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RICE UNIVERSITY Handle crushing harmonic maps between surfaces by Andy C. Huang A Tupsrs Sue\¿Irrpt rN Penuel FUIpILLMENT oF THE RnqurneMENTS FoR THE Dpcnpp Doctor of Philosophy AppRovpo, Tunsrs Covvtlrrpp: Michael Wolf, Professor of Mathematics Robert M. Hardt, W.L. Moody Professor of Mathematics Béatrice M. R,ivière, Professor of Computational and Applied Mathematics, Noah G. Harding Chair HousroN, TnxRS FpsRuRRv, 2076 Abstract Handle crushing harmonic maps between surfaces by Andy C. Huang In this thesis, we construct polynomial growth harmonic maps from once-punctured Rie- mann surfaces of any finite genus to any even-sided, regular, ideal polygon in the hyper- bolic plane. We also establish their uniqueness within a class of maps which differ by exponentially decaying variations. Previously, harmonic maps from C (which are conformally once-punctured spheres) to H2 have been parameterized by holomorphic quadratic differentials on C ([WA94], [HTTW95], [ST02]). Our harmonic maps, mapping a genus g > 1 punctured surface to a k-sided polygon, correspond to meromorphic quadratic differentials with one pole of order (k + 2) at the puncture and (4g + k − 2) zeros (counting multiplicity). In this way, we can associate to these maps a holomorphic quadratic differential on the punctured Rie- mann surface domain. As an example, we explore a special case of our theorems: the unique harmonic map from a punctured square torus to an ideal square. We use the symmetries of the map to deduce the three possibilities for its Hopf differential. i Acknowledgments I owe so much to my parents, whose unconditional love and support without which I would not have gotten to where I am today. For everything they have provided me, and especially for encouraging me to be ever curious, I am thankful. I’d like thank my thesis advisor, Dr. Mike Wolf, for persuading me to “think like a harmonic map” when I needed inspiration, enduring many long discussions, and parsing my scribbles and drawings on his chalkboard. Thanks also to my thesis defense committee members Dr. Robert Hardt, for his kindness and mentorship, and Dr. Beatrice Riviere, who showed me an appreciation of numerical methods for thinking about PDEs, and the math department staff Marie Magee, Ligia Leismer, and Bonnie Hausman for pleasant conversations, administrative help, and encouragement. My academic siblings Qiongling Li and Jorge Acosta have been tirelessly enthusiastic and endlessly helpful. Another academic sibling, Zeno Huang, felt at times more like a second advisor and faithful supporter even though he was so far away. Without these people, my academic experience at Rice would have been incomplete. The graduate program has allowed me to experience so much and connect personally with so many. In no particular order, I thank Diego Vela, Timur Takhtaganov, Charles Puelz, and too many ultimate frisbee, climbing, cycling, and Rice Bikes shop friends to name here, for sharing their time with me in grad school, without whom I’d probably finish sooner but in poorer mental and physical health. ii Contents Abstract . .i Acknowledgments . ii List of Figures . vii 1 Introduction 1 1.1 Introduction and Main results . .1 1.1.1 Background . .5 1.2 Overview of thesis . 12 2 Mathematical preliminaries 13 2.1 Riemannian geometry concepts . 13 2.1.1 Tensors . 14 2.1.2 Connections . 17 2.1.3 Metrics on vector bundles with connections . 19 2.2 Harmonic map formulae . 20 2.2.1 The Dirichlet energy . 20 2.2.2 Existence and uniqueness . 23 2.2.3 Examples of harmonic maps . 25 2.2.4 Properties of harmonic maps . 26 2.3 Harmonic maps from surfaces . 27 iii 2.3.1 Regularity . 28 2.3.2 Specialization of harmonic map formulae . 29 2.3.3 Conformal invariance of energy . 30 2.3.4 Hopf differential . 31 2.3.5 Courant-Lebesgue Lemma . 34 2.3.6 Bochner formulae and consequences . 36 2.4 Some recent results on harmonic maps . 37 2.4.1 Harmonic maps C ! H2 ......................... 37 2.4.2 Wood’s analysis of images of harmonic maps . 38 2.4.3 Dumas-Wolf’s uniqueness of orientation-preserving harmonic maps through Hopf differentials . 39 2.4.4 Cheng’s Lemma . 40 2.4.5 Choe’s iso-energy inequality . 41 3 Handle-crushing harmonic maps 42 3.0.6 Organization of section . 42 3.1 Construction: Proof of Theorem 1.1.1 . 45 3.1.1 Set-up for construction . 46 3.1.2 Conditions for sub-convergence . 50 3.1.3 A comparison map for hs away from the handles . 55 3.1.4 Putting it all together . 59 3.2 Uniqueness: Proof of Theorem 1.1.2 . 61 3.3 Example: punctured square torus squashed onto ideal square . 63 3.4 The toy problem and The Energy Estimate . 66 3.4.1 Parachute maps . 69 3.4.2 Geometry of images . 70 iv 3.4.3 Hopf differentials of parachute maps . 71 3.4.4 Bounded Core Lemma . 