\documentstyle{amsppt} \magnification = 1200 \hcorrection{.25in} \document %\baselineskip \topmatter \title {Divergence of projective structures and lengths of measured laminations} \endtitle \rightheadtext {Divergence of projective structures} \author Harumi Tanigawa \endauthor \address Graduate School of Polymathematics, Nagoya University, Nagoya 464-01, Japan \endaddress \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 $\operatorname {PSL} (2, \bold C)$-representations of surface groups. \endabstract \subjclass {1991 Mathematics Subject Classification Primary 32G15: Secondary 30F10} \endsubjclass \thanks {Research at MSRI is supported by NSF grant \#DMS--9022140} \endthanks \endtopmatter \head 1. Introduction \endhead Throughout this paper, $\Sigma_g$ denotes an oriented differentiable surface of genus $g >1$ and $P_g$ the space of $\operatorname {\bold CP}^1$-structures (or, projective structures) on $\Sigma_g$. Here, a $\operatorname {\bold CP}^1$-structure on a surface is a structure modelled on $(\operatorname {\bold CP}^1, \operatorname {PSL}(2,{\bold C}))$. (Here $\operatorname {PSL}(2,{\bold C}$) is identified with the projective automorphism group of $\operatorname {\bold CP}^1$.) As a M\"obius transformation is holomorphic, each $\operatorname {\bold CP}^1$-structure determines its underlying complex structure.\par There are two parametrizations of $P_g$ which are often quoted. One of them is by the bundle $\pi: Q_g \to T_g$ of holomorphic quadratic differentials on Riemann surfaces over Teichm\"uller space $T_g$. The canonical projection $\pi$ represents the correspondence from each $\operatorname {\bold CP}^1$-structure to its underlying complex structure. As for the detail about this parametrization, see Hejhal \cite {H}, for example. The other parametrization was introduced by Thurston, that says every $\operatorname {\bold CP}^1$-structure is obtained from some hyperbolic structure by grafting a measured lamination to it: $P_g$ is parametrized by $T_g \times \Cal {ML}$, where $\Cal {ML}$ stands for the space of measured laminations. As for a description of grafting and this parametrization, see \cite {KT} or \cite {Tg}. \par In this paper, we will investigate $\operatorname {\bold CP}^1$-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 \cite {Ka}, that the monodromies diverge as the $\operatorname {\bold CP}^1$-structures diverge on a fixed complex structure. \par The idea is to generalize the concept of realization of measured laminations in hyperbolic $3$-manifolds to that for $\operatorname {PSL}(2, {\bold C})$-representations of the fundamental group $\pi_1(\Sigma_g)$ which is not necessarily discrete, and then observe that the lengths of measured laminations are locally bounded in the space of $\operatorname {PSL}(2,\bold C)$-representations, while the length of some measured lamination diverges as $\operatorname {\bold CP}^1$-structures on the fixed complex structure diverge. \par 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. \head 2. Pleated surfaces and measured laminations for representations \endhead First, we recall the monodromy of a $\operatorname {\bold CP}^1$-structure. Given a $\operatorname {\bold CP}^1$-structure on $\Sigma_g$, we take its {\it developing map}: beginning with a local chart of the $\operatorname {\bold CP}^1$-structure, we take its analytic continuation along curves and have a multivalued meromorphic function (meromorphic with respect to the underlying complex structure of the $\operatorname {\bold CP}^1$-structure). The values over a point differ from each other by post-compositions of M\"obius transformations, which are determined by the closed curves along which the analytic continuation is carried out. Thus this procedure yields a homomorphism $\pi_1 \Sigma_g \to \operatorname {PSL}(2,{\bold C})$, which is determined up to conjugations by M\"obius transformations (depending on the local chart to begin the analytic continuation with). This homomorphism is called a {\it monodromy} of the $\operatorname {\bold CP}^1$-structure, or the {\it monodromy representation of } $\pi_1 \Sigma_g$.