Complex Projective Structures, Grafting, and eicThmüller ... · I must also thank Nathan Dun eld,...

Complex Projective Structures, Grafting, and TeichmüllerTheory

A thesis presentedby

David Dumas

toThe Department of Mathematics

in partial fulllment of the requirementsfor the degree of

Doctor of Philosophyin the subject of

Mathematics

Harvard UniversityCambridge, Massachusetts

May 2004

c©2004 - David DumasAll rights reserved.

Thesis advisor AuthorCurtis T. McMullen David Dumas

Complex Projective Structures, Grafting, and Teichmüller Theory

Abstract

We study the space P(S) of marked complex projective (CP1) structures on a compactsurface in terms of Teichmüller theory and hyperbolic geometry. In particular, we showthat the structure of this space as a bundle over the Teichmüller space T (S) of conformalstructures is compatible with Thurston's parameterization of P(S) using grafting, and weprovide an explicit description of the boundary of the ber P (X) over X ∈ T (S) in termsof an involution on the space PML (S) of projective measured laminations. This involutionencodes the conformal geometry of X via the orthogonality of the vertical and horizontalmeasured foliations of holomorphic quadratic dierentials.

We also apply these results to study the projective structures with Fuchsian holonomyon a xed Riemann surface X. We formulate a general conjecture comparing these Fuchsiancenters and associated Strebel dierentials on X, and then prove that this conjecture holdsalong rays in ML (S) supported on nitely many simple closed curves. This generalizesprevious results about Fuchsian centers obtained by Anderson.

The proofs of these results involve the theory of harmonic maps between Riemannsurfaces and from Riemann surfaces to R-trees. Specically, we show that the canonicalcollapsing and co-collapsing maps associated to a complex projective surface are nearlyharmonic, and then apply existing results about harmonic maps and their Hopf dierentials.

Contents

Title Page . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

1 Introduction 11.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Complex projective geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 The antipodal map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4 Fuchsian centers and Strebel dierentials . . . . . . . . . . . . . . . . . . . . . 41.5 Harmonic Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Preliminaries 82.1 Grafting and pruning. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2 Conformal metrics and tensors on Riemann surfaces . . . . . . . . . . . . . . 92.3 Quadratic dierentials, foliations, and the antipodal map . . . . . . . . . . . . 112.4 R-trees and co-collapsing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.5 Complex projective geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.6 Projective Grafting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3 The antipodal map 183.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.2 Harmonic maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.3 Energy and grafting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.4 Convergence to the harmonic map . . . . . . . . . . . . . . . . . . . . . . . . 223.5 Measurable quadratic dierentials . . . . . . . . . . . . . . . . . . . . . . . . . 253.6 The antipodal map extends pruning . . . . . . . . . . . . . . . . . . . . . . . 273.A Appendix: Asymmetry of Teichmüller geodesics . . . . . . . . . . . . . . . . . 29

4 Fuchsian centers and Strebel dierentials 334.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.2 Holonomy of projective structures . . . . . . . . . . . . . . . . . . . . . . . . . 354.3 Fuchsian centers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.4 Grafting annuli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384.5 The developing map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

iv

Contents v

4.6 Finite rays and Strebel dierentials . . . . . . . . . . . . . . . . . . . . . . . . 415 Comments and questions 43

5.1 Open questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435.2 Connectedness and computer experiments . . . . . . . . . . . . . . . . . . . . 44

Bibliography 47

Acknowledgments

First of all, I would like to thank my advisor, Curt McMullen, for all he has taughtme over the last ve years, and for his sound advice and inspiring clarity in all thingsmathematical.

Special thanks are also due to Yum-Tong Siu and Cli Taubes for reading drafts of thisthesis.

I must also thank Nathan Duneld, Maryam Mirzakhani, Laura DeMarco, Danny Cale-gari, and Daniel Allcock for many helpful and inspiring conversations.

I am grateful to the graduate students in the Mathematics Department for creating apleasant and helpful working environment, and to the Department, Harvard University, andthe 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 andsupport through good times and bad. I am especially indebted to Aravind Asok, JulieNorseen, Jacob Hartman, and Deepee Khosla for helping me keep my sanity when it seemedimpossible.

Finally, I cannot imagine how I would have made it through this most dicult lastyear without the love and support of Sarah Hutton. She has seen me at my worst, and stilllaughs at all of my stupid jokes and listens to all of my boring stories. This is more than Icould reasonably expect from anyone.

Chapter 1

Introduction

1.1 OverviewIn this thesis we study complex projective Riemann surfaces, which are geometric

objects that incorporate aspects of both two-dimensional conformal geometry and three-dimensional hyperbolic geometry. This dual nature arises from the simultaneous action ofthe group PSL2 C of Möbius transformations on CP1 by holomorphic automorphisms, andon the three-dimensional hyperbolic space H3 by orientation-preserving isometries. Whilecomplex projective geometry has long been studied as a topic in complex analysis, the rich-ness of its connection with hyperbolic geometry has only recently come to light, beginningwith 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 andcomplex-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 ona compact dierentiable surface S, which is a bundle over the Teichmüller space T (S) ofconformal structures. The ber P (X) over X ∈ T (S) is the space of all complex projectivestructures with underlying conformal structure X, and is naturally identied with the spaceQ(X) of holomorphic quadratic dierentials on X.

Each quadratic dierential φ ∈ Q(X) determines a horizontal foliation of X, whichis 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 7→ −φ)of Q(X), which interchanges the vertical and horizontal foliations of φ, to an involutioniX : ML (S) → ML (S). By homogeneity, there is an associated antipodal involution:

iX : PML (S) → PML (S).

For each λ ∈ ML (S) and Y ∈ T (S), there is a Riemann surface X = grλ Y , thegrafting of Y along λ, which is obtained by thickening the geodesic realization of λ on Yby inserting Euclidean strips. In a sense, this procedure mixes the hyperbolic geometry ofY and the Euclidean geometry of a thickened lamination, and X is like the midpoint ofthese two objects.

1

2 Chapter 1: Introduction

For each λ ∈ ML (S), there is a unique Y = prλX ∈ T (S), the pruning of X along λ,such that X = grλ Y . Just as xing the midpoint of a pair of points forces them to move inopposite directions, we show that the large-scale behavior of the correspondence λ 7→ prλXis given by the antipodal map on the boundary:3 The pruning map with basepoint X, (λ 7→ prλX), extends continuously to the boundaryPML (S), where it agrees with the antipodal involution iX : PML (S) → PML (S).

Within P (X) ' Q(X), there is a countably innite discrete set of projective structuresthat have Fuchsian holonomy representations for example, 0 ∈ Q(X) corresponds to theFuchsian uniformization of X by H. In Chapter 4, we further relate the analytic and geo-metric aspects of projective structures by estimating the positions of these Fuchsian centerscγ ∈ Q(X), which are indexed by integral laminations γ ∈ ML Z(S). The comparison isbased on the associated Strebel dierentials sγ ∈ Q(X), which are holomorphic dierentialswith closed trajectories corresponding to γ:3 For each X ∈ T (S), γ ∈ ML Z(S), and k ∈ N, we have

‖2ckγ − skγ‖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-pointprojection from a domain in CP1 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 thesemaps are very nearly harmonic (for the collapsing map, this is due to Tanigawa [35]), and asa result they have the same large-scale behavior as the harmonic maps between the relevantspaces. We then deduce the main theorems by applying existing results about harmonicmaps 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 thenbriey discuss the techniques involved in the proofs.

1.2 Complex projective geometryA complex projective structure is an atlas of charts on a Riemann surface taking val-

ues in CP1 with the property that the transition functions are Möbius transformations.Since (CP1,PSL2 C) compacties (H3,PSL2 C), complex projective geometry is the naturalboundary geometry for 3-dimensional hyperbolic geometry.

The moduli space P(S) of marked complex projective structures on a compact smoothsurface S of genus g > 1 is a contractible complex manifold of dimension 6g− 6; the naturalmap π : P(S) → T (S) to Teichmüller space that records the underlying conformal structuremakes P(S) into a complex ane vector bundle. The Schwarzian derivative S can be usedto identify P(S) with the cotangent bundle to Teichmüller space, i.e. the bundle Q(S) ofholomorphic quadratic dierentials.

Alternately, Thurston has shown that every complex projective structure can be uniquelydescribed by grafting, a cut-and-paste operation on hyperbolic Riemann surfaces. This leadsto a homeomorphism Gr : ML (S)×T (S) → P(S).

Chapter 1: Introduction 3

The typical example of a complex projective structure is the boundary at innity ofa complete hyperbolic manifold M . Each geometrically nite end of such a manifold is acomplex projective Riemann surface; Thurston's grafting coordinates in this case are thebending lamination and intrinsic hyperbolic metrics of the faces of the convex core. Ingeneral, a kind of local convex hull construction gives rise to a locally convex pleatedsurface whose metric and bending recover the grafting data.

A complex projective structure on a Riemann surface X can also be dened via aholomorphic immersion f : X → CP1 that intertwines the action of π1(X) by deck transfor-mations with some representation ρ : π1(X) → PSL2(C). The latter is the holonomy of theprojective structure, and is well-dened as an element of

V (S) = Homirr(π1S,PSL2 C)/(PSL2 C conjugacy).To summarize, P(S) is linked to other standard objects in Teichmüller theory via an

ensemble of maps:

ML (S)×T (S) Gr'

//

gr

((QQQQQQQQQQQQQQQQQP(S)

π

η // V (S)

T (S)

where3 ML (S) is the space of measured laminations on S,3 GrλX is the grafted projective surface obtained from X by removing the lamination λ

and inserting Euclidean strips according to the measure,3 π is the forgetful map that sends a CP1 structure to its underlying complex structure,3 gr = π Gr is the conformal grafting map, where the projective structure on the grafted

surface is weakened to a conformal structure,3 η is the holonomy map, which records the failure of the projective charts to be globally

dened as an element of3 V (S) = Homirr(π1S,PSL2 C)/(PSL2 C conjugacy), the representation variety.

1.3 The antipodal mapScannell and Wolf have shown (extending work of Tanigawa) that for each lamination

λ ∈ ML (S), the conformal grafting map grλ : T (S) → T (S) is a homeomorphism, thusthere is an inverse or pruning map prλ : T (S) → T (S) [31].

We now describe the large-scale behavior of the grafting data for complex projectivestructures on X. The description is based on the map

Λ : Q(X) → ML (S)


which records the measured lamination equivalent to the horizontal foliation of a holomorphicquadratic dierential. Hubbard and Masur showed that Λ is a homeomorphism, so we canuse it to transport the involution (φ 7→ −φ) of Q(X) to an involutive homeomorphismiX : ML (S) → ML (S). Since the map Λ is homogeneous, iX descends to an involutionon PML (S) = ML (S)/R+,

iX : PML (S) → PML (S),

which we call the antipodal involution with respect to X, since it is conjugate by Λ to theactual antipode map on the vector space Q(X).

