Handle Crushing Harmonic Maps Between Surfaces

RICE UNIVERSITY

Handle crushing harmonic maps between surfaces

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

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 hyperbolic 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 Φ on Σ1 ...... 6

3.1 Identiﬁcation space model for Σg ...... 47

3.2 The Scherk map w : C → H2 with Φw = −z2dz2...... 48

3.3 Example Pk for k = 4, 6, and 8...... 49

3.4 Lines of symmetries for h : Σ1 → P4 from Corollary 1.1.3...... 64

h 3.5 Possible locations for zeroes of Φ of h : Σ1 → P4 handle crusher...... 64

4.1 Harmonic self map of unit disk exhibiting a cusp...... 100

B.1 Foliations for Φu = z2dz2 on C ...... 110

C.1 Suspension of u : C → H2 with Φu(z) = z2dz2.† ...... 115

∗Produced using Ken Brakke’s Surface Evolver program. †Produced 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 compact 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 ∈ C (Σg, Σh) between compact hyperbolic surfaces (Σg, σ)

and (Σh, ρ) of any genera induces a homomorphism u∗ : π1(Σg) → π1(Σh) between their

fundamental groups. Conversely, since compact surfaces are K(π1, 1) spaces, any homo-

morphism µ between their fundamental groups can be induced by a continuous map f

between them, i.e., f∗ ≡ µ. Negative curvature of the target metric ρ 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∗ π1(Σg) = e is induced. We state this correspondence as

∞ 1−1 {[u] |u ∈ C (Σg, Σh), harmonic, non-constant} ←→ u∗ ∈ Hom π1(Σg), π1(Σh) \{e} .

1 ∗ Furthermore, any smooth map u deﬁnes a symmetric 2-tensor u (ρ) on Σg by pulling

back the target metric. We can use the complex structure induced by σ on Σg to extract the

(2, 0)-component of this pullback metric, called the Hopf differential of u,

( ) Φu := (u∗(ρ)) 2,0 .

Hopf observed that the harmonicity of u implies holomorphicity of Φu with respect to this

complex structure [Hop51]. Thus, there is a map

∞ u u ∈ C (Σg, Σh) −→ Φ ∈ QD(Σg),

where QD(Σg) is the vector space of holomorphic quadratic differentials on Σg.

In this way, the complex analytic object Φu captures some of the differential topology

of the harmonic map u and equivalently the algebraic information of the homomorphism

u∗ between their fundamental groups π1(Σg) and π1(Σh).

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 Σg and for any regular

2 2 ideal k-sided polygon Pk in H for k > 2 even, there exists a harmonic map h : Σg → H whose

image h(Σg) is the interior of Pk and whose closure h(Σg) = Pk. For this harmonic map, the Hopf differential has (4g + k − 2) zeros and one pole of order (k + 2) at the puncture.

Here, we say Pk is regular if it has a dihedral isometry group Dk of order 2k.

2 Our proof of Theorem 1.1.1 is by approximation on compacta of a punctured Riemann surface. The evenness of k ensures a convenient reﬂectional symmetry. To obtain convergence of the approximating sequence, we introduce and analyze parachute maps, each of which is a harmonic map from a compact cylinder and each of which solves a toy harmonic mapping problem constrained by a partially free boundary condition. The regularity of Pk

gives us rotational symmetries to help us in their analysis through their Hopf differentials.

The key techniques in our proof lie in the comparison of the energy of a harmonic

map on a compactum of the punctured surface to the energy of a parachute map on the

appropriate cylinder. We exploit the non-injectivity of the parachute map to bound the

image distance in terms of the pointwise norm of its Hopf differential. We bound the Hopf

differential, in turn, through studying its Laurent series.

As a by-product of this comparison to the parachute maps, we are also able to show

that the energy densities for the harmonic maps from Theorem 1.1.1 have polynomial

growth (whose degrees determine the ﬁnite order pole and the number of sides of Pk).

Furthermore, recognizing the parabolic nature (in the complex function theoretic sense) of

the punctured Riemann surface domains, we also address uniqueness for these harmonic

maps:

2 Theorem 1.1.2. (Uniqueness) Suppose h, v : Σg → H are two harmonic maps. Denote their

pointwise distance function by d(z) := distH2 (h(z), v(z)). If we have

p cosh ◦ d − 1 ∈ L (Σg) for some some p ∈ (1, +∞], then d(z) ≡ 0, i.e., the maps v and h must agree pointwise.

As an example application, we specialize Theorems 1.1.1 and 1.1.2 for a square torus mapped to an ideal square. From the uniqueness provided by Theorem 1.1.2 and the large amount of symmetry, we are able to deduce the possibilities for the Hopf differential of the resulting harmonic map. In particular, we show that:

3 02 02 04 06

02 ∞6 01 01 ∞6 ∞6

(a) (b) (c)

Figure 1.1: The possible Hopf differential conﬁgurations for Corollary 1.1.3

Corollary 1.1.3. There exists a harmonic map h from the punctured square torus Σ1 to the ideal

2 h square P4 in H . Its Hopf differential Φ has exactly one pole of order 6 and has one of the the

following three arrangements of zeros, as depicted in Figure 1.1:

(a) three zeros of order 2,

(b) or one zero of order 4 and two of order 1,

In arrangement (a), the Hopf differential is a square of a holomorphic one-form. Furthermore, two

of these double zeroes lie an a subset where J > 0 and orientation is preserved, while the third double zero lies on a subset where J < 0 and the orientation is reversed. In the arrangements (b) and (c), the Hopf differential is not the square of a one-form.

By the square punctured torus, we mean the conformally unique punctured torus with a dihedral D4 symmetry group of order 8 which ﬁxes the puncture. The Scherk map is the harmonic map w : C → H2 with Hopf differential Φw(z) = z2dz2 and whose image lies in an ideal square. It is described in more detail in Theorem 2.4.1. For visualization, it is helpful to realize it as the projection of the Scherk type minimal surface in H2 × R, discovered by Nelli and Rosenberg, onto H2 (c.f. section 4 of [NR02]). We approximate it numerically in Figure C.1 in Appendix C.

4 From only the Existence and Uniqueness Theorems 1.1.1 and 1.1.2, we will not be able to conclude precisely which of these three possibilities is the unique Hopf differential realized. Nonetheless, we conjecture:

Conjecture 1.1.4. The Hopf differential Φh of Corollary 1.1.3 has divisor data described by ar-

rangement (a).

Observe that this conﬁguration has a pole of order 6 at the puncture, ﬁxed by the D4

symmetry of the square torus, and three zeroes of order 2, each on a line of reﬂectional

symmetry. Thus, we can express Φh in terms of the Weierstrass ℘ and ℘0 functions, and in

fact as a square of a holomorphic 1-form. Then, we have:

h 2 Corollary 1.1.5. If the Hopf differential Φ of the map h : Σ1 → H has divisor data as in

conﬁguration (a) described in Corollary 1.1.3, then there exists a harmonic function f : Σ1 → R for

h f 2 which Φ = −Φ . In particular, this provides a minimal suspension h˜ := (h, f ) : Σ1 → H × R

2 with h˜(Σ1) a genus one minimal surface in H × R.

Its minimal suspension (see Appendix C for deﬁnition) is numerically approximated

in Figure 1.2. A numerical depiction of the horizontal and vertical foliations is produced

in Figure 1.3. This approximation of the minimal suspension provides numerical evidence

for conﬁguration (a).

1.1.1 Background

Let us take a moment to describe the development of the theory for harmonic maps from

Riemann surfaces, so that we can frame the peculiarities of our problem and also put our

theorems in context. We will precisely state the theorems discussed below in Chapter 2.

5 Figure 1.2: Approximate minimal suspension of map from Corollary 1.1.5.∗

(a) FH (b) FV

h Figure 1.3: Arrangement (a) horizontal and vertical foliations for Φ on Σ1

Why study harmonic maps?

The “best” map from an interval I into a manifold N is arguably an arc-length parame-

terized geodesic γ : I → N, since it locally realizes the shortest path between two points

on a Riemannian manifold. In fact, for manifolds N of non-positive curvature, a geodesic segment globally realizes the unique shortest path between its endpoints. Furthermore, the geometry of a geodesic is simple to describe: it is auto-parallel, which is to say its tangent

6 vector is covariantly parallel along itself,

∇γ˙ γ˙ = 0.

Reinterpreting this equation reveals that geodesics can be variationally characterized. The auto-parallel condition coincides with the Euler-Lagrange equation for the Dirichlet energy functional E(I, γ) on the space of C1 maps I → N,

Z 1 2 E(I, γ) = ||γ˙ (s)||γ∗(TN)ds. 2 I

Thus, the best maps from one-dimensional domains, the parameterized geodesics, are exactly the minimizers/critical points of the Dirichlet energy functional.

This leads us to consider the direct generalization as a candidate for the best map from a surface into a target manifold. We call a critical point for the Dirichlet energy functional for a map from a surface (with compactly supported variations, when the surface has boundary) a harmonic map. Note that an energy minimizer among homotopic maps from surfaces is not guaranteed to exist, in contrast with the guaranteed existence of a minimizer for energy among maps from intervals into complete Riemannian manifolds.

It turns out that conformal changes of the domain metric leave the energy invariant

- analogous to geodesic paths remaining geodesic under reparameterization. Harmonic maps from closed, compact surfaces also enjoy many existence and uniqueness properties.

For example, when the target manifold is non-positively curved, any continuous map can be homotopically deformed into a harmonic map [ES64] and furthermore, that harmonic map is unique in its homotopy class up to geodesic translation [Har67].

Applications drive the study of harmonic maps from surfaces. When we require a candidate continuous map from a surface into a target manifold, it is prudent to choose a harmonic representative when it is unique in its homotopy class of continuous maps.

Harmonic maps from surfaces also enjoy many regularity properties, so that existence is established in many settings [ES64], [SU82], [Lab91].

7 Surface harmonic mapping history

Harmonic maps between compact Riemann surfaces of the same topological complexity have been well studied. Since compact surfaces are K(G, 1) spaces, homomorphisms between their fundamental groups are in bijection with homotopy classes of smooth maps.

Furthermore, there exists a unique harmonic representative in each homotopy class of smooth maps by the existence and uniqueness theorems for harmonic maps from [ES64] and [JK79]. These harmonic maps are particularly well-behaved - for example, any such harmonic map which is isotopic to the identity is in fact a diffeomorphism [SY78], [Sam78].

Harmonic maps between surfaces of the same genus have also found applications in

Teichmuller¨ theory. Starting with a harmonic diffeomorphism between Riemann surfaces, smoothly deforming the target Riemann surface structure induces a smooth variation of the harmonic map [EL81]. Wolf showed that, under these deformations, the associated

Hopf differential on the domain Riemann surface is uniquely determined [Wol89] and, furthermore, that the Teichmuller¨ space Tg of a genus g compact Riemann surface is home- omorphic to the vector space QD(Σg) of holomorphic quadratic differentials on the ﬁxed

Riemann surface domain. Preceding this, Sampson observed that the mapping of the target hyperbolic metric to its corresponding Hopf differential is injective [Sam78], and Hitchin constructed hyperbolic metrics for each holomorphic quadratic differential [Hit87].

For harmonic maps between surfaces of different genera, less has been said. At a local scale, Wood gave a classiﬁcation of admissible folds and cusps for harmonic maps between compact surfaces through a description of the admissible germs of a harmonic map [Woo77]. On a global scale, some examples have been obtained. Non-holomorphic harmonic maps from positive genus surfaces to the sphere CP1 have been produced by

cutting and gluing methods [Lem78]. A qualitative example of a harmonic map from a

genus two surface to the torus C/Z2 was described by integrating a pair of Abelian dif-

ferentials with appropriate periods [Woo74]. Less is known for harmonic maps between

8 compact negatively curved Riemann surfaces of different genera.

Somewhat tangentially, harmonic maps from C to H2 have recently received a lot of attention [Wan92], [WA94], [HTTW95], [ST02]. Sparked by Schoen’s conjecture on the nonexistence of a harmonic diffeomorphism C → H2 [Sch93], considerable effort has been put toward studying the Bochner equation on C, a semi-linear elliptic partial differential equation necessarily satisﬁed by the holomorphic energy density of a harmonic map from

C. It is remarkable that furnishing a solution to this equation satisfying some geometric constraints provides sufﬁcient data to produce a harmonic diffeomorphismof C into an ideal polygon of H2, called the Scherk maps, through the methods of [WA94], [HTTW95], and [ST02].

This produces for us many examples of harmonic diffeomorphisms from a simply connected domain onto negatively curved surfaces. In a sense, these maps can be seen as local models for harmonic maps from a punctured sphere to a negatively curved target which are only diffemorphisms. Our task is to extend these models to produce handle collapses, harmonically mapping a punctured Riemann surface to the hyperbolic plane.

This paper establishes the existence and uniqueness of harmonic maps from once- punctured Riemann surfaces with arbitrary but ﬁnite topology to the interior of a regular ideal polygon in the hyperbolic plane. The existence of these maps can be viewed both as an extension and application of the harmonic mapping theory of C → H2. On the one hand, we generalize the punctured sphere domain C to a punctured Riemann surface domain Σ with positive genus. On the other hand, the construction will involve a local blow-up argument; this blow-up forces us to analyze the behavior of these C → H2

Scherk maps.

9 Remark on approach

As mentioned, a common approach to exhibiting the existence of a harmonic diffeomorphism from a planar domain is through constructing solutions to the Bochner equation.

It is worth noting that the Bochner equation on a punctured Riemann surface has many solutions. The development of these solutions to produce folding, non-injective harmonic maps is not straight-forward, since it is not clear where and how to induce the folding.

For a concrete example of this phenomenon and difﬁculty, it is enough to compare two harmonic maps whose holomorphic energy densities solve the same Bochner equation: Fix a Riemann surface domain Σg. Choose any harmonic map v : Σg → Σh which

v is non-injective, and denote its Hopf differential by Φ . Since QD(Σg) ≈ Tg, there exists

0 a Riemann surface Σg also of genus g and for which there exists a harmonic diffeomor-

0 u v phism u : Σg → Σg with Hopf differential Φ ≡ Φ . Since the Hopf differentials agree,

their Bochner equations are the same, so we cannot foresee the folding behavior from the

information of the holomorphic energy densities (the solutions to the Bochner equation)

alone.

So, we abandon the approach of solving for and developing a solution of the Bochner

equation, and choose instead the naive approach of proving existence by compact exhaus-

tion. The main difﬁculties which arise come from pinching the energy of the harmonic

maps from above and below on each compact subset. Our energy estimate of the handle

crushing harmonic map on a compactum revolves around comparing two other harmonic

maps to each other - the Scherk maps and the parachute maps (introduced in §3.4).

Heuristically, we describe the reasoning of our approach as follows: We remove a disk

from the Scherk map (which is to say: an open disk of C and its image under the Scherk

map), and deform this mapping of an annulus to the unique energy minimizer in its ho-

motopy class with the same ideal boundary conditions (the parachute map). We show that

the boundary of the removed disk does not need to move very far relative to its image

10 under the Scherk map. This suggests that, if we replaced the disk with a compact surface with non-trivial topology, the map would be nearly harmonic.

Remark on technique

This paper presents at least four methods for dealing with proper non-injective harmonic maps from non-compact surfaces:

1. To obtain a lower bound on the energy of a harmonic map, we observe that a partially-

free boundary value harmonic mapping problem solution has less energy than its

completely Dirichlet boundary value harmonic mapping problem counterpart (see

the Doubling Lemma 3.1.5).

2. To estimate the length of the image of the free boundary for harmonic maps solving

a partially-free boundary value problem, we relate the pointwise Hopf differential

norm on the free boundary to the energy density on the free boundary (see Proposi-

tion 3.4.6).

In our proof, we need to show that a sequence of harmonic mappings from a sequence of nested compact domains to a regular ideal polygon of H2 (solving a sequence of Dirich- let boundary value problems) have locally bounded Hopf differentials. We achieve this through an analysis of the Laurent series expansions for their sequence of Hopf differentials (see Proposition 3.4.2):

3. We bound the inﬁnite tails of the Laurent series (in norm) by applying the Cauchy

integral formula along the prescribed Dirichlet boundaries. This uniformly bounds

all but ﬁnitely many coefﬁcients, so that divergence of the Hopf differentials could

only occur by divergence of these central coefﬁcients.