\par Now we review quickly the two parametrization of $P_g$ mentioned in the introduction. \par First we recall the parametrization by the bundle of quadratic differentials $\pi : Q_g \to T_g$. Given a $\operatorname {\bold CP}^1$-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 \cite {Th2}.) Therefore, we have a Riemann surface $X \in T_g$ 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 $\operatorname {\bold CP}^1$-structure on $X$ whose Schwarzian derivative of the developing map is $q$. Note that for each point $X \in T_g$, the space of $\operatorname {\bold CP}^1$-structures on $X$ is the fiber $Q_g(X)$ of $\pi : Q_g \to T_g$ over $X$, which is complex $3g-3$-dimensional Banach space. \par On the other hand, Thurston's parametrization theorem is as follows. \proclaim {Theorem 2.1 (Thurston, unpublished)} The space of $\operatorname {\bold CP}^1$-structures on $\Sigma_g$ is parametrized by $T_g \times \Cal {ML}$: every $\operatorname {\bold CP}^1$-structure is obtained by grafting a measured lamination to a hyperbolic surface. \endproclaim As for a full proof of this parametrization theorem, see section 2 of \cite {KT},or, for a short discription, section 2 of \cite {Tg}. His idea is as follows: Given a hyperbolic surface $X$ and a measured geodesic lamination $\lambda$, first, embed the universal cover $\operatorname {\bold H}^2$ of $X$ into $\operatorname {\bold H}^3$. Then, lift $\lambda$ to $\operatorname {\bold H}^2$ and then bend $\operatorname {\bold H}^2$ along the lift of $\lambda$ so that the bending measure is the measure of $\lambda$. This bent structure can be ``pushed forward" to the sphere at infinity to determin a $\operatorname {\bold CP}^1$-structure. \par Note that the projection from $T_g \times \Cal {ML}$ to its first coordinate $T_g$ is {\it not} the correspondence between $\operatorname {\bold CP}^1$-structures and their underlying complex structures. On the contrary, the underlying complex structure of any $\operatorname {\bold CP}^1$-structure represented by $(X , \mu) \in T_g \times \Cal {ML}$, differs from $X$, unless $\mu = 0$. \remark {Remark} In relation to this parametrization, Labourie \cite {L} gave a new parametrization for $P_g$. \endremark \par Now we define the pleated maps and the lengths of measured laminations for $\operatorname {PSL}(2,{\bold C})$-representations of $\pi_1 \Sigma_g$. We begin with ``pleated surfaces". It is defined in a parallel way to that of $3$-manifolds. . \definition {Definition 2.2} Let $\phi : \pi_1 \Sigma_g \to \operatorname {PSL}(2,{\bold C})$ be a representation. A {\it pleated mapping} for $\phi$ is a continuous mapping $f : \operatorname {\bold H}^2 \to \operatorname {\bold H}^3$ with the following properties: \roster \item there exists a Fuchsian group $\Gamma$ acting on $\operatorname {\bold H}^2$ isomorphic to $\pi_1 \Sigma_g$ such that $f \circ \gamma = \phi (\gamma) \circ f $ for all $\gamma \in \Gamma$. \item for every $z \in \operatorname {\bold H}^2$, there is an open interval of hyperbolic line containing $z$ which is mapped by $f$ to a straight line segment. \item $f$ maps every geodesic segment in $\operatorname {\bold H}^2$ to a rectifiable arc in $\operatorname {\bold H}^3$ with the same length. \endroster \enddefinition This is the same as the usual definition of pleated surfaces when $\phi (\pi_1 \Sigma_g)$ is discrete. We shall call the image of a pleated map a {\it pleated surface}.