The main result of Chapter 3 is that the antipodal map governs the large-scale behaviorof 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 7→ prλX, extendscontinuously to a map ML (S) → T (S) whose boundary values are exactly the antipodalmap iX : PML (S) → PML (S).

Let P (X) denote the space of projective structures with underlying conformal structureX. Then Theorem 1.1 provides an explicit description of the boundary of P (X) when viewedas 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 graphof the antipodal involution iX , i.e.

∂P (X) = ([λ], [iXλ]) ∈ 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 thatof the geodesic involution in a symmetric space. Indeed, the antipodal map relative toX exchanges projective measured laminations ([λ], [µ]) dening Teichmüller geodesics thatpass through X, just as the geodesic involution exchanges endpoints (at innity) of geodesicsthrough a point in a symmetric space.

However, we provide an example showing that this analogy does not work, becauseTeichmüller geodesics are badly behaved with respect to the Thurston compactication.Specically, in 3.A we construct a pair of Teichmüller geodesics through a single point inT (S) that are asymptotic to each other in one direction, but which have distinct limit pointsin the other direction. This precludes the existence of any map of the Thurston boundarythat plays the role of a Teichmüller geodesic involution.

1.4 Fuchsian centers and Strebel dierentialsIn 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 puncturedtorus that have discrete holonomy representations.1 The point marked at the center of

1The image in Figure 1.1 was created using the author's software package Bear, which is available on theworld wide web at http://bear.sourceforge.net/. See also 5.2.


Figure 1.1: The regionK(X) ⊂ Q(X) ' C is shown for the hexagonal punctured torus. FourFuchsian centers corresponding to the empty lamination and the three shortest hyperbolicgeodesics are marked. The central component is Bers' embedding of the Teichmüller spaceT (S); the other three marked components represent exotic projective structures on X.


the picture is the origin in Q(X), which corresponds to the Fuchsian uniformization of Xby H. It lies at the center of the Bers embedding B(X) of Teichmüller space, a boundedcontractible open set in Q(X) consisting of quasifuchsian groups that have a domain ofdiscontinuity with quotient conformal structure X. This island is known by work of Shigaand Tanigawa to be a connected component of the interior of K(X).

Surrounding B(X) are other exotic islands of quasifuchsian holonomy, some of whichcontain Fuchsian centers projective structures with Fuchsian holonomy. A Fuchsian centercγ ∈ Q(X) is uniquely specied by an integral lamination γ ∈ ML Z(S) (i.e. a weightedmulticurve); in Figure 1.1, three exotic centers are marked, corresponding to the threeshortest closed hyperbolic geodesics on X. The three-fold symmetry of the picture reectsthe fact that this particular punctured torus is obtained by identifying opposite edges of aregular hexagon and removing one cycle of vertices.

For each integral lamination, γ ∈ ML Z(S), there is also the Strebel dierential sγ ∈Q(X), which has closed trajectories representing the curves in the support of γ, and wherethese closed trajectories foliate Euclidean cylinders whose heights are given by the weightsof the curves in 2πγ.

We conjecture that when properly normalized, the Fuchsian centers lie within a boundeddistance of the associated Strebel dierentials:Conjecture 1.3. For each X ∈ T (S) and all γ ∈ ML Z, we have

‖sγ − 2cγ‖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 lowerbound 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,

‖skγ − 2ckγ‖1 ≤ C(X, γ).

We actually prove a stronger result about dierentials corresponding to nite raysin ML (S), i.e. those that are supported on simple closed curves with arbitrary positiveweights. Theorem 1.4 is the resulting statement for integral laminations.

Note that the Strebel dierentials skγ lie on a line in the vector space Q(X), whilethe Fuchsian centers ckγ lie within islands of quasifuchsian holonomy. Thus Theorem 1.4implies that each ray of integral Strebel dierentials follows the course of a sequence ofdistinct islands of quasifuchsian holonomy in Q(X). In comparison to the Bers embeddingB(X), little is known about these other islands of quasifuchsian holonomy. We thereforeview Theorem 1.4 as a rst attempt at understanding their distribution by means of theFuchsian centers.


1.5 Harmonic MapsThe 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 brieydescribe how this theory is connected to grafting and complex projective geometry, and howthis connection is used to prove Theorem 1.1 in Chapter 3. In Chapter 4, the same basicprinciples 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 un-derlying conformal structure. Using Thurston's parameterization, each of these is obtainedby grafting, and therefore has the form Grλ Y for some Y ∈ T (S) and λ ∈ ML (S).

The projective structure Grλ Y ∈ P (X) is made up of hyperbolic pieces that come fromY and a Euclidean part that results from thickening λ ⊂ Y . There is a canonical collapsingmap κ : X → Y that collapses the grafted part back to its geodesic representative in Y , andwhich is an isometry on the hyperbolic part of Grλ Y . In Thurston's theory of projectivegrafting, this lift of the collapsing map to X is realized as the projection from CP1 down toa locally convex pleated plane p : H2 → H3.

There is a dual object κ, which we call the co-collapsing map, which collapses thehyperbolic part of X and remembers only the leaves of the natural geodesic foliation of theEuclidean part by parallels of λ:

κ : X → Tλ

The image of this map is a one-dimensional object Tλ which can be interpreted as the setof supporting hyperplanes of the pleated plane p : H2 → H3. For example, if λ is supportedon a set of simple closed curves, then p has a locally nite pattern of bending lines, andTλ is the dual simplicial tree. For more general laminations, Tλ is the R-tree dual to thelamination λ.

It follows from results of Wolf that as Y →∞, the Hopf dierential Φ(h) of the harmonicmap h : X → Y detects the Thurston limit of Y via its vertical foliation [37]. Similarly, theHopf dierential Φ(h) of the harmonic map from h : X → Tλ detects the lamination λ viaits vertical foliation.

The key observation about the collapsing and co-collapsing maps is that they are bothvery nearly harmonic, in the sense that the energy of either map exceeds that of the asso-ciated harmonic map by at most 2π|χ(S)|; for the collapsing map, this was rst noticed byMcMullen and Tanigawa. Furthermore, the Hopf dierentials Φ(κ) and Φ(κ) are exactlyopposite, i.e.

Φ(κ) = −Φ(κ),

and therefore dene orthogonal foliations of the grafted part of Grλ Y .Since κ and κ 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 verticalfoliations of Φ(κ) and −Φ(κ), respectively, which means exactly that they are related by theantipodal involution iX .

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 hyperbolicgeodesics on any hyperbolic surface Y ∈ T (S) are in one-to-one correspondence with thefree homotopy classes of simple closed curves on S; therefore, when a particular hyperbolicmetric is under consideration, we will use these objects interchangeably.

Fix Y ∈ T (S) and γ, a simple closed hyperbolic geodesic on Y . Grafting is theoperation of removing γ from Y and replacing it with a Euclidean cylinder γ × [0, t], asshown in Figure 2.1. The resulting surface is called the grafting of Y along the weightedgeodesic tγ, written grtγ Y .

Associated to each grafted surface grtγ Y is a canonical map κ : grtγ Y → Y , thecollapsing map, that collapses the grafted cylinder γ× [0, t] back onto the geodesic γ. Thereis also a natural C1 conformal metric ρTh on grtγ Y , the Thurston metric, that unites thehyperbolic metric on Y with the Euclidean metric of the cylinder γ × [0, t]. The collapsingmap is distance non-increasing with respect to the hyperbolic metric on Y and the Thurstonmetric on grtγ Y .

We now dene a generalization of a weighted simple closed curve that is compatiblewith grafting.

Let C denote the set of free homotopy classes of simple closed curves. The geometric(unsigned) intersection number

i : C × C → Z≥0

Ygrtγ

t

Y

γ

Figure 2.1: The basic example of grafting.

8

Chapter 2: Preliminaries 9

denes an embedding of R× C into RC viatγ 7→ t i(γ, ·).

The space ML (S) of measured geodesic laminations (or measured laminations) on S is theclosure of the image of R≥0 × C under this map. The intersection number extends to acontinuous 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 Gλ ofa closed subset of Y by complete hyperbolic geodesics, and associates to each transversalτ : [0, 1] → Y a positive Borel measure in a way that is compatible with maps betweentransversals induced by transversality-preserving isotopy. We call the resulting object thegeodesic realization of λ on Y , and the set of geodesics in Gλ is the support of λ.

For example, the geodesic realization of a weighted simple closed curve tγ ∈ R+×C onY is the closed hyperbolic geodesic freely homotopic to γ, and the measure on a transversalis 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 lamina-tions, and thus denes a map

gr : ML (S)×T (S) → T (S) where (λ, Y ) 7→ grλ Y.

Morally, grλ Y is obtained from Y by thickening the geodesic realization of λ in a mannerdetermined by the transverse measure. As in the simple closed curve case, there is a col-lapsing map κ : grλ Y → Y that collapses the grafted part A ⊂ grλ Y (which is no longera union of annuli) onto the geodesic realization of λ on Y , and a conformal metric ρTh ongrλ Y that is hyperbolic on grλ Y −A. One can show that ρTh is of class C1,1 on grλ Y , andthus its curvature is dened almost everywhere [20].

Scannell and Wolf have shown that for each λ ∈ ML (S), the map grλ : 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) 7→ prλX = gr−1λ X.

In other words, for each X ∈ T (S) and λ ∈ ML (S), there is a unique way to present X asa grafting of some Riemann surface Y = gr−1

λ (X) along λ, and pruning is the operation ofrecovering Y from the pair (λ,X).

We will be interested in the pruning map when the surface X is xed, i.e. the mapλ 7→ prλX from ML (S) to T (S). Theorem 3.1 describes the asymptotic behavior of thismap in terms of the conformal geometry of X.

2.2 Conformal metrics and tensors on Riemannsurfaces

Fix a hyperbolic Riemann surface X and let S(X) denote the space of measurablecomplex-valued quadratic forms on TX. A form β ∈ S(X) can be decomposed according to

10 Chapter 2: Preliminaries

the complex structure on X:β = β2,0 + β1,1 + β0,2 where βi,j ∈ Γ((T 1,0X)⊗i ⊗ (T 0,1X)⊗j).

Then |β1,1| is a conformal metric on X, which can be thought of as a circular average of|β|,

|β1,1(v)| = 12π

∫ 2π

0|β(Rθv,Rθv)|dθ,

where v ∈ TxX and Rθ ∈ Aut(TxX) is the rotation by angle θ dened by the conformalstructure of X. The area of X with respect to |β1,1|, when it is nite, denes a natural L1

norm‖β‖L1(X) =

∫X|β1,1|,

which we will abbreviate to ‖β‖1 if the domain X is xed. We also call ‖β‖1 the energy ofβ.

