The above result holds for a general τ. Proof. Let S be a surface in f(c τ ) P g, P a circle packing on S with nerve τ, and H = π(s), λ = β(s). The measured lamination λ can be pulled back to a lamination with transverse measure by blowing up the atomic leaves on H to parallel leaves in S with stretched transverse measure in a canonical way. Hence we regard S as a surface with this measured lamination µ. To see the boundedness of β(f(c τ )), it is sufficient to show that the measure along each edge of τ is uniformly bounded since τ generates the fundamental group of Σ g and µ collapses to λ. We can choose a reference point for each circle in P to represent the vertex v of τ such that the supporting maximal disk of the point representing v contains the circle and take their preimage in P to get equivariant reference vertices. Let C 1 and C 2 be contact circles in P, and D i the supporting maximal disk of the reference point v i of C i. D i contains C i for both i = 1, 2, so these form a π roof over the pleated hyperbolic surface in H 3 locally corresponding to the developed image of S, see Figure 4. So the total transverse measure along a path ẽ between v 1 and v 2 contained in D 1 D 2 is bounded above by π. C 1 C 2 D 1 D 2 Figure 4

9 CIRCLE PACKINGS AND UNIFORMIZATION 9 4. Holonomy Representations In this section, we restrict ourselves to the case where τ has one vertex, and see how the holonomy representations behave. Our main concern is the space of representations of π 1 (Σ g ) in PGL 2 (C) up to conjugacy, which is unfortunately non-hausdorff, and hence we take further algebro-geometric quotient, to obtain X g = Hom(π 1 (Σ g ), PGL 2 (C))//, which can be identified with the space of characters. To each projective Riemann surface, assign its holonomy representation and we obtain the map hol : P g X g. The map hol is known to be a local homeomorphism by Hejhal [5]. We first show Lemma 4.1. Suppose that τ has only one vertex. Then there is a finite subset F of the fundamental group π 1 (Σ g ) such that if {c n } is a sequence of points in C τ which escapes away from the compact sets, then there is an element g F and a subsequence of {c n } for which tr(ρ n (g)), where ρ n = hol(f(c n )). In particular, the composition hol f : C τ X g is proper. Remark. Although the image of holonomy lies in PGL 2 (C) and the trace makes sense only up to signs, its absolute value is well-defined. Proof. Since τ has one vertex, each edge e i of τ starts and ends at the same vertex v, so corresponds to a pair of elements g i ±1 π 1 (Σ g ), and the set {g i ±1 } forms a generating set for π 1 (Σ g ). F is defined to be the set of all words of length 2 in this generating set. Passing to a subsequence, by Lemma 2.2, we may assume that the real value c n (e) of some fixed edge e of τ approaches either 0 or as n, and then by using (1), (2) and (4), we may further assume that in fact, it approaches. For each S n = f(c n ), consider the developed image of S n and in particular the configuration of 6 circles C 1,...,C 6 in P n with corresponding vertices v 1,...,v 6 of τ as given in Figure 5a. Here, the edge e 3 = v 1 v 3 is the one whose assigned real value c n (e 3 ) approaches as n. Note that the configuration of C j and v j depends on n. For each n, we may normalize the developed image so that it is given by Figure 5b, where the tangency point p 13 between C 1 and C 3 is 1. The concatenation of the two directed edges e 2 = v 2 v 1 and e 4 = v 1 v 4 which are the neighboring edges of e 3 about the vertex v 1 corresponds to an element g = g 4 g2 1 F. Its holonomy image ϕ n := ρ n (g) = ρ n (g 4 g2 1 ) is an element of PGL 2 (C) mapping C 2 to C 4. Note that in the normalized picture, the radius of C 4 approaches zero as n since c n (e 3 ). Hence we can represent the images of 0, 1 and, all lying on C 2, under ϕ n by 1 + ε 1,

10 10 S. KOJIMA, S. MIZUSHIMA, AND S. P. TAN 1 + ε2 and 1 + ε 3 respectively, where ε 1, ε 2, ε 3 depend on n and they all approach 0 as n. Note that the point 1 and its image are not necessarily contact points. Letting K n = ε 2 ε 3 ε 2 ε 1, ϕ 1 n is then described by ϕ 1 z ( 1 + ε 1 ) n : z K n z ( 1 + ε 3 ), and ( 1 Kn K M n = n ( ) 1 + ε 1 ) Kn (ε 1 ε 3 ) 1 ( SL 1 + ε 3 ) 2 (C) is a matrix representative of ϕ 1 n. We have tr(m n ) = 1 ε1 ε 3 ( K n 1 Kn ( 1 + ε 3 )) and tr(ϕ n ) = tr(ϕ 1 n ) = tr(m n ). If tr(m n ), we are done, otherwise, we must have ( K n 1 Kn ( 1 + ε 3 )) 0 since (ε 1 ε 3 ) 0 as n. This implies that K n approaches 1 since ε 3 0 too. Geometrically, this means that the angle formed by 1 + ε 1, 1 + ε 2 and 1 + ε3 approaches π/2, and hence 1 + ε 1 and 1 + ε 3 approach diametrically opposite positions on the circle C 4. In other words, the arc α 1 = ϕ n ([, 0]) occupies half of the circle C 4 in the limit. On the other hand, the minor arc α 2 on C 4 connecting p 14 and p 34 (this is the arc on the circle joining p 14 to p 34 which does not contain p 45 and p 46 ) also occupies half of C 4 in the limit since the radius of C 4, which we denote by rad(c 4 ), approaches 0, where p jk is the tangency point between C j and C k. 1 C 2 p 23 p 34 p 12 p 13 p C C6 (a) C 3 C 1 C 5 C 3 C 2 = p 13 p 12 = 0 (b) p 14 p 34 2 C5 C 4 C C 6 1 Figure 5 For each circle, the contact points with the neighboring circles cuts it into arcs meeting only at the endpoints. We have Claim: α 1 and α 2 are the images under ϕ n of two distinct, non-adjacent arcs of C 2 and thus are distinct non-adjacent arcs on C 4.

