Complex Projective Structures, Grafting, and Teichmüller Theory

Transcription

1 Complex Projective Structures, Grafting, and Teichmüller Theory A thesis presented by David Dumas to The Department of Mathematics in partial fulllment of the requirements for the degree of Doctor of Philosophy in the subject of Mathematics Harvard University Cambridge, Massachusetts May 2004

2 c David Dumas All rights reserved.

3 Thesis advisor Curtis T. McMullen Author David Dumas Complex Projective Structures, Grafting, and Teichmüller Theory Abstract We study the space P(S) of marked complex projective (CP 1 ) structures on a compact surface in terms of Teichmüller theory and hyperbolic geometry. In particular, we show that the structure of this space as a bundle over the Teichmüller space T (S) of conformal structures is compatible with Thurston's parameterization of P(S) using grafting, and we provide an explicit description of the boundary of the ber P (X) over X T (S) in terms of an involution on the space PML (S) of projective measured laminations. This involution encodes the conformal geometry of X via the orthogonality of the vertical and horizontal measured foliations of holomorphic quadratic dierentials. We also apply these results to study the projective structures with Fuchsian holonomy on a xed Riemann surface X. We formulate a general conjecture comparing these Fuchsian centers and associated Strebel dierentials on X, and then prove that this conjecture holds along rays in ML (S) supported on nitely many simple closed curves. This generalizes previous results about Fuchsian centers obtained by Anderson. The proofs of these results involve the theory of harmonic maps between Riemann surfaces and from Riemann surfaces to R-trees. Specically, we show that the canonical collapsing and co-collapsing maps associated to a complex projective surface are nearly harmonic, and then apply existing results about harmonic maps and their Hopf dierentials.

4 Contents Title Page Abstract Table of Contents Acknowledgments i iii iv vi 1 Introduction Overview Complex projective geometry The antipodal map Fuchsian centers and Strebel dierentials Harmonic Maps Preliminaries Grafting and pruning Conformal metrics and tensors on Riemann surfaces Quadratic dierentials, foliations, and the antipodal map R-trees and co-collapsing Complex projective geometry Projective Grafting The antipodal map Introduction Harmonic maps Energy and grafting Convergence to the harmonic map Measurable quadratic dierentials The antipodal map extends pruning A Appendix: Asymmetry of Teichmüller geodesics Fuchsian centers and Strebel dierentials Introduction Holonomy of projective structures Fuchsian centers Grafting annuli The developing map iv

5 Contents v 4.6 Finite rays and Strebel dierentials Comments and questions Open questions Connectedness and computer experiments Bibliography 47

6 Acknowledgments First of all, I would like to thank my advisor, Curt McMullen, for all he has taught me over the last ve years, and for his sound advice and inspiring clarity in all things mathematical. Special thanks are also due to Yum-Tong Siu and Cli Taubes for reading drafts of this thesis. I must also thank Nathan Duneld, Maryam Mirzakhani, Laura DeMarco, Danny Calegari, and Daniel Allcock for many helpful and inspiring conversations. I am grateful to the graduate students in the Mathematics Department for creating a pleasant and helpful working environment, and to the Department, Harvard University, and the National Science Foundation for their generosity in supporting my studies nancially. I would like to thank my family and friends for oering so much understanding and support through good times and bad. I am especially indebted to Aravind Asok, Julie Norseen, Jacob Hartman, and Deepee Khosla for helping me keep my sanity when it seemed impossible. Finally, I cannot imagine how I would have made it through this most dicult last year without the love and support of Sarah Hutton. She has seen me at my worst, and still laughs at all of my stupid jokes and listens to all of my boring stories. This is more than I could reasonably expect from anyone.

7 Chapter 1 Introduction 1.1 Overview In this thesis we study complex projective Riemann surfaces, which are geometric objects that incorporate aspects of both two-dimensional conformal geometry and threedimensional hyperbolic geometry. This dual nature arises from the simultaneous action of the group PSL 2 C of Möbius transformations on CP 1 by holomorphic automorphisms, and on the three-dimensional hyperbolic space H 3 by orientation-preserving isometries. While complex projective geometry has long been studied as a topic in complex analysis, the richness of its connection with hyperbolic geometry has only recently come to light, beginning with the work of Thurston on grafting, and continuing with more recent results of Tanigawa, McMullen, Scannell, Wolf, and others. Despite this considerable progress, the geometric and complex-analytic theories of complex projective geometry have remained mostly separate, and it is the compatibility between these perspectives that we address in our main results. Specically, we consider the space P(S) of marked complex projective structures on a compact dierentiable surface S, which is a bundle over the Teichmüller space T (S) of conformal structures. The ber P (X) over X T (S) is the space of all complex projective structures with underlying conformal structure X, and is naturally identied with the space Q(X) of holomorphic quadratic dierentials on X. Each quadratic dierential φ Q(X) determines a horizontal foliation of X, which is equivalent to a unique measured geodesic lamination Λ(φ) ML (S). The map Λ : Q(X) ML (S) is a homeomorphism and can be used to transport the involution (φ φ) of Q(X), which interchanges the vertical and horizontal foliations of φ, to an involution i X : ML (S) ML (S). By homogeneity, there is an associated antipodal involution: i X : PML (S) PML (S). For each λ ML (S) and Y T (S), there is a Riemann surface X = gr λ Y, the grafting of Y along λ, which is obtained by thickening the geodesic realization of λ on Y by inserting Euclidean strips. In a sense, this procedure mixes the hyperbolic geometry of Y and the Euclidean geometry of a thickened lamination, and X is like the midpoint of these two objects. 1

8 2 Chapter 1: Introduction For each λ ML (S), there is a unique Y = pr λ X T (S), the pruning of X along λ, such that X = gr λ Y. Just as xing the midpoint of a pair of points forces them to move in opposite directions, we show that the large-scale behavior of the correspondence λ pr λ X is given by the antipodal map on the boundary: The pruning map with basepoint X, (λ pr λ X), extends continuously to the boundary PML (S), where it agrees with the antipodal involution i X : PML (S) PML (S). Within P (X) Q(X), there is a countably innite discrete set of projective structures that have Fuchsian holonomy representations for example, 0 Q(X) corresponds to the Fuchsian uniformization of X by H. In Chapter 4, we further relate the analytic and geometric aspects of projective structures by estimating the positions of these Fuchsian centers c γ Q(X), which are indexed by integral laminations γ ML Z (S). The comparison is based on the associated Strebel dierentials s γ Q(X), which are holomorphic dierentials with closed trajectories corresponding to γ: For each X T (S), γ ML Z (S), and k N, we have 2c kγ s kγ 1 C(X, γ) where C(X, γ) is a constant depending only on X and γ (but not k). The proofs of the main results in this thesis rely on a careful study of the collapsing and co-collapsing maps associated to a grafted surface, which generalize the nearest-point projection from a domain in CP 1 to the boundary of its hyperbolic convex hull and its dual, the supporting hyperplane map of the convex hull boundary. We show that both of these maps are very nearly harmonic (for the collapsing map, this is due to Tanigawa [35]), and as a result they have the same large-scale behavior as the harmonic maps between the relevant spaces. We then deduce the main theorems by applying existing results about harmonic maps between Riemann surfaces and from surfaces to R-trees (see [38], [39], and [40]). In the rest of this chapter, we formulate the main results in greater detail, and then briey discuss the techniques involved in the proofs. 1.2 Complex projective geometry A complex projective structure is an atlas of charts on a Riemann surface taking values in CP 1 with the property that the transition functions are Möbius transformations. Since (CP 1, PSL 2 C) compacties (H 3, PSL 2 C), complex projective geometry is the natural boundary geometry for 3-dimensional hyperbolic geometry. The moduli space P(S) of marked complex projective structures on a compact smooth surface S of genus g > 1 is a contractible complex manifold of dimension 6g 6; the natural map π : P(S) T (S) to Teichmüller space that records the underlying conformal structure makes P(S) into a complex ane vector bundle. The Schwarzian derivative S can be used to identify P(S) with the cotangent bundle to Teichmüller space, i.e. the bundle Q(S) of holomorphic quadratic dierentials. Alternately, Thurston has shown that every complex projective structure can be uniquely described by grafting, a cut-and-paste operation on hyperbolic Riemann surfaces. This leads to a homeomorphism Gr : ML (S) T (S) P(S).

9 Chapter 1: Introduction 3 The typical example of a complex projective structure is the boundary at innity of a complete hyperbolic manifold M. Each geometrically nite end of such a manifold is a complex projective Riemann surface; Thurston's grafting coordinates in this case are the bending lamination and intrinsic hyperbolic metrics of the faces of the convex core. In general, a kind of local convex hull construction gives rise to a locally convex pleated surface whose metric and bending recover the grafting data. A complex projective structure on a Riemann surface X can also be dened via a holomorphic immersion f : X 1 CP that intertwines the action of π 1 (X) by deck transformations with some representation ρ : π 1 (X) PSL 2 (C). The latter is the holonomy of the projective structure, and is well-dened as an element of V (S) = Hom irr (π 1 S, PSL 2 C)/(PSL 2 C conjugacy). To summarize, P(S) is linked to other standard objects in Teichmüller theory via an ensemble of maps: Gr ML (S) T (S) P(S) η V (S) where gr π T (S) ML (S) is the space of measured laminations on S, Gr λ X is the grafted projective surface obtained from X by removing the lamination λ and inserting Euclidean strips according to the measure, π is the forgetful map that sends a CP 1 structure to its underlying complex structure, gr = π Gr is the conformal grafting map, where the projective structure on the grafted surface is weakened to a conformal structure, η is the holonomy map, which records the failure of the projective charts to be globally dened as an element of V (S) = Hom irr (π 1 S, PSL 2 C)/(PSL 2 C conjugacy), the representation variety. 1.3 The antipodal map Scannell and Wolf have shown (extending work of Tanigawa) that for each lamination λ ML (S), the conformal grafting map gr λ : T (S) T (S) is a homeomorphism, thus there is an inverse or pruning map pr λ : T (S) T (S) [31]. We now describe the large-scale behavior of the grafting data for complex projective structures on X. The description is based on the map Λ : Q(X) ML (S)

10 4 Chapter 1: Introduction which records the measured lamination equivalent to the horizontal foliation of a holomorphic quadratic dierential. Hubbard and Masur showed that Λ is a homeomorphism, so we can use it to transport the involution (φ φ) of Q(X) to an involutive homeomorphism i X : ML (S) ML (S). Since the map Λ is homogeneous, i X descends to an involution on PML (S) = ML (S)/R +, i X : PML (S) PML (S), which we call the antipodal involution with respect to X, since it is conjugate by Λ to the actual antipode map on the vector space Q(X). The main result of Chapter 3 is that the antipodal map governs the large-scale behavior of pruning with basepoint X. Let ML (S) denote the natural compactication of ML (S) by PML (S), and let T (S) denote the Thurston compactication of Teichmüller space, which also has boundary PML (S). Theorem 1.1 (Antipodal limit). The pruning map with basepoint X, λ pr λ X, extends continuously to a map ML (S) T (S) whose boundary values are exactly the antipodal map i X : PML (S) PML (S). Let P (X) denote the space of projective structures with underlying conformal structure X. Then Theorem 1.1 provides an explicit description of the boundary of P (X) when viewed as a subset of ML (S) T (S). Theorem 1.2 (Boundary P (X)). For each X T (S), the boundary of P (X) is the graph of the antipodal involution i X, i.e. P (X) = {([λ], [i X λ]) PML PML } (ML (S) T (S)). In particular, Theorem 1.2 implies that the closure of P (X) is a ball of dimension 6g 6, where g is the genus of S. It is tempting to compare the role of the antipodal involution in Theorem 1.1 to that of the geodesic involution in a symmetric space. Indeed, the antipodal map relative to X exchanges projective measured laminations ([λ], [µ]) dening Teichmüller geodesics that pass through X, just as the geodesic involution exchanges endpoints (at innity) of geodesics through a point in a symmetric space. However, we provide an example showing that this analogy does not work, because Teichmüller geodesics are badly behaved with respect to the Thurston compactication. Specically, in Ÿ3.A we construct a pair of Teichmüller geodesics through a single point in T (S) that are asymptotic to each other in one direction, but which have distinct limit points in the other direction. This precludes the existence of any map of the Thurston boundary that plays the role of a Teichmüller geodesic involution. 1.4 Fuchsian centers and Strebel dierentials In Chapter 4 we develop results about the holonomy of complex projective structures. Figure 1.1 shows the set K(X) P (X) C of projective structures on a xed punctured torus that have discrete holonomy representations. 1 The point marked at the center of 1 The image in Figure 1.1 was created using the author's software package Bear, which is available on the world wide web at See also Ÿ5.2.