We write S2,0(X) for the space of measurable quadratic dierentials on X, which arethose quadratic forms φ ∈ S(X) such that φ = φ2,0. Within S2,0(X), there is the spaceQ(X) of holomorphic quadratic dierentials, i.e. holomorphic sections of T ∗X⊗2. By theRiemann-Roch theorem, if X is compact and has genus g,

dimCQ(X) = 3g − 3.

If X is given a conformal metric σ, then we can also dene the Lp and L∞ norms:

‖β‖Lp(X,σ) =(∫

X

(|β1,1|σ

)p

dσ

)1/p

‖β‖L∞(X,σ) = supz∈X

|β1,1(z)|σ(z)

We use ‖β‖Lp(X) and ‖β‖L∞(X) as abbreviations for these norms when σ is the hyperbolicmetric uniformizing X.

Fixing a conformal metric also allows us to dene the unit tangent bundleSX = (x, v) ∈ TX | ‖v‖σ = 1 ,

in which case the interpretation of β1,1 as the circular average of β yields another expressionfor ‖β‖L1(X):

‖β‖L1(X) =12π

∫X

∫SxX

|β(v)|dθ(v)dσ(x)

The Hopf dierential Φ(β) of β ∈ S(X) is the (2, 0) part of its decomposition,Φ(β) = β2,0,

which (along with Φ(β)) measures the failure of β to be compatible with the conformalstructure of X. For example, there is a function b : X → C such that β = bσ if and only ifΦ(β) = Φ(β) = 0.


Let f : X → (M,ρ) be a smooth map from X to a Riemannian manifold (M,ρ). Thenthe energy E (f) and Hopf dierential Φ(f) of f are dened to be those of the pullbackmetric f∗(ρ):

E (f) = ‖f∗(ρ)‖1 =∫

X|f∗(ρ)1,1|

Φ(f) = [f∗(ρ)]2,0(2.1)

Since f∗(ρ) is a real quadratic form, at points where Df is nondegenerate f is conformal ifand only if Φ(f) = 0. Thus E (f) is a measure of the average stretching of the map f , whileΦ(f) records its anisotropy.

2.3 Quadratic dierentials, foliations, and theantipodal map

In this section we briey recall the identications between the spaces of measured folia-tions, measured geodesic laminations, and holomorphic quadratic dierentials on a compactRiemann surface.

We will consider foliations on Riemann surfaces which have certain singularities. Foreach k ∈ Z+, the foliation of C by horizontal lines pulls back via the map (z 7→ z(k/2)) to yielda foliation F (k) of C∗ which has a singularity at the origin. For our purposes, a singularfoliation of a surface is a foliation that is dened except at a discrete set of points xi, andwhere the foliation of a punctured neighborhood of xi is dieomorphic to a neighborhood of0 in F (ki) for some ki ∈ Z+.

A measured foliation on S is a singular foliation F and an assignment of a Borel mea-sure µF on [0, 1] to each arc [0, 1] → S that is everywhere transverse to F , subject to thecondition that the measure should be invariant under transversality-preserving isotopy. Thenotation MF (S) is used for the quotient of the set of measured foliations by the equiva-lence relation generated by isotopy and Whitehead moves (e.g., collapsing leaves connectingsingularities).

The typical example of a measured foliation comes from a holomorphic quadratic dif-ferential φ ∈ Q(X). The measured foliation F (φ) determined by φ is the pullback of thehorizontal 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 ∈ TxX 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 foliationsand X ∈ T (S) there is a unique holomorphic quadratic dierential φX(ν) ∈ Q(X) such that

F (φX(ν)) = ν.

Furthermore, the map φX : MF (S) → Q(X) is a homeomorphism.


Note that the transverse measure of φ ∈ Q(X) is dened using √φ, and so for c > 0,F (cφ) = c

12 F (φ).

As a result, the Hubbard-Masur map φX has the following homogeneity property:φX(Cν) = C2φX(ν) for all C > 0 (2.2)

One can also view measured foliations as diuse versions of measured laminations, inthat every measured foliation is associated to a unique measured lamination with the sameintersection properties.

More formally, a class of measured foliations F ∈ MF (S) is uniquely determinedby the point IF ∈ RC , where IF (γ) is the minimum total measure associated to a closedtransversal homotopic to γ, i.e.

IF (γ) = infτ∼γ

∫τµF

The image of MF (S) in RC is exactly the set of measured geodesic laminations, inducinga homeomorphism

MF (S) ∼−→ ML (S).

Using this homeomorphism implicitly, we can consider the Hubbard-Masur map φX tohave 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 7→ (−φ) of Q(X) to an involutivehomeomorphism iX : ML (S) → ML (S), i.e.

iX(λ) = Λ(−φX(λ)).

Since F (φ) and F (−φ) are orthogonal foliations, we say that λ, µ ∈ ML (S) are orthogonalwith respect to X if iX(λ) = µ.

The resulting homeomorphism depends on X ∈ T (S) in an essential way, just as theorthogonality 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 iX : PML (S) → PML (S) that is topologically conjugate tothe antipodal map (−1) : P+Q(X) ' S2n−1 → S2n−1. We call iX the antipodal involutionwith respect to X.

2.4 R-trees and co-collapsingAn 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 intervalin R. Thus an R-tree is like a simplicial tree, but there is no distinction between branching


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 coverS ' H2. Dene a pseudometric dF on S:

dF (x, y) = inf i(F , γ) | γ : [0, 1] → S, γ(0) = x, γ(1) = y (2.3)Then the quotient metric space TF = S/(x ∼ y if dF (x, y) = 0) is an R-tree whose isometrytype depends only on the measure equivalence class of F . Alternately, TF is the space ofleaves of F with metric induced by the transverse measure, where we consider all of theleaves that emanate from a singular point of F to be a single point of TF . The action ofπ1(S) on S by deck transformations descends to an action on TF by isometries. If F is ameasured foliation associated to the measured lamination λ, then the resulting R-tree Tλ

with metric dλ is called the dual R-tree of λ.If λ is supported on a family of simple closed curves, then the R-tree Tλ is actually a

simplicial tree with one vertex for each lift of a complementary region of λ to S, and wherean edge connecting two adjacent complementary regions has length equal to the weight ofthe geodesic that separates them.

A slight generalization of this construction arises naturally in the context of grafting.The grafting locus A ⊂ grλ Y has a natural foliation FA by Euclidean geodesics that mapisometrically onto λ, with a transverse measure induced by the Euclidean metric in theorthogonal direction. The associated pseudometric on grλ Y ,

dFA(x, y) = inf i(FA, γ) | γ : [0, 1] → grλ Y , γ(0) = x, γ(1) = y,

yields a quotient R-tree isometric to Tλ and a mapκ : grλ Y → Tλ,

which we call the co-collapsing map. While the collapsing map κ : grλ Y → Y compressesthe entire grafted part back to its geodesic representative, the co-collapsing map collapseseach connected component of (grλ Y − A), the complement of the grafted part of grλ Y(generically a hyperbolic ideal triangle), and each leaf of FA to a single point. Here we canthink of (grλ Y − A) as a thickened version of the graph of leaves of a measured foliationincident on its singular points.

2.5 Complex projective geometryA complex projective structure on a compact dierentiable surface S is a maximal atlas

of charts on S with values in CP1 and Möbius transition functions. By analogy with the Te-ichmüller space T (S) of marked conformal structures, let P(S) denote the space of markedcomplex projective structures on S with the topology induced by uniform convergence ofcharts.

Since Möbius transformations are holomorphic, each complex projective structure de-termines a conformal structure, and there is an associated projection (forgetful map):

π : P(S) → T (S).


If Z is a projective structure with π(Z) = X ∈ T (S), we will say that Z is a projectivestructure on the Riemann surface X.

A projective structure on X denes a developing map δ : X → C and holonomyrepresentation η : π1(S) → PSL2(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 projectivechart (for the projective structure of X lifted from X).

2. For all γ ∈ π1(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 obtaina locally univalent holomorphic map; η(γ) is then dened to be the Möbius transformationwith germ δγx γ δ−1

x at δ(x) ∈ CP1.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 projectivestructure on X.

For example, every hyperbolic structure X ∈ T (S) determines a standard Fuchsianprojective structure whose developing map is a Riemann map δ : X → H ⊂ CP1. The asso-ciated holonomy representation maps π1(S) to a Fuchsian group π1(X) such that H/π1(X)is isometric to X.

Möbius transformations preserve the set of round circles in CP1, and so one way tothink of a complex projective structure on a Riemann surface X is that it provides a naturalnotion of roundness. We say that a path τ on X is round if projective charts map portionsof τ to arcs of round circles. Because the hyperbolic geodesics of H2, when realized as thedisk ∆ or upper half-plane H, are arcs of circles, geodesics for the hyperbolic metric of Xare round with respect to the standard Fuchsian projective structure.

The Schwarzian derivative can be used to identify the space P(S) of marked projectivestructures on S with the cotangent bundle T ∗T (S) of Teichmüller space as follows: LetF (z) be a locally univalent meromorphic function on a domain Ω ⊂ C. The Schwarzianderivative S (F ) is the holomorphic quadratic dierential:

S (F )(z) =

[(F ′′(z)F ′(z)

)′− 1

2

(F ′′(z)F ′(z)

)2]dz2

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 oscF : Ω →PSL2(C) that associates to a point z ∈ Ω the best approximating (osculating) Möbiustransformation of the germ Fz (see [36]). From this description, both of the above propertiesare easily derived.

It follows from these properties that the Schwarzian derivative is invariant under post-composition with Möbius transformations. Using this invariance, we associate a holomorphicquadratic dierential to a projective structure on X as follows:


First, uniformize the universal cover of X by H, and view the developing map of theprojective structure as a locally univalent holomorphic map δ : H → CP1. The Schwarzianderivative φ = S (δ) is invariant under the deck action of π1(X) on H, because for allγ ∈ π1(X), the germs δz and δγz dier by post-composition with a Möbius transformation.Therefore, φ descends to a holomorphic quadratic dierential φ on X. For brevity we call φthe Schwarzian of the projective structure dened by δ.

Given φ ∈ Q(X), there is an inverse construction that yields a projective structure onX. Its developing map is a function δ : H→ CP1 such that S (δ) = φ, where φ is the lift ofφ to H. This function can be constructed as the ratio of two linearly independent solutionsto the linear ODE:

u′′(z) +12φ(z)u(z) = 0

Therefore, the space P (X) = π−1(X) of projective structures on X is identied with thevector space Q(X). This is the Poincaré parameterization.

Allowing X to vary, this gives an identication of P(S) with the cotangent bundleT ∗T , since T ∗XT is naturally isomorphic to Q(X) by Teichmüller theory.