4. We control the central terms of the Laurent series using a blow-up argument on large

disks in the Hopf differential norm.

11 Using these methods, we are able to compare the energy of a harmonic map on a compactum of the punctured Riemann surface to the difference between energy of two different harmonic maps, the Scherk map and the parachute map, on an annular domain. We are able to bound this difference because the energy of the parachute map can be estimated in terms of its energy density on the free boundary, which in turn we are able to bound by showing that the Laurent series for its Hopf differential is bounded.

1.2 Overview of thesis

The main contributions of this thesis are contained in Chapter 3. Before proceeding to the proofs of Theorems 1.1.1 and 1.1.2, we review some mathematical preliminaries in Chapter

2. This preliminary material is deﬁnitional or classical, but sets our notation. It ends with a collection of recent results on harmonic maps, which will be used throughout Chapter 3.

Chapter 3 contains the proofs of Theorems 1.1.1 and 1.1.2.

Chapter 4 contains a discussion of future directions, where handle crushing maps can be applied, and where we can understand these maps better.

Chapter 5 describes some tangential questions related to handle crushing maps.

12 Chapter 2

Mathematical preliminaries

In this chapter, we review the deﬁnitions of objects that will be encountered throughout this thesis. After recalling some concepts from Riemannian geometry, we will be in a position to deﬁne harmonic maps and provide some examples of harmonic maps. We then discuss formulae and quantities associated to a harmonic map in order to give a sense of their qualitative behavior. Next, we specialize our treatment of harmonic maps to those maps which have two-dimensional domain and range. This chapter concludes with more recent results on harmonic maps, all of which will be used in the proof of Theorem 1.1.1.

2.1 Riemannian geometry concepts

We begin with a brief review of some Riemannian geometry concepts which will be uti- lized throughout our discussion. For the purposes of this chapter, (Mm, σ) and (Nn, ρ) are smooth Riemannian manifolds of dimensions m and n, respectively. When we discuss maps between manifolds, we will treat M as the domain and N as the range. As necessary, we will use local coordinates x1,..., xi,..., xm on M with Latin placeholder indices and local coordinates y1,..., yα,..., yn on N with Greek placeholder indices. These are not necessarily geodesic normal coordinates, although we will indicate when they should

13 be chosen so. We adopt the Einstein summation convention, so it is understood that an expression with the same index appearing both in the subscript and the superscript indi- cates a summation of the expression ranging over all possible values for that index. For example, ∂ m ∂ ∂ n ∂ i ≡ i α ≡ i X i ∑ X i and Y α ∑ Y α . ∂x i=1 ∂x ∂y α=1 ∂y

2.1.1 Tensors

For any smooth Riemannian manifold M, there are two naturally associated vector bun-

dles: the tangent bundle TM and the cotangent bundle T∗ M of M. The tangent bundle is a

vector bundle over M with ﬁbers isomorphic to Rm. At a point p ∈ M, a basis for the tangent space Tp M at p is given by the m derivations

∂ ∂ 1 ,..., m ∂x p ∂x p

∗ and a basis for the cotangent space Tp M at p is given by the m differential forms

1 m dx |p,..., dx |p.

To avoid notational clutter, we will often abbreviate the notation for tangent vectors when there is no risk for confusion, so that the following symbols are interchangeable:

∂ ∂ ≡ ∂ i ≡ . i x ∂xi

There are natural projection maps π : TM → M and π : T∗ M → M which are smooth,

and make TM and T∗ M vector bundles over M.

A smooth section σ of a vector bundle π : E → M is a smooth map

σ : M → E

∞ which satisﬁes σ ◦ π = idM. We denote the C vector space of smooth sections of E by

Γ(E). We call elements of Γ(TM) vector ﬁelds and elements of Γ(T∗ M) differential forms.

14 From the tangent and cotangent bundles, we can form their tensor products. We re- serve special notation for the space of smooth sections of E when E is a tensor product of

∗ k the tangent and cotangent spaces TM and T M of M.A mixed tensor of type l is a section h ⊗ ⊗ i∗ λ of the vector bundle (TM) k ⊗ (T∗ M) l . In other words, it is a multi-linear map

⊗ ⊗ λ : Γ (TM) k ⊗ (T∗ M) l → R.

It evaluates on k vector ﬁelds and l forms.

0 In particular, tangent vectors are mixed tensors of the type 1 and differential forms

1 k are mixed tensors of the type 0 . In general, we call mixed tensors of type 0 a k-tensor

0 and mixed tensors of type l an l-distribution.

A differential k-form is a section ω of ∧kT∗ M which is alternating in its components. In other words, it is a multi-linear map

ω : Γ(∧kTM) → R on the oriented k-planes of TM. In this way, we think of it as dual to the k-distributions.

The metric σ on M is a smooth section of the bundle T∗ M ⊗ T∗ M which is positive deﬁnite, bilinear, and symmetric. It is thus a map

σ : Γ (TM ⊗ TM) → R which is C∞-linear map in each of its components. In coordinates, it is expressible in a symmetric tensor product basis:

j j σ = σijdx dx .

Pull-back tensors with values in a vector bundle E

k Let V be the 0 tensor bundle over N. Suppose a map u : M → N, and a tensor σ ∈

k Γ(V ⊗ E) which is a l -tensor with values in E where π : E → N is one of the vector

15 bundles formed above, we can express σ locally as

σ(x) = s(x) ⊗ ω|x i1 ik = si1···il (x) ⊗ dy ⊗ · · · ⊗ dy where s itself is a section of E, i.e., a smooth map s : M → E.

In this local coordinate, we write u(x) = (u1(x),..., un(x)), and the pullback of σ is

given by the formula

u∗(σ) :=s(u(x)) ⊗ u∗(ω) ∗ i1 ∗ ik =si1···il ⊗ u dy ⊗ · · · ⊗ u dy

i1 ∂u k ∗ i =s ··· (u(x)) ⊗ dx ⊗ · · · ⊗ u dy k i1 il ∂dxk

Thus, u∗(σ) is a section of u∗(E) ⊗ u∗(V).

Total derivative of a map

Adopting the same local coordinates as above for a smooth map u : (M, g) → N, h) be-

tween Riemannian manifolds, its total derivative is deﬁned as the tensor

∂uα ∂ ⊗ dxi ∈ Γ (u∗ (TN) ⊗ TM) , ∂xi ∂yα which is a section of the pull back of the target tangent bundle along u tensored with the domain tangent bundle.

Trace and contraction of tensors

∗ ∗ For tensors σ ∈ Γ (T M ⊗ T M), we call the trace of σ over (M, g) the function trM(σ) on

M evaluated by m i i trM(σ)(p) = ∑ σ(e |p, e |p) i=1 i for each p ∈ M, where {e |p} ∈ Tp M forms an orthonormal basis for Tp M with respect to the metric g on M.

16 Analogously, we can form the trace of different tensors with pairs of arguments in

TM ⊗ TM or T∗ M ⊗ T∗ M, resulting in a tensor with fewer arguments (and not necessarily a function).

If we have two tensors σ ∈ Γ(TM) and λ ∈ Γ(T∗ M), we can express them locally as

i j σ = σ ∂xi and λ = λjdx .

We call the contraction of σ and λ the function deﬁned by

i j σ λjdx (∂xi ) .

Analogously, we can form the contraction of different tensors σ and λ with arguments in T∗ M and TM, resulting in a tensor with 2 fewer total arguments. Note that the contrac-

−1 tion of σ and λ trM(σ ⊗ λ) is equivalent to either trM((g ⊗ σ) ⊗ λ) or trM(σ ⊗ (g ⊗ λ)),

with the trace taken over the appropriate arguments.

2.1.2 Connections

Connection on a vector bundle

A linear connection on a vector bundle π : E → M is a map ∇ on the space of vector ﬁelds

over M and the space of smooth sections Γ(E) of E

∇ : Γ(TM) × Γ(E) → Γ(E)

(X, σ) 7→ ∇Xσ

such that the following hold:

∞ ∞ 1. ∇ is C (M)-linear in X, i.e., ∇( f ·X)σ = f · (∇Xσ) for all f ∈ C (M)

2. ∇ is R-linear in σ, i.e., ∇X (λσ) = λ · (∇Xσ) for all λ ∈ R

3. ∇ respects the Leibniz rule, i.e., ∇X ( f · σ) = (∇X f ) · σ + f · (∇Xσ) for all smooth

functions f ∈ C∞(M) and all smooth sections σ ∈ Γ(E)

17 where ∇X f ≡ X f is the directional derivative of f along X. In this way, we generalize the notion of a directional derivative of functions along vector ﬁelds to one for sections. We call ∇Xσ the covariant derivative of σ in the direction X.

Direct sum and tensor product connections

If πE : E → M and πF : F → M are two vector bundles over M with connections ∇E and

∇F, respectively, we can deﬁne connections on the vector bundles

πE⊕F : E ⊕ F → M

πE⊗F : E ⊗ F → M over M as follows, using the natural identiﬁcations of the spaces of sections Γ(E) ⊕ Γ(F) =∼

Γ(E ⊕ F) and Γ(E) ⊗ Γ(F) =∼ Γ(E ⊗ F):

∇E⊕F : Γ(E ⊕ F) × Γ(TM) → Γ(E ⊕ F)

E F (σ ⊕ ξ, X) 7→ ∇Xσ ⊕ ∇Xσ

∇E⊗F : Γ(E ⊗ F) × Γ(TM) → Γ(E ⊗ F)

E F (σ ⊗ ξ, X) 7→ ∇Xσ ⊗ ξ + σ ⊗ ∇Xξ

Note that the tensor product connection is rigged to satisfy the Leibniz product rule.

Dual connection

If π : E → M is a vector bundle over M with a conenction ∇, we can deﬁne a natural

dual connection ∇∗ on the dual bundle π∗ : E∗ → M. Recall that the dual vector bundle is

deﬁned as E∗ := Hom(E, R) and elements λ ∈ E∗ are functionals:

λ : Γ(E) → R

σ 7→ λ(σ)

18 for all σ ∈ Γ(E). We deﬁne ∇∗ dually by the following formula, so that the equation holds for all σ ∈ Γ(E) and for all λ ∈ Γ(E∗):

∗ (∇Xλ) σ = ∇X (λ(σ)) − λ (∇Xσ)

Observe that this deﬁning equation is necessarily satisﬁed if we demand the tensor product

∗ bundle R =∼ E∗ ⊗ E with its tensor product connection ∇E ⊗E satisﬁes the Leibniz rule, as the identiﬁcation R =∼ E∗⊗ is given by λ(σ) =∼ λ ⊗ σ and

C∞(M) E∗⊗E ∇X (λ(σ)) = ∇X (λ ⊗ σ)

∗ = (∇Xλ) σ + λ (∇Xσ)

Higher order exterior connections

The connections above induce connections on the following:

pth-power tensor bundle ⊗p E → M

p-power exterior bundle ∧p E → M

symmetric pth-power bundle p E → M

2.1.3 Metrics on vector bundles with connections

A Riemannian structure on a vector bundle E with a connection ∇E is given by a section

g ∈ Γ 2E∗ such that, for all σ, η ∈ Γ(E) and for all X ∈ Γ(TM), we have

E E ∇X g(σ, η) = g(∇Xσ, η) + g(σ, ∇Xη).

E Note that this is equivalent to ∇X g = 0 for all vector ﬁelds X ∈ Γ(TM).

19 Curvatures of a connection

The curvature R of a connection ∇ is the map

R : ∧2Γ(TM) × Γ(E) → Γ(E)

(X, Y, σ) 7→ −∇X∇Yσ + ∇Y∇Xσ + ∇[X,Y]σ

Note that R(X, Y) = −R(X, Y) as an map Γ(E) → Γ(E), and that R is C∞(M)-linear in each of its arguments X, Y, and σ.

We call the sectional curvature Riemp(X ∧ Y) of the oriented plane X ∧ Y ∈ Γ(TM ∧ TM) spanned by orthonormal vectors at p if Xp and Yp the value given by

Riem(X ∧ Y) = g(R(X, Y)X, Y).

We say that M has non-negative sectional curvature if Riem(X ∧ Y) ≥ 0 for all X ∧ Y.

We call the Ricci curvature Ricp(X, Y) the value of

Ricp(X, Y) = tr(Z 7→ R(X, Z)Y)

We say that M has non-negative Ricci curvature if Ric(X, X) ≥ 0 for all X.

2.2 Harmonic map formulae

We are ﬁnally at a position to deﬁne harmonic maps. All of this material is classical, and a good reference for these concepts is contained in the aptly titled Two Reports on Harmonic maps [EL95].

2.2.1 The Dirichlet energy

The Dirichlet energy is a functional on the space of C2 maps between Riemannian manifolds

(M, σ) and (N, ρ). For a map u ∈ C2(M, N), we deﬁne the energy of u as

Z 1 2 ||∇u||u−1(TN)⊗T∗ Mdvolg. M 2

20 Notice that ∇u is a C1 section of the bundle u−1(TN) ⊗ T∗ M over M, so its norm is deﬁned

with respect to the metric u∗(ρ) ⊗ σ∗ on u−1(TN) ⊗ T∗ M. The integrand is called the energy

density of u, and it has various forms:

u 1 2 e (z) := ||∇u|| − 2 u 1(TN)⊗TM 1 = tr (u∗(ρ))) 2

u Observe that the energy density measure e (z)dvolg is expressible as

1 tr (u∗(ρ) ⊗ σ) 2 M

Thus, the energy density is simply half the sum of the square of the eigenvalues of the ﬁrst

fundamental form u−1(ρ) on M. Note that the total derivative ∇u is a section of the bundle

u−1 (TN) ⊗ TM, and that the norm deﬁned on this metric is deﬁned through pullback and

over tensor product. In coordinates, the energy density of u is given by the formula

1 ∂uα ∂uβ eu ≡ gij h 2 ∂xi ∂xj αβ where uα(x1,..., xn) is the yα coordinate of the map u, and the metric coefﬁcients are given

ij i j by g = hdx , dx iT∗ M and hαβ = h∂yα , ∂yβ iTN.

First variation: harmonic map deﬁnition

We will intrinsically calculate the ﬁrst variational formula and then reformulate it in local coordinates. Consider a smooth one parameter variation ut = expu(x)(tη) of the map

21 −1 u0 = u, so that ∇∂tut = η ∈ u (TN) is a smooth vector ﬁeld along u.

d d 1 Z E(ut) = hDut, Dutiu−1(TN)⊗TMdvolM dt t=0 dt t=0 2 M

1 Z d = hDut, Dutiu−1(TN)⊗TMdvolM 2 M dt t=0 1 Z − = 2h∇∂t (Dut), Dutiu 1(TN)⊗TM dvolM 2 M t=0 Z − = hD(∇∂t ut), Dutiu 1(TN)⊗TM dvolM M t=0 Z = hDη, Duiu−1(TN)⊗TMdvolM M Z = − hη, trM(∇Du)iu−1(TN)dvolM M

Hence, the Euler-Lagrange equation for the Dirichlet energy is given by

trM(∇Du) = 0

Next, in local coordinates, the energy density is given by

∂uα ∂uβ eu = gij h ∂xi ∂xj αβ

γ ( ) = γ and the tension ﬁeld τ x τ ∂y u(x) is given by

∂uγ ∂uγ ∂uγ ∂uα ∂uβ τγ = gij − MΓk + NΓγ ∂xi ∂xj ij ∂xk αβ ∂xi ∂xj α β γ ij N γ ∂u ∂u = ∆( )u + g Γ M,g αβ ∂xi ∂xj

The tension field defines a system of second-order elliptic semi-linear partial differential equation, called the harmonic map equations. The energy density and tension fields are most easily understood as: at each point x ∈ M and its image u(x) ∈ N, choose geodesic normal

1 m 1 n ij j M k β coordinates x ,..., x ∈ Tx M and y ,..., y ∈ Tu(x) N, so that g = δi , Γij = 0, hαβ = δα ,

N γ u and Γαβ = 0; then e (x) is simply the sum of squares of the stretches in these orthogonal directions, m u 1 i 2 e (x) = ∑ ||Du(e )||TN, 2 i=1

22 and the tension ﬁeld is the Euclidean Laplacian,

m ∂2uγ γ = τ ∑ i i i=1 ∂x ∂x

Second variation: Jacobi ﬁelds and harmonic map stability

Let u : M → N be a harmonic map, and consider a two-parameter family variation us,t of

u, for which u = u0,0 and the initial variational vector ﬁelds are

= ∇ | = ∇ | η ∂s us,t (s,t)=(0,0) and µ ∂t us,t (s,t)=(0,0)

Then the second variational formula of the energy is given by computing

2 Z ∂ h D N Ei E(us,t) = h∇η, ∇µi − trM R (Du, η)Du, µ dvolM ∂s∂t (s,t)=(0,0) M Z D 2 N E = − tr∇uη + trM R (Du, η)Du, η , µ dvolM M Z = hJuη, µi dvolM = 0 M

where we denote the Jacobi operator deﬁning the second variational formula by

2 N Juη := −tr∇uη − trM R (Du, η)Du, η .

We call solutions to Juη = 0 the Jacobi ﬁelds along u. Note that, if us,t is a variation of

harmonic maps with variational vector ﬁelds η and µ, and if Juη = 0, then exp(u(x)(tη) is

harmonic for small t. In particular, we can deduce from the second variational formula:

Proposition 2.2.1. Suppose u : M → N is a harmonic map. If RiemN < 0, then the Hessian of

the energy functional is positive deﬁnite, i.e., for all variations η, we have hJuη, ηi > 0.