\par A typical example of a pleated surface is the ``bent surface" defining a $\operatorname {\bold CP}^1$-structure in Thurston's geometric parametrization theorem. It is a pleated map for the monodromy of the $\operatorname {\bold CP}^1$-structure.(see \cite {EM}, \cite {KT} and \cite {Tg}). \par Now we define the lengths of measured laminations for $\operatorname {PSL}(2, {\bold C})$-representations of $\pi_1 \Sigma_g$. \definition {Definition 2.3} Let $\phi : \pi_1 \Sigma_g \to \operatorname {PSL}(2, {\bold C})$ be a non-elementary representation and let $f : \operatorname {\bold H}^2 \to \operatorname {\bold H}^3$ be a pleated map equivariant with respect to some Fuchsian group $\Gamma$ isotopic to $\pi_1 \Sigma_g$ acting on $\operatorname {\bold H}^2$ and $\phi (\pi_1 \Sigma_g)$. Such a mapping determines a hyperbolic structure on $\Sigma_g$. Then let $l_f(\mu)$ be the total mass of the product of the transverse measure of $\mu$ and the length along the lines of the support of $\mu$. The length $l_\phi(\mu)$ is the infimum of $l_f(\mu)$ where the infimum is taken over all Fuchsian groups $\Gamma$ isotopic to $\pi_1 \Sigma_g$ and pleated maps $f$. \enddefinition \remark {Remark} As we will see in Lemma 2.5, for any non-elementary $\operatorname {PSL (2, \bold C)}$-rep\-res\-en\-ta\-tion of $\Sigma_g$, there exist at least one pleated map as above. \endremark As Definition 2.2, when the $\operatorname {PSL}(2, {\bold C})$-representation of $\pi_1 \Sigma_g$ 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. \par 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 $\operatorname {PSL}(2, {\bold C})$-representation of $\pi_1 \Sigma_g $.) However, we do not need that strong property for our purpose. What we need is local boundedness (Lemma 2.5 bellow). \par Kapovich (\cite {Ka}) has already mentioned the concept of pleated maps and showed the existence for any non-elementary $\operatorname {SL}(2,{\bold C})$-representation of $\pi_1 \Sigma_g$ briefly. We use these examples of pleated maps observed by Kapovich.\par Here, for convenience, we will exibit somewhat detailed argument about them. \par The following decomposition is the key. \proclaim {Theorem 2.4 (\cite {Ka})} For any non-elementary representation $$\phi : \pi_1\Sigma_g \to \operatorname {PSL}(2,{\bold C})$$ there exists a pants decomposition $\Sigma_g = P_1 \cup ...\cup P_{2g-2}$ such that \roster \item $\phi (\pi_1(P_i))$ is non-elementary and \item $\phi (\gamma)$ is loxodromic for each boundary curve $\gamma$ of $P_i$ for $ i = 1,...,2g-2$. \endroster \endproclaim \proclaim {Lemma 2.5} The length function of measured laminations is locally bounded in the space of non-elementary $\operatorname {PSL}(2,\bold C)$-representations of $\pi_1\Sigma_g$. \endproclaim \demo {Proof} Given a non-elementary representation $\phi : \pi_1\Sigma_g \to \operatorname {PSL}(2,{\bold C})$ take a pants decomposition $\Sigma_g = P_1 \cup ...\cup P_{2g-2}$ as in the above theorem. Decompose each pants into two ideal triangles by adding leaves spiraling around the boundaries. \par Take a continuous equivariant map $f : \operatorname {\bold H}^2 \to \operatorname {\bold H}^3$ (with respect to any Fuchsian group isotopic to $\Sigma_g$ and $\phi (\Sigma_g)$) 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 $\phi$ to the fundamental group of each pants is isomorphic. Thus we have a pleated surface with pleating locus the geodesics defining the ideal triangulation.