There is a natural complex structure on P(S) (see [12]), as for the deformation spaceof any suciently rigid holomorphic geometric structure, but this is not the same as thecomplex structure induced by the identication P(S) ' T ∗T described above. The problemis that the Fuchsian groups uniformizing Riemann surfaces do not vary holomorphically withX ∈ T ; equivalently, the section s : T → P(S) assigning the standard Fuchsian projectivestructure to each complex structure is not holomorphic, even though it corresponds to thezero section of the bundle T ∗T .

2.6 Projective GraftingWe now describe a construction of Thurston that assigns a canonical projective structure

to a grafted Riemann surface grλ 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 inversesof a Fuchsian uniformization of Y by H; similarly, the innite cylinder γ × R has a naturalprojective structure whose charts are local inverses of the map f : C→ C∗,

f(z) = exp(2πiz/`(γ, Y )),

where (γ × R) is isometric to C∗ equipped with the conformal metric `(γ, Y ) |dz|2π|z| . Here

`(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 (2.5),and which is locally projectively isomorphic to a contractible neighborhood of R+ in C∗. Asa result, there is a natural complex projective structure on grtγ Y that joins the standardFuchsian structure on (Y − γ) and the projective structure described above on γ × [0, t] ⊂γ × R. We write Grtγ Y for this complex projective structure on grtγ Y .


κH3

CP1

H (H3)

κ

Figure 2.2: The collapsing and co-collapsing maps associated to a projective structure cor-respond to the projection of the developing map to a locally convex pleated plane and itsset of supporting hyperplanes, respectively.

As before the grafting map extends continuously to general measured laminations yield-ing a map Gr : ML (S)×T (S) → P(S), which ts into a commutative diagram:

ML (S)×T (S) Gr'

//

gr))SSSSSSSSSSSSSSSST (S)

π

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 isa unique choice of Y ∈ T (S) and λ ∈ ML (S) such that Grλ Y = Z. Kamishima and Tanrecently gave a complete proof of this theorem [15].

The key to the proof of Thurston's theorem is the construction of a locally convexpleated plane p : Y → H3 that is equivariant with respect to a representation η : π1(S) →PSL2(C), and such that the bending of p is given by the lift λ ⊂ Y of λ. The developingmap δλ of Grλ Y can then be normalized so that for each z ∈ Grλ Y , p(κ(z)) is the nearestpoint retraction of δλ(z) to a convex neighborhood of p(κ(z)) in p(Y ), where κ : Grλ Y → Yis the lift of the collapsing map κ.

The pleated plane p is a kind of local convex hull for the developing map δλ, and thelift κ is a generalization of the nearest-point retraction of a set Ω ⊂ CP1 onto the boundaryof its convex hull ∂CH(Ω) ⊂ H3 (see Figure 2.2. Here the co-collapsing map can also beunderstood in terms of hyperbolic geometry; for each z ∈ Grλ Y let h(z) denote the uniquesupporting hyperplane of the locally convex pleated surface p(Y ) that contains p(κ(z)) andwhose normal vector there is tangent to the geodesic ray landing at δλ(z) ∈ CP1 ' ∂H3.

Then h is a map from Grλ Y to the Lorentzian manifold H (H3) of hyperplanes in H3;in fact h is Lipschitz, and its (a.e. dened) tangent vector has positive norm, reecting thefact that nearby supporting hyperplanes of a locally convex surface must intersect. The


induced metric on the image of h gives it the structure of an R-tree isometric to Tλ, and themap h coincides with the co-collapsing map, i.e.

κ : Grλ Y → Tλ.

Chapter 3

The antipodal map

3.1 IntroductionIn this chapter we prove that the large-scale behavior of the pruning map with basepoint

X is governed by the antipodal involution iX :Theorem 3.1. The pruning map based at X, λ 7→ prλX, extends continuously to a home-omorphism ML (S) → T (S), where T (S) is the Thurston compactication. The boundaryvalues of this homeomorphism give the antipodal involution with respect to X:

iX : PML (S) → PML (S).

Recall that the projective grafting map Gr : ML (S)×T (S) '−→ P(S) and the confor-mal 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 surfacesGrλ Y where grλ Y = X, or equivalently, Y = prλX. Therefore, Theorem 3.1 allows for thefollowing 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], [iXλ]), [λ] ∈ ML (S) ⊂ PML (S)× PML (S) ⊂ ∂(ML (S)×T (S)).

3.2 Harmonic mapsWe now consider the energy functional E (f) for maps f : X → (M,ρ) from a Riemann

surface X to a Riemannian manifold M . In 2.2, we dened the energy for smooth maps,but the natural setting in which to work with the energy functional is the Sobolev spaceW 1,2(X,M) of maps with L2 distributional derivatives. There is some technical dicultyin making this denition precise, as the W 1,2 condition does not imply continuity even inthe classical setting of C-valued functions on Riemannian manifolds; see [17] for details.

18

Chapter 3: The antipodal map 19

While we prefer to phrase our results in this appropriate level of generality, we will onlyneed to consider the energy functional for Lipschitz continuous maps, which are dierentiablealmost everywhere (hence (2.1) still makes sense).

A stationary point of the energy functional E : W 1,2(X,M) → R≥0 is a harmonicmap; it follows from the Euler-Lagrange equation for E (f) and Weyl's lemma that the Hopfdierential

Φ(f) = f∗(ρ)2,0

is holomorphic (Φ(f) ∈ Q(X)) if f is harmonic. In particular, the maximal and minimalstretch directions of a harmonic map are realized as a pair of orthogonal foliations by straightlines in the singular Euclidean metric |Φ(f)|. Ishihara has shown that a map f : X → Mto a Riemannian manifold is harmonic if and only if f pulls back germs of convex functionson 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 considerharmonic maps from Riemann surfaces to R-trees. The main references for this theory arethe papers of Wolf ([38], [39], [40]); a much more general theory of harmonic maps to metricspaces is discussed by Korevaar and Schoen in [17] and [18].

Even though R-trees are somewhat badly behaved spaces (they are not in general locallycompact about any point), they are natural objects to consider when examining the limitingbehavior of sequences of hyperbolic structures Yi ∈ T (S) that leave all compact subsets:Theorem 3.3 (Wolf [38]). Let X,Yi ∈ T (S) and µ ∈ ML (S) be such that Yi → [µ] ∈PML (S) in the Thurston compactication. Then after rescaling the hyperbolic metrics ρi onYi appropriately, the sequence of metric spaces (Yi, ρi) converges in the equivariant Gromov-Hausdor sense to the R-tree Tµ.

The equivariant Gromov-Hausdor topology is a natural setting in which to considerconvergence of metric spaces equipped with isometric group actions. The application of thistopology to the Thurston compactication is due to Paulin [29]; for other perspectives onthe connection between R-trees and Teichmüller theory, see [3], [28], [4].

Remarkably, the theory of harmonic maps is well-adapted to this generalization fromsmooth surfaces to metric spaces like R-trees; for example, the convergence statement ofTheorem 3.3 can be extended to a family of harmonic maps from a xed Riemann surfaceX:Theorem 3.4 (Wolf [38]). For X ∈ T (S) and λ ∈ ML (S), let πλ : X → Tλ denote theprojection onto the leaves of F (φX(λ)), where φX(λ) is the Hubbard-Masur dierential forλ. Then:

1. πλ is harmonic, meaning that it pulls back germs of convex functions on Tλ to germsof subharmonic functions on X.

20 Chapter 3: The antipodal map

2. If Yi ∈ T (S) is a sequence such that Yi → [λ] ∈ PML (S), then the lifts of harmonicmaps hi : X → Yi converge in the Gromov-Hausdor sense to πX : 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, Tλ) of equivariant maps with L2

distributional derivatives. Once again, we defer to Korevaar and Schoen for details, andnote that the maps we consider are Lipschitz [17].

By denition, the metric dλ on Tλ is isometric to the standard Euclidean metric on Ralong each geodesic segment. Thus the pullback of this metric via an equivariant W 1,2 mapf : X → Tλ is a well-dened (possibly degenerate) quadratic form on TX which is invariantunder the action of π1(X). This allows us to dene the energy E (f) and Hopf dierentialΦ(f) of such an equivariant map as the L1 norm and (2, 0) part of the induced quadraticform on TX.

For later use we record the following calculations relating the Hopf dierential andenergy of the projection πλ to the Hubbard-Masur dierential φX(λ); details can be foundin [38].Lemma 3.5. The Hopf dierential of πλ : X → Tλ is

Φ(πλ) = φX(12λ) =

14φX(λ),

and the energy of πλ is given byE (πλ) =

12E(λ,X) =

12‖φX(λ)‖.

3.3 Energy and graftingIn [35], Tanigawa shows that for a xed lamination λ, the grafting map grλ : T (S) →

T (S) is proper; the proof relies on the following inequality relating the geometry of a graftedsurface to the energy of a harmonic map:Lemma 3.6 (Tanigawa [35]). Let X = grλ Y , where X,Y ∈ T (S) and λ ∈ ML (S), andlet h (resp. E (h)) denote the harmonic map h : X → Y compatible with the markings (resp.its energy). Then

12`(λ, Y ) ≤ 1

2`(λ, Y )2

E(λ,X)≤ E (h) ≤ 1

2`(λ, Y ) + 2π|χ(S)|,

where `(λ, Y ) is the hyperbolic length of λ on Y , and E(λ,X) is the extremal length of λ onX.Note. The middle part of Tanigawa's inequality, i.e.

12`(λ, Y )2

E(λ,X)≤ E (h)

is due to Minsky, and holds for any harmonic map between nite-area Riemann surfaces ofthe same type and any measured lamination λ [26].


For our purposes, one of the most important consequences of Lemma 3.6 is a relationshipbetween the hyperbolic length of the grafting lamination on Y and its extremal length ongrλ Y :Lemma 3.7. Let X = grλ Y , where Y ∈ T (S). Then we have

`(λ, Y ) = E(λ,X) +O(1)

where the implicit constant depends only on |χ(S)|.Proof. The lower bound

`(λ, Y ) ≥ E(λ,X)

is immediate from the left-hand side of the inequality of Lemma 3.6, and we also have12`(λ, Y )2

E(λ,X)≤ 1

2`(λ, Y ) + 2π|χ(S)|

and therefore,`(λ, Y )2

12`(λ, Y ) +O(1)

≤ E(λ,X).

Solving for `(λ, Y ) yields

`(λ, Y ) ≤ 12E(λ,X) +

12(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 geometryof grafting will be used in the proof of Theorem 3.1; however, we can already deduce aweak form of compatibility between the grafting coordinates for P(S) and the projectionπ : 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 (Grλ Y ) 7→ Yto P (X) is proper.

2. The sequence of projective surfaces GrλiYi ∈ P (X) diverges if and only if both λi and

Yi diverge (in ML (S) and T (S), respectively).Proof.