Thus, harmonic maps u : M → N into negatively curved manifolds N are a priori

energy minimizers, if they exist.

2.2.2 Existence and uniqueness

Let us turn to discussing the existence and uniqueness of harmonic maps.

23 Existence

Harmonic maps from surfaces have been shown to exist in various settings. The semi- nal result was obtained by Eells-Sampson [ES64] using a heat ﬂow approach to decrease energy:

1 2 Theorem 2.2.2. Suppose f ∈ C (M , N), with N compact and RiemN ≤ 0. Then f is homotopic to a harmonic map (which minimizes energy in its homotopy class).

It was realized that singularities could form along the heat ﬂow, and produce a degenerate solution in the limit. An example of such a phenomenon is the bubbling of a harmonic

1 1 map from a surface: the one parameter family of harmonic maps hλ : CP → CP deﬁned

by [1, z] 7→ [1, λz] does not converge to a harmonic map as λ % ∞.

Nonetheless, Sacks-Uhlenbeck found topological conditions on the target manifold for which a harmonic homotopic representative can be found [SU82]:

Theorem 2.2.3. If π2(N) = 0 (e.g., if RicN ≤ 0), then there exists an energy minimizing harmonic map in every homotopy class of maps in C0(M2, N).

Uniqueness

Al’ber and Hartman established a sense of uniqueness for harmonic maps [Al’64], [Har67]:

Theorem 2.2.4. If M is compact and u0, u1 : M → N are homotopic harmonic maps, then they are smoothly homotopic through harmonic maps, and the energy is constant on any arcwise connected set of harmonic maps. Furthermore, we can choose a smooth homotopy ut for 0 ≤ t ≤ 1 through

harmonic maps so that each path s 7→ us(x) is a geodesic segment parameterized proportionally to

arc-length for each x ∈ M.

As an example application, this yields uniqueness of harmonic maps from a compact

set to a negatively curved Riemannian manifold.

24 2.2.3 Examples of harmonic maps

To convey the properties of a harmonic map, we provide some examples.

Some simple examples of harmonic maps include:

1. The identity map on any Riemannian manifold is harmonic.

2. The constant map from any Riemannian manifold to a point (of, say, another Rie-

mannian manifold) is harmonic.

3. For N = R, the harmonic map equation simply reads ∆Mu = 0, and so the concept

of a harmonic map coincides with that of a harmonic function.

4. For M = R, the tension ﬁeld only has one component, and it measures acceleration

along the path, i.e., geodesic curvature of the parameterized path. Hence, a harmonic

map parameterizes a constant speed (zero acceleration) geodesic.

Some more complicated examples of harmonic maps are:

1. Any totally geodesic map u : M → N is harmonic. We say u is totally geodesic when

∇Du ≡ 0 identically. These maps have the geometric description that geodesics are

mapped onto geodesics (although the speed of the geodesic parameterizations can

vary).

2. Between Kahler¨ manifolds M and N, all holomorphic maps u : M → N are har-

monic.

3. For u : M → N conformal and v : N → P harmonic, the composition v ◦ u : M → P

is a harmonic map. We say u is conformal if u∗(ρ) = λσ, where λ is a non-vanishing

continuous function λ : M → R.

4. For u : M1 → N1 and v : M2 → N2 harmonic maps between Riemannian manifolds,

the map (u, v) : M1 × M2 → N1 × N2 is also harmonic.

25 k n 5. A coordinate function xi restricted to a minimal k-dimensional submanifold Σ ⊂ R

k is a harmonic map xi : Σ → R. In fact, minimal sub-manifolds in Euclidean space

are equivalent to conformal, harmonic maps from their parameterizing domain. So,

the restriction of the coordinate function onto the minimal sub-manifold can be real-

ized as the composition of the parameterizing map with the totally geodesic projec-

tion of Rn onto a coordinate axis.

2.2.4 Properties of harmonic maps

To complement the concrete examples of harmonic maps above, we discuss some a priori local and global analytic properties of harmonic maps.

Relation to convex and sub-harmonic functions

There is a useful analytic characterization of harmonic maps observed by Ishihara [Ish79], expressed as a relationship between the space of convex functions on the target manifold and the space of sub-harmonic functions on the domain:

Theorem 2.2.5. A map u : (M, g) → (N, h) is a harmonic map if and only if, for any open subset

U of N, every convex function f on U pulls back to a sub-harmonic function f ◦ u on f −1(U).

This characterization is particularly useful for visualizing minimal surfaces in R3. For

example, consider any convex domain D ⊂ R3 disjoint from a minimal surface Σ ⊂ R3,

with a nearest point p ∈ Σ. The distance function to D is convex on R3 \ D, so restricts to

a sub-harmonic function on any neighborhood U ⊂ Σ of p. By the maximum principle for

subharmonic functions, this implies that Σ must “bend away” from D near p, i.e., there are points of ∂U farther away from D than p is from D.

26 Geometric maximum principle

The relationship between convex and sub-harmonic functions by harmonic maps can be geometrically reinterpreted. The following were established by Sampson [Sam78]:

Theorem 2.2.6. If u : M → N is a non-constant harmonic map and S ⊂ N is a hypersurface with deﬁnite second fundamental form at a point q = u(p), then no neighborhood of p ∈ M is mapped entirely on the concave side of S.

Furthermore, harmonic maps have very nice local-to-global properties because of their real analyticity. There is a unique continuation theorem:

Theorem 2.2.7. Let u, v : M → N be two harmonic maps whose inﬁnite order jets agree at some

∞ ∞ point p ∈ M, i.e., jp (u) = jp (v). Then u ≡ v. In particular, if u and v agree on an open subset of M, then u ≡ v.

And, along the same vein:

Theorem 2.2.8. Suppose u : M → N is a harmonic map from a connected Riemannian manifold

M to a Riemannian manifold N which has rank equal to r on an open subset U ⊂ M. If r = 0, then

u(M) is a point. If r = 1, then u maps M into a geodesic arc of N; furthermore, if M is compact, then u maps M onto a closed geodesic. Finally, if u : M → N maps an open subset U ⊂ M into a complete, totally geodesic submanifold N0 of N, then u(M) ⊂ N0.

2.3 Harmonic maps from surfaces

We now restrict our attention to harmonic maps from a Riemann surface (Σ, σ) and describe the attractive analytic consequences of restricting to surface domains. We will often abuse notation and denote the target metric by ρ(u)du du whenever we are dealing with a map u : (Σ, σ) → (N, ρ). Throughout this section, let z = x + iy be a local conformal

27 parameter on Σ1. In particular, the partial derivatives of u satisfy the following identities:

1 u ≡ u − iu z 2 x y 1 u ≡ u + iu z 2 x y

Let us ﬁrst discuss regularity for harmonic maps from surfaces.

2.3.1 Regularity

A map u ∈ C0((Σ, σ), (N, ρ)) ∩ W1,2((Σ, σ), (N, ρ)) is called weakly harmonic if, for all test

0 1,2 maps v ∈ C ((Σ, σ), (N, ρ)) ∩ W0 ((Σ, σ), (N, ρ)), we have

Z ρu uzvz − uzuzv idz dz = 0. Σ ρ

Weakly harmonic maps are simpler to ﬁnd by the direct method of energy minimization.

Proposition 2.3.1. If u ∈ C0((Σ, σ), (N, ρ)) ∩ W1,2((Σ, σ), (N, ρ)) is a minimum for E(u), then

u is weakly harmonic. If u ∈ C2((Σ, σ), (N, ρ)), then u is harmonic.

For maps from surface domains to negatively curved targets, the Weyl Lemma for ellip-

tic partial differential equations implies that weakly harmonic maps are in fact harmonic.

Theorem 2.3.2. Let u : (Σ1, σ) → (Σ2, ρ) be a weakly harmonic map of ﬁnite energy between

Riemann surfaces. Suppose that ρ is a metric of constant negative curvature. Then u is smooth,

i.e., u satisﬁes the harmonic map equation.

Next, we specialize some of the formulae for harmonic maps from the preceding sec-

tion to the case in which both the domain and range are surfaces. This discussion is also

somewhat classical, but more recent. We collect some computations from the books Two-

dimensional geometric variational problems [Jos91], Compact Riemann Surface [Jos06], and Lec-

tures on Harmonic maps [SY97].

28 2.3.2 Specialization of harmonic map formulae

Energy recast on a surface

2 For a map u ∈ C ((Σ1, σ), (Σ2, ρ)), the energy of u is given by

1 Z E(u) ≡ uxux + uyuy ρ(u)dz dz 2 Σ1 Z i = [uzuz + uzuz] ρ(u) dz dz Σ1 2 1 Z h i = ||∂u||2 + ||∂u||2 dz dz 2 Σ1 where the ﬁnal integrand is a sum of the holomorphic energy Hu and anti-holomorphic energy

Lu of u, given respectively by

ρ Hu := ||∂u||2 = |u |2 σ z ρ Lu := ||∂u||2 = |u |2 σ z

Harmonic map equation

For maps between surfaces, the ﬁrst variational formula of the Dirichlet energy functional

reduces to a simple formula.

Proposition 2.3.3. The harmonic map equation for a map u : (Σ1, σ) → (Σ2, ρ) between Riemann surfaces is given by ρ u + u u u = 0. zz ρ z z

Proof. In local coordinates, we can express a variation ut of u as

u + tµ

0 1,2 for some µ ∈ C ∩ W0 (Σ1, Σ2), which in turn can be expresses as

η µ = . ρ(u)

29 In this coordinate system, the Euler-Lagrange equation can be directly computed:

d 0 = E(u + tµ) dt t=0 d Z = [(u + tµ)z (u + tµ)z + (u + tµ)z (u + tµ)z] ρ(u + tµ)idz dz dt t=0 Z = [(uzµz + uzµz + uzµz + uzµz) · ρ(u) + (uzuz + uzuz)(ρuµ + ρuµ)] idz dz Z η η = u η − (ρ u + ρ u ) + u η − (ρ u + ρ u ) z z ρ(u) u z u z z z ρ(u) u z u z η η + u η − (ρ u + ρ u ) + u η − (ρ u + ρ u ) z z ρ(u) u z u z z z ρ(u) u z u z 1 + (u u + u u )(ρ η + ρ η) · ρ(u)idz dz z z z z u u ρ(u) Z ρ Z ρ =2R u η − u u u η idz dz + 2R u η − u u u η idz dz z z ρ z z z z ρ z z where R denotes the real part of its argument. Now, we can integrate by parts, or use a molliﬁcation argument if necessary, obtaining:

Z ρ Z ρ 0 =R u + u u u ηidz dz + R u + u u u ηidz dz zz ρ z z zz ρ z z Z ρ =2R u + u u u ηidz dz zz ρ z z

Since µ (and hence η) can be chosen arbitrarily, this forces

ρ u + u u u = 0 zz ρ z z

2.3.3 Conformal invariance of energy

The situation with a surface domain is special. Suppose that z(w) is a conformal change of parameter, and that u(z) : (Σ1, σ) → (Σ2, ρ) is a harmonic map. Then the energy of u and

30 the energy of v(w) := u ◦ z(w) are equal, since

1 Z E(v) := [vwvw + vwvw] ρ(v)idw dw 2 Σ1 Z dz dz = [uzzw · uzzw + uzzw · uzzw] i Σ1 zw zw 1 Z = [uzuz + uzuz] ρ(u(z))idz dz = E(u) 2 Σ1

2.3.4 Hopf differential

A powerful feature of restriction to a surface domain (Σ, σ) is the possibility for discussing complex analytic objects prescribed to harmonic maps from (Σ, σ). For, a surface (Σ, σ) in-

herits a natural complex structure z on Σ induced by its Riemann surface structure. Using

this complex structure z, the boundary operator d : T∗Σ → T∗Σ on the deRham cohomol-

ogy of Σ decomposes as

d = dz ∧ ∇∂z + dz ∧ ∇∂z .

This allows us to decompose the vector bundles of symmetric tensors of any order over Σ into types. In particular, we have the decomposition

T∗Σ T∗Σ = hdz dzi ⊕ hdz dzi ⊕ hdz dzi into the types (2, 0), (1, 1), and (0, 2), where hdz dzi denotes the C∞(Σ) span of dz dz in T∗Σ T∗Σ, etc. The tensor dz dz is oftentimes labeled as dz2.

We call a section λ holomorphic if ∇∂z λ = 0 and anti-holomorphic if ∇∂z λ = 0. Note that

a (2, 0) tensor ﬁeld is not necessarily holomorphic. For example, f (z)dz2 is not a holomorphic (2, 0) tensor for any non-constant holomorphic function f (z) on Σ with respect to z, since

2 2 2 ∇∂z f (z)dz = (∇∂z f (z)) dz + f (z) ∇∂z dz

2 = fz(z)dz + 0

31 We can use this complex structure z on (Σ, σ) to deﬁne the Hopf differential Φu of any C2

map u : Σ → N, where N can be any Riemannian manifold of any dimension:

Φu(z) := (u∗(ρ))(2,0).

Observe that Φu(z) is the (2, 0)-part of the pullback metric, whereas the energy density

u 1 ∗ of u is half of the trace of the pullback metric, e (z) = 2 trΣ(u (ρ)). The full relationship between eu, Φu, Hu, and Lu is obtained from the expression for the pullback metric u∗(ρ):

u∗(ρ)(z) ≡ ρ(u(z)) · d(u(z)) d(u(z))

= (ρ ◦ u) · uzuzdz dz + ρ ◦ u (uzuz + uzuz) dz dz + (ρ ◦ u) · uzuzdz dz

≡ φ(z)dz dz + (Hu + Lu) · σ(z) · dz dz + φ(z)dz dz

We deduce the following equations relating quantities associated to the harmonic map u:

||Φu||2 = Hu ·Lu · σ

eu(z) = Hu + Lu

J u = Hu − Lu where J u denotes the functional Jacobian of the map u.

Analytic properties detected by the Hopf differential

Some analytic attributes of u are captured by its Hopf differential Φu.

Proposition 2.3.4. Suppose u : (Σ, σ) → (N, ρ) is either a C1 harmonic map or a stationary map with respect to smooth variations of the domain Σ. Then its Hopf differential Φu is holomorphic with respect to the complex structure induced by σ.

Proposition 2.3.5. Conversely, let u : (Σ, σ) → (N, ρ) be a C1 diffeomorphism onto its image or a C2 map. If its Hopf differential Φu is holomorphic with respect to the complex structure induced by σ, then u is a harmonic map.

32 Furthermore, the Hopf differential Φu detects conformality of u.

Proposition 2.3.6. Let u : (Σ, σ) → (N, ρ) be a C1 map. Then u is conformal, i.e., there exists a

function λ ∈ C0(Σ) such that u∗(ρ) = λ · σ, if and only if its Hopf differential Φu vanishes.