11 CIRCLE PACKINGS AND UNIFORMIZATION 11 Proof of Claim. It suffices to show that the image of [, 0] on C 1 by ρ n (g2 1 ) and the image of the arc α 2 by ρ n (g4 1 ) are non-adjacent arcs of C 1. Here we review intersecting and non-intersecting triples described in [8], 4. For a packing with one circle, an edge of the nerve τ corresponds to the self contact points of the unique circle on Σ g and hence to the pair of contact points on a circle in the developed image where the contact points are developed images of the prescribed contact point on Σ g. Furthermore, a triangle of the nerve τ appears as a triple of such pairs of contact points, where the six contact points consist of three pairs of neighboring contact points at the same time. There are two possible cases as shown in Figure 6, which are called an intersecting triple and a non-intersecting triple. (a) intersecting (b) non-intersecting Figure 6 The pair of edges (e 2, e 3 ) are either part of an intersecting or non-intersecting triple of edges, since they are adjacent. Similarly, the pair of edges (e 3, e 4 ) are also part of an intersecting or non-intersecting triple. For all possibilities, ρ n (g2 1 )[, 0] and ρ n(g4 1 )(α 2) are non-adjacent arcs of C 1, see Figure 7 (a),(b) and (c). From the claim, the subtended angle between p 45 and p 34 on C 4 in Figure 5 (b) approaches zero, as does the angle between p 46 and p 14. Thus, the ratios of the radius of C 5 and C 6 to that of C 4, rad(c 5) rad(c 4 ) and rad(c 6) rad(c 4 ) approach zero as n. This in turn implies that the values of c n at the two edges v 1 v 4 and v 3 v 4 both approach. This can be seen by scaling the picture in Figure 5 (b) so that rad(c 4 ) = 1 for each n, then C 1 tends to a straight line parallel to C 3 and the radii of C 5 and C 6 tends to 0 and hence the values of c n at the two edges v 1 v 4 and v 3 v 4 both approach (see [8] proposition 2.8). Repeating the argument with the roles of C 2 and C 4 reversed, we see that either tr(ϕ n ) or the values of c n at v 1 v 2 and v 2 v 3 both approach. In other words, either tr(ϕ n ) or the values of c n at all neighboring edges approach. By induction, using the edge e 4 instead of e 3 and repeating the above argument, we see that either some element g F has holonomy with diverging trace, or the values of c n at all the

12 12 S. KOJIMA, S. MIZUSHIMA, AND S. P. TAN C 3 C 4 α 2 ρ n (g 1 4 )(α 2) C 3 C 4 C 1 C 2 C 1 C 2 [, 0] ρ n (g2 1 )([, 0]) (a) Two non-intersecting (b) Intersecting and triples non-intersecting triples C 4 C 3 C 2 C 1 (c) Two intersecting triples Figure 7 edges of τ diverge to. However, in the latter case, the (2, 2) term d m of the matrix corresponding to the word W m defined in 2 has a dominating term x 1 x 2 x m and diverges to. This contradicts the fact that d m = 1 for all points of the sequence by condition (1) in 2. This completes the proof. Lemma 4.1 implies Lemma 4.2. Suppose that τ has only one vertex. The composition π f : C τ s(t g ) of the forgetting map with the collapsing map π : P g s(t g ) in the Thurston parameterization is proper. In particular, the restriction of π to f(c τ ) is proper. Proof. Suppose that {c n } is a sequence in C τ which escapes away from the compact sets. By passing to a subsequence, we may assume by lemma 4.1 that there is an element g π 1 (Σ g ) with holonomy ϕ n = ρ n (g) for which tr(ϕ n ). g corresponds to a closed curve γ on Σ g and the length l n (γ) of the geodesic representative of γ on the collapsed surface H n = π(s n ) satisfies the inequality l n (γ) d H 3(z n, ϕ n (z n )) where z n is any point on the axis of ϕ n in H 3 and d H 3(z n, ϕ n (z n )) is the hyperbolic distance between z n and ϕ n (z n ). Since tr(ϕ n ), d H 3(z n, ϕ n (z n ))