1. Suppose GrλiYi diverges but Yi remain in a compact set in Teichmüller space.

Since Gr : ML (S) × T (S) → P(S) is a homeomorphism, a divergent sequence inP (X) has the form Grλi

Yi where either λi →∞, Yi →∞, or both. Since Yi is assumedto be bounded, we conclude that λi →∞.


Therefore `(λi, Yi) →∞, and by Lemma 3.6, we haveE (hi) ≥

12`(λi, Yi) →∞,

where hi : X → Yi 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 (hi) →∞, we conclude Yi →∞, whichis a contradiction.

2. Once again, Gr : ML (S) × T (S) → P(S) is a homeomorphism, so divergence of asequence Grλi

Yi is equivalent to divergence of at least one of the two sequences λi orYi. We therefore need only show that if the sequence lies in P (X), then divergence ofeither grafting coordinate implies divergence of both.We observed in the proof of (1) that for Grλi

Yi ∈ P (X),(Yi →∞) ⇒ (λi →∞).

Suppose that instead λi →∞; then E(λi, X) →∞ and by Lemma 3.7 `(λi, Yi) →∞.As before, this implies the divergence of the energy, and Yi. Therefore

(λi →∞) ⇒ (Yi →∞).

3.4 Convergence to the harmonic mapLet X,Yi ∈ T (S) and suppose Yi → ∞; let ρi denote a hyperbolic metric on Yi, and

hi : X → Yi the harmonic map (with respect to ρi) compatible with the markings.We say that a sequence of maps fi ∈ W 1,2(X,Yi) compatible with the markings of X

and Yi is a minimizing sequence iflimi→∞

E (fi)E (hi)

= 1.

Since the harmonic map hi is the unique energy minimizer in its homotopy class, a min-imizing sequence asymptotically minimizes energy. In this section we will show that allminimizing sequences have the same asymptotic behavior, in a precise sense:Theorem 3.9. Let X and Yi be as above, and suppose limi→∞ Yi = [µ] ∈ PML (S) in theThurston compactication.

Then for any minimizing sequence fi : X → Yi, the measurable quadratic dierentials[f∗i (ρi)](2,0) converge projectively in the L1 sense to a holomorphic quadratic dierentialΦ ∈ Q(X) such that

[F (−Φ)] = [µ],

i.e. there are constants ci > 0 such thatlimi→∞

ci[f∗i (ρi)](2,0) = Φ.


Note. The vertical foliation F (−Φ) appears in Theorem 3.9 because the Thurston limit is alamination whose intersection number provides an estimate of hyperbolic length; directionsorthogonal to F (−Φ) (that is, tangent to F (Φ)) are maximally stretched by a map withHopf 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 isbased.Theorem 3.10 (Wolf, [37]). Let X,Yi ∈ T (S), and let Ψi denote the Hopf dierential ofthe unique harmonic map hi : X → Yi respecting markings, where Yi is given the hyperbolicmetric ρi. Then

limi→∞

Yi = [µ] ∈ PML (S)

if and only iflimi→∞

[F (−Ψi)] = [µ].

In other words, if one compacties Teichmüller space according to the limiting behaviorof the Hopf dierential of the harmonic map from a xed conformal structure X, thenthe vertical foliation map F (−1) : P+Q(X) → PMF (S) ' PML (S) identies thiscompactication with the Thurston compactication.

Theorem 3.10 is actually a consequence of the convergence of harmonic maps hi to theharmonic projection πµ 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 harmonicmap by means of the pullback metrics. The following theorem is the main technical tool:Theorem 3.11. Let f ∈ W 1,2(X,Y ), where X,Y ∈ T (S) and Y is given the hyperbolicmetric ρ. Let h be the harmonic map homotopic to f . Then

‖f∗(ρ)− h∗(ρ)‖1 ≤ 2 (E (f)− E (h)) ,

and in particular‖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 =

12π

∫X

∫SxX

∣∣∣‖f∗v‖2ρ − ‖h∗v‖2ρ∣∣∣dθ(v)dσ(x) (3.1)

For x ∈ X, let m(x) denote the midpoint of the geodesic segment γx 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 mapm : X → Y . By the quadrilateral inequality in hyperbolic space ([30] or 2.1 of [17], forv ∈ TxX,

‖m∗v‖2ρ ≤12‖h∗v‖2ρ +

12‖f∗v‖2ρ −

14

∣∣∣‖f∗v‖ρ − ‖h∗v‖ρ

∣∣∣2.Informally, this means that the midpoint of a geodesic segment in a negatively curvedRiemannian manifold is rather insensitive to movement of the endpoints, especially if oneendpoint is moved faster than the other.


Applying this inequality to the norm dierence estimate (Equation 3.1), we obtain:

‖f∗(ρ)− h∗(ρ)‖1 ≤∫

X

∫SxX

(2‖h∗v‖2ρ + 2‖f∗v‖2ρ − 4‖m∗v‖2ρ

)dθ(v)dAσ(x)

= 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 ofTheorem 3.11 also follows.

When Theorem 3.11 is combined with Wolf's result on the convergence of harmonicmaps of surfaces to harmonic projections to R-trees (Theorem 3.4), we obtain the followingcorollary:Corollary 3.12. Let f ∈ W 1,2(X, Tλ) be a π1-equivariant map, where X ∈ T (S) andλ ∈ ML (S). Then

‖4Φ(f)− φX(λ)‖1 ≤ 2 (E (f)− E (πλ))

Note. The role of the midpoint mapm in the proof of Theorem 3.11 is part of a more generaltheory of convexity of the energy functional E when two maps into negatively curved spacesare connected by a geodesic homotopy. This, in turn, relies on the convexity of the distancefunction between geodesics in negatively curved spaces. For details, see [17].Proof of Theorem 3.9. Clearly the sequence of harmonic maps hi : X → Yi is a minimizingsequence. Applying Theorem 3.10 to the sequence Yi we nd that the Hopf dierentialsconverge projectively:

limi→∞

[Φ(hi)] = [Φ∞], where F (−[Φ∞]) = [µ].

To prove Theorem 3.9, we therefore need only show that (measurable) the Hopf dier-entials Φ(fi) of any minimizing sequence have the same projective limit as the holomorphicHopf dierentials Ψi = Φ(hi) of the harmonic maps.

Applying Theorem 3.11 to such a sequence, we nd‖f∗i (ρi)− h∗i (ρi)‖1 ≤ 2 (E (fi)− E (hi)) = o(E (hi)),

and solimi→∞

‖f∗i (ρi)− h∗i (ρi)‖1E (hi)

= 0.

Since the Hopf dierential is the (2, 0) part of the pullback metric, we havelimi→∞

[Φ(fi)] = limi→∞

[Ψi] = [Φ], where [F (−Φ)] = [µ].


3.5 Measurable quadratic dierentialsFor 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

ω(φ, ψ) =∫

X| Im

(√α√β)|.

Theni(F (φ),F (ψ)) = ω(φ, ψ).

However, the quantity ω(α, β) makes sense for L1 quadratic dierentials α and β, holomor-phic or not.

While a measurable dierential α does not dene a measured foliation, it does have ahorizontal line eld L (α) consisting of directions v ∈ TX 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 → prλX, and for simplicity let us rst supposeλ is supported on a single simple closed hyperbolic geodesic γ ⊂ prλX, i.e. λ = tγ. Thenthe grafting locus A ⊂ X is the Euclidean cylinder γ × [0, t], and the collapsing map is theprojection onto the geodesic γ. Just as the Hopf dierential of the orthogonal projection ofC onto R is

Φ(z 7→ Re(z)) = [dx2]2,0 =14dz2,

the Hopf dierential of κ on A is the pullback of 14dz

2 via local Euclidean charts that takeparallels of γ to horizontal lines. This dierential is holomorphic on A, and corresponds tothe measured foliation 1

2λ. Thus the Euclidean metric on A, which is the restriction of theThurston metric of X, is given by |4Φ(κ)|.

On the complement of the grafting locus, the collapsing map is conformal and thus theHopf dierential is zero. Therefore Φ(κ) is a piecewise holomorphic dierential on X whosehorizontal line eld is the natural foliation of the grafting locus by parallels of the graftinglamination, with half of the measure of λ. This analysis extends by continuity to the caseof a general lamination λ ∈ ML (S).

It follows that the line eld L (Φ(κ)) represents the measured lamination 12λ, in that

for all ψ ∈ Q(X),ω(Φ(κ), ψ) =

12i(λ,F (ψ)). (3.2)

We therefore use the notationΦX(λ) = Φ(κ : X → prλX)

for the Hopf dierential of the collapsing map, which is somewhat like φX(12λ) in that it

is a quadratic dierential whose foliation is a distinguished representative for the measuredfoliation class of 1

2λ. The Hopf dierential ΦX(λ) is not holomorphic, however, though wewill later see (3.6) that is is nearly so.

For now, we will simply show that L1 convergence of Hopf dierentials ΦX(λ) to aholomorphic limit implies convergence of the laminations λ:


Lemma 3.13. Let X ∈ T (S) and λi ∈ ML (S). If[ΦX(λi)] → [ψ], where ψ ∈ Q(X)

then[λi] → [F (ψ)] ∈ PML (S).

Here [ΦX(λi)] is the image of ΦX(λi) in PS2,0(X).Proof. First, we can choose ci > 0 such that

c2i ΦX(λi) → ψ.

It is well known that there are nitely many simple closed curves νk, k = 1 . . . N ,considered as measured laminations with unit weight, such that the map I : ML (S) → RN ,I(λ) = i(λ, νk) is a homeomorphism onto its image. Recall that φX(νk) ∈ Q(X) is theunique holomorphic quadratic dierential satisfying F (φX(νk)) = νk.

Since ω(·, νk) : S2,0(X) → R is evidently a continuous map, we conclude from (3.2) andthe hypothesis ciΦX(λi) → ψ that

ω(c2i ΦX(λi), φX(νk)) =ci2i(λi, νk) →

12i(F (ψ), νk).

and therefore ciλi → 12F (ψ).

Using the above description of ΦX(λ), we can also compute its norm, and the energyof the collapsing map:Corollary 3.14. The L1 norm of ΦX(λ) is given by

‖ΦX(λ)‖1 =14`(λ,prλX) =

14E(λ,X) +O(1),

and the energy of the collapse map κ : X → prλX is

E(κ) =12`(λ,prλX) + 2π|χ(S)|.

Proof. We have seen that |4ΦX(λ)| induces the Thurston metric on the grafted part A ⊂ Xand is zero elsewhere. The area of A with respect to the Thurston metric is `(λ,prλX), andtherefore,

‖ΦX(λ)‖1 =14`(λ,prλX).

On the other hand, it follows from Lemma 3.6 that`(λ,prλX) = E(λ,X) +O(1),

which yields the rst statement in Corollary 3.14.