11 Chapter 1: Introduction 5 Figure 1.1: The region K(X) Q(X) C is shown for the hexagonal punctured torus. Four Fuchsian centers corresponding to the empty lamination and the three shortest hyperbolic geodesics are marked. The central component is Bers' embedding of the Teichmüller space T (S); the other three marked components represent exotic projective structures on X.

12 6 Chapter 1: Introduction the picture is the origin in Q(X), which corresponds to the Fuchsian uniformization of X by H. It lies at the center of the Bers embedding B(X) of Teichmüller space, a bounded contractible open set in Q(X) consisting of quasifuchsian groups that have a domain of discontinuity with quotient conformal structure X. This island is known by work of Shiga and Tanigawa to be a connected component of the interior of K(X). Surrounding B(X) are other exotic islands of quasifuchsian holonomy, some of which contain Fuchsian centers projective structures with Fuchsian holonomy. A Fuchsian center c γ Q(X) is uniquely specied by an integral lamination γ ML Z (S) (i.e. a weighted multicurve); in Figure 1.1, three exotic centers are marked, corresponding to the three shortest closed hyperbolic geodesics on X. The three-fold symmetry of the picture reects the fact that this particular punctured torus is obtained by identifying opposite edges of a regular hexagon and removing one cycle of vertices. For each integral lamination, γ ML Z (S), there is also the Strebel dierential s γ Q(X), which has closed trajectories representing the curves in the support of γ, and where these closed trajectories foliate Euclidean cylinders whose heights are given by the weights of the curves in 2πγ. We conjecture that when properly normalized, the Fuchsian centers lie within a bounded distance of the associated Strebel dierentials: Conjecture 1.3. For each X T (S) and all γ ML Z, we have s γ 2c γ 1 C(X), where C(X) is a constant that depends only on X. Conjecture 1.3 would imply that the norms of the centers grow quadratically as the measure on γ is rescaled. Previously, Anderson showed that there is a quadratic lower bound for the norms of centers corresponding to simple closed curves [2]. In Chapter 4, we show that the above conjecture holds for rays in ML Z (S): Theorem 1.4. Fix X T (S) and γ ML Z (S). Then there exists a constant C(X, γ) such that for all k N, s kγ 2c kγ 1 C(X, γ). We actually prove a stronger result about dierentials corresponding to nite rays in ML (S), i.e. those that are supported on simple closed curves with arbitrary positive weights. Theorem 1.4 is the resulting statement for integral laminations. Note that the Strebel dierentials s kγ lie on a line in the vector space Q(X), while the Fuchsian centers c kγ lie within islands of quasifuchsian holonomy. Thus Theorem 1.4 implies that each ray of integral Strebel dierentials follows the course of a sequence of distinct islands of quasifuchsian holonomy in Q(X). In comparison to the Bers embedding B(X), little is known about these other islands of quasifuchsian holonomy. We therefore view Theorem 1.4 as a rst attempt at understanding their distribution by means of the Fuchsian centers.

13 Chapter 1: Introduction Harmonic Maps The proofs of the results mentioned above use techniques from the theory of harmonic maps between Riemann surfaces and from surfaces to R-trees. In this section we briey describe how this theory is connected to grafting and complex projective geometry, and how this connection is used to prove Theorem 1.1 in Chapter 3. In Chapter 4, the same basic principles are combined with the theory of univalent functions to prove Theorem 1.4. We x X T (S) and consider P (X), the space of projective structures with this underlying conformal structure. Using Thurston's parameterization, each of these is obtained by grafting, and therefore has the form Gr λ Y for some Y T (S) and λ ML (S). The projective structure Gr λ Y P (X) is made up of hyperbolic pieces that come from Y and a Euclidean part that results from thickening λ Y. There is a canonical collapsing map κ : X Y that collapses the grafted part back to its geodesic representative in Y, and which is an isometry on the hyperbolic part of Gr λ Y. In Thurston's theory of projective grafting, this lift of the collapsing map to X is realized as the projection from CP 1 down to a locally convex pleated plane p : H 2 H 3. There is a dual object ˆκ, which we call the co-collapsing map, which collapses the hyperbolic part of X and remembers only the leaves of the natural geodesic foliation of the Euclidean part by parallels of λ: ˆκ : X Tλ The image of this map is a one-dimensional object T λ which can be interpreted as the set of supporting hyperplanes of the pleated plane p : H 2 H 3. For example, if λ is supported on a set of simple closed curves, then p has a locally nite pattern of bending lines, and T λ is the dual simplicial tree. For more general laminations, T λ is the R-tree dual to the lamination λ. It follows from results of Wolf that as Y, the Hopf dierential Φ(h) of the harmonic map h : X Y detects the Thurston limit of Y via its vertical foliation [37]. Similarly, the Hopf dierential Φ(ĥ) of the harmonic map from ĥ : X T λ detects the lamination λ via its vertical foliation. The key observation about the collapsing and co-collapsing maps is that they are both very nearly harmonic, in the sense that the energy of either map exceeds that of the associated harmonic map by at most 2π χ(s) ; for the collapsing map, this was rst noticed by McMullen and Tanigawa. Furthermore, the Hopf dierentials Φ(κ) and Φ(ˆκ) are exactly opposite, i.e. Φ(κ) = Φ(ˆκ), and therefore dene orthogonal foliations of the grafted part of Gr λ Y. Since κ and ˆκ closely approximate the respective harmonic maps, the Thurston limit of Y and the projective limit of λ are approximated by (and in the limit, equal to) the vertical foliations of Φ(κ) and Φ(κ), respectively, which means exactly that they are related by the antipodal involution i X.

14 Chapter 2 Preliminaries 2.1 Grafting and pruning. Let S be a compact oriented surface of genus g > 1, and T (S) the Teichmüller space of marked conformal (equivalently, hyperbolic) structures on S. The simple closed hyperbolic geodesics on any hyperbolic surface Y T (S) are in one-to-one correspondence with the free homotopy classes of simple closed curves on S; therefore, when a particular hyperbolic metric is under consideration, we will use these objects interchangeably. Fix Y T (S) and γ, a simple closed hyperbolic geodesic on Y. Grafting is the operation of removing γ from Y and replacing it with a Euclidean cylinder γ [0, t], as shown in Figure 2.1. The resulting surface is called the grafting of Y along the weighted geodesic tγ, written gr tγ Y. Associated to each grafted surface gr tγ Y is a canonical map κ : gr tγ Y Y, the collapsing map, that collapses the grafted cylinder γ [0, t] back onto the geodesic γ. There is also a natural C 1 conformal metric ρ Th on gr tγ Y, the Thurston metric, that unites the hyperbolic metric on Y with the Euclidean metric of the cylinder γ [0, t]. The collapsing map is distance non-increasing with respect to the hyperbolic metric on Y and the Thurston metric on gr tγ Y. We now dene a generalization of a weighted simple closed curve that is compatible with grafting. Let C denote the set of free homotopy classes of simple closed curves. The geometric (unsigned) intersection number i : C C Z 0 Y gr tγ Y γ t Figure 2.1: The basic example of grafting. 8

15 Chapter 2: Preliminaries 9 denes an embedding of R C into R C via tγ t i(γ, ). The space ML (S) of measured geodesic laminations (or measured laminations) on S is the closure of the image of R 0 C under this map. The intersection number extends to a continuous map i : ML (S) ML (S) R 0. Another denition of a measured geodesic lamination uses a hyperbolic metric Y T (S). Then a measured geodesic lamination λ ML (S) determines a foliation G λ of a closed subset of Y by complete hyperbolic geodesics, and associates to each transversal τ : [0, 1] Y a positive Borel measure in a way that is compatible with maps between transversals induced by transversality-preserving isotopy. We call the resulting object the geodesic realization of λ on Y, and the set of geodesics in G λ is the support of λ. For example, the geodesic realization of a weighted simple closed curve tγ R + C on Y is the closed hyperbolic geodesic freely homotopic to γ, and the measure on a transversal is the sum of atoms of weight t at each of its intersection points with the geodesic. Thurston has shown that grafting extends continuously to arbitrary measured laminations, and thus denes a map gr : ML (S) T (S) T (S) where (λ, Y ) gr λ Y. Morally, gr λ Y is obtained from Y by thickening the geodesic realization of λ in a manner determined by the transverse measure. As in the simple closed curve case, there is a collapsing map κ : gr λ Y Y that collapses the grafted part A gr λ Y (which is no longer a union of annuli) onto the geodesic realization of λ on Y, and a conformal metric ρ Th on gr λ Y that is hyperbolic on gr λ Y A. One can show that ρ Th is of class C 1,1 on gr λ Y, and thus its curvature is dened almost everywhere [20]. Scannell and Wolf have shown that for each λ ML (S), the map gr λ : T (S) T (S) is a homeomorphism. Thus there is the inverse, or pruning map pr : ML (S) T (S) T (S) where (λ, X) pr λ X = gr 1 λ X. In other words, for each X T (S) and λ ML (S), there is a unique way to present X as a grafting of some Riemann surface Y = gr 1 λ (X) along λ, and pruning is the operation of recovering Y from the pair (λ, X). We will be interested in the pruning map when the surface X is xed, i.e. the map λ pr λ X from ML (S) to T (S). Theorem 3.1 describes the asymptotic behavior of this map in terms of the conformal geometry of X. 2.2 Conformal metrics and tensors on Riemann surfaces Fix a hyperbolic Riemann surface X and let S(X) denote the space of measurable complex-valued quadratic forms on T X. A form β S(X) can be decomposed according to

16 10 Chapter 2: Preliminaries the complex structure on X: β = β 2,0 + β 1,1 + β 0,2 where β i,j Γ((T 1,0 X) i (T 0,1 X) j ). Then β 1,1 is a conformal metric on X, which can be thought of as a circular average of β, β 1,1 (v) = 1 2π β(r θ v, R θ v) dθ, 2π 0 where v T x X and R θ Aut(T x X) is the rotation by angle θ dened by the conformal structure of X. The area of X with respect to β 1,1, when it is nite, denes a natural L 1 norm β L 1 (X) = β 1,1, which we will abbreviate to β 1 if the domain X is xed. We also call β 1 the energy of β. We write S 2,0 (X) for the space of measurable quadratic dierentials on X, which are those quadratic forms φ S(X) such that φ = φ 2,0. Within S 2,0 (X), there is the space Q(X) of holomorphic quadratic dierentials, i.e. holomorphic sections of T X 2. By the Riemann-Roch theorem, if X is compact and has genus g, X dim C Q(X) = 3g 3. If X is given a conformal metric σ, then we can also dene the L p and L norms: ( β L p (X,σ) = X β L (X,σ) = sup z X ( β 1,1 ) p 1/p dσ) σ β 1,1 (z) σ(z) We use β L p (X) and β L (X) as abbreviations for these norms when σ is the hyperbolic metric uniformizing X. Fixing a conformal metric also allows us to dene the unit tangent bundle SX = {(x, v) T X v σ = 1}, in which case the interpretation of β 1,1 as the circular average of β yields another expression for β L 1 (X): β L 1 (X) = 1 β(v) dθ(v)dσ(x) 2π X S xx The Hopf dierential Φ(β) of β S(X) is the (2, 0) part of its decomposition, Φ(β) = β 2,0, which (along with Φ(β)) measures the failure of β to be compatible with the conformal structure of X. For example, there is a function b : X C such that β = bσ if and only if Φ(β) = Φ(β) = 0.

