Abstract On geometry along grafting rays in Teichm¨uller space by Renee A. Laverdiere In this work, we investigate the mid-range behavior of geometry along a grafting ray in Teichm¨uller space. The main technique is to describe the hyperbolic metric σt at a point along the grafting ray in terms of a conformal factor gt times the Thurston (grafted) metric and study solutions to the linearized Liouville equation. We give a formula that describes, at any point on a grafting ray, the change in length of a sum of distinguished curves in terms of the hyperbolic geometry at the point. We then make precise the idea that once the length of the grafting locus is small, local behavior of the geometry for grafting on a general manifold is like that of grafting on a cylinder. Finally, we prove that the sum of lengths of is eventually monotone decreasing along grafting rays. Acknowledgments First and foremost, this thesis would not have been possible without the support, encouragement, and patience of my advisor, Mike Wolf. With his guidance, I have learned the rhythm of dancing between seeing the forest and seeing the trees. Thank you. In my daily work, I have been fortunate to be surrounded by others both gener- ous with their time and with enthusiasm for my work. In particular, I would like to thank Casey Douglas, Andy Huang, Frank Jones, Evelyn Lamb, and Qiongling Li for listening, nodding, smiling, and asking insightful questions. I am grateful to Bernard and Carolyn Aresu, Matteo Pasquali, and Marie-Nathalie Contou-Carrere for helping me establish a sense of community at Rice that bouyed me through darker days. I owe many smiles, meals, and the belief that there is a light at the end of the tunnel to Anna Grassini and John McDevitt. I will always be grateful to my parents for fostering an appreciation of mathe- matics in our home and for encouraging me to pursue my dreams. I am especially thankful for the gift of Father Nelligan, the canine love of my life. Nelligan deserves his own thanks, for reminding me on a daily basis that a dose of love, a few treats, and a breath of fresh air are the foundation for happiness. I owe my deepest gratitude to my husband, David. The words “thank you” are iv insufficient for managing life when my mind has been otherwise occupied, coping with uncapped pens and scattered stacks of loose paper, and (most importantly) believing in me when I did not believe in myself. Dedication For David. Without you, and the promise that you kept, this dream may have remained precisely that - a dream. Contents Abstract..................................... ii Acknowledgments . iii Dedication.................................... v List of Figures . 1 1 Introduction 2 2 Background 5 2.1 Klein’s perspective . 5 2.2 Fundamentals: Teichm¨uller space and complex projective structures . 8 2.2.1 (X, G)-structures ......................... 8 2.2.2 Complex and complex projective structures . 9 2.3 Parameterization of (S)........................12 P 2.3.1 Analytic parameterization . 12 2.3.2 Grafting parameterization . 16 3 Grafting applications 27 3.1 Polar Coordinates for Teichm¨uller space . 27 vi vii 3.2 Recent developments . 30 4 A surface geometry approach to grafting 36 4.1 Introduction . 36 4.2 Introduction to technique . 38 4.2.1 Representing the metrics along the grafting ray: two approaches. 39 4.2.2 Length............................... 44 4.2.3 Prescribed curvature equation. 46 4.2.4 Applying Formula 4.10 . 49 4.3 A geometric formula . 56 4.3.1 First variation of length . 57 4.3.2 Variational vector field . 61 4.3.3 Relating the variational formula to the complement of the ↵-collar 63 4.3.4 A general formula . 66 4.4 Observations . 70 4.5 An application . 72 4.5.1 Positive terms . 73 4.5.2 Small terms . 74 4.5.3 Summary . 87 List of Figures 2.1 Triangle bounded by circular arcs with angles ↵,β,⌫.......... 6 2.2 Polar attachment on the left; lateral attachment on the right. 7 2.3 Inserting a Euclidean annulus in the grafting construction. 17 2.4 The region A˜t ............................... 18 2.5 Locally, a geodesic lamination can look like a cantor set on the surface. 19 2.6 Grafting along a weighted multicurve. 21 2.7 Three possibilities for C(@ Dz)..................... 23 1 2.8 The Thurston metric is flat on the intersection D00. This picture is precisely the image for grafting. 24 2.9 Bending the hyperbolic surface CH(@D) along `............ 25 3.1 Grafting by t + s along the top line versus iterated grafting. 