We include the proofs of these statements in Appendix A.1. Because of these facts, the

Hopf differential is an indispensable tool for studying harmonic maps, conformal maps, minimal sub-manifolds, and constant mean curvature sub-manifolds. In Rn, these objects are intimately related through the Gauss map. Given a k-dimensional sub-manifold Σk of

n k k R , its Gauss map Π : Σ → Gr(k, n) is deﬁned by Π(p) = Ap(TpΣ ) ∈ Gr(k, n), where

k n k TpΣ ⊂ R is the tangent k-plane to Σ at p, Ap is the translation of p to the origin 0, and

Gr(k, n) is the Grassmannian manifold of k-dimensional planes through the origin in Rn.

Observe that Gr(n − 1, n) =∼ Sn−1. The following theorem is due to Ruh-Vilms [RV70]:

Theorem 2.3.7. Let Π : Mk → Gr(k, n) be the Gauss map of a k-dimensional sub-manifold of Rn.

Then τΠ = ∇H, where H is the mean curvature function of Mk. In particular, if Mk is a minimal sub-manifold (satisfying ∇H = 0), then Π is a harmonic map.

Thus, the following are equivalent notions for a surface Σ of R3: Σ is a minimal surface,

Σ has a conformal harmonic immersion parameterization, and Σ has a conformal or anti- conformal Gauss map Π : Σ → S2. The following are also equivalent: Σ is a constant mean curvature surface, and Σ has a harmonic Gauss map Π : Σ → S2.

As an example application, Hopf provided an alternative proof to Alexandroff’s theorem that the only embedded constant mean curvature topological sphere in R3 must be the round sphere. Whereas Alexandroff developed the method of moving planes to prove this, exploiting the maximum principle for constant mean curvature hypersurfaces, Hopf used the fact that the only holomorphic quadratic differential on a sphere must be constant.

33 Geometric properties described by the Hopf differential

We can relate the image of a harmonic diffeomorphism u : Σ → N from a surface Σ to the

Hopf differential Φu. We have the following [Wol91]:

Proposition 2.3.8. Let u : (Σ, σ) → (N, ρ) be a harmonic map. In the natural coordinate system

ζ = ξ + iη in which Φu = dζ2, valid away from the zeros of Φu, we have the identity

σ · e σ · e u∗(ρ(u)du du) = + 2 dξ2 + − 2 dη2. |Φu| |Φu|

Proposition 2.3.9. Let u : (Σ, σ) → (N, ρ) be a harmonic map. The image of the horizontal leaf

γ(t) = (t, η0) has geodesic curvature

1 1 σ · e 2 ∂ σ · e κ (t) = − − 2 · log + 2 . γ 2 |Φu| ∂η |Φu|

For completeness, these statements are proven in Appendix A.2.

2.3.5 Courant-Lebesgue Lemma

Another quality of a studying maps from a surface domain is that the equicontinuity of

2 2 a family of maps ut : Σ → N, for t ∈ [0, 1] from a surface domain Σ to a Riemannian manifold N can be obtained immediately from a uniform energy upper bound, i.e., if there exists M > 0 such that E(ut) < M for all t. The following was proven in [Cou37]:

Theorem 2.3.10. (Courant-Lebesgue Lemma) Consider a C2 map u : Σ2 → N with energy √ 2 EΣ2 (u) ≤ K. Then, for all p ∈ Σ and δ ∈ (0, 1) there exists ρ ∈ (δ, δ) so that, for all

2 q 8πK x, y ∈ {q ∈ Σ |dΣ2 (q, p) = ρ}, we have dN(u(x), u(y)) ≤ 1 , where dN denotes the log( δ ) intrinsic distance on N.

Classically, harmonic maps from a ﬁxed surface domain into a compact target space have been shown to exist by the direct method: consider a sequence of maps with decreas- ing energy in a ﬁxed homotopy class of continuous maps, and show that the limit exists.

34 The uniform bound on the energy of the maps leads to a uniform bound on the modulus of continuity by the Courant-Lebesgue Lemma.

For our non-compact setting, we will look at a sequence of harmonic maps on a compact exhaustion. In order to get them to uniformly sub-converge, it will be required to restrict them onto a common compact domain. We will apply the Courant-Lebesgue Lemma to obtain equicontinuity of these restrictions.

For completeness, we provide a proof here.

2 Proof. Let Bp(e) ⊂ Σ be the intrinsic disk of radius e such that x, y ∈ ∂Bp(e), for any

e ∈ (0, 1). Since u(∂Bp(e)) contains a parameterized path from u(x) to u(y), after adopting

(r, θ) ∈ [0, e] × [0, 2π) polar coordinates for Bp(e), it is clear for any e ∈ (0, 1) that

Z ∂ d (u(x), u(y)) ≤ Du| ds. N (e,θ) ∂θ ∂Bp(e)

Applying the Cauchy-Schwarz inequality to the right-hand side and squaring reveals

2 Z ∂ d2 (u(x), u(y)) ≤ 2π · Du| ds. N (e,θ) ∂θ ∂Bp(e) √ Observe that, by the Intermediate Value Theorem, there exists some ρ ∈ (δ, δ) so that √ = = 2 2 Z r δ 1 Z θ 2π ∂ ZZ 1 ∂ dr · Du|(ρ,θ) dθ ≤ √ Du dr dθ. r=δ r θ=0 ∂θ Bp( δ) r ∂θ

Rearranging terms, this implies that

= 2 2 Z θ 2π ∂ 2 ZZ ∂ Du|(ρ,θ) dθ ≤ · √ Du|(ρ,θ) dr dθ = ∂θ 1 ( ) ∂θ θ 0 log( δ ) Bp δ and hence, since ρ ∈ (0, 1), we have

ZZ 2 2 4π 1 ∂ dN(u(x), u(y)) ≤ · √ Du dr dθ. 1 ( ) r ∂θ log( δ ) Bp δ

Finally, the polar coordinate expression for the energy density measure is given by

" 2 2# ∂ 1 ∂ eu(z) dA = Du + Du r dr dθ, ∂r r2 ∂θ

35 so it follows that the energy upper bound EΣ2 (u) ≤ K implies

2 1 ZZ 1 ∂ Du dr dθ ≤ K 2 √ r ∂θ Bp( δ)\Bp(δ)

∂ by ignoring the non-negative integrand summand involving Du ∂r .

Thus, we are left with 8πK 2 ( ( ) ( )) ≤ dN u x , u y 1 . log( δ )

2.3.6 Bochner formulae and consequences

Let u : (M2, σ) → (N, ρ) be a harmonic map between a Riemann surface and a Riemannian manifold. Recall that the harmonic map equation reads

uzz + (logρ)u uzuz = 0 by using a complex coordinate z for M. We have the following formulae:

4 ∂2 ∆ ≡ M σ ∂z ∂z 1 KM = − ∆ log(σ) 2 M 1 KN = − ∆ log(ρ) 2 N

These curvatures are related to the holomorphic and anti-holomorphic energies of a harmonic map. Recall that J u = Hu − Lu. We have the following:

Proposition 2.3.11. (Bochner formulae for harmonic maps) Let u : (M2, σ) → (N, ρ) be a harmonic map from a surface (M2, σ) to a Riemannian manifold (N, ρ). Then

u N u M ∆M log (H ) = −K J + K

u N u M ∆N log (L ) = K J + K

36 We remark that orientation preserving harmonic maps u can be analyzed a priori through

their Bochner equations by imposing the ansatz J u ≥ 0. Using the change of variables

1 u w = 2 log (H ), this yields a desirable sign on the holomorphic energy Bochner equation

2w 2 −2w ∆Mw = e − ||Φ|| e so that a version of the Maximum Principle holds. This Maximum Principle has been widely applied (c.f. [SY78], [Sam78], [Wol89], [WA94], [TD13]). Our handle crushing harmonic maps cannot be orientation preserving, so we have to develop different techniques for their analysis.

2.4 Some recent results on harmonic maps

We will apply the following in our proof of Theorem 1.1.1.

2.4.1 Harmonic maps C → H2

The Scherk maps are the basis for our investigation. We will use them to provide appropriate boundary conditions for harmonic maps on compacta of a punctured Riemann surface.

Theorem 2.4.1. (Scherk maps) Let Φ be a polynomial holomorphic quadratic differential of degree

k on C. If Φ is not identically zero, then there exists a unique C∞ solution to

∆W = e2W − |Φ|2e−2W for which e2W − |Φ|2e−2W ≥ 0 and e2W |dz|2 is a complete Riemannian metric on C. Furthermore, these solutions are in bijection with harmonic maps w : C → H2 for which the Hopf differential

Φw of w is identically Φ. Furthermore, w(C) forms the interior of an ideal (k + 2)-gon.

This statement is an application of theorems from [WA94], [HTTW95], and [ST02].

In [WA94], the Bochner equation is solved to produce constant mean curvature cuts in

37 Minkowski space. These are graphical, totally space-like surfaces of constant meant curvature in R2,1 parameterized by W : C → R, where we identify C =∼ {(x, y, 0)|x, y ∈ R} ⊂

R2,1. Their generalized Gauss maps w : C → H2 have image in the space-like sphere

H2 = {(x, y, z)|x2 + y2 − z2 = −1}. Furthermore, w is harmonic precisely when the graph

of W has constant mean curvature [Mil83].

The proof of Theorem 2.4.1 exploits the fact that the partial differential equation solved

by W coincides with the Bochner equation for the holomorphic energy of a harmonic map

w. It is remarkable that a solution to the Bochner equation is sufﬁcient to produce a har-

2 w 1 monic map w : C → H , since this H = 2 log||∂w|| necessarily solves the Bochner equation when w is harmonic.

In [HTTW95], the closure of the image of a Scherk map is characterized to be an ideal polygon. In fact, they also prove that any harmonic map C → H2 × R with image lying in the convex hull of ﬁnitely many ideal points (i.e., an ideal polygon) must have a polynomial Hopf differential.

2.4.2 Wood’s analysis of images of harmonic maps

The following two results are contained in [Woo74], but we recall their cleaner statements from different sources. The following was adapted from the proof of Theorem 5.1 (and its corollary) of [Woo77], and places restrictions on the Euler characteristic of the image of a harmonic map:

Theorem 2.4.2. (Gauss-Bonnet formula for images of harmonic maps) Let u : (M, σ) →

(N, ρ) be a harmonic map between real analytic surfaces, where M may have boundary ∂M. In the case ∂M 6= ∅, we only assume u|∂M is smooth on the boundary and that u|int(M) is harmonic in

the interior of M. The total curvature K(u(M)), the geodesic curvature of the boundary image,

38 and the Euler characteristic χ(u(M)) of the image of u are related by the inequality

K(u(M)) + ∑ κγ ≥ 2πχ(u(M)). γ∈∂u(M)

In particular, if ∑γ∈∂M κγ = 0 and if the metric ρ on N has non-positive curvature, then u(M)

cannot be contractible.

This adaptation is proven exactly as the original theorem, without omitting the pre-

scribed boundary value terms. The following was reproven in [SY78] and [Sam78], and

restate their formulation here:

Theorem 2.4.3. (Jacobian property) Suppose that M and N are smooth surfaces. If u : M → N

is a harmonic map deﬁned on a connected open subset D ⊂ M satisfying J u ≥ 0 on D, then

either J u is identically 0 or the zeros of J u are isolated. Moreover, if there is a number ` so that

# u−1(q) ≤ ` for each regular value q ∈ N of u, then each isolated zero of J u is a nontrivial

branch point of u.

2.4.3 Dumas-Wolf’s uniqueness of orientation-preserving harmonic maps through

Hopf differentials

We will consider parachute maps to study the energies of the harmonic maps from com-

2 pacta Σs to H . In studying the behavior of the quadratic differentials of these parachute

maps, we will use a blow-up analysis of the parachute maps, leading to harmonic maps

from C → H2. We will characterize the possibilities for those blow-ups using the following

result which holds in the simply connected domain setting (Theorem 5.3 of [DW15]):

Theorem 2.4.4. For any polynomial holomorphic differential Φ of degree k, there is a unique com-

plete and non-positively curved metric σ(z)|dz|2 on C for which the curvature satisﬁes

2 Kσ = −1 + |Φ|σ

39 This theorem leads to a characterization of harmonic maps because the pull-back metric of a harmonic map from C to H2 has a metric with Gaussian curvature expressed by this formula.

2.4.4 Cheng’s Lemma

The classical Liouville Theorem states that a bounded entire function on C must be constant. Yau extended the Liouville theorem by generalizing the domain of the harmonic function: any bounded harmonic function on a manifold of non-negative Ricci curvature must be constant [Yau75]. Cheng takes this a step further by generalizing the target manifold: if M is a Riemannian manifold on non-negative Ricci curvature, and N is a simply

connected Riemannian manifold of non-positive sectional curvature, then any harmonic

map u : M → N whose image u(M) lies in a compact subset of N must be constant

[Che80].

This ﬁnal generalization was obtained in [Che80] as a consequence of a gradient esti-

mate for harmonic maps:

Theorem 2.4.5. (An interior gradient estimate) Let M be an m-dimensional Riemannian man-

ifold with Ricci curvature bounded below by −K ≤ 0. Let Ba(x0) be the closed geodesic ball of

radius a and center x0 in M, and let r be the distance function from x0. Let f : M → N be a har-

monic map into N which is simply connected and having non-positive sectional curvature. Let y0

lie outside of f (Ba(x0)) and let ρ denote the distance from y0. Let b > sup{ρ( f (x))|x ∈ Ba(x0)}.

Then we have on Ba(x0):

(a2 − r2)2|∇ f |2 Ka4 a2(1 + Ka) a2b2 ≤ c max , , (b2 − ρ2 ◦ f )2 m β β β

2 2 where cm is a constant depending only on the dimension of M and β = in f {b − ρ ◦ f (x)|x ∈

Ba(x0)}.

40 2.4.5 Choe’s iso-energy inequality

The parachute maps we will introduce solve a partially-free boundary value harmonic mapping problem. In order to estimate its 1-dimensional energy on the free boundary, we can relate this energy to the energy of a harmonic map from a disk “ﬁlling in” the free boundary image. We will be able to do this via (Theorem 2.4 of [Cho98]):

Theorem 2.4.6. (Iso-energy inequality) Let u be harmonic map from a ball B in Rn into a non- positively curved manifold. Let E(u, B) denote the energy of u and E(u|∂B, ∂B) denote the energy of u|∂B. Then

(n − 1)E(u, B) ≤ E(u|∂B, ∂B).

41 Chapter 3

Handle-crushing harmonic maps

This chapter contains the proofs for our Theorems 1.1.1 and 1.1.2. Throughout this chapter, we will assume that g > 1 and k > 2 an even positive integer have been ﬁxed. We also assume the deﬁnitions and results from Chapter 2.

3.0.6 Organization of section

Throughout this section, we will assume that g > 1 and k > 2 an even positive integer have been ﬁxed.

In §3.1, we prove Theorem 1.1.1 (Existence). This section begins with a brief outline of the proof. We will assume the Energy Estimate (Lemma 3.4.7) and the results on harmonic maps recalled in §??. The Energy Estimate will be proven in §3.4,

In §3.2, we prove Theorem 1.1.2 (Uniqueness), describing the senses in which the harmonic maps from Theorem 1.1.1 are unique.

In §3.3, we specialize Theorems 1.1.1 and 1.1.2 to the case of a punctured square torus mapped to an ideal square. We investigate the Hopf differential of this map, deducing

Corollary 1.1.3.

In §3.4, we study a toy problem: does there exist a symmetric harmonic map from an inﬁnite cylinder to the hyperbolic plane? After a brief outline, we pose and solve this

42 problem, whose solutions we call parachute maps. The section closes with a derivation of the Energy Estimate (3.4.7).

Glossary of notation

Throughout the construction, we will compare so many maps to each other that it may be useful to have a dossier for the maps, their domains, and some associated quantities. We collect below the many symbols used throughout our discussion.