83 3.4.5 The Energy Estimate: Parachute maps tied to Scherk map . 86 4 Future directions 90 4.1 The whole picture: handle crushers between compact surfaces . 91 4.1.1 Limit of compact maps to a diffeomorphism and a handle crushing model . 91 4.1.2 Bridging harmonic diffeomorphisms with handle crushers . 92 4.2 Minimal surfaces in H2 × R ............................ 94 4.3 Deforming handle crushers . 95 4.3.1 Deformations changing the image polygon . 95 4.3.2 Deformations preserving the image set . 95 4.4 Combinatorial data from Hopf differential . 96 4.5 Germs of harmonic maps . 98 4.5.1 Space of harmonic polynomials . 101 5 Tangential questions 103 5.1 Surface group representation domination by Fuchsian representations . 103 5.2 Discrete, non-faithful representations and branched hyperbolic structures . 104 A Riemannian geometry computations 105 A.1 Concerning Hopf differentials . 105 A.2 Geometry of harmonic mappings via their Hopf differentials . 107 B Holomorphic quadratic differential structures 109 B.1 Measured foliations . 109 B.2 Harmonic maps to R-trees . 111 v C Minimal suspensions in H2 × R 113 vi List of Figures 1.1 The possible Hopf differential configurations for Corollary 1.1.3 . .4 1.2 Approximate minimal suspension of map from Corollary 1.1.5.∗ .......6 h 1.3 Arrangement (a) horizontal and vertical foliations for F on S1 .......6 3.1 Identification space model for Sg ......................... 47 3.2 The Scherk map w : C ! H2 with Fw = −z2dz2................. 48 3.3 Example Pk for k = 4, 6, and 8. 49 3.4 Lines of symmetries for h : S1 !P4 from Corollary 1.1.3. 64 h 3.5 Possible locations for zeroes of F of h : S1 !P4 handle crusher. 64 4.1 Harmonic self map of unit disk exhibiting a cusp. 100 B.1 Foliations for Fu = z2dz2 on C ........................... 110 C.1 Suspension of u : C ! H2 with Fu(z) = z2dz2.y ................ 115 ∗Produced using Ken Brakke’s Surface Evolver program. yProduced using Ken Brakke’s Surface Evolver program. vii Chapter 1 Introduction 1.1 Introduction and Main results This thesis begins an investigation of the question: what is the shape of a harmonic map between surfaces from higher topological complexity to lower topological complexity? In order to focus this question, we consider the special class of harmonic maps between com- pact hyperbolic surfaces of different genera. This family of harmonic maps has the ap- pealing property that they are in bijection with homomorphisms between fundamental groups. ¥ For, any smooth map u 2 C (Sg, Sh) between compact hyperbolic surfaces (Sg, s) and (Sh, r) of any genera induces a homomorphism u∗ : p1(Sg) ! p1(Sh) between their fundamental groups. Conversely, since compact surfaces are K(p1, 1) spaces, any homo- morphism m between their fundamental groups can be induced by a continuous map f between them, i.e., f∗ ≡ m. Negative curvature of the target metric r ensures a unique harmonic representative exists in the homotopy class of that continuous map, if the map is non-constant. In the case the map is constant, the trivial homomorphism u∗ p1(Sg) = e is induced. We state this correspondence as ¥ 1−1 f[u] ju 2 C (Sg, Sh), harmonic, non-constantg ! u∗ 2 Hom p1(Sg), p1(Sh) n feg . 1 ∗ Furthermore, any smooth map u defines a symmetric 2-tensor u (r) on Sg by pulling back the target metric. We can use the complex structure induced by s on Sg to extract the (2, 0)-component of this pullback metric, called the Hopf differential of u, ( ) Fu := (u∗(r)) 2,0 . Hopf observed that the harmonicity of u implies holomorphicity of Fu with respect to this complex structure [Hop51]. Thus, there is a map ¥ u u 2 C (Sg, Sh) −! F 2 QD(Sg), where QD(Sg) is the vector space of holomorphic quadratic differentials on Sg. In this way, the complex analytic object Fu captures some of the differential topology of the harmonic map u and equivalently the algebraic information of the homomorphism u∗ between their fundamental groups p1(Sg) and p1(Sh). It is our present aim to create a local model to study the handle-crushing harmonic maps from higher genus to lower genus surfaces, and then characterize their Hopf dif- ferentials. Such handle crushing maps can be locally described as mapping a surface of positive genus and with boundary onto a disk, non-injectively taking interior to interior and monotonically taking boundary to boundary. These induce the trivial homomorphism on fundamental groups of punctured surfaces. We produce a model of handle crushing harmonic maps in the following: Theorem 1.1.1. (Existence) For any once punctured genus g ≥ 0 surface Sg and for any regular 2 2 ideal k-sided polygon Pk in H for k > 2 even, there exists a harmonic map h : Sg ! H whose image h(Sg) is the interior of Pk and whose closure h(Sg) = Pk.
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