17 Chapter 2: Preliminaries 11 Let f : X (M, ρ) be a smooth map from X to a Riemannian manifold (M, ρ). Then the energy E (f) and Hopf dierential Φ(f) of f are dened to be those of the pullback metric f (ρ): E (f) = f (ρ) 1 = f (ρ) 1,1 X (2.1) Φ(f) = [f (ρ)] 2,0 Since f (ρ) is a real quadratic form, at points where Df is nondegenerate f is conformal if and only if Φ(f) = 0. Thus E (f) is a measure of the average stretching of the map f, while Φ(f) records its anisotropy. 2.3 Quadratic dierentials, foliations, and the antipodal map In this section we briey recall the identications between the spaces of measured foliations, measured geodesic laminations, and holomorphic quadratic dierentials on a compact Riemann surface. We will consider foliations on Riemann surfaces which have certain singularities. For each k Z +, the foliation of C by horizontal lines pulls back via the map (z z (k/2) ) to yield a foliation F (k) of C which has a singularity at the origin. For our purposes, a singular foliation of a surface is a foliation that is dened except at a discrete set of points x i, and where the foliation of a punctured neighborhood of x i is dieomorphic to a neighborhood of 0 in F (k i ) for some k i Z +. A measured foliation on S is a singular foliation F and an assignment of a Borel measure µ F on [0, 1] to each arc [0, 1] S that is everywhere transverse to F, subject to the condition that the measure should be invariant under transversality-preserving isotopy. The notation MF (S) is used for the quotient of the set of measured foliations by the equivalence relation generated by isotopy and Whitehead moves (e.g., collapsing leaves connecting singularities). The typical example of a measured foliation comes from a holomorphic quadratic differential φ Q(X). The measured foliation F (φ) determined by φ is the pullback of the horizontal line foliation of C under integration of the locally dened holomorphic 1-form φ. The measure on transversals is obtained by integrating the length element Im φ. Equivalently, a vector v T x X is tangent to F (φ) if and only if φ(v) > 0. The foliation F (φ) is called the horizontal foliation of φ. Since φ = i φ, F (φ) and F ( φ) are orthogonal, and F ( φ) is called the vertical foliation of φ. Hubbard and Masur proved that measured foliations and quadratic dierentials are essentially equivalent notions: Theorem 2.1 (Hubbard and Masur, [11]). For each ν MF (S) of measured foliations and X T (S) there is a unique holomorphic quadratic dierential φ X (ν) Q(X) such that F (φ X (ν)) = ν. Furthermore, the map φ X : MF (S) Q(X) is a homeomorphism.

18 12 Chapter 2: Preliminaries Note that the transverse measure of φ Q(X) is dened using φ, and so for c > 0, F (cφ) = c 1 2 F (φ). As a result, the Hubbard-Masur map φ X has the following homogeneity property: φ X (Cν) = C 2 φ X (ν) for all C > 0 (2.2) One can also view measured foliations as diuse versions of measured laminations, in that every measured foliation is associated to a unique measured lamination with the same intersection properties. More formally, a class of measured foliations F MF (S) is uniquely determined by the point I F R C, where I F (γ) is the minimum total measure associated to a closed transversal homotopic to γ, i.e. I F (γ) = inf µ F τ γ τ The image of MF (S) in R C is exactly the set of measured geodesic laminations, inducing a homeomorphism MF (S) ML (S). Using this homeomorphism implicitly, we can consider the Hubbard-Masur map φ X to have domain ML (S), and we write Λ for its inverse, Λ : Q(X) ML (S). We can also use Λ to transport the linear involution φ ( φ) of Q(X) to an involutive homeomorphism i X : ML (S) ML (S), i.e. i X (λ) = Λ( φ X (λ)). Since F (φ) and F ( φ) are orthogonal foliations, we say that λ, µ ML (S) are orthogonal with respect to X if i X (λ) = µ. The resulting homeomorphism depends on X T (S) in an essential way, just as the orthogonality of foliations or tangent vectors depends on the choice of a conformal structure. Since Λ is homogeneous, it also induces a homeomorphism between projective spaces: Λ : P + Q(X) = Q(X)/R + PML (S) = ML (S)/R +. Thus we obtain an involution i X : PML (S) PML (S) that is topologically conjugate to the antipodal map ( 1) : P + Q(X) S 2n 1 S 2n 1. We call i X the antipodal involution with respect to X. 2.4 R-trees and co-collapsing An R-tree (or real tree) is a complete geodesic metric space in which there is a unique embedded path joining every pair of points, and each such path is isometric to an interval in R. Thus an R-tree is like a simplicial tree, but there is no distinction between branching

19 Chapter 2: Preliminaries 13 points and edges. We will mainly consider R-trees that arise from the following construction (a more detailed treatment can be found in [16]): Let F MF (S) be a measured foliation of S and F its lift to the universal cover S H 2. Dene a pseudometric d F on S: d F (x, y) = inf {i(f, γ) γ : [0, 1] S, γ(0) = x, γ(1) = y} (2.3) Then the quotient metric space T F = S/(x y if d F (x, y) = 0) is an R-tree whose isometry type depends only on the measure equivalence class of F. Alternately, T F is the space of leaves of F with metric induced by the transverse measure, where we consider all of the leaves that emanate from a singular point of F to be a single point of TF. The action of π 1 (S) on S by deck transformations descends to an action on T F by isometries. If F is a measured foliation associated to the measured lamination λ, then the resulting R-tree T λ with metric d λ is called the dual R-tree of λ. If λ is supported on a family of simple closed curves, then the R-tree T λ is actually a simplicial tree with one vertex for each lift of a complementary region of λ to S, and where an edge connecting two adjacent complementary regions has length equal to the weight of the geodesic that separates them. A slight generalization of this construction arises naturally in the context of grafting. The grafting locus A gr λ Y has a natural foliation F A by Euclidean geodesics that map isometrically onto λ, with a transverse measure induced by the Euclidean metric in the orthogonal direction. The associated pseudometric on gr λ Y, d FA (x, y) = inf {i(f A, γ) γ : [0, 1] gr λ Y, γ(0) = x, γ(1) = y}, yields a quotient R-tree isometric to T λ and a map ˆκ : gr λ Y T λ, which we call the co-collapsing map. While the collapsing map κ : gr λ Y Y compresses the entire grafted part back to its geodesic representative, the co-collapsing map collapses each connected component of ( gr λ Y Ã), the complement of the grafted part of gr λ Y (generically a hyperbolic ideal triangle), and each leaf of F A to a single point. Here we can think of ( gr λ Y Ã) as a thickened version of the graph of leaves of a measured foliation incident on its singular points. 2.5 Complex projective geometry A complex projective structure on a compact dierentiable surface S is a maximal atlas of charts on S with values in CP 1 and Möbius transition functions. By analogy with the Teichmüller space T (S) of marked conformal structures, let P(S) denote the space of marked complex projective structures on S with the topology induced by uniform convergence of charts. Since Möbius transformations are holomorphic, each complex projective structure determines a conformal structure, and there is an associated projection (forgetful map): π : P(S) T (S).

20 14 Chapter 2: Preliminaries If Z is a projective structure with π(z) = X T (S), we will say that Z is a projective structure on the Riemann surface X. A projective structure on X denes a developing map δ : X Ĉ and holonomy representation η : π 1 (S) P SL 2 (C). This pair is characterized by two conditions: 1. The restriction of δ to any domain U X on which it is univalent gives a projective chart (for the projective structure of X lifted from X). 2. For all γ π 1 (S) and x X, δ(γx) = η(γ)δ(x). The pair (δ, η) can be constructed by analytically continuing projective charts for X to obtain a locally univalent holomorphic map; η(γ) is then dened to be the Möbius transformation with germ δ γx γ δx 1 at δ(x) CP 1. The pair (δ, η) is well-dened up to simultaneous conjugation of δ and post-composition of η with Möbius transformations. Conversely, any such pair (δ, η) denes a projective structure on X. For example, every hyperbolic structure X T (S) determines a standard Fuchsian projective structure whose developing map is a Riemann map δ : X H CP 1. The associated holonomy representation maps π 1 (S) to a Fuchsian group π 1 (X) such that H/π 1 (X) is isometric to X. Möbius transformations preserve the set of round circles in CP 1, and so one way to think of a complex projective structure on a Riemann surface X is that it provides a natural notion of roundness. We say that a path τ on X is round if projective charts map portions of τ to arcs of round circles. Because the hyperbolic geodesics of H 2, when realized as the disk or upper half-plane H, are arcs of circles, geodesics for the hyperbolic metric of X are round with respect to the standard Fuchsian projective structure. The Schwarzian derivative can be used to identify the space P(S) of marked projective structures on S with the cotangent bundle T T (S) of Teichmüller space as follows: Let F (z) be a locally univalent meromorphic function on a domain Ω Ĉ. The Schwarzian derivative S (F ) is the holomorphic quadratic dierential: S (F )(z) = [ (F ) (z) F 1 (z) 2 ( F ) ] (z) 2 F dz 2 (z) The signicance of this dierential operator stems from two key properties: 1. S (A) = 0 if A is (the restriction of) a Möbius transformation. 2. S (F G) = S (G) + G S (F ), the cocycle property. The Schwarzian derivative can be constructed as the derivative of the map osc F : Ω PSL 2 (C) that associates to a point z Ω the best approximating (osculating) Möbius transformation of the germ F z (see [36]). From this description, both of the above properties are easily derived. It follows from these properties that the Schwarzian derivative is invariant under postcomposition with Möbius transformations. Using this invariance, we associate a holomorphic quadratic dierential to a projective structure on X as follows:

21 Chapter 2: Preliminaries 15 First, uniformize the universal cover of X by H, and view the developing map of the projective structure as a locally univalent holomorphic map δ : H CP 1. The Schwarzian derivative φ = S (δ) is invariant under the deck action of π 1 (X) on H, because for all γ π 1 (X), the germs δ z and δ γz dier by post-composition with a Möbius transformation. Therefore, φ descends to a holomorphic quadratic dierential φ on X. For brevity we call φ the Schwarzian of the projective structure dened by δ. Given φ Q(X), there is an inverse construction that yields a projective structure on X. Its developing map is a function δ : H CP 1 such that S (δ) = φ, where φ is the lift of φ to H. This function can be constructed as the ratio of two linearly independent solutions to the linear ODE: u (z) φ(z)u(z) = 0 Therefore, the space P (X) = π 1 (X) of projective structures on X is identied with the vector space Q(X). This is the Poincaré parameterization. Allowing X to vary, this gives an identication of P(S) with the cotangent bundle T T, since TX T is naturally isomorphic to Q(X) by Teichmüller theory. There is a natural complex structure on P(S) (see [12]), as for the deformation space of any suciently rigid holomorphic geometric structure, but this is not the same as the complex structure induced by the identication P(S) T T described above. The problem is that the Fuchsian groups uniformizing Riemann surfaces do not vary holomorphically with X T ; equivalently, the section s : T P(S) assigning the standard Fuchsian projective structure to each complex structure is not holomorphic, even though it corresponds to the zero section of the bundle T T. 2.6 Projective Grafting We now describe a construction of Thurston that assigns a canonical projective structure to a grafted Riemann surface gr λ Y. First suppose that λ is supported on a single simple closed curve γ. The hyperbolic surface Y has the standard Fuchsian projective structure whose chart maps are local inverses of a Fuchsian uniformization of Y by H; similarly, the innite cylinder γ R has a natural projective structure whose charts are local inverses of the map f : C C, f(z) = exp(2πiz/l(γ, Y )), where (γ R) is isometric to C equipped with the conformal metric l(γ, Y ) dz 2π z. Here l(γ, Y ) is the hyperbolic length of the simple closed geodesic γ. These two projective structures are compatible in the sense that they induce the same projective structure on a neighborhood of γ (γ {0}) one in which γ is round (Ÿ2.5), and which is locally projectively isomorphic to a contractible neighborhood of R + in C. As a result, there is a natural complex projective structure on gr tγ Y that joins the standard Fuchsian structure on (Y γ) and the projective structure described above on γ [0, t] γ R. We write Gr tγ Y for this complex projective structure on gr tγ Y.