\par Thus, for each non-elementary $\operatorname {PSL}(2, {\bold C})$-representation $\phi$, the length of any measured lamination is finite. Here, note that when we take an ideal triangulation as above for $\phi$, we can take the same one for $\operatorname {PSL}(2, {\bold C})$-representations which are sufficiently close to $\phi$. As the $\operatorname {PSL}(2, {\bold C})$-representations changing continuously near $\phi$, we can take hyperbolic structures on $\Sigma_g$ 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. \qed \enddemo \remark {Remark} Let $mon : P_g \to \operatorname {Hom} (\Sigma_g, \operatorname {PSL}(2, {\bold C}))$ be the monodromy map, namely, the map $mon$ sends each $\operatorname {\bold CP}^1$-structure to its monodromy representation. Hejhal \cite {H} showed that $mon$ is a local diffeomorphism. Therefore, the image $mon(P_g)$ is a region of $\operatorname {Hom} (\Sigma_g, \operatorname {PSL}(2, {\bold C}))$. 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 \cite {GKM} says that every non-elementary $\operatorname {SL}(2, {\bold C})$-representation of $\Sigma_g$ is obtained as the monodromy of a $\operatorname {\bold CP}^1$-structure. Therefore, if we would use that result, the local boundedness on the entire space of $\operatorname {\bold CP^1}$-structures would follow immediately. However, we exibited a direct proof rather than using their theorem. \endremark \head 3. Divergence of projective structures \endhead Our goal is to give a geometric proof for the following: \proclaim {Theorem 3.1 (\cite {Ka})} For any compact set $K$ of $T_g$ and any diverging sequence $\{q_n\}$ of $\operatorname {\bold CP}^1$-structures with $\{\pi (q_n)\} \subset K$, their monodromies $mon (q_n)$ diverge. Here, $\pi(q_n)$ stands for the underlying complex structure of $q_n$. \endproclaim 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 $l_X(\mu)$ and $E_{gr_\mu(X)}(\mu)$ is in the same order when they are large. Here, $gr_\mu(X)$ denote the Riemann surface obtained by grafting $\mu$ to $X$ (namely, the underlying structure of the $\operatorname {\bold CP}^1$-structure given by Thurston's parameter $(X, \mu)$), $l_X(\mu)$ is the hyperbolic length of $\mu$ on $X$ and $E_{gr_\mu (X)}(\mu)$ is the extremal length of $\mu$ on the grafted surface $gr_\mu(X)$. Furthermore, by the definition of bendinging, $l_X(\mu)$ is the length of $\mu$ for the monodromy of the $\operatorname {\bold CP}^1$-structure. Therefore, for a diverging sequence $q_n = (X_n, \mu_n)$ as in the assumption of Theorem 3.1, $l_{X_n}(\mu)$ had to grow ``too fast" and we can not maintain the monodromy in a bounded set of $\operatorname {PSL}(2, \bold C)$-representation as $n \to \infty$, considering Lemma 2.5.\par \proclaim {Theorem 3.2(\cite {Tg})} Let $X$ be a hyperbolic surface and $\mu$ be a measured lamination. Let $h : gr_\mu(X) \to X$ be the harmonic map with respect to the hyperbolic metric on $X$ and $\Cal {E} (h)$ be its energy. Then $$\frac {1}{2} l_X(\mu) \le \frac {1}{2}\frac{l_X(\mu)^2}{E_{gr_\mu (X)}(\mu)} \le \Cal {E} (h) \le \frac {1}{2} l_X(\mu) + 4\pi (g-1).$$ \endproclaim Recall that in Remark 1 in \cite {Tg}, we observed that when $l_X(\mu)$ 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. \par The following fact follows easily from the above inequality (cf. \cite {Tg}). \proclaim {Corollary 3.3} For any $X \in T_g$, the mapping $gr_\cdot (X) : \Cal {ML} \to T_g$ is a proper map. For any $\mu \in \Cal {ML}$, the mapping $gr_\mu(\cdot) : T_g \to T_g $ is a proper map. \endproclaim \demo {Proof} The properness of $gr_\mu(\cdot): T_g \to T_g $ was shown in \cite {Tg}. For the properness of $gr_\cdot (X) : \Cal {ML} \to T_g$, note that $$ \frac {E_{gr_\mu (X)}(\mu)}{E_X (\mu)} \le \frac {l_X(\mu)}{E_X (\mu)}, $$ by Theorem 3.2. When $X$ is fixed and $\mu$ tends to infinity, the right term tends to $0$, therefore, so is the left term. It follows that the Riemann surface $gr_\mu (X)$ tends to infinity.