DIVERGENCE OF PROJECTIVE STRUCTURES

AND LENGTHS OF MEASURED LAMINATIONS

Harumi Tanigawa

Abstract. Given a complex structure, we investigate diverging sequences of projective structures on the fixed complex structure in terms of Thurston’s parametrization. In particular, we will give a geometric proof to the theorem by Kapovich stating that as the projective structures on a fixed complex structure diverge so do their monodromies. In course of arguments, we extend the concept of realization of laminations for PSL(2, C)-representations of surface groups.

1. Introduction

Throughout this paper, Σ_{g} denotes an oriented differentiable surface of genus g > 1 and P_{g} the space of CP^{1}-structures (or, projective structures) on Σ_{g}. Here, a CP^{1}-structure on a surface is a structure modelled on (CP^{1}, PSL(2, C)). (Here PSL(2, C) is identified with the projective automorphism group of CP^{1}.) As a Möbius transformation is holomorphic, each CP^{1}-structure determines its underlying complex structure.

There are two parametrizations of P_{g} which are often quoted. One of them is by the bundle π : Q_{g} → T_{g} of holomorphic quadratic differentials on Riemann surfaces over Teichmüller space T_{g}. The canonical projection π represents the correspondence from each CP^{1}-structure to its underlying complex structure. As for the detail about this parametrization, see Hejhal [H], for example. The other parametrization was introduced by Thurston, that says every CP^{1}-structure is obtained from some hyperbolic structure by grafting a measured lamination to it: P_{g} is parametrized by T_{g} × ML, where ML stands for the space of measured laminations. As for a description of grafting and this parametrization, see [KT] or [Tg].

In this paper, we will investigate CP^{1}-structures on a fixed complex structure in terms of Thurston’s parametrization. In particular, we will give a geometric proof for the fact, originally proved by Kapovich [Ka], that the monodromies diverge as the CP^{1}-structures diverge on a fixed complex structure.

The idea is to generalize the concept of realization of measured laminations in hyperbolic 3-manifolds to that for PSL(2, C)-representations of the fundamental group π_{1}(Σ_{g}) which is not necessarily discrete, and then observe that the lengths of measured laminations are locally bounded in the space of PSL(2, C)-representations,

Typeset by AMS-TEX

1991 Mathematics Subject Classification. 1991 Mathematics Subject Classification Primary 32G15: Secondary 30F10.

Research at MSRI is supported by NSF grant #DMS–9022140

while the length of some measured lamination diverges as CP^{1}-structures on the fixed complex structure diverge.

The author would like to thank William Thurston for inspiring conversations. This work was done at Mathematical Sciences Research Institute. The author is grateful to their hospitality.

2. Pleated surfaces and measured laminations for representations

First, we recall the monodromy of a CP^{1}-structure. Given a CP^{1}-structure on Σ_{g}, we take its developing map: beginning with a local chart of the CP^{1}-structure, we take its analytic continuation along curves and have a multivalued meromorphic function (meromorphic with respect to the underlying complex structure of the CP^{1}-structure). The values over a point differ from each other by post-compositions of Möbius transformations, which are determined by the closed curves along which the analytic continuation is carried out. Thus this procedure yields a homomorphism π_{1}Σ_{g} → PSL(2, C), which is determined up to conjugations by Möbius transformations (depending on the local chart to begin the analytic continuation with). This homomorphism is called a monodromy of the CP^{1}-structure, or the monodromy representation of π_{1}Σ_{g}.

Now we review quickly the two parametrization of P_{g} mentioned in the introduction.

First we recall the parametrization by the bundle of quadratic differentials π : Q_{g} → T_{g}. Given a CP^{1}-structure, we take its developing map f. Then its Schwarzian derivative, defined by (f′′/f′)^{2} − 1/2(f′′/f′)′, is a quadratic differential holomorphic with respect to the underlying complex structure. (For geometric meanings of Schwarzian derivatives see Thurston [Th2].) Therefore, we have a Riemann surface X ∈ T_{g} and a holomorphic quadratic differential on X. Conversely, it is known that given a Riemann surface X and a holomorphic quadratic differential q there is a CP^{1}-structure on X whose Schwarzian derivative of the developing map is q. Note that for each point X ∈ T_{g}, the space of CP^{1}-structures on X is the fiber Q_{g}(X) of π : Q_{g} → T_{g} over X, which is complex 3g − 3-dimensional Banach space.

On the other hand, Thurston’s parametrization theorem is as follows.

Theorem 2.1 (Thurston, unpublished). The space of CP^{1}-structures on Σ_{g} is parametrized by T_{g} × ML: every CP^{1}-structure is obtained by grafting a measured lamination to a hyperbolic surface.