22 16 Chapter 2: Preliminaries H (H 3 ) ˆκ CP 1 κ H 3 Figure 2.2: The collapsing and co-collapsing maps associated to a projective structure correspond to the projection of the developing map to a locally convex pleated plane and its set of supporting hyperplanes, respectively. As before the grafting map extends continuously to general measured laminations yielding a map Gr : ML (S) T (S) P(S), which ts into a commutative diagram: Gr ML (S) T (S) T (S) π gr T (S) Thurston's theorem (unpublished) states that the projective grafting map Gr : ML (S) T (S) P(S) is a homeomorphism. In other words, for each projective surface Z, there is a unique choice of Y T (S) and λ ML (S) such that Gr λ Y = Z. Kamishima and Tan recently gave a complete proof of this theorem [15]. The key to the proof of Thurston's theorem is the construction of a locally convex pleated plane p : Ỹ H3 that is equivariant with respect to a representation η : π 1 (S) PSL 2 (C), and such that the bending of p is given by the lift λ Ỹ of λ. The developing map δ λ of Gr λ Y can then be normalized so that for each z Gr λ Y, p( κ(z)) is the nearest point retraction of δ λ (z) to a convex neighborhood of p( κ(z)) in p(ỹ ), where κ : Gr λ Y Ỹ is the lift of the collapsing map κ. The pleated plane p is a kind of local convex hull for the developing map δ λ, and the lift κ is a generalization of the nearest-point retraction of a set Ω CP 1 onto the boundary of its convex hull CH(Ω) H 3 (see Figure 2.2. Here the co-collapsing map can also be understood in terms of hyperbolic geometry; for each z Gr λ Y let h(z) denote the unique supporting hyperplane of the locally convex pleated surface p(ỹ ) that contains p( κ(z)) and whose normal vector there is tangent to the geodesic ray landing at δ λ (z) CP 1 H 3. Then h is a map from Gr λ Y to the Lorentzian manifold H (H 3 ) of hyperplanes in H 3 ; in fact h is Lipschitz, and its (a.e. dened) tangent vector has positive norm, reecting the fact that nearby supporting hyperplanes of a locally convex surface must intersect. The

23 Chapter 2: Preliminaries 17 induced metric on the image of h gives it the structure of an R-tree isometric to T λ, and the map h coincides with the co-collapsing map, i.e. ˆκ : Gr λ Y T λ.

24 Chapter 3 The antipodal map 3.1 Introduction In this chapter we prove that the large-scale behavior of the pruning map with basepoint X is governed by the antipodal involution i X : Theorem 3.1. The pruning map based at X, λ pr λ X, extends continuously to a homeomorphism ML (S) T (S), where T (S) is the Thurston compactication. The boundary values of this homeomorphism give the antipodal involution with respect to X: i X : PML (S) PML (S). Recall that the projective grafting map Gr : ML (S) T (S) P(S) and the conformal grafting map gr : ML (S) T (S) T (S) satisfy π Gr = gr, and so the ber P (X) = π 1 (X) P(S) corresponds to the set of all grafted surfaces Gr λ Y where gr λ Y = X, or equivalently, Y = pr λ X. Therefore, Theorem 3.1 allows for the following description of the boundary of P (X) in ML (S) T (S): Theorem 3.2 (Boundary P (X)). For each X T (S), the boundary of P (X) in ML (S) T (S) is the graph of the antipodal involution with respect to X, i.e. P (X) = {([λ], [i X λ]), [λ] ML (S)} PML (S) PML (S) (ML (S) T (S)). 3.2 Harmonic maps We now consider the energy functional E (f) for maps f : X (M, ρ) from a Riemann surface X to a Riemannian manifold M. In Ÿ2.2, we dened the energy for smooth maps, but the natural setting in which to work with the energy functional is the Sobolev space W 1,2 (X, M) of maps with L 2 distributional derivatives. There is some technical diculty in making this denition precise, as the W 1,2 condition does not imply continuity even in the classical setting of C-valued functions on Riemannian manifolds; see [17] for details. 18

25 Chapter 3: The antipodal map 19 While we prefer to phrase our results in this appropriate level of generality, we will only need to consider the energy functional for Lipschitz continuous maps, which are dierentiable almost everywhere (hence (2.1) still makes sense). A stationary point of the energy functional E : W 1,2 (X, M) R 0 is a harmonic map; it follows from the Euler-Lagrange equation for E (f) and Weyl's lemma that the Hopf dierential Φ(f) = f (ρ) 2,0 is holomorphic (Φ(f) Q(X)) if f is harmonic. In particular, the maximal and minimal stretch directions of a harmonic map are realized as a pair of orthogonal foliations by straight lines in the singular Euclidean metric Φ(f). Ishihara has shown that a map f : X M to a Riemannian manifold is harmonic if and only if f pulls back germs of convex functions on M to germs of subharmonic functions on X (see [13]). For any pair of compact hyperbolic surfaces X, Y and nontrivial homotopy class of maps [f] : X Y, there is a unique harmonic map h : X Y that is smooth and homotopic to f; furthermore, h minimizes energy among such maps [9]. In particular, for each X, Y T (S) there is a unique harmonic map h : X Y that is compatible with the markings; dene E (X, Y ) = E (h : X Y ). For the proof of Theorem 3.15 and for the results of Chapter 4, we will need to consider harmonic maps from Riemann surfaces to R-trees. The main references for this theory are the papers of Wolf ([38], [39], [40]); a much more general theory of harmonic maps to metric spaces is discussed by Korevaar and Schoen in [17] and [18]. Even though R-trees are somewhat badly behaved spaces (they are not in general locally compact about any point), they are natural objects to consider when examining the limiting behavior of sequences of hyperbolic structures Y i T (S) that leave all compact subsets: Theorem 3.3 (Wolf [38]). Let X, Y i T (S) and µ ML (S) be such that Y i [µ] PML (S) in the Thurston compactication. Then after rescaling the hyperbolic metrics ρ i on Y i appropriately, the sequence of metric spaces (Y i, ρ i ) converges in the equivariant Gromov- Hausdor sense to the R-tree T µ. The equivariant Gromov-Hausdor topology is a natural setting in which to consider convergence of metric spaces equipped with isometric group actions. The application of this topology to the Thurston compactication is due to Paulin [29]; for other perspectives on the connection between R-trees and Teichmüller theory, see [3], [28], [4]. Remarkably, the theory of harmonic maps is well-adapted to this generalization from smooth surfaces to metric spaces like R-trees; for example, the convergence statement of Theorem 3.3 can be extended to a family of harmonic maps from a xed Riemann surface X: Theorem 3.4 (Wolf [38]). For X T (S) and λ ML (S), let π λ : X Tλ denote the projection onto the leaves of F (φ X (λ)), where φ X (λ) is the Hubbard-Masur dierential for λ. Then: 1. π λ is harmonic, meaning that it pulls back germs of convex functions on T λ to germs of subharmonic functions on X.

26 20 Chapter 3: The antipodal map 2. If Y i T (S) is a sequence such that Y i [λ] PML (S), then the lifts of harmonic maps h i : X Y i converge in the Gromov-Hausdor sense to π X : X Tλ. Much like the case of maps between Riemann surfaces, it is most natural to work with the energy functional on a Sobolev space W 1,2 ( X, T λ ) of equivariant maps with L 2 distributional derivatives. Once again, we defer to Korevaar and Schoen for details, and note that the maps we consider are Lipschitz [17]. By denition, the metric d λ on T λ is isometric to the standard Euclidean metric on R along each geodesic segment. Thus the pullback of this metric via an equivariant W 1,2 map f : X Tλ is a well-dened (possibly degenerate) quadratic form on T X which is invariant under the action of π 1 (X). This allows us to dene the energy E (f) and Hopf dierential Φ(f) of such an equivariant map as the L 1 norm and (2, 0) part of the induced quadratic form on T X. For later use we record the following calculations relating the Hopf dierential and energy of the projection π λ to the Hubbard-Masur dierential φ X (λ); details can be found in [38]. Lemma 3.5. The Hopf dierential of π λ : X Tλ is and the energy of π λ is given by 3.3 Energy and grafting Φ(π λ ) = φ X ( 1 2 λ) = 1 4 φ X(λ), E (π λ ) = 1 2 E(λ, X) = 1 2 φ X(λ). In [35], Tanigawa shows that for a xed lamination λ, the grafting map gr λ : T (S) T (S) is proper; the proof relies on the following inequality relating the geometry of a grafted surface to the energy of a harmonic map: Lemma 3.6 (Tanigawa [35]). Let X = gr λ Y, where X, Y T (S) and λ ML (S), and let h (resp. E (h)) denote the harmonic map h : X Y compatible with the markings (resp. its energy). Then 1 2 l(λ, Y ) 1 l(λ, Y ) 2 2 E(λ, X) E (h) 1 l(λ, Y ) + 2π χ(s), 2 where l(λ, Y ) is the hyperbolic length of λ on Y, and E(λ, X) is the extremal length of λ on X. Note. The middle part of Tanigawa's inequality, i.e. 1 l(λ, Y ) 2 2 E(λ, X) E (h) is due to Minsky, and holds for any harmonic map between nite-area Riemann surfaces of the same type and any measured lamination λ [26].

27 Chapter 3: The antipodal map 21 For our purposes, one of the most important consequences of Lemma 3.6 is a relationship between the hyperbolic length of the grafting lamination on Y and its extremal length on gr λ Y : Lemma 3.7. Let X = gr λ Y, where Y T (S). Then we have l(λ, Y ) = E(λ, X) + O(1) where the implicit constant depends only on χ(s). Proof. The lower bound l(λ, Y ) E(λ, X) is immediate from the left-hand side of the inequality of Lemma 3.6, and we also have 1 l(λ, Y ) 2 2 E(λ, X) 1 l(λ, Y ) + 2π χ(s) 2 and therefore, Solving for l(λ, Y ) yields l(λ, Y ) 2 1 E(λ, X). 2l(λ, Y ) + O(1) l(λ, Y ) 1 2 E(λ, X) (E(λ, X))1/2 (E(λ, X) + O(1)) 1/2 = E(λ, X) + O(1) Detailed consideration of the relationship between harmonic maps and the geometry of grafting will be used in the proof of Theorem 3.1; however, we can already deduce a weak form of compatibility between the grafting coordinates for P(S) and the projection π : P(S) T (S) to the underlying complex structure: Lemma 3.8. Let X T (S) and P (X) = π 1 (X) P(S). Then 1. The restriction of the map P(S) ML (S) T (S) T (S) dened by (Gr λ Y ) Y to P (X) is proper. 2. The sequence of projective surfaces Gr λi Y i P (X) diverges if and only if both λ i and Y i diverge (in ML (S) and T (S), respectively). Proof. 1. Suppose Gr λi Y i diverges but Y i remain in a compact set in Teichmüller space. Since Gr : ML (S) T (S) P(S) is a homeomorphism, a divergent sequence in P (X) has the form Gr λi Y i where either λ i, Y i, or both. Since Y i is assumed to be bounded, we conclude that λ i.