13 CIRCLE PACKINGS AND UNIFORMIZATION 13 and hence l n (γ). It follows that {H n } escapes away from the compact sets in s(t g ). 5. Proof of Theorem We recall the following result from [13], the notation has been modified slightly to fit into our framework. Theorem 5.1 ([13], theorem 3.4). Let S = (H, λ) be a projective Riemann surface homeomorphic to Σ g (g 2) where H = π(s) s(t g ) and λ = β(s) ML g and let X be the underlying Riemann surface. Let h : X H denote a harmonic map with respect to the hyperbolic metric on H and E(h) its energy. Then (5) l H (λ) l H(λ) 2 E X (λ) 2E(h) l H(λ) + 8π(g 1), where l H (λ) is the hyperbolic length of λ on H, and E X (λ) is the extremal length of λ on X. We are now ready to prove the theorem. Proof of Theorem 1.1. Assuming that τ has one vertex, we follow the argument of Tanigawa in [13]. Let {c n } be a sequence of points in C which escapes away from the compact sets, f(c n ) = S n = (H n, λ n ) s(t g ) ML g and X n = u(s n ) the corresponding underlying Riemann surface. By lemma 4.2, we may assume that {H n } escapes away from the compact sets in T. Furthermore, by lemma 3.1, since {λ n } lies in a compact subset of ML g, we may assume by taking a subsequence if necessary that λ n λ for some fixed measured lamination λ. Let h n be the harmonic map from X n to H n with respect to a hyperbolic metric on H n. If E(h n ) is bounded, then {X n } escapes away from the compact sets as desired, since otherwise, we have a contradiction with a result of M. Wolf [16]. If not, the diverging rate of all terms in the inequalities (5) is the same as that of l Hn (λ n ) and hence lim E X n n (λ) = lim E X n n (λ n ) = lim (l H n n (λ n ) + O(1)) =. This implies that {X n } escapes away from the compact sets as well and we are done. References [1] L. Ahlfors, Complex Analysis. An introduction to the theory of analytic functions of one complex variable, McGraw-Hill Book Company, Inc., New York-Toronto-London, xii+247 pp. [2] R. Bers, Finite dimensional Teichmüller spaces and generalizations, Bull. Amer. Math. Soc. 5 (1981),

14 14 S. KOJIMA, S. MIZUSHIMA, AND S. P. TAN [3] G. Faltings, Real projective structures on Riemann surfaces, Compositio Math. 48 (2) (1983), [4] A. Fathi, F. Laudenbach and V. Poenaru et al., Travaux de Thurston sur les surfaces, Astérisque (1979), [5] D. Hejhal, Monodromy groups and linearly polymorphic functions, Acta Math. 135 (1975), [6] Y. Kamishima and S.P. Tan, Deformation Spaces associated to Geometric Structures, Advanced Studies in Pure Mathematics 20 (1992), [7] L. Keen and C. Series, Pleating coordinates for the Maskit embedding of the Teichmüller space of a punctured torus, Topology 32 (1993), [8] S. Kojima, S. Mizushima and S. P. Tan, Circle packings on surfaces with projective structures, J. Differential Geom. 63 (2003), [9] R. Miller, Geodesic laminations from Nielsen s viewpoint, Adv. in Math. 45 (1982), [10] S. Mizushima, Circle packings on complex affine tori, Osaka J.Math. 37 (2000), [11] C.T. McMullen, Complex earthquakes and Teichmüller theory, J. Amer. Math. Soc. 11 (1998), no. 2, [12] K.P.Scannell and M. Wolf, The grafting map of Teichmüller space, J. Amer. Math. Soc. 15 (2002), no. 4, [13] H. Tanigawa, Grafting, harmonic maps, and projective structures on surfaces, J. Differential Geom. 47 (3) (1997), [14] W. P. Thurston, The geometry and topology of 3-manifolds, Lecture Notes, Princeton Univ., 1977/78. [15] W. P. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. 19 (1988), [16] M. Wolf, The Teichmüller theory of harmonic maps, J. Differential Geom. 29 (1989), Received Received date / Revised version date Department of Mathematical and Computing Sciences Tokyo Institute of Technology Ohokayama, Meguro Tokyo Japan address: sadayosi@is.titech.ac.jp Department of Mathematical and Computing Sciences Tokyo Institute of Technology Ohokayama, Meguro Tokyo Japan address: mizusima@is.titech.ac.jp Department of Mathematics National University of Singapore Singapore , Singapore address: mattansp@nus.edu.sg

15 CIRCLE PACKINGS AND UNIFORMIZATION 15 The third author gratefully acknowledge support from the National University of Singapore academic research grant R and the hospitality of the Department of Mathematical and Computing Sciences, Tokyo Institute of Technology.