As for a full proof of this parametrization theorem, see section 2 of [KT],or, for a short discription, section 2 of [Tg]. His idea is as follows: Given a hyperbolic surface X and a measured geodesic lamination λ, first, embed the universal cover H^{2} of X into H^{3}. Then, lift λ to H^{2} and then bend H^{2} along the lift of λ so that the bending measure is the measure of λ. This bent structure can be “pushed forward” to the sphere at infinity to determin a CP^{1}-structure.

Note that the projection from T_{g} × ML to its first coordinate T_{g} is not the correspondence between CP^{1}-structures and their underlying complex structures. On the contrary, the underlying complex structure of any CP^{1}-structure represented by (X, μ) ∈ T_{g} × ML, differs from X, unless μ = 0.

Remark. In relation to this parametrization, Labourie [L] gave a new parametrization for P_{g}.

Now we define the pleated maps and the lengths of measured laminations for PSL(2, C)-representations of π_{1}Σ_{g}. We begin with “pleated surfaces”. It is defined in a parallel way to that of 3-manifolds. .

Definition 2.2. Let ϕ : π_{1}Σ_{g} → PSL(2, C) be a representation. A pleated mapping for ϕ is a continuous mapping f : H^{2} → H^{3} with the following properties:

(1) there exists a Fuchsian group Γ acting on H^{2} isomorphic to π_{1}Σ_{g} such that f ∘ γ = ϕ(γ) ∘ f for all γ ∈ Γ.

(2) for every z ∈ H^{2}, there is an open interval of hyperbolic line containing z which is mapped by f to a straight line segment.

(3) f maps every geodesic segment in H^{2} to a rectifiable arc in H^{3} with the same length.

This is the same as the usual definition of pleated surfaces when ϕ(π_{1}Σ_{g}) is discrete. We shall call the image of a pleated map a pleated surface.

A typical example of a pleated surface is the “bent surface” defining a CP^{1}- structure in Thurston’s geometric parametrization theorem. It is a pleated map for the monodromy of the CP^{1}-structure.(see [EM], [KT] and [Tg]).

Now we define the lengths of measured laminations for PSL(2, C)-representations of π_{1}Σ_{g}.

Definition 2.3. Let ϕ : π_{1}Σ_{g} → PSL(2, C) be a non-elementary representation and let f : H^{2} → H^{3} be a pleated map equivariant with respect to some Fuchsian group Γ isotopic to π_{1}Σ_{g} acting on H^{2} and ϕ(π_{1}Σ_{g}). Such a mapping determines a hyperbolic structure on Σ_{g}. Then let l_{f}(μ) be the total mass of the product of the transverse measure of μ and the length along the lines of the support of μ. The length l_{ϕ}(μ) is the infimum of l_{f}(μ) where the infimum is taken over all Fuchsian groups Γ isotopic to π_{1}Σ_{g} and pleated maps f.

Remark. As we will see in Lemma 2.5, for any non-elementary PSL(2, C)-repres-entation of Σ_{g}, there exist at least one pleated map as above.

As Definition 2.2, when the PSL(2, C)-representation of π_{1}Σ_{g} is discrete, the above definition coincides with that of usual length of a measured lamination. Also, given a homotopy class of an equivariant map, the uniqueness of the realization of a lamination holds by the same reason as in the discrete case.

The length function is a function of two variables: measured laminations and representations. We might as well expect the continuity of the length function with respect to the both variables. (In fact, for a simple closed curve, the length is continuous with respect to PSL(2, C)-representation of π_{1}Σ_{g}.) However, we do not need that strong property for our purpose. What we need is local boundedness (Lemma 2.5 bellow).

Kapovich ([Ka]) has already mentioned the concept of pleated maps and showed the existence for any non-elementary SL(2, C)-representation of π_{1}Σ_{g} briefly. We use these examples of pleated maps observed by Kapovich.

Here, for convenience, we will exibit somewhat detailed argument about them. The following decomposition is the key.

Theorem 2.4 ([Ka]). For any non-elementary representation

ϕ : π_{1}Σ_{g} → PSL(2, C)

there exists a pants decomposition Σ_{g} = P_{1} ∪ ... ∪ P_{2g−2} such that (1) ϕ(π_{1}(P_{i})) is non-elementary and

(2) ϕ(γ) is loxodromic for each boundary curve γ of P_{i} for i = 1, ..., 2g − 2.

Lemma 2.5. The length function of measured laminations is locally bounded in the space of non-elementary PSL(2, C)-representations of π_{1}Σ_{g}.