28 22 Chapter 3: The antipodal map Therefore l(λ i, Y i ), and by Lemma 3.6, we have E (h i ) 1 2 l(λ i, Y i ), where h i : X Y i is the harmonic map compatible with the markings. On the other hand, a result of Wolf (see [37]) states that for any xed X T (S), E (X, ) is a proper function on T (S). Since E (h i ), we conclude Y i, which is a contradiction. 2. Once again, Gr : ML (S) T (S) P(S) is a homeomorphism, so divergence of a sequence Gr λi Y i is equivalent to divergence of at least one of the two sequences λ i or Y i. We therefore need only show that if the sequence lies in P (X), then divergence of either grafting coordinate implies divergence of both. We observed in the proof of (1) that for Gr λi Y i P (X), (Y i ) (λ i ). Suppose that instead λ i ; then E(λ i, X) and by Lemma 3.7 l(λ i, Y i ). As before, this implies the divergence of the energy, and Y i. Therefore (λ i ) (Y i ). 3.4 Convergence to the harmonic map Let X, Y i T (S) and suppose Y i ; let ρ i denote a hyperbolic metric on Y i, and h i : X Y i the harmonic map (with respect to ρ i ) compatible with the markings. We say that a sequence of maps f i W 1,2 (X, Y i ) compatible with the markings of X and Y i is a minimizing sequence if E (f i ) lim i E (h i ) = 1. Since the harmonic map h i is the unique energy minimizer in its homotopy class, a minimizing sequence asymptotically minimizes energy. In this section we will show that all minimizing sequences have the same asymptotic behavior, in a precise sense: Theorem 3.9. Let X and Y i be as above, and suppose lim i Y i = [µ] PML (S) in the Thurston compactication. Then for any minimizing sequence f i : X Y i, the measurable quadratic dierentials [f i (ρ i)] (2,0) converge projectively in the L 1 sense to a holomorphic quadratic dierential Φ Q(X) such that [F ( Φ)] = [µ], i.e. there are constants c i > 0 such that lim c i[fi (ρ i )] (2,0) = Φ. i

29 Chapter 3: The antipodal map 23 Note. The vertical foliation F ( Φ) appears in Theorem 3.9 because the Thurston limit is a lamination whose intersection number provides an estimate of hyperbolic length; directions orthogonal to F ( Φ) (that is, tangent to F (Φ)) are maximally stretched by a map with Hopf dierential Φ, so the intersection number with F ( Φ) provides such a length estimate. Before giving the proof of Theorem 3.9, we recall a theorem of Wolf upon which it is based. Theorem 3.10 (Wolf, [37]). Let X, Y i T (S), and let Ψ i denote the Hopf dierential of the unique harmonic map h i : X Y i respecting markings, where Y i is given the hyperbolic metric ρ i. Then lim i Y i = [µ] PML (S) if and only if lim [F ( Ψ i)] = [µ]. i In other words, if one compacties Teichmüller space according to the limiting behavior of the Hopf dierential of the harmonic map from a xed conformal structure X, then the vertical foliation map F ( 1) : P + Q(X) PMF (S) PML (S) identies this compactication with the Thurston compactication. Theorem 3.10 is actually a consequence of the convergence of harmonic maps h i to the harmonic projection π µ to an R-tree (combining Theorem 3.3 and Theorem 3.4), though in [37] Wolf provides an elementary and streamlined proof of this result. We will compare the Hopf dierentials of a minimizing sequence to those of a harmonic map by means of the pullback metrics. The following theorem is the main technical tool: Theorem Let f W 1,2 (X, Y ), where X, Y T (S) and Y is given the hyperbolic metric ρ. Let h be the harmonic map homotopic to f. Then and in particular f (ρ) h (ρ) 1 2 (E (f) E (h)), Φ(f) Φ(h) 1 2 (E (f) E (h)) Proof. Recall the denition of the norm on S(X): f (ρ) h (ρ) 1 = 1 f v 2 ρ h v 2 ρ dθ(v)dσ(x) (3.1) 2π X S xx For x X, let m(x) denote the midpoint of the geodesic segment γ x from f(x) to h(x) that is in the same class as the path dened by a homotopy of f to h; this denes a map m : X Y. By the quadrilateral inequality in hyperbolic space ([30] or Ÿ2.1 of [17], for v T x X, m v 2 ρ 1 2 h v 2 ρ f v 2 ρ 1 2 f v ρ h v ρ. 4 Informally, this means that the midpoint of a geodesic segment in a negatively curved Riemannian manifold is rather insensitive to movement of the endpoints, especially if one endpoint is moved faster than the other.

30 24 Chapter 3: The antipodal map Applying this inequality to the norm dierence estimate (Equation 3.1), we obtain: ( f (ρ) h (ρ) 1 2 h v 2 ρ + 2 f v 2 ρ 4 m v 2 ) ρ dθ(v)daσ (x) X S xx = 2E (h) + 2E (f) 4E (m) Since h is energy-minimizing, E (m) E (h), and f (ρ) h (ρ) 1 2 (E (f) E (h)). Since the Hopf dierential is the (2, 0) part of the pullback metric, the second statement of Theorem 3.11 also follows. When Theorem 3.11 is combined with Wolf's result on the convergence of harmonic maps of surfaces to harmonic projections to R-trees (Theorem 3.4), we obtain the following corollary: Corollary Let f W 1,2 ( X, T λ ) be a π 1 -equivariant map, where X T (S) and λ ML (S). Then 4Φ(f) φ X (λ) 1 2 (E (f) E (π λ )) Note. The role of the midpoint map m in the proof of Theorem 3.11 is part of a more general theory of convexity of the energy functional E when two maps into negatively curved spaces are connected by a geodesic homotopy. This, in turn, relies on the convexity of the distance function between geodesics in negatively curved spaces. For details, see [17]. Proof of Theorem 3.9. Clearly the sequence of harmonic maps h i : X Y i is a minimizing sequence. Applying Theorem 3.10 to the sequence Y i we nd that the Hopf dierentials converge projectively: lim [Φ(h i)] = [Φ ], where F ( [Φ ]) = [µ]. i To prove Theorem 3.9, we therefore need only show that (measurable) the Hopf dierentials Φ(f i ) of any minimizing sequence have the same projective limit as the holomorphic Hopf dierentials Ψ i = Φ(h i ) of the harmonic maps. Applying Theorem 3.11 to such a sequence, we nd and so f i (ρ i ) h i (ρ i ) 1 2 (E (f i ) E (h i )) = o(e (h i )), fi lim (ρ i) h i (ρ i) 1 i E (h i ) = 0. Since the Hopf dierential is the (2, 0) part of the pullback metric, we have lim i [Φ(f i)] = lim i [Ψ i ] = [Φ], where [F ( Φ)] = [µ].

31 Chapter 3: The antipodal map Measurable quadratic dierentials For holomorphic quadratic dierentials φ, ψ Q(X), the intersection number of their measured foliations can be expressed in terms of the dierentials (see [7]); dene ( α ) ω(φ, ψ) = Im β. Then X i(f (φ), F (ψ)) = ω(φ, ψ). However, the quantity ω(α, β) makes sense for L 1 quadratic dierentials α and β, holomorphic or not. While a measurable dierential α does not dene a measured foliation, it does have a horizontal line eld L (α) consisting of directions v T X such that α(v) > 0. Then ω(α, β) measures the average transversality (sine of twice the angle) between the line elds L (α) and L (β), averaged with respect to the measure α 1/2 β 1/2. Now consider the collapsing map κ : X pr λ X, and for simplicity let us rst suppose λ is supported on a single simple closed hyperbolic geodesic γ pr λ X, i.e. λ = tγ. Then the grafting locus A X is the Euclidean cylinder γ [0, t], and the collapsing map is the projection onto the geodesic γ. Just as the Hopf dierential of the orthogonal projection of C onto R is Φ(z Re(z)) = [dx 2 ] 2,0 = 1 4 dz2, the Hopf dierential of κ on A is the pullback of 1 4 dz2 via local Euclidean charts that take parallels of γ to horizontal lines. This dierential is holomorphic on A, and corresponds to the measured foliation 1 2λ. Thus the Euclidean metric on A, which is the restriction of the Thurston metric of X, is given by 4Φ(κ). On the complement of the grafting locus, the collapsing map is conformal and thus the Hopf dierential is zero. Therefore Φ(κ) is a piecewise holomorphic dierential on X whose horizontal line eld is the natural foliation of the grafting locus by parallels of the grafting lamination, with half of the measure of λ. This analysis extends by continuity to the case of a general lamination λ ML (S). It follows that the line eld L (Φ(κ)) represents the measured lamination 1 2λ, in that for all ψ Q(X), We therefore use the notation ω(φ(κ), ψ) = 1 i(λ, F (ψ)). (3.2) 2 Φ X (λ) = Φ(κ : X pr λ X) for the Hopf dierential of the collapsing map, which is somewhat like φ X ( 1 2 λ) in that it is a quadratic dierential whose foliation is a distinguished representative for the measured foliation class of 1 2 λ. The Hopf dierential Φ X(λ) is not holomorphic, however, though we will later see (Ÿ3.6) that is is nearly so. For now, we will simply show that L 1 convergence of Hopf dierentials Φ X (λ) to a holomorphic limit implies convergence of the laminations λ:

32 26 Chapter 3: The antipodal map Lemma Let X T (S) and λ i ML (S). If [Φ X (λ i )] [ψ], where ψ Q(X) then [λ i ] [F (ψ)] PML (S). Here [Φ X (λ i )] is the image of Φ X (λ i ) in PS 2,0 (X). Proof. First, we can choose c i > 0 such that c 2 i Φ X (λ i ) ψ. It is well known that there are nitely many simple closed curves ν k, k = 1... N, considered as measured laminations with unit weight, such that the map I : ML (S) R N, I(λ) = i(λ, ν k ) is a homeomorphism onto its image. Recall that φ X (ν k ) Q(X) is the unique holomorphic quadratic dierential satisfying F (φ X (ν k )) = ν k. Since ω(, ν k ) : S 2,0 (X) R is evidently a continuous map, we conclude from (3.2) and the hypothesis c i Φ X (λ i ) ψ that and therefore c i λ i 1 2 F (ψ). ω(c 2 i Φ X (λ i ), φ X (ν k )) = c i 2 i(λ i, ν k ) 1 2 i(f (ψ), ν k). Using the above description of Φ X (λ), we can also compute its norm, and the energy of the collapsing map: Corollary The L 1 norm of Φ X (λ) is given by Φ X (λ) 1 = 1 4 l(λ, pr λ X) = 1 E(λ, X) + O(1), 4 and the energy of the collapse map κ : X pr λ X is E(κ) = 1 2 l(λ, pr λ X) + 2π χ(s). Proof. We have seen that 4Φ X (λ) induces the Thurston metric on the grafted part A X and is zero elsewhere. The area of A with respect to the Thurston metric is l(λ, pr λ X), and therefore, Φ X (λ) 1 = 1 4 l(λ, pr λ X). On the other hand, it follows from Lemma 3.6 that l(λ, pr λ X) = E(λ, X) + O(1), which yields the rst statement in Corollary 3.14.