28 4.1 The surface with cylinder of length t + t0 (bottom) di↵ers from that of length t0 (top) by a linear stretch along the cylinder. 40 4.2 The boundary curves, ↵±, and the geodesic as imagined on the grafted (top image) and hyperbolic (bottom image) surfaces. 54 4.3 ↵+ is a natural boundary for comp+................... 64 viii 1 4.4 If A = 0 then the pre-image of the curve ⇠ = A must intersect γ⇤... 78 6 0 4.5 The ↵-rectangle is the smallest Euclidean sub-cylinder containing the ↵-collar. .................................. 82 4.6 The skinny tubes T↵± bound the wandering of the ⇠=constant curves. 85 Chapter 1 Introduction In this paper, we examine a natural geometric deformation of hyperbolic structures, grafting.LetS be a compact, orientable surface of genus g 2. Then a complex ≥ structure X on S is a maximal atlas of charts from S to C with biholomorphic transi- tion functions. A complex projective structure (or CP 1-structure) Z on S is a maximal atlas of charts from open subsets of S to CP 1 that are restrictions of M¨obius maps. Of primary interest is the Teichm¨uller space (S) and the space of CP 1-structures T (S). These are isotopy classes of complex and CP 1-structures, respectively. In par- P ticular, because every M¨obius map is holomorphic, every CP 1-structure is in fact a complex structure, and this underlying structure makes S a Riemann surface. This defines the forgetful projection map ⇡ : (S) (S). P !T There are two parameterizations of (S). The first, essentially complex-analytic P in nature, associates to every Z (S) a holomorophic quadratic di↵erential via the 2P Schwarzian derivative of the developing map. The second, due to Thurston, shows 2 3 that (S) is homeomorphic to (S) (S), where (S) is the space of measured P ML ⇥P ML geodesic laminations. This homeomorphism Gr : (S) (S) (S) is called ML ⇥T !P the projective grafting map. The composition gr = ⇡ Gr : (S)x (S) (S) ◦ ML T !T is called the conformal grafting map. The simplest picture of conformal grafting is as follows: choose a hyperbolic metric X (S) and γ a simple closed X-geodesic. 2T t t Then the grafting of X by tγ is obtained by replacing γ by the flat cylinder [ − , ] γ. 2 2 ⇥ This gives a new conformal structure, whic we denote by grtγX. Via uniformization, there is a new hyperbolic metric in the conformal class of gr X. Fixing γ (S), tγ 2ML we call the set of conformal structures grtγX : t R a grafting ray. We are inter- { 2 } ested in the evolution of geometric invariants of the hyperbolic surfaces along grafting rays in Teichm¨uller space. In first chapter, we develop the background material and sketch Thurston’s proof that the projective grafting map Gr : (S) (S) (S) is a homeomorphism. ML ⇥T !P The second chapter is a review of current research involving grafting. We begin the third chapter by introducing a technique for studying geometry at any point along a grafting ray generated by a simple closed geodesic, assuming that the geodesic in the homotopy class of the grafting locus does not escape the grafting cylinder. In section 3.2, we derive a formula that expresses the change in two geometric values in terms of the geometry of the hyperbolic surface in the conformal class of the grafted surface. More specifically, for any curve δ on the surface, denote by `(t, δ) the length of the δ with respect to the hyperbolic metric in the conformal class of gr(t+t0)γX. Then the formula of section 3.2 can be paraphrased to say the following: 4 + Theorem 4.1. Let γ be a separating curve on S. Choose t0 R . Let σ0 be 2 the hyperbolic metric in the conformal class of grt0γX. Then if γ0 is the hyperbolic + geodesic in the homotopy class of γ, and ↵0 and ↵0− are curves that are parallel to γ0 with respect to σ0, then + `(0,↵0 ) d + `(0,↵0−) d d 2 `(t, ↵ ) 2 `(t, ↵−)+4 `(t, γ ) − ` dt 0 − ` dt 0 dt t t=0 t=0 t=0 is given in terms of t0, the Euclidean length of the grafting cylinder, the conformal 2 + factor g, where σ = g(z) dz , distance of ↵ and ↵− from γ , and the average 0 | | 0 0 0 angle between ⇠=constant, a curve of constant σ0 (hyperbolic) geodesic curvature, and x=constant, a curve of constant Euclidean geodesic curvature. In section 3.3, we observe that if the surface S is actually a cylinder and γ is the core geodesic for a hyperbolic metric X on S, then the geometry along the grafting ray is particularly simple to understand. We restate the formula given in section 3.2 in such a way that shows the defect of the grafting operation on the general surface from being like that of grafting on a cylinder.