Proof. Given a non-elementary representation ϕ : π_{1}Σ_{g} → PSL(2, C) take a pants decomposition Σ_{g} = P_{1} ∪ ... ∪ P_{2g−2} as in the above theorem. Decompose each pants into two ideal triangles by adding leaves spiraling around the boundaries.

Take a continuous equivariant map f : H^{2} → H^{3} (with respect to any Fuchsian group isotopic to Σ_{g} and ϕ(Σ_{g})) and homotope f so that f sends each leaf of the ideal triangulation onto a straight line, then straighten the map in each ideal triangle. This is possible because by Theorem 2.4 the restriction of ϕ to the fundamental group of each pants is isomorphic. Thus we have a pleated surface with pleating locus the geodesics defining the ideal triangulation.

Thus, for each non-elementary PSL(2, C)-representation ϕ, the length of any measured lamination is finite. Here, note that when we take an ideal triangulation as above for ϕ, we can take the same one for PSL(2, C)-representations which are sufficiently close to ϕ. As the PSL(2, C)-representations changing continuously near ϕ, we can take hyperbolic structures on Σ_{g} determined by the pleated maps with the fixed ideal triangulation so that they change continuously. From the continuity of the lengths of measured laminations on surfaces, we have a local upper bound of the lengths of measured laminations by the lengths on the hyperbolic structures. It follows that the lengths for the representations are locally bounded. □

Remark. Let mon : P_{g} → Hom(Σ_{g}, PSL(2, C)) be the monodromy map, namely, the map mon sends each CP^{1}-structure to its monodromy representation. Hejhal [H] showed that mon is a local diffeomorphism. Therefore, the image mon(P_{g}) is a region of Hom(Σ_{g}, PSL(2, C)). Therefore, obviously the lengths of measured laminations are locally bounded on the region mon(P_{g}) by Theorem 2.1. A theorem by Gallo, Kapovich and Marden [GKM] says that every non-elementary SL(2, C)- representation of Σ_{g} is obtained as the monodromy of a CP^{1}-structure. Therefore, if we would use that result, the local boundedness on the entire space of CP^{1}- structures would follow immediately. However, we exibited a direct proof rather than using their theorem.

3. Divergence of projective structures

Our goal is to give a geometric proof for the following:

Theorem 3.1 ([Ka]). For any compact set K of T_{g} and any diverging sequence {q_{n}} of CP^{1}-structures with {π(q_{n})} ⊂ K, their monodromies mon(q_{n}) diverge. Here, π(q_{n}) stands for the underlying complex structure of q_{n}.

The idea is as follows: The extremal length is the square order of the hyperbolic length when hyperbolic structures stay in a compact set of T_{g}. On the other hand, Theorem 3.2 bellow will show that l_{X}(μ) and E_{gr_{μ}(X)}(μ) is in the same order when they are large. Here, gr_{μ}(X) denote the Riemann surface obtained by grafting μ to X (namely, the underlying structure of the CP^{1}-structure given by Thurston’s parameter (X, μ)), l_{X}(μ) is the hyperbolic length of μ on X and E_{gr_{μ}(X)}(μ) is the extremal length of μ on the grafted surface gr_{μ}(X). Furthermore, by the definition of bendinging, l_{X}(μ) is the length of μ for the monodromy of the CP^{1}-structure. Therefore, for a diverging sequence q_{n} = (X_{n}, μ_{n}) as in the assumption of Theorem 3.1, l_{X_{n}}(μ) had to grow “too fast” and we can not maintain the monodromy in a bounded set of PSL(2, C)-representation as n → ∞, considering Lemma 2.5.

Theorem 3.2([Tg]). Let X be a hyperbolic surface and μ be a measured lamination. Let h : gr_{μ}(X) → X be the harmonic map with respect to the hyperbolic metric on X and E(h) be its energy.

Then

\frac{1}{2}l_{X}(μ) ≤ \frac{1}{2} \frac{l_{X}(μ)^{2}}{E_{gr_{μ}(X)}(μ)} ≤ E(h) ≤ \frac{1}{2}l_{X}(μ) + 4π(g − 1).

Recall that in Remark 1 in [Tg], we observed that when l_{X}(μ) is very large, that is, when the grafted structure is far from X, grafting is very close to the inverse of the harmonic map h and the pleated surface defining the grafting is close to the image of the harmonic equivariant map for the monodromy representation.

The following fact follows easily from the above inequality (cf. [Tg]).

Corollary 3.3. For any X ∈ T_{g}, the mapping gr_{·}(X) : ML → T_{g} is a proper map. For any μ ∈ ML, the mapping gr_{μ}(·) : T_{g} → T_{g} is a proper map.