33 Chapter 3: The antipodal map 27 For the energy computation, we follow Tanigawa (see [35]): On the grafting locus, κ collapses directions orthogonal to the parallels of the grafting lamination, while it maps directions tangent to such parallels isometrically. Thus this part of the surface contributes E (κ A ) = 1 2 Area(A) = 1 2 l(λ, pr λ X) to the total energy. Since κ is an isometry on the complement of the grafting locus, E (κ X A ) = Area(X A) = Area(pr λ X) = 2π χ(s). The formula for E (κ) is obtained by adding these two contributions. We can apply the same analysis to the Hopf dierential of the co-collapsing map ˆκ : Gr λ Y T λ. Though the co-collapsing map is dened on the universal cover of the grafted surface, its Hopf dierential is invariant under the action of π 1 (S) and therefore descends to a measurable quadratic dierential Φ(ˆκ) S 2,0 (X). The co-collapsing map ˆκ is piecewise constant on the complement of the grafting locus, so (like Φ(κ)) its Hopf dierential is identically zero there. Within the grafting locus it is modeled on the orthogonal projection of C onto ir (where the leaves of F A correspond to horizontal lines in C). Since Φ(z Im(z)) = [dy 2 ] 2,0 = 1 4 dz2, we conclude that the Hopf dierentials Φ(ˆκ) and Φ(κ) are inverses, i.e. Φ(ˆκ) = Φ(ˆκ) = Φ X (λ) (3.3) Remark. The relationship between κ and ˆκ and their Hopf dierentials is reminiscent of the minimal suspension technique introduced by Wolf; for details, see [41]. 3.6 The antipodal map extends pruning In this section we apply Theorem 3.9 to the collapsing maps κ i : X Y i to prove Theorem 3.1 (see Ÿ3.1). Proof of Theorem 3.1. Let Z i = Gr λi Y i P (X), and suppose Z i. By Lemma 3.8, Y i, λ i, and E (X, Y i ). We need to show that if Y i [µ] PML (S) and [λ i ] [λ] PML (S) then i X ([λ]) = [µ], or equivalently, that [λ] and [µ] are the horizontal and vertical measured laminations of a single holomorphic quadratic dierential on X. By Lemma 3.6, while by Corollary 3.14, 1 2 l(λ, Y i) E (X, Y i ) 1 2 l(λ, Y i) + 2π χ(s) (3.4) E (κ i ) = 1 2 l(λ, Y i) + 2π χ(s), (3.5)

34 28 Chapter 3: The antipodal map so κ i is a minimizing sequence. Applying Theorem 3.9, we conclude that [Φ X (λ i )] [Φ] P + Q(X), where [F ( Φ)] = [µ]. On the other hand, by Lemma 3.13, this implies that [F (Φ)] = [λ], and i X ([λ]) = [µ]. While Theorem 3.1 is an asymptotic statement, the ideas used in the proof above also yield the following nite version of the comparison between the Hopf dierentials of collapsing map to the holomorphic dierential φ X (λ) representing the grafting lamination: Theorem Let X T (S) and λ ML (S), and let h : X pr λ X denote the harmonic map compatible with the markings. Then 4Φ X (λ) φ X (λ) 1 16π χ(s). Proof. Recall from (3.3) that the Hopf dierential of the co-collapsing map ˆκ is and its energy is Φ(ˆκ) = Φ X (λ) E (ˆκ) = 1 2 Area(A) = 1 2 l(λ, pr λ X), where A X is the grafting locus. Recall that by Lemma 3.6, 1 2 l(λ, pr λ X) = 1 E(λ, X) + O(1), 2 while by Lemma 3.5 we know that the energy and Hopf dierential of the harmonic projection π λ : X T λ are: E (π λ ) = 1 E(λ, X) 2 Φ(π λ ) = φ X ( 1 2 λ) = 1 4 φ X(λ) Therefore, we nd E (ˆκ) E (π λ ) 2π χ(s), and Corollary 3.12 implies that The Theorem then follows by algebra. Φ X (λ) 1 4 φ X(λ) 1 4π χ(s).

35 Chapter 3: The antipodal map 29 (a) (b) 1 M l 1/ log(m) M = 2M l 1/ log(2m) 1 M /M = 2 l /l 1 as M Figure 3.1: (a) Euclidean cylinders with moduli M and 2M correspond to (b) hyperbolic cylinders whose core geodesics have approximately the same length. This phenomenon leads to Teichmüller rays for distinct Strebel dierentials that converge to the same point in the Thurston boundary of Teichmüller space. 3.A Appendix: Asymmetry of Teichmüller geodesics Since the antipodal involution i X is a homeomorphism of PML (S) to itself, it seems natural to look for an involutive homeomorphism of T (S) that has i X as its boundary values. In fact, an obvious candidate is the Teichmüller geodesic involution I X : T (S) T (S) that is the pushforward of ( 1) : Q(X) Q(X) via the Teichmüller exponential map τ : Q(X) T (S). However, we now sketch an example showing that I X does not extend continuously to the Thurston compactication of T (S), leading us to view the pruning map based at X, λ pr λ X, as a kind of substitute for the Teichmüller geodesic involution that does extend continuously to the antipodal map of PML. By a theorem of Masur, if φ is a Strebel dierential on X whose trajectories represent homotopy classes (α 1,..., α n ), then the Teichmüller ray determined by φ converges to the point [α α n ] PML (S) in the Thurston compactication [23]. Note that the limit point corresponds to a measured lamination in which each curve α i has the same weight, independent of the relative sizes of the cylinders on X determined by φ. This happens because the Thurston boundary reects the geometry of hyperbolic geodesics on the surface, and hyperbolic length is approximated by the reciprocal of the logarithm of a cylinder's height, as in Figure 3.1. Now suppose φ and ψ are holomorphic quadratic dierentials on a Riemann surface X such that each of ±φ, ±ψ is Strebel, where the trajectories of φ and ψ represent dierent sets of homotopy classes, while those of φ and ψ represent the same homotopy classes. Then by Masur's theorem, the Teichmüller geodesics corresponding to φ and ψ converge to

36 30 Chapter 3: The antipodal map Figure 3.2: Horizontal (solid) and vertical (dashed) trajectories of φ 0 and ψ 0 on the square torus. the same point on PML (S) in the negative direction, while in the positive direction they converge to distinct points. If I X were to extend to a continuous map of the Thurston boundary, then any pair of Teichmüller geodesics that are asymptotic in one direction would necessarily be asymptotic in both directions, thus no such extension exists. One can explicitly construct such (X, φ, ψ) as follows: Let X 0 denote the square torus C/(2Z 2iZ), and let ψ 0 = ψ 0 (z)dz 2 be a meromorphic quadratic dierential on X 0 with simple zeros at z = ± 1+ɛ 2 and simple poles at z = ± 1 2 and such that ψ 0(x) R for x R. Such a dierential exists by the Abel-Jacobi theorem, and in fact is given by ψ 0 (z)dz 2 = (z) c 0 (z) c 1 for suitable constants c i, where (z) is the Weierstrass function for X 0. Let φ 0 = dz 2, a holomorphic quadratic dierential on X 0. Let X be the surface of genus 2 obtained as a 2-fold cover of X 0 branched over ± 1 2 ; then ψ 0 and φ 0 determine holomorphic quadratic dierentials ψ and φ on X, where the lift of ψ 0 is holomorphic because the simple poles at ± 1 2 are branch points of the covering map X X 0. Both φ and ψ have closed vertical and horizontal trajectories as in the construction above (see Figure 3.2). Specically, let γ and η denote the free homotopy classes of simple closed curves on X 0 that arise as the quotients of R and ir, respectively; both γ and η have two distinct lifts (γ ± and η ±, respectively) to X. Let α denote the separating curve on X that covers [ 1/2, 1/2], and let β denote the simple closed curve on X that is the union of the two lifts of [ i, i]. Then: 1. the trajectories of φ represent (γ +, γ ),

37 Chapter 3: The antipodal map the trajectories of ψ represent (γ +, γ, α), and 3. the trajectories of both φ and ψ represent (η +, η, β).

38 32 Chapter 3: The antipodal map a) β γ + γ η + η b) β γ + η + α η γ Figure 3.3: Two constructions of a surface of genus two with a Strebel dierential: (a) Two tori are glued along a segment of a leaf of a Euclidean foliation. (b) Two tori are glued to the ends of a Euclidean cylinder. Below each example, the homotopy classes represented by the horizontal and vertical trajectories are shown (as solid and dashed lines, respectively). When the tori and cylinder are chosen correctly, this construction produces an example of Teichmüller geodesics that are asymptotic in one direction while converging to distinct endpoints in the opposite direction.

39 Chapter 4 Fuchsian centers and Strebel dierentials 4.1 Introduction In this chapter we will study the parameterization of projective structures on a xed Riemann surface X via the Schwarzian derivative. After formulating a general conjecture about this parameterization (Conjecture 4.2) we prove a special case that has applications to the study of projective structures with Fuchsian holonomy. It follows from work of Shiga, Tanigawa, Goldman and Anderson that Q(X) P (X) contains a countable discrete set of projective structures with Fuchsian holonomy. We call these Fuchsian centers because they provide canonical center points for islands of quasifuchsian holonomy surrounding the Bers embedding of Teichmüller space (Ÿ4.2). Each center c γ Q(X) is labeled by unique integral lamination γ ML Z, and corresponds to a projective structure on X with grafting lamination 2πγ. Each Fuchsian center has an associated Strebel dierential s γ = φ X (2πγ), with L 1 norm s γ 1 = E(2πγ, X). The Strebel dierential is fairly easy to understand in terms of the conformal or hyperbolic geometry of X, while the Fuchsian center c γ is comparatively mysterious. Nevertheless, we show that s γ Q(X) is a good estimate for the position of c γ : Theorem 4.1. For each γ ML Z (S) and all k N, we have 2c kγ s kγ 1 = O(1), where the implicit constant depends on X and γ but not k. In particular, the Fuchsian centers c kγ lie within bounded distance of the associated line of Strebel dierentials in Q(X), and their norms grow quadratically with k. Previously, Anderson used complex-analytic techniques to obtain a quadratic lower bound for the norms of Fuchsian centers where γ is supported on a single simple closed curve (see Chapter 6 of [2]); this bound also follows from Theorem 4.1. In order to prove Theorem 4.1, we consider a more general problem about two dierent ways to obtain a measured lamination from a holomorphic quadratic dierential. The rst, 33

40 34 Chapter 4: Fuchsian centers and Strebel dierentials and more familiar, is the map Λ that sends φ Q(X) to the lamination represented by its horizontal foliation: Λ : Q(X) ML (S) The content of the Hubbard-Masur theorem (Ÿ2.3) is that this map is a homeomorphism, and we use φ X to denote its inverse. The map Λ is homogeneous of degree 1 2, i.e. Λ(t2 φ) = tλ(φ) for t > 0, and in particular sends rays in Q(X) to rays in ML (S). The second map is less familiar, and comes from the interpretation of Q(X) P (X) as the space of complex projective structures on X (Ÿ2.5). By Thurston's theorem, the projective structure corresponding to φ Q(X) arises from grafting in a unique way, i.e. (X, φ) = Gr λ Y for Y T (S) and λ ML (S). Equivalently, λ = β(φ) is the bending lamination of the locally convex pleated surface in H 3 corresponding to (X, φ). The resulting map β : Q(X) ML (S) is somewhat exotic compared to Λ for example, while Λ(0) = 0 (see Ÿ3.3), it is not homogeneous. It follows from the work of Scannell and Wolf that β is a homeomorphism, and we use ξ X to denote its inverse. It is technically simpler to compare φ X and ξ X (rather than their inverses, Λ and β) because they take values in the Banach space Q(X). The results in this chapter can be seen as evidence supporting the following conjecture: Conjecture 4.2. The dierence between φ X and 2ξ X is bounded by a constant that depends only on X, φ X (λ) 2ξ X (λ) 1 = O(1). Remark. It seems unlikely that the dierence between φ X and 2ξ X could be bounded uniformly for all X T (S) because the trajectories of holomorphic quadratic dierentials can be very badly behaved with respect to the hyperbolic metric on a surface with short geodesics. We will show that the map ξ X, while lacking the homogeneity properties of φ X, behaves as predicted by Conjecture 4.2 along the ray R + λ, where λ is a nite lamination, i.e. one supported on a nite collection of simple closed curves. Theorem 4.3. Fix X T (S) and let λ ML (S) be a nite lamination. Then for all t > 0, we have φ X (tλ) 2ξ X (tλ) 1 = O(1), where the implicit constant depends on X and λ but not t. The main result on Fuchsian centers, Theorem 4.1, follows easily from Theorem 4.3 since c γ = ξ X (2πγ) and s γ = φ X (2πγ).