43 Domains

Σ := the punctured Riemann surface

Σs := a compactum of a compact exhaustion for the punctured Riemann surface

Ωs := the disk of radius s

Ωr,s := the annulus Ωs \ Ωr (identiﬁed with Σr,s := Σs \ Σr) for r >> 1

γs := the boundary ∂Ωs

Γs := the image of γs under w

2 Pk := the k-sided regular ideal polygon in H

Functions and associated quantities

2 τf := the tension ﬁeld of a C map f

f 2 Φ (z) := φf (z)dz = the Hopf differential of the map f

H f = the holomorphic energy of the map f

L f = the anti-holomorphic energy of the map f

e f (z) = the (pointwise) energy density of the map f

E(D, f ) = the energy of the map f |D on the domain D

Auxilliary harmonic maps (in order of appearance)

w := the Scherk map C → H2 with an ideal square image

2 hs := a harmonic map Σs → H with Dirichlet boundary conditions

2 us := the parachute map Ωr,s → H solving a partially free BVP

44 3.1 Construction: Proof of Theorem 1.1.1

In this section, we prove Theorem 1.1.1, assuming the The Energy Estimate (Lemma 3.4.7),

along with the previous results on harmonic maps from Chapter 2. The proof proceeds by

producing a sub-converging sequence of harmonic maps from compact domains exhaust-

ing a genus g punctured Riemann surface to an ideal k-sided polygon in H2. To organize

ideas, we structure the proof in stages. Each sub-section is devoted to a stage of the proof:

§3.1.1: We will deﬁne harmonic maps hs on compacta of a punctured Riemann surface Σ.

1. Deﬁne a genus g Riemann surface Σ and a compact exhaustion ∪sΣs of Σ.

2 2. Deﬁne a k-sided ideal polygon Pk ⊂ H .

2 3. Deﬁne harmonic maps hs from these compacta Σs to Pk ⊂ H .

§3.1.2: We will determine conditions for sub-convergence for these maps hs.

1. Aiming to diagonalize, ﬁx r > 0 and restrict all maps hs (for s > r) to the domain

Σr.

2. Bound the energies E(Σ , h ) of these restrictions h | in terms of the energies r s s Σr

on annuli Ωr,s of the Scherk map and the hs away from the handles.

3. Deduce a sufﬁcient condition for uniform sub-convergence of the restrictions,

expressed as a difference between the energies of the restrictions h | and the s Σr

Scherk map w on annuli Ωr,s.

§3.1.3: We will produce comparison maps us to obtain a more amenable Relaxed Sufﬁcient

Condition (Lemma 3.1.6), so that we only need to bound the difference between the

energies of the us and the Scherk map w on annuli Ωr,s. Then, we cite the Energy Esti-

mate to verify that the Relaxed Sufﬁcient Condition this is satisﬁed by the restrictions

h | . s Σr

45 1. Deﬁne the parachute maps us. Their analysis is deferred to §3.4.

2. The energies of these u are compared to the energies of the restrictions h | via s s Σr

the Doubling Lemma (Lemma 3.1.5).

3. Use the Energy Estimate (Lemma 3.4.7) to estimate the energy of the parachute

maps.

§3.1.4: We will provide a proof of Theorem 1.1.1.

1. Use the estimates from §3.1.3 to show that the restrictions h | satisfy the Re- s Σr

laxed Sufﬁcient Condition (Lemma 3.1.6) for sub-convergence from §3.1.2.

2. Check for non-collapsing of the limit.

3. Analyze the Hopf differential and ﬁnish proof of 1.1.1.

3.1.1 Set-up for construction

Notation and set-up

Let us begin by establishing some notation for various domain and range spaces involved,

and collect some required known results. For H2, we will use the Poincare´ unit disk model ! 2 2 H = {|ζ| < 1|ζ ∈ C}, p dζ . 1 − |ζ|2

Scherk map: There is a harmonic map w : C → H2 having Hopf differential Φw(z) =

zk−2dz2 whose image is the ideal k-gon in H2 and which is symmetric with respect to the

x-axis, in the sense that pre- and post- composition with the x-axis reﬂectional symmetries

on C and H2 leaves the map invariant (existence by Theorem 2.4.1).

This map can be realized as the composition of the projection of the Scherk-type min-

imal surface in H2 × R onto H2 following the parameterization from C of the minimal

surface [Wol07]. The minimal surface is conformally of type C because it has ﬁnite total

curvature. The Scherk surface exists [MRR11] and has a polynomial Hopf differential [? ].

46 ag a0 a1 ag-1 ag

a0 d1 c1 c1 d1 ··· dg cg cg dg b0

bg b0 b1 bg-1 bg

- + - + H H Hg Hg L 1 1

Figure 3.1: Identiﬁcation space model for Σg

That its Hopf differential is −zk−2dz2 follows by study of symmetric maps in [ST02] and

[AW05].

Domain data: The punctured Riemann surface serving as our domain is constructed as an identiﬁcation space, and is denoted by Σ = Σg. The conformal model for Σ used in this construction is obtained by identifying 2g + 1 subsets of C by translation:

1 1 i 1 1 i L = C \ − , + ∪ − , − 2 2 2 2 2 2 j 1 1 i i H = z ∈ C − ≤ Re(z) ≤ , − ≤ Im(z) ≤ + 2 2 2 2 j 1 1 i i H = −z ∈ C − ≤ Re(z) ≤ , − ≤ Im(z) ≤ − 2 2 2 2 for 1 ≤ j ≤ g. The identifications are depicted in Figure 3.1, with decorations indicat- ing opposite sides of a geodesic segment in the identification. We will denote by Σs the compact subset g h j j i Σs := (Ωs ∩ L) ∪ ∪ H+ ∪ H− j=1 of Σ obtained by the same identifications. Note that

∞ Σ = ∪ Σs s=1 provides a compact exhaustion of Σ.

Other domains: Let Ωr be the disk {z | |z| ≤ r} ⊂ C, so that it has boundary γr = ∂(Ωr).

Let Γs denote the image of γs under w. The curves Γs := w(γs) parameterized by w|γs will

47 serve as boundary data for the sequence of harmonic mappings hs on compacta of the punctured Riemann surface Σ. The Scherk map is deﬁned on Ωr, and the parachute map will be deﬁned on the cylinder Ωr,s. We will also identify Σs \ Σr by Ωr,s for all s > r.A depiction of the Scherk map is provided in Figure 3.2.

Γs γs

Figure 3.2: The Scherk map w : C → H2 with Φw = −z2dz2.

Target data: The regular ideal k-sided polygon Pk we will use as the target for our harmonic maps is given by the convex hull of the the ideal points

2 {ξ0, ξ2,..., ξk−1} ⊂ ∂H where we are using the Poincare disk model for H2 and

2πi 1 2πi j ξj := e 2k · e k .

It is ﬁxed set-wise by the involution reﬂecting over the x-axis and also by the rotations

1 2π 2πi 2k about the origin by integer multiples of k . Note the factor of e to create the reﬂectional

symmetry over the x-axis. The polygon Pk is depicted in Figure 3.3 for various values of k.

Comparison model (the parachute maps): Along the way, we will study harmonic maps us

2 from cylinders into H which have comparable energy to the harmonic mappings hs of Σs on the subsets Ωr,s ≡ Σs \ Σr (for r < s). This allows us to pass from studying the (energy of the) harmonic mapping of Ωs to harmonic mappings of Ωr,s - reducing the consideration

of a punctured torus domain to an analytically and topologically simpler annular domain.

48 (a) P4 (b) P6 (c) P8

Figure 3.3: Example Pk for k = 4, 6, and 8.

2 Sequence {hs : Ωs → H } of harmonic mappings on compacta

1 When r > 2 , the domain Ωr is a neighborhood of the handles formed by identifying

g h j j i 1 ∪j=1 H+ ∪ H− to L. So, ﬁx r >> 2 ; the precise size of r will be chosen later, when it matters in estimating the geometry of the boundary values Γr we will prescribe.

For each s > r, we pose the Dirichlet harmonic mapping problem

2 hs : Ωs → H (Ds)

hs|γs = w|γs noting the parameterized boundary condition. This boundary condition makes sense to

impose because, although the domains for hs and w|Ωs (being Σs and Ωs) are different, their charts have γs as identical boundaries. There is a unique solution hs to (Ds) which minimizes energy in its homotopy class ﬁxing the boundary, shown to exist by Lemaire in

[Lem82].

This sequence of harmonic mappings

2 hs : Σs → H is the focus of our construction, from which we will extract a sub-convergent sequence. We note that this sequence of harmonic maps has an increasing amount of energy E(Σs, hs) %

+∞, so that it is not immediate that there should exist a convergent subsequence.

49 3.1.2 Conditions for sub-convergence

Since we wish to show that hs converges uniformly on compact subsets, we now restrict the maps hs to a common compact domain Σr, for a ﬁxed r and for all s > r. We appeal

to the Arzela-Ascoli Theorem to guarantee a convergent subsequence of {hs|Σr } on each

Σr, i.e., it sufﬁces to show that {hs|Σr } is uniformly bounded and equicontinuous. These

will both follow from an application of the Courant-Lebesgue Lemma ?? and a geometric

consideration of hs. We can pin down one point so that its image does not escape to inﬁnity

along the entire subsequence:

Proposition 3.1.1. For all s, we have dH2 (hs(0), O) ≤ C(k), where the constant C(k) is given

by π π ! 1 + sec k − tan k C(k) := log π π 1 − sec k + tan k and mathcalO is the origin in H2. In particular, using the Poincare disk model for H2, we have that 0 ∈ L maps to a point of {x + i · 0 | |x| < C(k)} ⊂ H2.

Proof. Recall that the polygon Pk has the ideal points

2 {ξ0, ξ2,..., ξk−1} ⊂ ∂H

where

2πi 1 2πi j ξj := e 2k · e k .

Indexed this way, the involution

ι : H2 → H2

p 7→ p

takes ξj 7→ ξ(k−j) mod k. Note that k is even, so there are no ideal ﬁxed points.

Observe that the maps hs(z) and hs(z) agree pointwise for all z ∈ γs = Σs. Since there is a unique energy minimizing solution to the Dirichlet boundary value problem (Ds), it

50 must be that

hs(z) = hs(z) on the interior of Σs as well. Thus, we must have

hs(0) = hs(0).

This implies that hs(0) is real. Since hs(Σs) ⊂ Pk, the image of 0 ∈ L must lie in the geodesic segment

Pk ∩ {x + i · 0|x ∈ (−1, 1)} π π π π ≡ tan − sec , sec − tan × {0} ⊂ H2 k k k k along the x-axis is ﬁxed by the isometry i. That the endpoints are these follows from an elementary exercise in hyperbolic geometry. Finally, note that the length of this line segment is π π ! 1 + sec k − tan k 2 log π π , 1 − sec k + tan k and that O is the midpoint of this line segment.

We now deduce a sufﬁcient condition for sub-convergence of our sequence of restrictions h | : s Σr

Proposition 3.1.2. If E(Ωr, hs) < M(r) is bounded for all s > r, then {hs|Ωr } is precompact.

In the following sections, the bulk of our work will be toward obtaining an energy bound of the form E(Ωr, hs) < M(r) for all s > r.

Proof. With the energy bound in hand, the Courant-Lebesgue Lemma implies that hs|Ωr is

equicontinuous and uniformly bounded, because E(Ωr, hs) < M gives us the inequality s 8πM d 2 (h (z), h (w)) ≤ . H s s log (µ−1)

51 By applying the triangle inequality and Proposition 3.1.1, the uniform bound

dH2 (hs(z), O) = dH2 (hs(z), hs(0)) + dH2 (hs(0), O) s 8πM ≤ + C(k) log (µ−1)

for all z, w ∈ Ωr, and where the constants µ = µ(r) is deﬁned as in Theorem ?? depending

only on the diameter of Σr and C(k) is deﬁned in Proposition 3.1.1. Thus, we can apply the

Arzela-Ascoli theorem to the family {hs|Ωr }.

Relating sub-convergent condition to the Scherk maps

For now, let us ﬁrst consider hs on its full domain Ωs, on which we can obtain a crude energy bound.

Proposition 3.1.3. For all s > 1, we have the energy upper bound

E(Σs, hs) ≤ E(Ωs, w) + K where the constant K = K(g, k) depends only on g and k (through w) and is given by

1 K = 2g · E(H+, w) < ∞

2 Proof. To estimate E(Ωs, hs), let us construct a map vs : Ωs → H satisfying the boundary conditions for the Dirichlet harmonic mapping problem (Ds), and whose energy we can compute. Since vs will be a candidate to the mapping problem (Ds), and hs minimizes energy among all candidates to (Ds), we will have

E(Ωs, hs) ≤ E(Ωs, vs).

To deﬁne an appropriate vs, recall that the conformal model for Σ (and Σs by restriction) is

52 given by identifying the 2g + 1 pieces:

1 1 i 1 1 i L = C \ − , + ∪ − , − 2 2 2 2 2 2 j 1 1 i i H = z ∈ C − ≤ Re(z) ≤ , − ≤ Im(z) ≤ + 2 2 2 2 j 1 1 i i H = −z ∈ C − ≤ Re(z) ≤ , − ≤ Im(z) ≤ − 2 2 2 2 for 1 ≤ j ≤ g. All of these can be considered as subsets of C, the domain for the Scherk

2 j map w. So, we can deﬁne a map vs : Ωs → H piecewise by restricting w to L, H+, and

j H−. Respecting changes in orientation, this means, for 1 ≤ j ≤ g:

vs|L(z) := w|L(z)

vs| j (z) := w| j (z) H+ H+

vs| j (z) := w| j (−z) H− H−

0 0 To be clear, vs is deﬁned so that vs |Ωs = vs for all s > s. Hence,

g h j j i E(Ωs, vs) = E(L ∩ Ωs, w) + ∑ E(H+, w) + E(H−, w) j=1

⇒ E(Ωs, vs) ≤ E(Ωs, w) + K where the constant K is deﬁned by

g h j j i K := ∑ E(H+, w) + E(H−, w) j=1

1 = 2g · E(H+, w) < ∞

g h j j i is simply the energy of the Scherk map on the pieces that form the handles ∪j=1 H+ ∪ H− for the domain Ωs. Thus, we have

E(Σs, hs) ≤ E(Σs, vs)

≤ E(Ωs, w) + K

53 Recall that in order to obtain a convergent subsequence, we need to verify the hypoth-

esis for Proposition 3.1.2: for an arbitrary ﬁxed r, the restrictions {hs|Σr }s≥r have uniformly bounded energy. However, we only have an energy estimate for hs on its entire domain of deﬁnition Σs by Proposition 3.1.3. We do not have a priori estimates on the energies of the restricted maps.

Our ﬁrst step to assuage this problem is by producing an a priori bound on E(Σs, hs) in terms of the energies of other harmonic maps which we can better understand - the Scherk

map w and the restriction hs|Σr,s . By decomposing the domain Σs as Σs = Σr ∪ Σr,s, and applying the Proposition 3.1.3, the energies of the restrictions can be bounded above by comparing maps of annuli:

Proposition 3.1.4. There exists a non-negative continuous function G(r) so that, for all s > r, we have:

E(Σr, hs) ≤ E(Ωr,s, w) − E(Σr,s, hs) + G(r).

Proof. The proof is entirely algebraic. Observing that Σs = Σr ∪ Σr,s, we deduce:

E(Σs, hs) = E(Σr, hs) + E(Σr,s, hs) since Σs = Σr ∪ Σr,s

⇒ E(Σr, hs) = E(Σs, hs) − E(Σr,s, hs) by rearranging terms

≤ [E(∆s, w) + K] − E(Σr,s, hs) by Proposition 3.1.3

≤ [E(∆r, w) + E(Ωr,s, w) + K] − E(Σr,s, hs) since ∆s = ∆r ∪ Ωr,s

⇒ E(Σr, hs) ≤ E(Ωr,s, w) − E(Σr,s, hs) + G(r)

Taking G(r) := E(Ωr, w) + K, where K was deﬁned in Proposition 3.1.3, we have our desired result.

This inequality estimates the growth of E(Ωr, hs) for all s > r. Observe that the right hand side compares the energies of the Scherk map w and the Dirichlet solution hs, both restriction to the common domain Ωr,s. Note that Ωr,s ≡ Σs \ Σr = Σr,s, so we will begin

54 to use these symbols interchangeably. In the following, we work toward bounding the difference

E(Ωr,s, w) − E(Ωr,s, hs).

3.1.3 A comparison map for hs away from the handles

We will now construct a comparison map whose energy will bound E(Σr,s, hs) from below.

This is desirable because such a lower bound in turn bounds E(Ωr,s, w) − E(Ωr,s, hs) from above.

Loosely speaking, we will show that h | has almost as much energy as w| (or, s Ωr,s Ωr,s said differently: we are trying to show that hs does not concentrate most of its energy in Σr around the handles, and that it spreads out its energy as well as w does over the annulus

Σs \ Σr ≡ Ωr,s).