For the energy computation, we follow Tanigawa (see [35]): On the grafting locus, κcollapses directions orthogonal to the parallels of the grafting lamination, while it mapsdirections tangent to such parallels isometrically. Thus this part of the surface contributes

E (κ |A) =12

Area(A) =12`(λ,prλX)

to the total energy. Since κ is an isometry on the complement of the grafting locus,E (κ |X−A) = Area(X −A) = Area(prλX) = 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 κ :

Grλ Y → Tλ. Though the co-collapsing map is dened on the universal cover of the graftedsurface, its Hopf dierential is invariant under the action of π1(S) and therefore descendsto a measurable quadratic dierential Φ(κ) ∈ S2,0(X).

The co-collapsing map κ 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 ismodeled on the orthogonal projection of C onto iR (where the leaves of FA correspond tohorizontal lines in C). Since

Φ(z 7→ Im(z)) = [dy2]2,0 = −14dz2,

we conclude that the Hopf dierentials Φ(κ) and Φ(κ) are inverses, i.e.Φ(κ) = −Φ(κ) = −ΦX(λ) (3.3)

Remark. The relationship between κ and κ and their Hopf dierentials is reminiscent of theminimal suspension technique introduced by Wolf; for details, see [41].

3.6 The antipodal map extends pruningIn this section we apply Theorem 3.9 to the collapsing maps κi : X → Yi to prove

Theorem 3.1 (see 3.1).Proof of Theorem 3.1. Let Zi = Grλi

Yi ∈ P (X), and suppose Zi → ∞. By Lemma 3.8,Yi → ∞, λi → ∞, and E (X,Yi) → ∞. We need to show that if Yi → [µ] ∈ PML (S) and[λi] → [λ] ∈ PML (S) then iX([λ]) = [µ], or equivalently, that [λ] and [µ] are the horizontaland vertical measured laminations of a single holomorphic quadratic dierential on X.

By Lemma 3.6,12`(λ, Yi) ≤ E (X,Yi) ≤

12`(λ, Yi) + 2π|χ(S)| (3.4)

while by Corollary 3.14,E (κi) =

12`(λ, Yi) + 2π|χ(S)|, (3.5)


so κi is a minimizing sequence.Applying Theorem 3.9, we conclude that

[ΦX(λi)] → [Φ] ∈ P+Q(X), where [F (−Φ)] = [µ].

On the other hand, by Lemma 3.13, this implies that [F (Φ)] = [λ], and iX([λ]) = [µ].

While Theorem 3.1 is an asymptotic statement, the ideas used in the proof abovealso yield the following nite version of the comparison between the Hopf dierentials ofcollapsing map to the holomorphic dierential φX(λ) representing the grafting lamination:Theorem 3.15. Let X ∈ T (S) and λ ∈ ML (S), and let h : X → prλX denote theharmonic map compatible with the markings. Then

‖4ΦX(λ)− φX(λ)‖1 ≤ 16π|χ(S)|.

Proof. Recall from (3.3) that the Hopf dierential of the co-collapsing map κ isΦ(κ) = −ΦX(λ)

and its energy isE (κ) =

12

Area(A) =12`(λ,prλX),

where A ⊂ X is the grafting locus. Recall that by Lemma 3.6,12`(λ,prλX) =

12E(λ,X) +O(1),

while by Lemma 3.5 we know that the energy and Hopf dierential of the harmonic projectionπλ : X → Tλ are:

E (πλ) =12E(λ,X)

Φ(πλ) = φX(12λ) =

14φX(λ)

Therefore, we ndE (κ)− E (πλ) ≤ 2π|χ(S)|,

and Corollary 3.12 implies that

‖ΦX(λ)− 14φX(λ)‖1 ≤ 4π|χ(S)|.

The Theorem then follows by algebra.


∼ 1

∼ 1

(a) (b)

`′ ∼ 1/ log(2M)

` ∼ 1/ log(M)

`′/`→ 1 as M →∞M ′/M = 2

M ′ = 2M

M

Figure 3.1: (a) Euclidean cylinders with moduli M and 2M correspond to (b) hyperboliccylinders whose core geodesics have approximately the same length. This phenomenon leadsto Teichmüller rays for distinct Strebel dierentials that converge to the same point in theThurston boundary of Teichmüller space.

3.A Appendix: Asymmetry of Teichmüller geodesicsSince the antipodal involution iX is a homeomorphism of PML (S) to itself, it seems

natural to look for an involutive homeomorphism of T (S) that has iX as its boundary values.In fact, an obvious candidate is the Teichmüller geodesic involution IX : T (S) → T (S)that is the pushforward of (−1) : Q(X) → Q(X) via the Teichmüller exponential mapτ : Q(X) ∼−→ T (S). However, we now sketch an example showing that IX does not extendcontinuously to the Thurston compactication of T (S), leading us to view the pruning mapbased at X, λ 7→ prλX, as a kind of substitute for the Teichmüller geodesic involution thatdoes extend continuously to the antipodal map of PML .

By a theorem of Masur, if φ is a Strebel dierential on X whose trajectories representhomotopy classes (α1, . . . , αn), then the Teichmüller ray determined by φ converges to thepoint [α1 + · · · + αn] ∈ PML (S) in the Thurston compactication [23]. Note that thelimit point corresponds to a measured lamination in which each curve αi has the sameweight, independent of the relative sizes of the cylinders on X determined by φ. Thishappens because the Thurston boundary reects the geometry of hyperbolic geodesics onthe surface, and hyperbolic length is approximated by the reciprocal of the logarithm of acylinder's height, as in Figure 3.1.

Now suppose φ and ψ are holomorphic quadratic dierentials on a Riemann surface Xsuch that each of ±φ,±ψ is Strebel, where the trajectories of φ and ψ represent dierentsets 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


Figure 3.2: Horizontal (solid) and vertical (dashed) trajectories of φ0 and ψ0 on the squaretorus.

the same point on PML (S) in the negative direction, while in the positive direction theyconverge to distinct points. If IX were to extend to a continuous map of the Thurstonboundary, then any pair of Teichmüller geodesics that are asymptotic in one direction wouldnecessarily be asymptotic in both directions, thus no such extension exists.

One can explicitly construct such (X,φ, ψ) as follows: Let X0 denote the square torusC/(2Z ⊕ 2iZ), and let ψ0 = ψ0(z)dz2 be a meromorphic quadratic dierential on X0 withsimple zeros at z = ±1+ε

2 and simple poles at z = ±12 and such that ψ0(x) ∈ R for x ∈ R.

Such a dierential exists by the Abel-Jacobi theorem, and in fact is given by

ψ0(z)dz2 =℘(z)− c0℘(z)− c1

for suitable constants ci, where ℘(z) is the Weierstrass function for X0. Let φ0 = dz2, aholomorphic quadratic dierential on X0.

Let X be the surface of genus 2 obtained as a 2-fold cover of X0 branched over ±12 ;then ψ0 and φ0 determine holomorphic quadratic dierentials ψ and φ on X, where the lift

of ψ0 is holomorphic because the simple poles at ±12 are branch points of the covering map

X → X0.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 simpleclosed curves on X0 that arise as the quotients of R and iR, respectively; both γ and η havetwo distinct lifts (γ± and η±, respectively) to X. Let α denote the separating curve on Xthat covers [−1/2, 1/2], and let β denote the simple closed curve on X that is the union ofthe two lifts of [−i, i]. Then:

1. the trajectories of φ represent (γ+, γ−),


2. the trajectories of ψ represent (γ+, γ−, α), and3. the trajectories of both −φ and −ψ represent (η+, η−, β).


α

a)

b)

η−η+

β

γ+ γ−

η+ η−

β

γ+ γ−

Figure 3.3: Two constructions of a surface of genus two with a Strebel dierential: (a) Twotori are glued along a segment of a leaf of a Euclidean foliation. (b) Two tori are glued tothe ends of a Euclidean cylinder. Below each example, the homotopy classes represented bythe horizontal and vertical trajectories are shown (as solid and dashed lines, respectively).When the tori and cylinder are chosen correctly, this construction produces an exampleof Teichmüller geodesics that are asymptotic in one direction while converging to distinctendpoints in the opposite direction.

Chapter 4

Fuchsian centers and Strebel

dierentials

4.1 IntroductionIn this chapter we will study the parameterization of projective structures on a xed

Riemann surface X via the Schwarzian derivative. After formulating a general conjectureabout this parameterization (Conjecture 4.2) we prove a special case that has applicationsto 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 callthese Fuchsian centers because they provide canonical center points for islands of quasifuch-sian holonomy surrounding the Bers embedding of Teichmüller space (4.2). Each centercγ ∈ Q(X) is labeled by unique integral lamination γ ∈ ML Z , and corresponds to a pro-jective structure on X with grafting lamination 2πγ.

Each Fuchsian center has an associated Strebel dierential sγ = φX(2πγ), with L1

norm ‖sγ‖1 = E(2πγ,X). The Strebel dierential is fairly easy to understand in terms ofthe conformal or hyperbolic geometry of X, while the Fuchsian center cγ is comparativelymysterious. Nevertheless, we show that sγ ∈ Q(X) is a good estimate for the position of cγ :Theorem 4.1. For each γ ∈ ML Z(S) and all k ∈ N, we have

‖2ckγ − skγ‖1 = O(1),

where the implicit constant depends on X and γ but not k. In particular, the Fuchsiancenters ckγ 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 lowerbound for the norms of Fuchsian centers where γ is supported on a single simple closedcurve (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 dierentways to obtain a measured lamination from a holomorphic quadratic dierential. The rst,

33

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 itshorizontal foliation:

Λ : Q(X) → ML (S)

The content of the Hubbard-Masur theorem (2.3) is that this map is a homeomorphism, andwe use φX to denote its inverse. The map Λ is homogeneous of degree 1

2 , i.e. Λ(t2φ) = 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 (2.5). By Thurston's theorem, theprojective structure corresponding to φ ∈ Q(X) arises from grafting in a unique way, i.e.(X,φ) = Grλ Y for Y ∈ T (S) and λ ∈ ML (S). Equivalently, λ = β(φ) is the bendinglamination of the locally convex pleated surface in H3 corresponding to (X,φ). The resultingmap

β : Q(X) → ML (S)

is somewhat exotic compared to Λ for example, while Λ(0) = 0 (see 3.3), it is nothomogeneous. It follows from the work of Scannell and Wolf that β is a homeomorphism,and we use ξX to denote its inverse.

It is technically simpler to compare φX and ξX (rather than their inverses, Λ and β)because they take values in the Banach space Q(X). The results in this chapter can be seenas evidence supporting the following conjecture:Conjecture 4.2. The dierence between φX and 2ξX is bounded by a constant that dependsonly on X,

‖φX(λ)− 2ξX(λ)‖1 = O(1).