41 Chapter 4: Fuchsian centers and Strebel dierentials Holonomy of projective structures Let V (S) denote the variety of irreducible parabolicity-preserving representations of π 1 (S) into PSL 2 (C) modulo conjugacy (briey, the representation variety), which is a complex manifold of dimension 3g 3. For a representation class [ρ] V (S), we let Γ ρ denote the image ρ(π 1 (S)) of a representative ρ : π 1 (S) PSL 2 (C). A representation [ρ] will be called discrete if Γ ρ is discrete. While the geometry of the set of discrete representations V disc (S) V (S) is known to be quite complicated, its interior can be described explicitly as follows. A representation [ρ] V (S) is quasifuchsian (resp. Fuchsian) if the limit set Λ ρ Ĉ of Γ ρ is a Jordan curve (resp. round circle). Fuchsian representations are precisely those which are conjugate into PSL 2 (R); quasifuchsian representations are those which are quasiconformally conjugate to a Fuchsian representation. Let QF(S) denote the set of quasifuchsian representations [ρ] V (S). Theorem 4.4. Int V disc (S) = QF(S) Proof. By Sullivan's stability theorem ([34]), Int V disc (S) consists of quasiconformally conjugate geometrically nite representations, which are therefore all faithful or all non-faithful. The latter is impossible because the set of non-faithful representations is a countable union of analytic subvarieties of V (S), which is nowhere dense. By a result of Maskit, these faithful representations of π 1 (S) are either quasifuchsian or degenerate [21]. Degenerate groups are not geometrically nite (see [22, ŸIX.G.19]), so the theorem follows. Compare [16, Ÿ8.3], [24, Ÿ4.3]. The space QF(S) of quasifuchsian representations is biholomorphic to the product T (S) T ( S), by associating to each [ρ] QF(S) the pair of surfaces Ω + /Γ ρ and Ω /Γ ρ, where Ω + Ω is the domain of discontinuity of Γ ρ [3]. In particular, QF(S) is connected and contractible. Here S denotes the surface S with its orientation reversed. Let Q(X, Ȳ ) be the quasifuchsian representation class corresponding to the pair (X, Ȳ ) T (S) T ( S). The Fuchsian representations are precisely those of the form Q(X, X). The association of a holonomy representation ρ to each projective structure denes a holonomy map hol : P(S) V (S). Theorem 4.5 (Hejhal [10]). hol is a holomorphic local dieomorphism. Remark. Hejhal's result applies to closed surfaces. Alternate approaches to this result appear in [5] and [12]. The generalization to punctured surfaces appears in [6]. Our object of study in this chapter is the set K (S) = hol 1 (V disc (S)) P(S), the set of projective structures with discrete holonomy. By Theorem 4.5, the structure of the interior of K (S) locally mirrors the situation in the representation variety. Corollary 4.6. Int K (S) = hol 1 (QF(S)) In fact, Int K (S) has many components, each of which is biholomorphic to QF(S) via the holonomy map. The components are labeled by integral laminations γ ML Z

42 36 Chapter 4: Fuchsian centers and Strebel dierentials ML (S); a lamination is integral if it is supported on simple closed curves and the weight of each closed curve is an integer. Integral grafting provides explicit examples of points in each component, thus connecting the Thurston parameterization (Ÿ2.1) with the study of holonomy. Theorem The interior of K (S) consists of countably many preimages of QF(S) under the holonomy map, each of which maps biholomorphically onto QF(S). The components are labeled by integral laminations, i.e. Int K (S) QF γ (S), γ ML Z where QF γ (S) P(S) and hol : QF γ (S) QF(S) is a biholomorphism. 2. For any Y T (S) and any integral lamination γ ML Z (S), we have hol(gr 2πγ Y ) = Q(Y, Ȳ ), i.e. Gr 2πγ Y has the same Fuchsian holonomy as the standard Fuchsian projective structure on Y. 3. Conversely, if Z P(S) has Fuchsian holonomy, then we have Z = Gr 2πγ Y for some γ ML Z (S) and Y T (S). 4. For each Y T (S) and γ ML Z (S), we have Sketch of proof. Gr 2πγ Y QF γ (S). 1. See [16]. Goldman has shown that there is a locally constant topological invariant θ : Int K (S) ML Z, the wrapping invariant, that separates components. Then QF γ (S) = θ 1 (γ). 2. The integral grafting Gr 2πγ Y, where γ ML Z (S), has the same holonomy group Q(Y, Ȳ ) as the standard Fuchsian structure of Y. This is because the developing map of Gr 2πγ Y is obtained by inserting 2πn-sectors into the hyperbolic plane Ĉ along all lifts of the curves in γ; since a 2πn-sector wraps completely around the sphere n times (see Figure 4.1), the holonomy of any closed loop in Y is unchanged. 3. See Theorem C of [8]. 4. This follows from Goldman's denition of the wrapping invariant θ : Int K (S) ML Z (S). It is reasonable that Y and Gr 2πγ Y do not lie in the same component of Int K (S) since the developing map is injective for Y and -to-one for Gr 2πγ Y. In fact, if Y and Gr 2πγ Y are joined by a path of projective structures, at the last moment when the developing map is univalent the limit set of the holonomy group cannot be a Jordan curve, and therefore the holonomy is not quasifuchsian [19].

43 Chapter 4: Fuchsian centers and Strebel dierentials 37 δ 2πγ X = gr 2πγ Y CP 1 Figure 4.1: The developing map of an integral grafting wraps a lift of the grafting annulus (e.g. one of the shaded regions at left) around the entire Riemann sphere. An example of this wrapping behavior for one lift is shown on the right. 4.3 Fuchsian centers In the previous section, we described the components of Int K (S) in terms of a discrete topological invariant (the wrapping lamination) and analytic moduli (the holonomy representation [ρ] QF(S)). We now investigate how Int K (S) intersects the bers P (X) in which the underlying complex structure of the projective surface is xed. Let K(X) = (P (X) K (S)), a subset of P (X) C 3g 3. While Int K (S) consists of quasifuchsian representations, there are 3g 3-dimensional families of singly degenerate representations in the boundary QF. It is therefore possible that K(X) could contain an open set of degenerate representations, and hence that Int K(X) (Int K (S)) P (X). The following theorem of Shiga and Tanigawa rules out such pathology: Theorem 4.8 (Shiga-Tanigawa [33]). Int K(X) consists of projective structures with quasifuchsian holonomy, and therefore Int K(X) = (Int K (S)) P (X). Notes. 1. Shiga-Tanigawa show that the quasiconformal deformations of holonomy groups can always be lifted to deformations of projective structures. 2. There exist projective structures with singly degenerate holonomy, and therefore there are 3g 3-dimensional families of such structures in P(S); Theorem 4.8 implies that these submanifolds are transverse to π : P(S) T (S). Knowing that the study of Int K(X) involves only quasifuchsian projective structures, it is natural to ask how each component QF γ (S) of Int K(S) intersects P (X). The rst observation is that this intersection is always nonempty; dene c γ Q(X) by (X, c γ ) = Gr 2πγ (pr 2πγ X)),

44 38 Chapter 4: Fuchsian centers and Strebel dierentials or equivalently, Then we have: c γ = ξ X (2πγ). Corollary 4.9 (of Theorem 4.7). For each X T (S) and γ ML Z (S), (X, c γ ) is the unique projective structure on X with Fuchsian holonomy and wrapping lamination γ. Proof. By part (3) of Theorem 4.7, Gr 2πγ (pr 2πγ X)) has Fuchsian holonomy and wrapping lamination is γ; uniqueness follows since the bending lamination map β : P (X) ML (S) is a homeomorphism. The origin c 0 = 0 is the standard Fuchsian structure on X. It is the Fuchsian center of QF 0 (S) P (X), the set of quasifuchsian projective structures on X with wrapping lamination 0, or equivalently, injective developing maps. The Bers slice B(X) Q(X) is dened as the set of quadratic dierentials that arise as Schwarzian derivatives of equivariant maps from X onto quasidisks in CP 1 (see [3], [24]). Thus the Schwarzian derivative identies QF 0 (S) with B(X). The Bers slice provides an embedding of the Teichmüller space of X, and is a connected, bounded, contractible subset of Q(X). This description of B X in terms of projective structures was rst given by Shiga [32]. A projective structure with quasifuchsian holonomy is called standard if its developing map is injective, and otherwise is called exotic. Since the Bers slice B(X) is the image of QF 0 (S) P (X) under the Schwarzian, we dene the exotic Bers slices B γ (X) = S (QF γ (S) P (X)) Q(X), γ ML Z (S) In contrast to the standard Bers slice, relatively little is known about B γ (X) for γ 0 (see Chapter 5). However, since c γ QF γ (S) P (X), the quadratic dierential c γ provides a distinguished Fuchsian center to the exotic Bers slice B γ (X). 4.4 Grafting annuli In an eort to understand the structure of the exotic Bers slices B γ (X) Q(X), we now study the arrangement of the rays ξ X (tγ) for nite laminations γ ML (S). The nite topological complexity of such laminations will allow us to apply tools from the theory of univalent functions to study the grafting annuli inserted along the corresponding geodesics. Fix X T (S) and a nite lamination γ = i h i [γ i ], γ i π 1 (S), and consider the family of projective structures Gr tγ (pr tγ X), t R +. If γ is integral, then this family includes the countably innite ray of Fuchsian centers c kγ = ξ X (2πkγ), k N. Recall that the grafting locus A = A(tγ) X is the union of annuli in the homotopy classes γ i. The goal of this section is to prove the following result about the geometry of the annuli A(tγ):

45 Chapter 4: Fuchsian centers and Strebel dierentials 39 Lemma There exists a constant m > 0 depending only on γ and X such that each component A i A(tγ) has modulus mod (A i ) m. Lemma 4.10 shows that the grafting annuli have bounded geometry in a complexanalytic sense. The following consequence of the modulus bound will be used in later sections. Corollary 4.11 (of Lemma 4.10). For each X T (S) and nite lamination γ, there is a constant ε(x, γ) > 0 such that for all t 0, each connected component of A(X, tγ) contains a hyperbolic disk of radius ε(x, γ). Using the modulus bound of Lemma 4.10, Corollary 4.11 is an easy application of the distortion theorem for univalent functions [1]. For a detailed proof, see [14]. Proof of Lemma The idea is to estimate the length of γ and γ i on X; from Lemma 3.7, we have l(tγ, pr tγ X) = E(tγ, X) + O(1). Using the homogeneity of hyperbolic length and extremal length on ML (S) (of degrees 1 and 2, respectively), we obtain l(γ, pr tγ X) = te(γ, X) + O(1/t). By the denition of the length of a nite lamination, l(γ, Z) = i h i l(γ i, Z), where the sum ranges over the nite set of curves in the support of γ, and h i R +. For t 0 we then have l(γ i, pr tγ X) 1 h min l(γ, pr tγ X) CtE(γ, X) where h min = min h i > 0 i and C depend on γ and X but not t. The Euclidean annulus A i in the homotopy class γ i has circumference l(γ i, pr tγ X) and height t h i, so its conformal modulus is Mod(A i ) = Using h i h min and the estimate above, we have Mod(A i ) t h i l(γ i, pr tγ X). t h i CtE(γ, X) h min CE(γ, X). The right hand side in this last inequality depends only on X and γ, so Lemma 4.10 follows.