2 Parachute maps: from compact annuli Ωr,s to H

Observe that hs|Ωr,s is a candidate to the partially-free Dirichlet boundary value harmonic mapping problem

2 us : Ωr,s → H

(PFDrs) us|γs = w|γs

us|γr free

It is well known (see, for example, [Ham75]) that critical points us of the energy functional

(which are critical with respect to variations of the map ﬁxing the Dirichlet boundary γs, without restriction on the free boundary γr) satisfy

(∇us)|γr (ν) = 0,

where ν is the outward pointing normal of Ωr,s along γr. A priori, it is not evident that hs satisﬁes this condition on γr.

55 In any case, we call the unique solution of our toy problem (PFDrs) a parachute map

us. Note that it is an energy minimizer among all candidates to the harmonic mapping

problem (PFDrs), so that the energy of hs|Ωr,s is no greater than that of us.

Here, we note that, but uniqueness of the solution to (PFDrs), the parachute maps respects the same Dk dihedral symmetries that the boundary curve Γs respects. In particular, we have the identities

ξ−j · us(z)us(ξjz)

us(z) = us(z)

2πi j for all z ∈ Ωr,s and for all ξj = e k , for j = 0, 1, . . . , k − 1.

Comparison models estimate EΩr,s (hs)

The crux of the argument is a doubling lemma which allows us to recast the partially- free/Dirichlet boundary value problem as a Dirichlet/Dirichlet boundary value problem on a doubled domain and appropriately reproduced boundary conditions (where the boundary conditions correspond to the inner/outer boundary components).

Lemma 3.1.5. (Doubling Lemma) Any critical point of (PFDrs) can be reflected to define a critical point of a Dirichlet harmonic mapping problem (defined below) that only has one critical point

(which also minimizes energy).

Hence, producing a critical point of (PFDrs), actually produces an energy minimizer among candidates to (PFDrs). We will apply this lemma after its proof.

Proof. This is proven by a series of observations.

1 (0) The domain Ω is conformally equivalent to Ω s under the scale . So, the rescaled r,s 1, r r mapping problem on the domain Ω s is equivalent to (PFDrs) by parameterizing the 1, r boundary appropriately under the rescaling, i.e., f |γs/r (r · z) = w|γs (z).

56 2 (1) The critical points f : Ω s → H of the equivalent (PFDrs) mapping problem 1, r

2 f : Ω s → H 1, r

f |γs/r = w|r·γs/r

f |γ1 free

are harmonic and satisfy (∇ f )|γ1 (ν) = 0, where ν denotes the outward normal along γ1.

The curve γ appears as the free boundary of Ω s and so, by the regularity of f on γ , f 1 1, r 1

can be reﬂected across γ onto Ω r to produce a map 1 s ,1

2 f˜ : Ω r s → H s , r using a Schwarz Reﬂection Principle.

(2) The map f˜ can be considered as a candidate for the Dirichlet harmonic mapping problem

2 f˜ : Ω r s → H s , r 1 f˜| (z) = w| · (Drs) γr/s r γs/r z ˜ f |γs/r (z) = w|r·γs/r (z).

Observe that the orientation is reversed on the boundary γr/s to preserve the dihedral symmetry (in the target) of the mapping problem. It is automatically satisfied by definition of f˜ by reflection of f across γ1.

(3) Observe that (Drs) has a unique solution F˜ which, in particular, minimizes energy

among all maps satisfying the Dirichlet boundary condition. Furthermore, it is the unique

critical point. So, it must be that f˜ = F˜.

(4) Thus, there is a unique critical point F of the (PFDrs) mapping problem. This critical

˜ point F minimizes energy, and can be realized as a rescaling by r of the restriction F|Ω s 1, r since the domains are conformal, i.e.,

Ω = r · Ω s . r,s 1, r

57 In other words, the solution to (PFDrs) is given by

˜ us(z) = F|Ω s (r · z). 1, r

since (∇us)|γ1 (ν) = 0. This completes the proof of the Doubling Lemma.

Let us return to studying the energies of the restrictions h | . Denote by u the solution s Σr s to (Drs). The Doubling Lemma 3.1.5 implies

EΩr,s (us) ≤ EΩr,s (hs).

Recall from Proposition 3.1.4 that

E(Σr, hs) ≤ E(Ωr,s, w) − E(Ωr,s, hs) + G(r).

Combining these, we ﬁnally arrive at

Lemma 3.1.6. (Relaxed Sufﬁcient Condition) For each s > r, and the for the maps hs, parachute map us, and Scherk map w deﬁned above, we have

E(Σr, hs) ≤ E(Ωr,s, w) − E(Ωr,s, us) + G(r), where G(r) is the function deﬁned in 3.1.4.

This inequality allows us to ﬁnally control the energies of the restrictions solely in terms of a comparison between the energies of the Scherk map w and the parachute maps us. We will now use the Energy Estimate to bound E(Ωr,s, w) − E(Ωr,s, us):

Lemma 3.4.7. (The Energy Estimate) There is a continuous increasing function F(r) such that, when r is large enough, for any s > r, the following inequality holds:

E(Ωr,s, w) − E(Ωr,s, us) < F(r)

We defer its proof to §3.4.5.

58 3.1.4 Putting it all together

With this in hand, we are ﬁnally able to diagonalize {hs}s≥r on the domains Ωr to obtain a convergent subsequence.

Proof. Fix r > 1 large enough to satisfy the hypotheses of the Energy Estimate (Lemma

3.4.7). For each s > r, the energy of the map { h | } can be bounded by s Σr s≥r

E(Σr, hs) ≤ E(Ωr,s, w) − E(Ωr,s, us) + G(r) by Proposition 3.1.6

≤ F(r) + G(r) by the Energy Estimate (Lemma 3.4.7) where F(r) is deﬁned in the Energy Estimate (Lemma 3.4.7) and G(r) is deﬁned in Propo- sition 3.1.6. Thus, by Proposition 3.1.2, the family { h | } is precompact. s Σr s≥r

Non-collapsing of the limit hs → h

With the uniform energy upper bound E(Ωr, hs) < M(r), we can take a limit of the harmonic maps using a diagonalization process. However, we face the possibility of hs collapsing a subset of the domain Ω. The classiﬁcation of local singularities for maps between surfaces by Wood is collected in his paper [Woo77], although the derivations are more thor- ough in his thesis [Woo74].

We have to rule out the degenerate possibility that the limiting map has an image which does not contain any open disk, i.e., that h is a piecewise constant map (with a

0−dimensional image) or maps to a union of geodesic segments. It sufﬁces to exhibit an energy growth rate of E(Σs, hs) bound from below.

Recall that the comparison models us satisfy:

E(Ωr,s, us) ≤ E(Ωr,s, hs) by Lemma 3.1.5

E(Ωr,s, w) − E(Ωr,s, us) ≤ F(r) by Lemma 3.4.7

59 Combining these, we see that there is a lower bound on E(Ωr,s, hs):

E(Ωr,s, w) − F(r) ≤ E(Ωr,s, hs)

since E(Ωr,s, w) is of order k polynomial growth in s. This, in turn, bounds E(Σs, hs) from below since E(Ωr,s, hs) ≤ E(Σs, hs). Finally, since the maps hs converge uniformly on compact subsets to h, the energy densities E(Σs, hs) converge to E(Ωr,s, h), and we deduce that

E(Σs, hs) is bounded from below as a function of s.

Hopf differential Φh and energy eh of h

Now, we turn to studying the limiting Hopf differential Φh. Recall that

ρ(h(z)) Hh := ||∂h||2 = |h |2 σ,ρ z σ(z) ρ(h(z)) Lh := ||∂h||2 = |h |2 σ,ρ z σ(z) respectively denote the holomorphic and anti-holomorphic energies of h. Observe that we have a pointwise inequality which helps us bound the norm of ||Φh||:

||Φh||2 = HhLh

≤ (Hh + Lh)2

≤ (eh)2

So, we have at most polynomial growth of

Z ||Φh||dvol Σr

of order k as a function of r. With the lower bound on energy growth rate from the non-

collapsing discussion above, we see that in fact the polynomial growth rate is k (otherwise,

the map must limit to strictly fewer than k ideal geodesics). This implies that Φh has a pole

of order k + 2.

60 Let us apply the Riemann-Roch theorem to deduce the number of zeros of Φh. The degree of the square of the canonical line bundle is equal to the number of zeros minus the number of poles of a non-vanishing meromorphic quadratic differential. Thus,

#{zeros} − #{poles} = 4g − 4 where g is the genus of the compact surface. Hence, there are (4g − 4) + (k + 2) zeros of

Φh on Σ. This proves the ﬁnal assertion of Theorem 1.1.1, describing the Hopf differential

h Φ of h : Σg → Pk.

3.2 Uniqueness: Proof of Theorem 1.1.2

Let us now turn to discussing the existence of the maps established in Theorem 1.1.1. We

prove:

2 Theorem 1.1.2. Suppose h, v : Σg → H are two harmonic maps. Denote their pointwise distance

function by d(z) := distH2 (h(z), v(z)). If we have

p (cosh ◦ d) − 1 ∈ L (Σg) for some some p ∈ (1, +∞], then d(z) ≡ 0, i.e., the maps v and h must agree pointwise.

Proof. Suppose we have two harmonic maps h, v : Σ → H2 and denote their pointwise distance function by d(z) := distH2 (h(z), v(z)). Our goal is to show that d ≡ 0. Observe that it is equivalent to show that the function

Q(z) := cosh(d(z)) − 1 vanishes identically on Σ. Since the target manifold H2 is non-positively curved and simply connected, Q is smooth as a function of z. We will now recall a computation from

[HW97] to obtain a sign on the Laplacian of Q, using the Riemannian chain rule.

61 Recall that the Riemannian chain rule states, for a map

h i g ∆M( f ◦ g)(z) = trM Hess( f )|g(z) ◦ ( Dg|z , Dg|z) + ∇N f |g(z) ( τ |z)

Let us choose a convenient basis for the computation. Given two points p and q in H2, there is a unique geodesic segment γpq between them. There are natural frames associ-

2 2 2 p ated to the tangent spaces TpH and TqH associated to γpq: TpH = span(Tanp, N ) and

2 q TqH = span(Tanq, N ), where Tanp and Tanq are unit tangent vectors to γpq at the points

p q p and q, respectively, and where N and N are unit normal vectors orthogonal to Tanp and Tanq, respectively.

p q Then, using the local orthonormal frames {e1, e2} for TΣ and {(Tanp, 0), (0, N ), (Tanq), (0, N )}

2 2 for T(p,q)H × H , we have:

HessH2×H2 cosh ◦ d = (cosh ◦ d)I4×4 + A where A is a 4 × 4 matrix whose only non-zero entries are −1 at ((Np, 0), (0, Np)) and

q q ((N , 0), (0, N )), and a −cosh ◦ d at ((Tanp, 0), (0, Tanq)) and ((Tanq, 0), (0, Tanp)).

Thus, the Laplacian of Q can be bounded:

2 ∆Q(z) ≥(cosh ◦ d(z)) h∇h, Tanh(z)i − h∇v, Tanv(z)i h i + (cosh ◦ d(z) − 1) h∇h, Nh(z)i2 + h∇v, Nv(z)i2

h v + hgradQ, (τ , τ )i(h,v)(z)

u(z) u(z) where h∇u, Tan i denotes the projection of du(e1) + d(e2) onto Tan .

Since h and v are harmonic, the vector ﬁeld (τh, τv) vanishes. This implies that the function Q(z) is subharmonic on Σ. Now, notice that Σg is non-compact and yet we have

∆Q(z) ≥ 0

Q(z) ≥ 0

Q(z) ∈ Lp(Σ)

62 Hence, we investigate the conditions under which d is forced to be a constant.

It is a classical fact from complex analysis that all bounded, non-negative sub-harmonic functions on a parabolic domain must be constant. In particular, Σg is a parabolic domain because its conformal model for the puncture can be described by C \{0}. So, if p = +∞, then Q must be constant. Since Q is also in Lp, it must be that Q vanishes identically.

For p ∈ (1, +∞), it was shown in [Yau75] that any non-negative sub-harmonic function on a complete Riemannian manifold must be constant. It similarly follows that, in this case,

Q vanishes identically.

3.3 Example: punctured square torus squashed onto ideal square

In the following, we will consider the special case of the square torus in which we can determine the location of the zeros of the Hopf differential, proving Corollary 1.1.3. After this section, the remainder of the thesis is devoted to the details of the energy estimate

Lemma 3.4.7.

Proof. Let g = 1 and k = 4 in the hypotheses of Theorem 1.1.1, and let h : Σ1 → P4 denote the harmonic handle crushing map of the square punctured torus to the ideal square. By the D4 dihedral symmetry of Σ1, each of the maps hs in the construction of h is symmetric with respect to the same D4 symmetries. So, the limiting map h also respects the D4

symmetry group. This is depicted in Figure 3.4; note the domain identiﬁcation space Σ1.

By the Riemann-Roch theorem, we establish that there are 6 zeroes, counting multiplic-

h ity, of Φ on Σ1. Observe that any point p not on a line of symmetry of Σ1 cannot be a zero

h h of Φ because the D4 symmetry of the map would yield 7 other zeros of Φ (the orbit of p

h under the 8 symmetries of D4 would yield 7 other zeroes). Hence, the zeroes of Φ must

lie on the lines of symmetry.

63 h

Figure 3.4: Lines of symmetries for h : Σ1 → P4 from Corollary 1.1.3.

P Q

h Figure 3.5: Possible locations for zeroes of Φ of h : Σ1 → P4 handle crusher.

Now, there are four kinds of possible locations for a zero of Φh. We draw the possibilities in Figure 3.5, in a triangular fundamental domain of a conformal model of Σ1 with respect to the D4 symmetries. We label the possible locations P, Q, R, and S.

If a zero P of order nP lies on a vertical line of symmetry, but not on the diagonal line of symmetry, and in the interior of the vertical line of symmetry (depicted on the vertical edge of the triangular fundamental domain), there would be 3 other zeroes of order nP by

the diagonal symmetries of D4.

If a zero Q of order nQ lies on a diagonal line of symmetry, but not on the vertical

and horizontal lines of symmetry, then there would be 3 other zeroes of order nQ by the

horizontal and vertical symmetries of D4.

64 If a zero R of order nR lies on the vertical or horizontal line of symmetry, but not on

the diagonal line of symmetry, and on the endpoint of the vertical or horizontal line of

symmetry (depicted at the right angle of the triangular fundamental domain), there would

be 1 other zero of order nR by the diagonal symmetries of D4.

The ﬁnal possible location for a zero S is at the ﬁxed point of the D4 symmetries (depicted at the top vertex of the triangular fundamental domain). Such a zero of order nS

h does not necessitate other zeroes of Φ by the D4 symmetry.

Hence, we must have 4nP + 4nQ + 2nR + nS = 6. Note that nP, nQ, nR, and nS must be non-negative integers. So, nP and nQ are at most 1. Observe that the order nS must be

h even, since the foliation of Φ near S must be ﬁxed by the D4 symmetries. Hence, nS is

either 0, 2, 4, or 6. By a case by case analysis, we must have one of the arrangements:

(a) three zeros of order 2, with (nP, nQ, nR, nS) = (0, 0, 2, 2)

(b) or one zero of order 4 and two of order 1, with (nP, nQ, nR, nS) = (0, 0, 1, 4),

This establishes the three possible arrangements.

Now, suppose Φh has the divisor given by arrangement (a). Let us study the orientation

of the map h at the zeroes of Φh. In arrangement (a), the Hopf differential Φh has a zero of

order two at each of the midpoints 0 in L, H+, and H−. Let us analyze the orientation of

the map at each of these zeros. Consider the curves

ηh := {(s, 0)| − 1 ≤ s ≤ 1} ∪ {(−s, 0)| − 1 ≤ s ≤ 1} ⊂ H+ ∪ H−

ηv := {(0, t)| − 1 ≤ t ≤ 1} ∪ {(0, −t)| − 1 ≤ t ≤ 1} ⊂ H− ∪ L

These curves generate the homology of Ω. Furthermore, by the symmetries of the map,

65 we have

2 h(ηh) ⊂ {(x, 0)} ⊂ H

2 h(ηv) ⊂ {(0, y)} ⊂ H

Since the diagonal isometry of Σ1 taking 0 ∈ L to 0 ∈ H+ and the diagonal isometry on

H2 are both orientation-reversing, and since pre- and post-composition of h by these maps

preserves h, the orientation of h at both of these points are the same. By this, we mean

J ≥ 0 at these points. Now, consider the Jacobian along the curve ηh.