Remark. It seems unlikely that the dierence between φX and 2ξX could be bounded uni-formly for all X ∈ T (S) because the trajectories of holomorphic quadratic dierentialscan be very badly behaved with respect to the hyperbolic metric on a surface with shortgeodesics.

We will show that the map ξX , while lacking the homogeneity properties of φX , behavesas predicted by Conjecture 4.2 along the ray R+ · λ, where λ is a nite lamination, i.e. onesupported 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 allt > 0, we have

‖φX(tλ)− 2ξX(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

sincecγ = ξX(2πγ) and sγ = φX(2πγ).

Chapter 4: Fuchsian centers and Strebel dierentials 35

4.2 Holonomy of projective structuresLet V (S) denote the variety of irreducible parabolicity-preserving representations of

π1(S) into PSL2(C) modulo conjugacy (briey, the representation variety), which is a com-plex manifold of dimension 3g − 3. For a representation class [ρ] ∈ V (S), we let Γρ denotethe image ρ(π1(S)) of a representative ρ : π1(S) → PSL2(C). A representation [ρ] will becalled discrete if Γρ is discrete. While the geometry of the set of discrete representationsVdisc(S) ⊂ V (S) is known to be quite complicated, its interior can be described explicitly asfollows.

A representation [ρ] ∈ V (S) is quasifuchsian (resp. Fuchsian) if the limit set Λρ ⊂C of Γρ is a Jordan curve (resp. round circle). Fuchsian representations are preciselythose which are conjugate into PSL2(R); quasifuchsian representations are those which arequasiconformally conjugate to a Fuchsian representation. Let QF(S) denote the set ofquasifuchsian representations [ρ] ∈ V (S).Theorem 4.4. IntVdisc(S) = QF(S)

Proof. By Sullivan's stability theorem ([34]), IntVdisc(S) consists of quasiconformally conju-gate 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 unionof analytic subvarieties of V (S), which is nowhere dense.

By a result of Maskit, these faithful representations of π1(S) are either quasifuchsianor degenerate [21]. Degenerate groups are not geometrically nite (see [22, IX.G.19]), sothe theorem follows.

Compare [16, 8.3], [24, 4.3].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+/Γρ and Ω−/Γρ,where Ω+ t Ω− is the domain of discontinuity of Γρ [3]. In particular, QF(S) is connectedand contractible. Here S denotes the surface S with its orientation reversed. Let Q(X, Y )be the quasifuchsian representation class corresponding to the pair (X, Y ) ∈ 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 aholonomy 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 appearin [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(Vdisc(S)) ⊂ P(S), theset of projective structures with discrete holonomy. By Theorem 4.5, the structure of theinterior of K (S) locally mirrors the situation in the representation variety.Corollary 4.6. IntK (S) = hol−1(QF(S))

In fact, IntK (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 ⊂


ML (S); a lamination is integral if it is supported on simple closed curves and the weightof each closed curve is an integer. Integral grafting provides explicit examples of points ineach component, thus connecting the Thurston parameterization (2.1) with the study ofholonomy.Theorem 4.7.

1. The interior of K (S) consists of countably many preimages of QF(S) under the holon-omy map, each of which maps biholomorphically onto QF(S). The components arelabeled by integral laminations, i.e.

IntK (S) '⊔

γ∈ML Z

QFγ(S),

where QFγ(S) ⊂ P(S) and hol : QFγ(S) → QF(S) is a biholomorphism.2. For any Y ∈ T (S) and any integral lamination γ ∈ ML Z(S), we have

hol(Gr2πγ Y ) = Q(Y, Y ),

i.e. Gr2πγ Y has the same Fuchsian holonomy as the standard Fuchsian projectivestructure on Y .

3. Conversely, if Z ∈ P(S) has Fuchsian holonomy, then we have Z = Gr2πγ Y for someγ ∈ ML Z(S) and Y ∈ T (S).

4. For each Y ∈ T (S) and γ ∈ ML Z(S), we haveGr2πγ Y ∈ QFγ(S).

Sketch of proof.1. See [16]. Goldman has shown that there is a locally constant topological invariantθ : Int K (S) → ML Z, the wrapping invariant, that separates components. ThenQFγ(S) = θ−1(γ).

2. The integral grafting Gr2πγ Y , where γ ∈ ML Z(S), has the same holonomy groupQ(Y, Y ) as the standard Fuchsian structure of Y . This is because the developing mapof Gr2πγ Y is obtained by inserting 2πn-sectors into the hyperbolic plane ∆ ⊂ C alongall lifts of the curves in γ; since a 2πn-sector wraps completely around the sphere ntimes (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 Gr2πγ Y do not lie in the same component of IntK (S) sincethe developing map is injective for Y and ∞-to-one for Gr2πγ Y . In fact, if Y and Gr2πγ Yare joined by a path of projective structures, at the last moment when the developing mapis univalent the limit set of the holonomy group cannot be a Jordan curve, and thereforethe holonomy is not quasifuchsian [19].


X = ˜gr2πγ Y CP1

δ2πγ

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 ofthis wrapping behavior for one lift is shown on the right.

4.3 Fuchsian centersIn the previous section, we described the components of IntK (S) in terms of a dis-

crete topological invariant (the wrapping lamination) and analytic moduli (the holonomyrepresentation [ρ] ∈ QF(S)). We now investigate how IntK (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) ' C3g−3. While IntK (S) consistsof quasifuchsian representations, there are 3g − 3-dimensional families of singly degeneraterepresentations in the boundary ∂QF. It is therefore possible that K(X) could contain anopen set of degenerate representations, and hence that IntK(X) 6= (Int K (S))∩P (X). Thefollowing theorem of Shiga and Tanigawa rules out such pathology:Theorem 4.8 (Shiga-Tanigawa [33]). IntK(X) consists of projective structures withquasifuchsian holonomy, and therefore IntK(X) = (Int K (S)) ∩ P (X).Notes.

1. Shiga-Tanigawa show that the quasiconformal deformations of holonomy groups canalways be lifted to deformations of projective structures.

2. There exist projective structures with singly degenerate holonomy, and therefore thereare 3g − 3-dimensional families of such structures in P(S); Theorem 4.8 implies thatthese submanifolds are transverse to π : P(S) → T (S).Knowing that the study of IntK(X) involves only quasifuchsian projective structures,

it is natural to ask how each component QFγ(S) of IntK(S) intersects P (X). The rstobservation is that this intersection is always nonempty; dene cγ ∈ Q(X) by

(X, cγ) = Gr2πγ(pr2πγ X)),


or equivalently,cγ = ξX(2πγ).

Then we have:Corollary 4.9 (of Theorem 4.7). For each X ∈ T (S) and γ ∈ ML Z(S), (X, cγ) is theunique projective structure on X with Fuchsian holonomy and wrapping lamination γ.Proof. By part (3) of Theorem 4.7, Gr2πγ(pr2πγ X)) has Fuchsian holonomy and wrappinglamination is γ; uniqueness follows since the bending lamination map β : P (X) → ML (S)is a homeomorphism.

The origin c0 = 0 is the standard Fuchsian structure on X. It is the Fuchsian center ofQF0(S)∩P (X), the set of quasifuchsian projective structures onX with wrapping lamination0, or equivalently, injective developing maps. The Bers slice B(X) ⊂ Q(X) is dened asthe set of quadratic dierentials that arise as Schwarzian derivatives of equivariant mapsfrom X ' ∆ onto quasidisks in CP1 (see [3], [24]). Thus the Schwarzian derivative identiesQF0(S) with B(X).

The Bers slice provides an embedding of the Teichmüller space of X, and is a con-nected, bounded, contractible subset of Q(X). This description of BX in terms of projectivestructures was rst given by Shiga [32].

A projective structure with quasifuchsian holonomy is called standard if its developingmap is injective, and otherwise is called exotic. Since the Bers slice B(X) is the image ofQF0(S) ∩ P (X) under the Schwarzian, we dene the exotic Bers slices

Bγ(X) = S (QFγ(S) ∩ P (X)) ⊂ Q(X), γ ∈ ML Z(S)

In contrast to the standard Bers slice, relatively little is known about Bγ(X) for γ 6= 0(see Chapter 5). However, since cγ ∈ QFγ(S)∩P (X), the quadratic dierential cγ providesa distinguished Fuchsian center to the exotic Bers slice Bγ(X).

4.4 Grafting annuliIn an eort to understand the structure of the exotic Bers slices Bγ(X) ⊂ Q(X), we

now study the arrangement of the rays ξX(tγ) for nite laminations γ ∈ ML (S). The nitetopological complexity of such laminations will allow us to apply tools from the theory ofunivalent functions to study the grafting annuli inserted along the corresponding geodesics.

Fix X ∈ T (S) and a nite laminationγ =

∑i

hi[γi], γi ∈ π1(S),

and consider the family of projective structures Grtγ(prtγ X), t ∈ R+. If γ is integral, thenthis family includes the countably innite ray of Fuchsian centers ckγ = ξX(2πkγ), k ∈ N.Recall that the grafting locus A = A(tγ) ⊂ X is the union of annuli in the homotopy classesγi.

The goal of this section is to prove the following result about the geometry of the annuliA(tγ):


Lemma 4.10. There exists a constant m > 0 depending only on γ and X such that eachcomponent Ai ⊂ A(tγ) has modulus

mod (Ai) ≥ m.

Lemma 4.10 shows that the grafting annuli have bounded geometry in a complex-analytic sense. The following consequence of the modulus bound will be used in latersections.Corollary 4.11 (of Lemma 4.10). For each X ∈ T (S) and nite lamination γ, thereis 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 thedistortion theorem for univalent functions [1]. For a detailed proof, see [14].Proof of Lemma 4.10. The idea is to estimate the length of γ and γi on X; from Lemma3.7, we have

`(tγ, prtγ X) = E(tγ,X) +O(1).

Using the homogeneity of hyperbolic length and extremal length on ML (S) (of degrees 1and 2, respectively), we obtain

`(γ,prtγ X) = tE(γ,X) +O(1/t).

By the denition of the length of a nite lamination,`(γ, Z) =

∑i

hi `(γi, Z),

where the sum ranges over the nite set of curves in the support of γ, and hi ∈ R+. Fort 0 we then have

`(γi,prtγ X) ≤ 1hmin

`(γ,prtγ X) ≤ CtE(γ,X)

wherehmin = min

ihi > 0

and C depend on γ and X but not t.The Euclidean annulus Ai in the homotopy class γi has circumference `(γi,prtγ X) and

height t hi, so its conformal modulus isMod(Ai) =

t hi

`(γi,prtγ X).

Using hi ≥ hmin and the estimate above, we haveMod(Ai) ≥

t hi

CtE(γ,X)≥ hmin

CE(γ,X).

The right hand side in this last inequality depends only on X and γ, so Lemma 4.10 follows.