Proof. The properness of gr_{μ}(·) : T_{g} → T_{g} was shown in [Tg]. For the properness of gr_{·}(X) : ML → T_{g}, note that

\frac{E_{gr_{μ}(X)}(μ)}{E_{X}(μ)} ≤ \frac{l_{X}(μ)}{E_{X}(μ)},

by Theorem 3.2. When X is fixed and μ tends to infinity, the right term tends to 0, therefore, so is the left term. It follows that the Riemann surface gr_{μ}(X) tends to infinity. □

Remark. In fact, in what follows we use only the leftmost inequality in Theorem 3.2. The others was necessary to show that gr_{μ}(·) is proper.

Now we can present a geometric proof of Theorem 3.1.

Proof of Theorem 3.1.

Let (X_{n}, μ_{n}) ∈ T_{g} × ML be the Thurston coordinate of the projective structure q_{n} and denote the underlying complex structure by π(q_{n}) = gr_{μ_{n}}(X_{n}). Note that in Corollary 3.3 in fact we can show that the mappings are “locally uniformly” proper with respect to each variable. Therefore, when π(q_{n}) stays in K and {q_{n}} tends to infinity, both of {X_{n}} and {μ_{n}} tend to infinity. Take a sequence ϵ_{n} converging to 0 such that ϵ_{n}μ_{n} converges to a non-zero measured lamination μ, taking a subsequence of {μ_{n}} if necessary.

By Theorem 3.1,

\frac{l_{X_{n}}(μ_{n})^{2}}{l_{X_{n}}(μ_{n}) + 8π(g − 1)} ≤ E_{gr_{μn} (X_{n})}(μ_{n}) ≤ l_{X_{n}}(μ_{n}).

Multiply each term of the above inequality by ϵ_{n}^{2}. Then we have

\frac{l_{X_{n}}(ϵ_{n}μ_{n})^{2}}{l_{X_{n}}(μ_{n}) + 8π(g − 1)} ≤ E_{gr_{μn} (X_{n})}(ϵ_{n}μ_{n}) ≤ ϵ_{n}l_{X_{n}}(ϵ_{n}μ_{n}).

Note that l_{X_{n}}(ϵ_{n}μ_{n}) is the length of ϵ_{n}μ_{n} for the monodromy representation mon(q_{n}) because the pleated surface defining the Thurston coordinate for q_{n} = (X_{n}, μ_{n}) has pleating locus μ_{n}. Therefore, if the representation mon(q_{n}) did not diverge, l_{X_{n}}(ϵ_{n}μ_{n}) would converge to a non-negative number by Lemma 2.5, taking a subsequence if necessary. Therefore, by the above inequality,

lim_{n→∞} E_{gr_{μn} (X_{n})}(ϵ_{n}μ_{n}) = 0

However, by taking a subsequence if necessary, we may assume that π(q_{n}) = gr_{μ_{n}}(X_{n}) converges to a point Y ∈ T_{g} by the assumption that {q_{n}} stays in the compact set K. Therefore E_{gr_{μn} (X_{n})}(ϵ_{n}μ_{n}) converges to E_{Y} (μ), which is a positive number. This is a contradiction. □

References

[EM] D. B. A. Epstein and A. Marden, Convex hulls in hyperbolic space, a theorem of Sullivan and measured pleated surfaces, London Mathematical Society Lecture Notes, vol. 111, Cambridge University Press, 1897, pp. 114-253.

[GKM] D. Gallo, M. Kapovich and A. Marden, The monodromy groups of Schwarzian equation on compact Riemann surfaces, (preprint).

[H] D. Hejhal, Monodromy groups and linearly polymorphic functions, Acta math. 135 (1975), 1-55.

[KT] Y. Kamishima and S. P. Tan, Deformation spaces on geometric structures, Aspects of Low Dimensional Manifolds, Advanced Studies in Pure Mathematics 20, Kinokuniya Co., 1992, pp. 263-299.

[Ka] M. Kapovich, On monodromy of complex projective structures, Invent. math. 119 (1995), 243-265.

[L] F. Labourie, Surfaces convexes dans l’espace hyperbolique et CP^{1}-structures, J. London Math. Soc. 45 (1992), 549-565.

[Tg] H. Tanigawa, Grafting, harmonic maps and hyperbolic surfaces, (preprint).

[Th1] W. Thurston, Geometry and Topology of 3-manifolds, Princeton University lecture notes (1979).

[Th2] W. Thurston, Zippers and Univalent Functions, The Bieberbach conjecture, West Lafayette Ind., 1985, pp. 185-197.

Graduate School of Polymathematics, Nagoya University, Nagoya 464-01, Japan