We argue that the sign at 0 ∈ H− is opposite to the sign at 0 ∈ H+ (and 0 ∈ L). Consider mapping half of ηh, which connects 0 ∈ H+ to 0 ∈ H− along the right side. It gets mapped into a curve along the positive real axis of H2 which is a loop at 0 ∈ H2. Tracing the derivative of this curve, we see that the tangent vector to the curve must change sign upon returning to 0 ∈ L. Hence, J h ≤ 0 here. Hence, the orientation of h at two of its zeroes

(each of order two) are the same, while the orientation of h at its other zero (of order two) is reversed.

3.4 The toy problem and The Energy Estimate

This section is devoted to studying the solution of a toy problem. This solution, the parachute map, was introduced in §3.1.3 to establish a priori bounds on the energy of the harmonic mapping of a compactum of the punctured Riemann surface. The energy bound hinges on the geometric quality that the free boundary of the parachute maps us stays in a bounded set independent of s.

In §3.4.2 and §3.4.3, we describe the geometry of the parachute map and also derive restrictions on the Laurent expansion of the Hopf differential. These will be applied in conjunction in §3.4.4, which is the heart of this section. We provide an outline of this section:

66 §3.4.1: Deﬁne the parachute maps.

1. Pose and solve the toy problem whose solutions are the parachute maps.

2. Describe symmetries of parachute maps.

3. Observe that the boundary condition at the core curve is free.

§3.4.2: Describe the injectivity of the maps.

1. Use Theorem 2.4.2 (Wood’s Gauss-Bonnet formula for the image of harmonic

maps) to show that the parachute maps are local diffeomorphisms.

2. Note that this lets us ﬁnd disks of increasing radius in the domain along the

sequence of parachute maps, from which the parachute maps are diffeomor-

phisms.

§3.4.3: Analyze Hopf differentials of the parachute maps through their symmetries.

1. Express the Hopf differential as a Laurent expansion.

2. Show that the coefﬁcients are symmetric with respect to the −2 index.

3. Observe that `2(C)-norm of the tails (indices outside [−k − 4, k]) is ﬁnite on the

core curve.

4. Deduce that the index −k − 4, −2, and k coefﬁcients determine boundedness of

|φs| on the core curve.

5. Obtain a uniform bound of the Hopf differential on the core curve (the free

boundary) by considering those three coefﬁcients.

§3.4.4: Show that the core curve mapped by any parachute map stays in a bounded set.

1. Observe that the Hopf differential norm equals the energy density on the core

curve.

67 2. Show that the sequence of Hopf differentials have bounded norm on the core

curve.

This involves an argument by contradiction: we use a blow up argument on a sequence

of disks of increasing radii in the Hopf differential norm, provided by §3.4.2.

3. Use the uniform bound on Hopf differentials on the core curve to obtain the

Bounded Core Lemma.

§3.4.5: Use this pointwise bound on the free boundary to derive the Energy Estimate (Lemma

3.4.7).

1. Use the Bounded Core Lemma to deduce that the pointwise distance between

the parachute map and the Scherk map is uniformly bounded (compared on

any common annular domain).

2. Obtain a gradient norm estimate for the parachute map along the core curve by

Cheng’s interior gradient estimate, which relies on the uniform bound between

the images of the parachute map and the Scherk map to choose appropriate

constants.

3. Observe that the parachute map energy density on the core curve reduces to

simply the tangential energy of the core curve, since the core curve is a free

boundary.

4. Bound the energy of an auxilliary harmonic map from a disk which “ﬁlls in”

the core curve using Choe’s Iso-energy inequality for harmonic maps (relating

interior 2-dimensional energy to 1-dimensional energy of the boundary).

5. Observe that the Scherk map on Ωs has less energy than a “ﬁlled in” harmonic

extension of u | into Ω . s Ωr,s r

6. Re-arrange the Scherk map energy bound to deduce the Energy Estimate (Lemma

3.4.7).

68 3.4.1 Parachute maps

Recall the partially free boundary value harmonic mapping problem

2 us : Ωr,s → H

(PFDrs) us|γs = w|γs

us|γr free whose solutions us we call parachute maps. These parachute maps were introduced in

§3.1.3 as comparison maps for h | , having the same image on the γ boundary compo- s Ωr,s s nent. In this section, we analyze the discrepancy of energy between the parachute maps and the Scherk map, E(us, Ωr,s) − E(w, Ωr,s), as a function of s, for all s greater than any

1 given ﬁxed r > 1. Without loss of generality, we rescale Ωr,s by a factor of r to avoid the

s factor of r in the following analysis.

The parachute maps us are closely tied to the Scherk map w because the Dirichlet

boundary condition on γs for us is obtained from w|γs . Note that the energy of us is less than that of w on their common domain of deﬁnition, since us solves the partially free boundary value problem. We are interested in the magnitude of this difference.

One can imagine the sequence of restrictions w|Ω1,s of the Scherk map converging (identically) to the Scherk map as s % ∞. On the other hand, the sequence of parachute maps

us (deﬁned on Ω1,s) agree with w|Ω1,s on the γs boundary component, but disagree on the

γ1 boundary component. For this reason, we chose the moniker parachute, since we are re- minded of the children’s toy parachute getting stretched along its outer boundary twoards inﬁnity, and the inner boundary moves freely to minimize the overall stretching of the fabric.

In comparing the energy of the parachute map to the Scherk map, then, we have a heuristic geometric description: this suggests their energies should be comparable from a geometric description: if us(γ1) remains a bounded distance from w(γ1) for all s, then the

69 difference in energy between the parachute maps and the Scherk map (on their common domains of deﬁnition) should be small; if us(γ1) approaches us(γs) as s % ∞, then the difference in energy will be great.

u Our analysis involves studying the Hopf differential Φ s of us. However, it is prudent to study the Hopf differential of the doubled parachute maps, since an extra symmetry imposes a convenient analytic condition (see Proposition 3.4.3). Recall that the Doubling

Lemma 3.1.5 realizes the parachute maps us as restrictions of the solutions of a Dirichlet-

Dirichlet boundary value harmonic mapping problem

2 us : Ω 1 → H s ,s

us|γs (z) = w|γs (z) 1 u | (z) = w| . s γ1/s γs z

We use the names us to mean either the parachute map or its double without ambiguity, since we will always specify the required domain of deﬁnition. We abbreviate Φs ≡ Φus .

The curve γ1 will be referred to as the core curve.

3.4.2 Geometry of images

In this section, we study the behavior of the parachute maps.

Proposition 3.4.1. The parachute maps us have a non-negative Jacobian on int (Ω1,s).

Proof. Observe that the cylinder Ω1,s has Euler characteristic 0. Let us denote the subsets

+ 0 − on which us has positive, zero, and negative functional Jacobian by Ω1,s, Ω1,s, and Ω1,s, + respectively. Note that us is orientation-preserving along the boundary γs, so Ω1,s is not − empty. Suppose for a contradiction that Ω1,s is non-empty.

0 Since Ω1,s is the zero set of a real analytic function, it is a collection of points and curves. + − Hence, Ω1,s and Ω1,s are multiply connected planar domains. In particular, this implies that

+ − χ us Ω1,s = χ us Ω1,s ∈ Z≥0

70 − Observe that the parachute map us restricts onto Ω1,s with a functional Jacobian with a − − deﬁnite sign. So, it is a local diffeomorphism from Ω1,s onto its image us Ω1,s . This im- − h − i plies that us Ω1,s is also a multiply connected planar domain. Note also that ∂ us Ω1,s is a collection of curves for which the geodesic curvature vector points outside of the region, by the maximum principle for harmonic maps [JK79]; this fact uses that we are con-

− + sidering Ω1,s and not Ω1,s.

Thus, by Theorem 2.4.2, we must have

− − K us Ω1,s ≥ 2πχ us Ω1,s ≥ 0.

− − We arrive at a contradiction, since us Ω1,s has negative Gauss curvature. Therefore Ω1,s

u must be empty, and J ≥ 0 on Ω1,s.

3.4.3 Hopf differentials of parachute maps

In this section, we analyze the pointwise norm of the Hopf differential of the parachute maps along the core curve. The aim of the analysis is to obtain the following:

Proposition 3.4.2. For the parachute maps us described above, there exists a positive constant

M < ∞ and a sequence of indices {s}% ∞ so that, for all z ∈ γ1, we have

|φs(z)| < M.

We will study the Hopf differential through its Laurent expansion. So, it will be useful to observe a symmetry of Φs induced by the reﬂectional symmetries:

1 Proposition 3.4.3. Each parachute map us respects the two reﬂectional symmetries us z =

s us(z) and us(z) = us(z). These symmetries induce a symmetry on the Hopf differential Φ (z) ≡

φs(z)dz2: 1 1 φs(z) = φs . z z4

71 Proof. Fix any s. We will drop the index s in this proof since it is not essential, and only clutters notation. First, note that φ(z) = uz(z) · uz(z). Thus, it is equivalent to show that

1 1 1 u (z) · u (z) = u · u · z z z z z z z4

We will proceed in polar coordinates. First, note that the symmetries re-expressed as:

1 1 u(z) = u ⇒ u reiθ = u eiθ z r u(z) = u(z) ⇒ u reiθ = u re−iθ

Combining these, we obtain one polar identity

1 u eiθ = u re−iθ r

Then, taking r and θ derivatives, we deduce the polar derivative identities:

1 1 − e−iθu eiθ = e−iθu re−iθ (3.1) r2 r r r 1 1 i u eiθ = −iru re−iθ (3.2) r θ r θ

∂ Note that the operator ∂z in polar coordinates becomes

∂ 1 ∂ 1 ∂ = e−iθ − i ∂z 2 ∂r r ∂θ

We will use these identities to verify the equality.

First, we compute uz(z) · uz(z) to evaluate the left-hand side: −iθ 1 −iθ 1 [uzuz]|z = e ur − iruθ · e ur − iruθ 2 z=reiθ 2 z=reiθ 1 1 = e−i2θ u (reiθ) − iru (reiθ) u (reiθ) − iru (reiθ) . 2 r θ 2 r θ

1 1 1 Second, we compute uz z , uz z , and z4 and combine them to evaluate the right- hand side: ∂ −iθ 1 u = e ur − iruθ ∂z 1 2 1 1 −iθ z z = r e 1 1 1 1 = eiθ u e−iθ − i u e−iθ 2 r r r θ r

72 and also

∂ −iθ 1 u = e ur − iruθ ∂z 1 2 1 1 −iθ z z = r e 1 1 1 1 = eiθ u e−iθ − i u e−iθ 2 r r r θ r

and of course

1 1 = e−i4θ z4 r4

Thus, their product forms the right-hand side:

1 1 −i2θ 1 1 −iθ 1 1 −iθ 1 1 −iθ 1 1 −iθ [uz · uz]| 1 = · e ur e − i uθ e · ur e − i uθ e z z4 r4 2 r r r 2 r r r 1 −1 1 1 1 1 −1 1 1 1 = e−i2θ u e−iθ + i u e−iθ · u e−iθ + i u e−iθ 2 r2 r r r3 θ r 2 r2 r r r3 θ r

Finally, we will verify that

1 1 1 u (z) · u (z) = u · u · z z z z z z z4 by using the polar derivative identities. Note that the identities (3.1) and (3.2) hold for all

r and θ, so in particular we can compare term by term:

1 1 −1 1 u reiθ = u e−iθ using ( 1 , θ) for (r, θ) in equation (3.1) 2 r 2 r2 r r r 1 1 −iru reiθ = i u eiθ using ( 1 , θ) for (r, θ) in equation (3.2) θ r3 θ r r 1 1 −1 1 u reiθ = u e−iθ using (r, θ) for (r, −θ) in equation (3.1) 2 r 2 r2 r r 1 1 −iru reiθ = i u e−iθ using (r, θ) for (r, −θ) in equation (3.2) θ r3 θ r

Thus, we see that: the ﬁrst factor of the left hand side coincides with the second factor

of the right hand side, and the second factor of the left hand side coincides with the ﬁrst

factor of the right hand side (with the equal summands of each factor appearing in the

same order).

Now we will prove Proposition 3.4.2. We recall its statement here:

73 Proposition 3.4.2. For the parachute maps us described above, there exists a positive constant

M < ∞ and a sequence of indices {s}% ∞ so that, for all z ∈ γ1, we have

|φs(z)| < M.

Proof. Consider the Laurent series for the coefﬁcients of the sequence of Hopf differentials:

∞ s s n φ (z) = ∑ cnz n=−∞ where each coefﬁcient is obtained by Cauchy’s integral formula along a contour γ

I φs(ζ) s ≡ | | cn n+1 d ζ γ ζ

s This expression is valid in the domain of holomorphicity of φ , which contains Ω 1 s ,s 2π Let us get a better handle on the coefﬁcients of these series. The k rotational symme-

2πi l tries of the map us imply that Φ(z) = Φ e k z for each l = 0, . . . , k − 1. Hence,

2 2 2πi l 2πi l 2 φ(z)dz = φ e k z e k dz

In terms of the Laurent coefﬁcients, by uniqueness of the Laurent expansion for φs(z), we must have

2πi 2πi s s n l 2 + l cn = cne k e k 2

2πi s + (n+2)l = cne k 2 for all l = 0, . . . , k − 1. In particular, for all indices n for which k does not divide n + 2, the

s coefﬁcient cn must vanish. This leaves us with an expansion given by

s c−(2k+2) c−(k+2) c−2 s k−2 2k−2 φ (z) = ··· + + + + c z + c − z + ··· z2k+2 zk+2 z2 k−2 2k 2 ∞ ! ∞ ! c c−(k+2) c− = −nk−2 + + 2 + cs zk−2 + c znk−2 ∑ nk+2 k+2 2 k−2 ∑ nk−2 n=2 z z z n=2

We separate the series into three parts as indicated by the parenthetical grouping: respec-

tively, the negative tail, the central terms being the sum of the index −(k + 2), −2, and

k − 2 terms, and the positive tail.

74 Our plan of attack will be to bound the norm of each of these parts, with the bounds holding for every point on the core curve and along a sequence of indices {sj}. Recall-

s ing the Cauchy integral expression for cn, we will choose appropriate contours to bound the norms of the inﬁnite tails. Then, we will control the central terms by analyzing the possibilities for those three coefﬁcients.

First, let us focus on the positive tail. For the coefﬁcients indexed by n > 0, we can

iθ adopt the contour γρ := {ρe |0 ≤ θ ≤ 2π} in their Cauchy integral expressions, valid for

1 ρ ∈ s , s , to obtain:

I φs(ζ) s ≡ | | cn n+1 d ζ γρ ζ I |φs(ζ)| ⇒ | s | ≤ cn n+1 dl γρ ρ

Multiplying both sides by 2ρ and integrating over the interval ρ ∈ (s − s`, s), we see that:

Z s Z s I |φs(ζ)| |cs |ρdρ ≤ 2 dl dρ ` n ` n s−s s−s γρ ρ 2 Z s I s 2 ` 2 s ⇒ |cn| s − s − s ≤ ` |φ (ζ)| dl dρ (s − s )n s−s` Z s I s 2 s ⇒ |cn| ≤ ` `− ` |φ (ζ)| dl dρ s (2s − s2 1) (s − s )n s−s` ` ∈ log(s−1) ` = log(s−1) − ` = valid for any 0, log(s) . Note that the value log(s) yields s s 1, and the

1 s integral on the right hand side is exactly the L -norm of Φ on Ω1,s.

s s Thus, noting that 2|φ | ≤ e and E us, Ωs−s`,s ≤ E (us, Ω1,s) ≤ E(w, Ωs), we establish

Z s I s 2 s |cn| ≤ ` `− ` |φ (ζ)| dl dρ s (2s − s2 1) (s − s )n s−s` 1 ≤ E(us, Ω ` ) s (2s` − s2`−1) (s − s`)n s−s ,s 1 ≤ E(us, Ω s) s (2s` − s2`−1) (s − s`)n 1, 1 ≤ E(w, Ω s) s (2s` − s2`−1) (s − s`)n 1,

The right hand side can be controlled because the energy density of the Scherk map w

75 obeys the estimate

ew(z) ≤ |zk−2| + J w(z).