4.5 The developing mapThe grafting construction provides a fairly explicit description of the developing map

(2.5). In this section we use this description to estimate the Schwarzian derivative.Lemma 4.12. Let γ =

∑i hi[γi] be a nite lamination and Y = prγ X. Let Ai denote the

connected component of the grafting locus in GrγY that is stabilized by γi. Normalize so thatthe holonomy map sends π1(S) to a Fuchsian group and γi to a dilation (z 7→ Kz), K > 1.Then the developing map on Ai is given by

δγ |Ai= gi(z)hi/π

where gi : Ai → H is biholomorphic and gi(Kz) = K ′gi(z) for some K ′ > 0.Proof. By the denition of the projective grafting map Grγ : T (S) → P(S), the developingmap of Grγ Y normalized as above sends Ai to C∗ (possibly many-to-one) such that eachEuclidean geodesic of Ai parallel to its boundary maps to a ray in C∗, and these rays turnthrough a total angle of hi.

Since Ai is simply connected, there is a unique branch of log(δγ |Ai) whose image is the

strip z | 0 < Im(z) < hi. The associated branch of (δγ)hi/π is therefore a univalent mapon Ai with the required property, and the lemma follows.

Recall from 3.5 that the Hopf dierential ΦX(γ) of the collapsing map κ : X → prγ Xis a measurable quadratic dierential, holomorphic on A(γ), whose horizontal foliation isthe union of the foliations of the grafting annuli by Euclidean geodesics parallel to theirboundaries.

Lemma 4.12 allows us to estimate ξX(γ), which is the Schwarzian of the developingmap δγ of Grγ(prγ X):Corollary 4.13 (of Lemma 4.12). For γ and Ai as above, we have

ξX(γ)|Ai = 2(π2 − h2

i

h2i

)ΦX(γ)|Ai + S (gi)

where as before gi : Ai → H is a Riemann map for Ai ⊂ X, with Schwarzian S (gi) whichdescends to a quadratic dierential on Ai.Proof. By Lemma 4.12 and the composition rule for the Schwarzian derivative (2.5),

S (δγ)|Ai= g∗i S (z 7→ zhi/π) + S (gi(z)). (4.1)

A calculation yieldsS (z 7→ zα) =

1− α2

2dz2

z2,

which is a holomorphic quadratic dierential on H with closed trajectories that are semi-circles centered at 0. Since gi intertwines the action of γi on X with a dilation on H, thepullback to Ai descends to a holomorphic dierential on Ai with trajectories orthogonal to


the natural foliation by circles. Therefore this pullback is a positive multiple of −ΦX(γ);comparing the induced metrics yields

(hi/π)2g∗i (dz2/z2) = 4ΦX(γ)|Ai .

Substituting this into (4.1), we have

ξX(γ)|Ai = S (δγ)|Ai = 2(

1− (hi/π)2

(hi/π)2

)ΦX(γ)|Ai + S (gi).

Estimating the Schwarzian of the developing map when restricted to the grafting locusas in Corollary 4.13 will be sucient for the proof of Theorem 4.3 because, by Corollary4.11, these annuli cover a denite fraction of the surface.

4.6 Finite rays and Strebel dierentialsIn this section we assemble the results about grafting annuli (4.4) and the developing

map (4.5) to prove Theorem 4.3 (see 4.1).Proof of Theorem 4.3. For X ∈ T (S) and a rational lamination γ, we need to show that

‖φX(tγ)− 2ξX(tγ)‖1 = O(1).

By Lemma 4.11, for t 0 each grafting annulus in X properly contains a hyperbolicball Bε(x) of radius ε = ε(X, γ). Fix a strip Ai ⊂ X covering one of the grafting annuli Ai

in X, and a hyperbolic disk Bε ⊂ (Ai/γi). Then by Corollary 4.13,

ξX(tγ)|Ai = 2(π2 − (t hi)2

(t hi)2

)ΦX(tγ)|Ai + S(gi) (4.2)

where gi is a Riemann map for gi. Since minhi > 0, for t 0 the coecient of ΦX(tγ)|Aiis (−2 +O(t−2)), while ‖ΦX(tγ)‖1 = O(t2) by Corollary 3.14. Therefore

2(π2 − (thi)2

(t hi)2

)ΦX(tγ)|Ai = −2ΦX(tγ) +O(1).

By Nehari's theorem (see [1]), any univalent function f on a simply connected domainΩ in H satises

‖S(f)‖L∞(Ω) ≤32

Since gi is such a map,‖S(gi)‖L∞(Ai) = ‖S(gi)‖L∞(Ai)

≤ 32.

By the Schwarz lemma, the inclusion Bε(x) → Ai is a contraction for the hyperbolic metric,thus

‖S(gi)‖L∞(Bε(x)) < ‖S(gi)‖L∞(Ai) ≤32,


which leads to a bound on the L1 norm on a ball of hyperbolic radius ε/2:‖S(gi)‖L1(Bε/2(x)) ≤ C, (4.3)

where C does not depend on ε.By Theorem 3.15, the Hopf dierential ΦX(tγ) of the collapsing map is close to the

Strebel dierential 14φX(tγ), i.e.

‖4ΦX(tγ)− φX(tγ)‖L1(X) ≤ 16π|χ(S)|

and in particular,‖4ΦX(tγ)− φX(tγ)‖L1(Bε/2(x)) ≤ 16π|χ(S)| (4.4)

Combining (4.3) and (4.4) with Corollary 4.13, we have‖2ξX(tγ)− φX(tγ)‖L1(Bε/2(x)) ≤

‖2ξX(tγ)− 4ΦX(tγ)‖L1(Bε/2(x)) + ‖4ΦX(tγ)− φX(tγ)‖L1(Bε/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 (2ξX(tγ) − φX(tγ)) lies in the nite-dimensionalspace Q(X) of holomorphic quadratic dierentials, so there is a constant M(ε,X) such thatfor all x ∈ X and ψ ∈ Q(X),

‖ψ‖L1(Bε/2(x))

‖ψ‖L1(X)≤M(ε,X).

Therefore‖2ξX(tγ)− φX(tγ)‖L1(X) ≤M(ε/2, X)‖2ξX(tγ)− φX(tγ)‖L1(Bε/2(x)) ≤ C(ε,X).

Since ε depends only on γ and X, Theorem 4.3 follows.

Chapter 5

Comments and questions

5.1 Open questionsWe 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 ofQ(X). In comparison to B(X), the exotic Bers slices Bγ(X) remain somewhat mysterious.For example, the following basic questions remain open:

3 Is Bγ(X) connected?3 Is Bγ(X) bounded?3 Does the topology of Bγ(X) depend on X?

Even if Bγ(X) is disconnected, it has a distinguished connected component that containsthe Fuchsian center cγ . Thus we can rephrase the question about connectedness of Bγ(X)as:

3 Does every connected component of IntK(X) contain a Fuchsian center?Further discussion of the connectedness issue can be found in 5.2.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 4.3:3 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, dothe 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 pointsin the compactication P(S) = T (S) × ML (S). McMullen has shown that there existlimiting slices B(λ) ⊂ ∂QF that generalize the basepoint of a Bers slice to a projectivelamination [λ] ∈ ∂T (S). For nite laminations, these slices sit within ane algebraicsubvarieties of V (S) (level sets of trace functions). It is therefore natural to ask:

43

44 Chapter 5: Comments and questions

3 Does the limit Bers slice B(λ) lie in the image of a properly embedded complex vectorspace Q(λ) → V (S)?

and further,3 Is there a geometric interpretation for Q(λ) as a space of complex projective struc-

tures 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 theuse of distributional objects.

In the case of a punctured torus, it may be possible to use such a theory of generalizedprojective structures and the theory of holomorphic motions to address the self-similarity ofBers' boundary of Teichmüller space:Conjecture 5.1 (McMullen, [25]). The boundary of Bers' embedding of the Teichmüllerspace of a once-punctured torus is C1+α-conformally self-similar about boundary points cor-responding to laminations λ+ and λ− xed by a pseudo-anosov mapping class ψ.

5.2 Connectedness and computer experimentsIn this nal section we further address the question of connectedness of exotic Bers

slices Bγ(X). Specically, we provide a heuristic argument suggesting that these slicesare sometimes disconnected when X is a punctured torus, and display computer-generatedimages 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 projectivestructures on punctured tori. This free software package can be obtained on the world wideweb at http://bear.sourceforge.net/.

The boundary of the standard Bers slice B0(X) of a punctured torus is a closed cuspycurve, i.e. a Jordan curve with a dense set of inward-pointing cusps [27]. A Jordan curveν has an inward-pointing cusp at a point x if there is a cardioid with its cusp at x that liesentirely within ν (see Figure 5.1). Each of the cusps on ∂B0(X) corresponds to a geomet-rically nite holonomy group with an accidental parabolic. Within B0(X) the holonomyis quasifuchsian, and one of the two domains of discontinuity has xed quotient conformalstructure X; thus, the accidental parabolics arise from pinching simple closed curves on theother conformal boundary component.

There is a qualitatively similar conjectural picture of the boundary of an exotic Bersslice Bγ(X). When γ 6= 0, the quasifuchsian holonomy groups in the exotic Bers slice donot have a xed quotient conformal structure on either domain of discontinuity, and thusaccidental parabolics can appear on Bγ(X) corresponding to a pinched curve on either (orboth) of the ends of the quasifuchsian manifold. Furthermore, these cusps seem to denselypopulate two cuspy curves, one for each of the two types of accidental parabolics, whichgenerically cross one another in several places. The points of Bγ(X) are those which liebetween these two curves (in such a way that the cusps are inward-pointing).

Chapter 5: Comments and questions 45

Figure 5.1: Part of the boundary of Maskit's embedding of the Teichmüller space of apunctured 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, resultingin connected components resembling the island at left. As the basepoint X ∈ T (S) ischanged, these curves may cross acquire new crossings (center), which often results in amultitude of smaller islands (right).

As X ∈ T (S) is varied, the boundary of the standard Bers slice B0(X) moves via aholomorphic motion [27]. Similarly, the two cuspy curves forming the boundary of an exoticBers slice Bγ(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 setof connected components of Bγ(X). Thus a single component of Bγ(X) may pinch o toform two or more components of Bγ(X ′) for X ′ near X; in fact, the cuspy nature of theseboundary curves tends to create a multitude of small islands near a point of crossing, asshown in Figure 5.2, leading us to suspect that Bγ(X) may have inntely many connectedcomponents for generic X ∈ T (S). Computer-generated images of K(X) that seem todisplay this behavior are shown in Figure 5.3.

46 Chapter 5: Comments and questions

Figure 5.3: Computer-generated images of the discreteness locus K(X) that seem to showthe behavior depicted in Figure 5.2. In the upper image, indiscrete representations areindicated by varying shades of gray according to the amount of computation required todetect this indiscreteness. In both images, points colored black lie in K(X).

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