\qed \enddemo \remark {Remark} In fact, in what follows we use only the leftmost inequality in Theorem 3.2. The others was necessary to show that $gr_\mu(\cdot)$ is proper. \endremark Now we can present a geometric proof of Theorem 3.1. \demo{Proof of Theorem 3.1} Let $(X_n, \mu_n) \in T_g \times \Cal {ML}$ be the Thurston coordinate of the projective structure $q_n$ and denote the underlying complex structure by $\pi (q_n) = gr_{\mu_n}(X_n)$. 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 $\pi(q_n)$ stays in $K$ and $\{q_n\}$ tends to infinity, both of $\{X_n\}$ and $\{\mu_n\}$ tend to infinity. Take a sequence $\epsilon_n$ converging to $0$ such that $\epsilon_n \mu_n$ converges to a non-zero measured lamination $\mu$, taking a subsequence of $\{\mu_n\}$ if necessary. \par By Theorem 3.1, $$ \frac{l_{X_n}(\mu_n)^2} {l_{X_n}(\mu_n) + 8\pi (g-1)} \le E_{gr_{\mu_n}(X_n)}(\mu_n) \le l_{X_n}(\mu_n). $$ Multiply each term of the above inequality by $\epsilon_n^2$. Then we have $$ \frac{l_{X_n}({\epsilon_n \mu_n})^2} {l_{X_n}({\mu_n}) + 8\pi (g-1)} \le E_{gr_{\mu_n}(X_n)}(\epsilon_n\mu_n) \le \epsilon_n l_{X_n}(\epsilon_n \mu_n). $$ Note that $l_{X_n}(\epsilon_n \mu_n)$ is the length of $\epsilon_n \mu_n$ for the monodromy representation $mon(q_n)$ because the pleated surface defining the Thurston coordinate for $q_n = (X_n, \mu_n)$ has pleating locus $\mu_n$. Therefore, if the representation $mon(q_n)$ did not diverge, $l_{X_n}(\epsilon_n \mu_n)$ would converge to a non-negative number by Lemma 2.5, taking a subsequence if necessary. Therefore, by the above inequality, $$\lim_{n \to \infty} E_{gr_{\mu_n}(X_n)}(\epsilon_n\mu_n) = 0$$ However, by taking a subsequence if necessary, we may assume that $\pi(q_n) = gr_{\mu_n}(X_n)$ converges to a point $Y \in T_g$ by the assumption that $\{q_n\}$ stays in the compact set $K$. Therefore $E_{gr_{\mu_n}(X_n)}(\epsilon_n\mu_n)$ converges to $E_Y(\mu)$, which is a positive number. This is a contradiction. \qed \enddemo \Refs \widestnumber\key{99999} \ref \key EM \by D. B. A. Epstein and A. Marden \paper Convex hulls in hyperbolic space, a theorem of Sullivan and measured pleated surfaces \inbook London Mathematical Society Lecture Notes \vol 111 \yr 1897 \pages 114-253 \publ Cambridge University Press \endref \ref \key GKM \by D. Gallo, M. Kapovich and A. Marden \paper The monodromy groups of Schwarzian equation on compact Riemann surfaces \jour (preprint) \endref \ref \key H \by D. Hejhal \paper Monodromy groups and linearly polymorphic functions \jour Acta math. \vol 135 \yr 1975 \pages 1-55 \endref \ref \key KT \by Y. Kamishima and S. P. Tan \paper Deformation spaces on geometric structures \inbook Aspects of Low Dimensional Manifolds, Advanced Studies in Pure Mathematics 20 \publ Kinokuniya Co. \yr 1992 \pages 263-299 \endref \ref \key Ka \by M. Kapovich \paper On monodromy of complex projective structures \jour Invent. math. \vol 119 \yr 1995 \pages 243-265 \endref \ref \key L \by F. Labourie \paper Surfaces convexes dans l'espace hyperbolique et $\operatorname {\bold CP}^1$-structures \jour J. London Math. Soc. \vol 45 \yr 1992 \pages 549-565 \endref \ref \key Tg \by H. Tanigawa \paper Grafting, harmonic maps and hyperbolic surfaces \jour (preprint) \endref \ref \key Th1 \by W. Thurston \paper Geometry and Topology of 3-manifolds \jour Princeton University lecture notes \yr 1979 \endref \ref \key Th2 \by W. Thurston \paper Zippers and Univalent Functions \inbook The Bieberbach conjecture \publ West La\-fayette Ind. \yr 1985 \pages 185-197 \endref \endRefs \enddocument