46 40 Chapter 4: Fuchsian centers and Strebel dierentials 4.5 The developing map The grafting construction provides a fairly explicit description of the developing map (Ÿ2.5). In this section we use this description to estimate the Schwarzian derivative. Lemma Let γ = i h i[γ i ] be a nite lamination and Y = pr γ X. Let Ãi denote the connected component of the grafting locus in Gr γ Y that is stabilized by γ i. Normalize so that the holonomy map sends π 1 (S) to a Fuchsian group and γ i to a dilation (z Kz), K > 1. Then the developing map on Ãi is given by δ γ Ãi = g i (z) h i/π where g i : à i H is biholomorphic and g i (Kz) = K g i (z) for some K > 0. Proof. By the denition of the projective grafting map Gr γ : T (S) P(S), the developing map of Gr γ Y normalized as above sends Ãi to C (possibly many-to-one) such that each Euclidean geodesic of à i parallel to its boundary maps to a ray in C, and these rays turn through a total angle of h i. Since Ãi is simply connected, there is a unique branch of log(δ γ Ãi ) whose image is the strip {z 0 < Im(z) < h i }. The associated branch of (δ γ ) h i/π is therefore a univalent map on Ãi with the required property, and the lemma follows. Recall from Ÿ3.5 that the Hopf dierential Φ X (γ) of the collapsing map κ : X pr γ X is a measurable quadratic dierential, holomorphic on A(γ), whose horizontal foliation is the union of the foliations of the grafting annuli by Euclidean geodesics parallel to their boundaries. Lemma 4.12 allows us to estimate ξ X (γ), which is the Schwarzian of the developing map δ γ of Gr γ (pr γ X): Corollary 4.13 (of Lemma 4.12). For γ and A i as above, we have ξ X (γ) Ai = 2 ( π 2 h 2 i h 2 i ) Φ X (γ) Ai + S (g i ) where as before g i : à i H is a Riemann map for Ãi X, with Schwarzian S (g i ) which descends to a quadratic dierential on A i. Proof. By Lemma 4.12 and the composition rule for the Schwarzian derivative (Ÿ2.5), S (δ γ ) Ãi = g i S (z z h i/π ) + S (g i (z)). (4.1) A calculation yields S (z z α ) = 1 α2 dz 2 2 z 2, which is a holomorphic quadratic dierential on H with closed trajectories that are semicircles centered at 0. Since g i intertwines the action of γ i on X with a dilation on H, the pullback to Ãi descends to a holomorphic dierential on A i with trajectories orthogonal to

47 Chapter 4: Fuchsian centers and Strebel dierentials 41 the natural foliation by circles. Therefore this pullback is a positive multiple of Φ X (γ); comparing the induced metrics yields (h i /π) 2 g i (dz 2 /z 2 ) = 4Φ X (γ) Ai. Substituting this into (4.1), we have ( 1 (hi /π) 2 ) ξ X (γ) Ai = S (δ γ ) Ai = 2 (h i /π) 2 Φ X (γ) Ai + S (g i ). Estimating the Schwarzian of the developing map when restricted to the grafting locus as in Corollary 4.13 will be sucient for the proof of Theorem 4.3 because, by Corollary 4.11, these annuli cover a denite fraction of the surface. 4.6 Finite rays and Strebel dierentials In this section we assemble the results about grafting annuli (Ÿ4.4) and the developing map (Ÿ4.5) to prove Theorem 4.3 (see Ÿ4.1). Proof of Theorem 4.3. For X T (S) and a rational lamination γ, we need to show that φ X (tγ) 2ξ X (tγ) 1 = O(1). By Lemma 4.11, for t 0 each grafting annulus in X properly contains a hyperbolic ball B ε (x) of radius ε = ε(x, γ). Fix a strip Ãi X covering one of the grafting annuli A i in X, and a hyperbolic disk B ε (Ãi/γ i ). Then by Corollary 4.13, ( π 2 (t h i ) 2 ) ξ X (tγ) Ai = 2 (t h i ) 2 Φ X (tγ) Ai + S(g i ) (4.2) where g i is a Riemann map for g i. Since min h i > 0, for t 0 the coecient of Φ X (tγ) Ai is ( 2 + O(t 2 )), while Φ X (tγ) 1 = O(t 2 ) by Corollary Therefore ( π 2 (th i ) 2 ) 2 (t h i ) 2 Φ X (tγ) Ai = 2Φ X (tγ) + O(1). By Nehari's theorem (see [1]), any univalent function f on a simply connected domain Ω in H satises S(f) L (Ω) 3 2 Since g i is such a map, S(g i ) L (A i ) = S(g i ) L (Ãi) 3 2. By the Schwarz lemma, the inclusion B ε (x) Ãi is a contraction for the hyperbolic metric, thus S(g i ) L (B ε(x)) < S(g i ) L (A i ) 3 2,

48 42 Chapter 4: Fuchsian centers and Strebel dierentials which leads to a bound on the L 1 norm on a ball of hyperbolic radius ε/2: S(g i ) L 1 (B ε/2 (x)) C, (4.3) where C does not depend on ε. By Theorem 3.15, the Hopf dierential Φ X (tγ) of the collapsing map is close to the Strebel dierential 1 4 φ X(tγ), i.e. 4Φ X (tγ) φ X (tγ) L 1 (X) 16π χ(s) and in particular, 4Φ X (tγ) φ X (tγ) L 1 (B ε/2 (x)) 16π χ(s) (4.4) Combining (4.3) and (4.4) with Corollary 4.13, we have 2ξ X (tγ) φ X (tγ) L 1 (B ε/2 (x)) 2ξ X (tγ) 4Φ X (tγ) L 1 (B ε/2 (x)) + 4Φ X (tγ) φ X (tγ) L 1 (B ε/2 (x)) C (4.5) where C depends only on χ(s) and ε = ε(x, γ) is independent of t. Now we observe that the dierence (2ξ X (tγ) φ X (tγ)) lies in the nite-dimensional space Q(X) of holomorphic quadratic dierentials, so there is a constant M(ɛ, X) such that for all x X and ψ Q(X), ψ L 1 (B ε/2 (x)) ψ L 1 (X) M(ε, X). Therefore 2ξ X (tγ) φ X (tγ) L 1 (X) M(ε/2, X) 2ξ X (tγ) φ X (tγ) L 1 (B ε/2 (x)) C(ɛ, X). Since ε depends only on γ and X, Theorem 4.3 follows.

49 Chapter 5 Comments and questions 5.1 Open questions We close by mentioning some open questions related to grafting and complex projective structures. The Bers slice B(X) has been studied extensively as an embedding of Teichmüller space into a complex vector space. Its image is a bounded, contractible, regular open subset of Q(X). In comparison to B(X), the exotic Bers slices B γ (X) remain somewhat mysterious. For example, the following basic questions remain open: Is B γ (X) connected? Is B γ (X) bounded? Does the topology of B γ (X) depend on X? Even if B γ (X) is disconnected, it has a distinguished connected component that contains the Fuchsian center c γ. Thus we can rephrase the question about connectedness of B γ (X) as: Does every connected component of Int K(X) contain a Fuchsian center? Further discussion of the connectedness issue can be found in Ÿ5.2. As for the distribution of the Fuchsian centers, the following question highlights the uniformity over dierent rays that is lacking in the statement of Theorem 4.3: On a punctured torus, do the Fuchsian structures corresponding to simple closed curves of slope 1/n, n = 1, 2,..., converge in P(S) as n? More generally, do the irrational pleating rays in P (X) converge in P(S)? It would also be interesting to study the geometry and topology of the space QF of quasifuchsian representations by nding a complex projective interpretation for the points in the compactication P(S) = T (S) ML (S). McMullen has shown that there exist limiting slices B(λ) QF that generalize the basepoint of a Bers slice to a projective lamination [λ] T (S). For nite laminations, these slices sit within ane algebraic subvarieties of V (S) (level sets of trace functions). It is therefore natural to ask: 43

50 44 Chapter 5: Comments and questions Does the limit Bers slice B(λ) lie in the image of a properly embedded complex vector space Q(λ) V (S)? and further, Is there a geometric interpretation for Q(λ) as a space of complex projective structures on the lamination λ, or perhaps its dual R-tree? The fact that the Schwarzian derivative of a quasiconformal deformation of a singly degenerate group in B(λ) exists as a distribution seems promising for this avenue of research; the singular nature of the space of leaves of a measured lamination is likely to require the use of distributional objects. In the case of a punctured torus, it may be possible to use such a theory of generalized projective structures and the theory of holomorphic motions to address the self-similarity of Bers' boundary of Teichmüller space: Conjecture 5.1 (McMullen, [25]). The boundary of Bers' embedding of the Teichmüller space of a once-punctured torus is C 1+α -conformally self-similar about boundary points corresponding to laminations λ + and λ xed by a pseudo-anosov mapping class ψ. 5.2 Connectedness and computer experiments In this nal section we further address the question of connectedness of exotic Bers slices B γ (X). Specically, we provide a heuristic argument suggesting that these slices are sometimes disconnected when X is a punctured torus, and display computer-generated images of the discreteness locus K(X) that seem to support this hypothesis. The computer experiments were conducted using the author's software package Bear, which computes and tests discreteness of holonomy representations of complex projective structures on punctured tori. This free software package can be obtained on the world wide web at The boundary of the standard Bers slice B 0 (X) of a punctured torus is a closed cuspy curve, i.e. a Jordan curve with a dense set of inward-pointing cusps [27]. A Jordan curve ν has an inward-pointing cusp at a point x if there is a cardioid with its cusp at x that lies entirely within ν (see Figure 5.1). Each of the cusps on B 0 (X) corresponds to a geometrically nite holonomy group with an accidental parabolic. Within B 0 (X) the holonomy is quasifuchsian, and one of the two domains of discontinuity has xed quotient conformal structure X; thus, the accidental parabolics arise from pinching simple closed curves on the other conformal boundary component. There is a qualitatively similar conjectural picture of the boundary of an exotic Bers slice B γ (X). When γ 0, the quasifuchsian holonomy groups in the exotic Bers slice do not have a xed quotient conformal structure on either domain of discontinuity, and thus accidental parabolics can appear on B γ (X) corresponding to a pinched curve on either (or both) of the ends of the quasifuchsian manifold. Furthermore, these cusps seem to densely populate two cuspy curves, one for each of the two types of accidental parabolics, which generically cross one another in several places. The points of B γ (X) are those which lie between these two curves (in such a way that the cusps are inward-pointing).

51 Chapter 5: Comments and questions 45 Figure 5.1: Part of the boundary of Maskit's embedding of the Teichmüller space of a punctured torus; this is a cuspy curve, containing a dense set of inward-pointing cusps. Cardioids lying entirely on one side of the curve are shown for several of the cusps. Figure 5.2: Pairs of cuspy curves seem to form the boundary of exotic Bers slice, resulting in connected components resembling the island at left. As the basepoint X T (S) is changed, these curves may cross acquire new crossings (center), which often results in a multitude of smaller islands (right). As X T (S) is varied, the boundary of the standard Bers slice B 0 (X) moves via a holomorphic motion [27]. Similarly, the two cuspy curves forming the boundary of an exotic Bers slice B γ (X) appear to move by independent holomorphic motions, and as a result, the location and number of crossings of these two curves change with X, as does the set of connected components of B γ (X). Thus a single component of B γ (X) may pinch o to form two or more components of B γ (X ) for X near X; in fact, the cuspy nature of these boundary curves tends to create a multitude of small islands near a point of crossing, as shown in Figure 5.2, leading us to suspect that B γ (X) may have inntely many connected components for generic X T (S). Computer-generated images of K(X) that seem to display this behavior are shown in Figure 5.3.

52 46 Chapter 5: Comments and questions Figure 5.3: Computer-generated images of the discreteness locus K(X) that seem to show the behavior depicted in Figure 5.2. In the upper image, indiscrete representations are indicated by varying shades of gray according to the amount of computation required to detect this indiscreteness. In both images, points colored black lie in K(X).