R w Here, C J dA = 2π(k − 2) since the area of w(C) is given by the area of an ideal k-gon.

Thus,

1 Z |cs | ≤ k−2dA + (k − ) n ` 2`−1 ` n ρ 2π 2 s (2s − s ) (s − s ) Ωs 1 2π ≤ sk + 2π(k − 2) s (2s` − s2`−1) (s − s`)n k

− log(s−1) 1 log(s) Note that lims%+∞ s = 1. Hence, there exists D > 0 so that, for s large enough, we have

s k−n−2 |cn| ≤ Ds .

This inequality can be used to obtain a bound on the norm of the positive tail on the core curve. We have:

∞ ∞ s nk−2 ≤ s ∑ cnk−2z ∑ cnk−2 n=2 n=2 ∞ ≤ D ∑ sk−[nk−2]−2 n=2 ∞ ≤ D ∑ s−k(n−1) n=2 1 ≤ D − 1 1 sk

Next, let’s consider the negative tail. By Proposition 3.4.3, since the parachute maps

1 satisfy the reﬂectional symmetries u z = u(z) and u(z) = u(z), the Hopf differential satisﬁes the identity 1 1 φs(z) = φs . z z4

Referring to the Laurent expansion for the Hopf differential, this reveals the identity

s s c−(n+2) = cn−2

76 among the coefﬁcients - relating them so that the sequence of coefﬁcients is symmetric about the n = −2 index. Thus, the norm of the negative tail can be bounded along the core curve:

∞ s ∞ c−nk− 2 ≤ cs ∑ nk+2 ∑ −nk−2 n=2 z n=2 ∞ s ≤ ∑ cnk−2 n=2 1 ≤ D − 1 1 sk as with the positive tail. Hence, for all s large enough, for any z ∈ γ1, we have bounded

s s c−(k+2) c | s(z)| ≤ D + + −2 + cs zk−2 φ k+2 2 k−2 z z

where D = D(k) is a positive constant given by the bound on the moduli of the tail coefﬁ-

cients.

s = s = Finally, let us turn to the central terms. We will relabel the coefﬁcients as C : c−(k+2)

s s s ck−2 and D = c−2. Thus far, we’ve deduced that the central terms have the form

Cs Ms 1 Ms + + Cszk−2 = Cs 1 + zk + z2k+2 zk+2 z2 zk+2 Cs

s s The symmetry us(z) = us(z) ensures C and M are real for all s. Thus, along the unit circle, we have:

C M − C M − + + Czk 2 ≤ + + Cszk 2 zk+2 z2 zk+2 z2

≤ 2|Cs| + |Ms|

From this expression, it is clear that we would like to obtain a sequence {sj} for which

sj sj both |C | and |M | are bounded uniformly. Thus, let {sj} be any sequence tending to

+∞. From this sequence, we will extract a sub-sequence for which both |Csj | and |Msj | are

bounded uniformly.

In the ﬁnal analysis, it will be convenient to have both Csj and Msj non-vanishing for

any sj. To sufﬁciently reduce the analysis to this hypothesis, we will simply drop all the

77 sj sj indices sj for which either C or M vanishes. This reduction requires us to delete either

ﬁnitely many or inﬁnitely many indices. If we are fortunate enough to require deleting

s inﬁnitely many indices, we can instead extract a sub-sequence {sl} for which: either C l or

s M l vanish identically for all sl.

Our process for producing the sub-sequence (and our plan of attack) is thus: produce

s s a sub-sequence (also labeled {sj}, by abuse of notation) along which |C l | and |M l | are bounded uniformly in each case.

1. If the reduction requires dropping inﬁnitely many terms, then from those terms we

can extract a subsequence {sj} for which one of the following must hold:

sj sj (a) for all sj, we have C = 0 and W = 0; or

sj sj (b) for all sj, we have C = 0 and W 6= 0; or

sj sj (c) for all sj, we have W = 0 and C 6= 0

2. or else, if the reduction only requires dropping ﬁnitely many terms, we reduce the

sequence so that

sj sj (d) for all sj, we have both C 6= 0 and W 6= 0

In case (a), with both coefficients vanishing identically, we trivially have |φs(z)| uniformly bounded by D(k) along the unit circle. For the (b) and (c) cases, we will argue by contradiction that there must be bounded subsequences of |Csj | and |Msj |; we’ll use two different geometric considerations to temper the norm of the non-vanishing coefficient. Fi- nally, we will conclude the analysis with case (d) by studying the ratios of the coefficients and the underlying foliation structure of its Hopf differential.

In case (b), we have Csj = 0, but Msj 6= 0. This suggests that having |Msj | unbounded would lead to the distance between the core curve and the nearest zero in the Φsj metric becoming unbounded. In particular, this allows us to obtain the absurd statement: we can

78 ﬁnd a circle’s worth of very large Φsj -disks, on each of which usj converges to a map of C onto a geodesic in H2, all while retaining its k-fold symmetry. Let us make this argument precise.

Suppose for a contradiction that |Msj | is unbounded. Then there exists a subsequence

s {sl} for which |M l | % +∞. Consider the family of Hopf differentials

1 Φsl . Msl

It forms a normal family (on compact subsets) since

1 1 1 1 Φs ≤ + D(k) |z| + Ms z2 Ms |z|

{ 1 sl } 1 2 Thus, Msl Φ converges pointwise, and hence uniformly on compact subsets, to z2 dz on

C∗ { 1 sl } 1 2 in sl. The zeroes of Msl Φ converge uniformly in sl to the zeroes of z2 dz - of which there are none. Hence, the original sequence {Φsl } contains Φsl -disks of size Msl .

Fix an angle θ. We can ﬁnd a ﬁxed disk Z(θ) ⊂ Ω1,sl so that Z(θ) has diameter at least

s M l sl 2 in the Φ -metric. In addition, we can choose it to lie away from the core curve γ1 and

with a center at an angle θ relative to the x-axis. By Proposition 3.4.1, it follows that the

Jacobian of us is non-negative on this disk. Hence, the sequence of induced maps

2 (Z(θ), dΦsl ) → H

converges to a map of C → H2, since the sequence of metrics induced by Φsl on Z(θ)

converges to the standard metric on C.

From this procedure, for each θ we obtain a harmonic map from C to the hyperbolic

plane. As each domain was obtained by using the Φsl -metric, the maps always have a

constant (depending on θ), non-zero Hopf differential. Here, we can apply Dumas-Wolf’s

uniqueness Theorem 2.4.4: by interpreting the theorem as a statement yielding uniqueness

of harmonic immersions from the plane into H2, we know that each of these maps must

be maps onto a geodesic.

79 Furthermore, we have a whole circle’s worth of maps from C to H2 whose images must be contained in the ideal k-sided polygon which bounds the images of usl . In particular, we can choose θ to lie in the direction of one of the ideal corners of the polygon. However, we cannot obtain such a limiting map whose image is a geodesic perpendicular to this direction (as the horizontal foliation lies orthogonal to this direction) since such a geodesic cannot be contained in the regular ideal polygon. Hence, there cannot exist a subsequence

s sj {sl} for which |M l | % +∞, and {|M |} must contain a bounded subsequence.

In case (c), we have Msj = 0, but Csj 6= 0. This suggests that having |Csj | unbounded

would lead to too much energy (when compared to the Scherk map). Let us make this

precise. Suppose for a contractiction that {|Csj |} does not contain a bounded subsequence,

so that it contains a subsequence with {|Csl |} % +∞ monotonically. Observe, then, that

s − C l 1 |Φsl (z)| ≥ Csl zk 2 − − D(k) ρ + zk+2 ρ 1 1 ≥ |Csl | ρk−2 − − D(k) ρ + ρk+2 ρ

Thus, it follows that the energy density of us can be bounded from below by

esl (z) ≥ |Φsl (z)| 1 1 ≥ |Csl | ρk−2 − − D(k) ρ + ρk+2 ρ where D(k) is the constant from the proof of Proposition 3.4.2.

This suggests that the energy density is growing at a rate of ρk−2, with a proportional constant |Csl |. However, comparing to the Scherk map, we necessarily have

E(Ω1,s, us) ≤ E(Ω1,s, w) + E(Ω1, w)

⇒ E(Ω1,s, us) ≤ E(Ω1,s, w) + T

where T := E(w, Ω1). Furthermore, the energy density of the Scherk map satisﬁes

ew(z) = ρk−2 + J (z)

80 for which J w > 0 and R J dA ≤ 2πk. Note that the energy estimate is supposed to Ω1,s

hold on the domain Ω1,s Hence, for all s > 1,

Z Z ew(z)dA − es(z)dA + T ≥ 0 Ω1,s Ω1,s

Z Z 1 1 ⇒ k−2 dA + k − |Csl | k−2 − − D(k) + dA + T ≥ ρ 2π ρ k+2 ρ 0 Ω1,s Ω1,s ρ ρ Z 1 Z 1 ⇒ ( − |Csl |) k−2 − |Csl | − D(k) + dA ≥ − k − T 1 ρ k+2 ρ 2π Ω1,s ρ Ω1,s ρ

The right hand side is a constant yet, for s large enough, the left hand side becomes arbitrarily large and negative since |Csl | % +∞. This is a contradiction. Hence, there cannot exist a subsequence with |Csl | % +∞. In other words, when the original subsequence {Csj }

falls into case (c), we can choose a subsequence of it for which {|Csl |} stays bounded.

Finally, we are left with considering the fourth possibility, for which Csj 6= 0 and Wsj 6=

0 for all sj. In this situation, we can consider the sequence of norms of the ratios of the

coefﬁcients. In Rd≥0 = R≥0 ∪ {+∞}, the possibilities are:

s C j (e) there exists a sequence {s }% +∞ along which s % +∞, or j M j

s M j (f) there exists a sequence {s }% +∞ along which s % +∞, or j C j

s C j (g) there exists a sequence {s }% +∞ along which s → D ∈ R+ j M j

We will proceed by contradiction to rule out the possibility of cases (e) and (f). The ﬂavor of these arguments will resemble the purely vanishing cases (b) and (c) above. The resolution of (g) will follow from the fact that, in this case, either both |Csj | and |MsJ | diverge or both stay bounded. The last step is to rule out their simultaneous divergence, again by ﬁnding a circle’s worth of large Φs-disks as in (b).

s C j In case (e), there exists a sequence {s }% +∞ along which s % +∞. Then there j M j s s s exists a subsequence {sl} so that |M l | < |C l | and |C l | % +∞. We employ an energy

81 estimate similar to that in case (c):

sj sj − C M 1 |Φsl (z)| ≥ Csj zk 2 − − − D(k) ρ + zk+2 z2 ρ 1 1 1 ≥ |Csl | ρk−2 − − |Msl | − D(k) ρ + ρk+2 ρ2 ρ 1 1 1 ≥ |Csl | ρk−2 − − |Csl | − D(k) ρ + ρk+2 ρ2 ρ

1 1 1 ⇒ esl (z) ≥ |Csl | ρk−2 − − − D(k) ρ + ρk+2 ρ2 ρ

At this point, we reach the same conclusion as in case (c) by contradicting the necessary

sl energy comparison E(u , Ω1,sl ) ≤ E(w, Ωsl ). Hence, no subsequence {sj} can be extracted which falls into case (e).

s M j In case (f), there exists a sequence {s }% +∞ along which s % +∞. Then there j C j s s s exists a subsequence {sl} so that |C l | < |M l | and |M l | % +∞. As in case (b), we can consider the family of holomorphic quadratic differentials

1 Φsl . Msl

1 sl This implies the family Msl Φ constitutes a normal family on compact subsets. Thus,

{ 1 sl } 1 2 C∗ Msl Φ converges pointwise, and hence uniformly on compact subsets, to z2 dz on in

sl. At this point, we reach the same conclusion as in case (b) by contradicting the existence of a circle’s worth of maps to geodesics. Hence, no subsequence {sj} can be extracted

which falls into case (f).

s C j In case (g), there exists a sequence {s }% +∞ along which s → D ∈ R+. Suppose j M j for a contradiction that |Csj | and |Msj | simultaneously diverge. Consider the family of

holomorphic quadratic differentials

1 Φsl Msl

82 on Ω1,s. Observe that we have the bound

sj 1 − C 1 1 1 1 Φsj (z) ≤ zk 2 + + + D(k) |z| + . Csj Msj z2 zk+2 Csj |z|

Hence, the family converges pointwise (and uniformly on compact subsets) to

1 1 zk−2 + D + dz2 z2 zk+2 on Ω1,s. Finally, observe that the coefﬁcient has the form

1 1 zk − N zk − dz2 zk+2 N where 1 N + = D N

Hence, in the Φsl metric, we can ﬁnd larger Φsl -disks away from the core curve. Again, we can ﬁnd a circle’s worth of maps to geodesics (like in case (b)) and obtain a contradiction.

sj sj sj C Thus, |C | and |M | cannot simultaneously diverge. Since we have s → D ∈ R+, it M j must be that |Csj | and |Msj | both stay bounded.

3.4.4 Bounded Core Lemma

We will call the curve γ1 the core curve, since it is ﬁxed by the reﬂectional symmetry on the domain Ω 1 . Note that it is the boundary component of Ω1,s on which the harmonic s ,s mapping problem (PFDrs) prescribes a free boundary.

Here, we state the Bounded Core Lemma: Consider the Scherk map restriction

w : Ω → H2.

If we “poke a hole of size Ωr” in the domain Ωs, and ﬂow the harmonic map on its changed domain to allow the hole to move freely, the Bounded Core Lemma purports that the hole does not move very far from the origin O ∈ H2, even though the hole itself may deform.

83 2 Lemma 3.4.4. (Bounded Core Lemma) Consider the parachute map us : Ω1,s → H as above.

Let γ1 := {|z| = 1} denote the core curve and O denote the origin in the Poincare disk model for

2 H . Then there exists a sequence of indices {sj ∈ R≥0}j with sj → ∞ so that we have

d(O, usj (z)) ≤ M

for any z ∈ γ1, for some M = M(k) depending only on the degree k of the Scherk map’s Hopf

differential.

Proposition 3.4.5. For each parachute map us deﬁned as above, there exists a constant B > 0

independently of s so that 1 " # 2 s d(O, us(z)) ≤ B · sup e (z) z∈γ1

Proof. The image of the core curve γ1 under the map ues inherits the same symmetries of the map ues, so

d(O, us(z)) ≤ L(us(γ1))

for all z ∈ γ1, where O is the origin of the hyperbolic plane and L(us(γ1)) denotes the

length of the image of the core curve under us. Hence, it sufﬁces to bound L(us(γ1)) for all

s. Note that

Z s L(us(γ1)) ≡ |Du (∂θ)|dl γ1 1 Z 2 1 2 ≤ (2π) 2 |Du(∂θ)| dl γ1 1 Z 2 1 2 ≤ (2π) 2 ||Du|| dl γ1 1 Z 2 1 s ≤ (2π) 2 e (z)dl γ1 1 " # 2 ≤ B · sup es(z) z∈γ1

1 1 where B = (2π) 2 · L(γ1) 2 is a constant independent of s.

84 1 1 2 Note that the initial rescaling of Ωr,s by a factor of r changes L(γ1) by a factor of r .

This only changes the constant appearing on the right-hand side of Proposition 3.4.5.

Proposition 3.4.6. For each parachute map us deﬁned as above, for each z ∈ γ1, we have

es(z) = 2|φs(z)|.

Proof. In general, we have an inequality relating the energy density, the Hopf differential, and the Jacobian:

2||Φ(z)|| ≤ e(z) ≤ 2||Φ(z)|| + |J (z)|.

These follow from the equations for the holomorphic and anti-holomorphic energies:

e = H + L

J = H − L √ ||Φ|| = HL

s Since the map us is symmetric by reﬂection across the core curve, the identity J (z) = 0

s s holds for all z ∈ γ1 and for all s. Thus, e (z) = 2||Φ (z)||. Furthermore, since σ(z) ≡ 1, we have simply

es(z) = 2|φs(z)|.

We now have enough to assemble the proof of the Bounded Core Lemma 3.4.4.

Proof. We aim to show that there exists M > 0 such that, for all s large enough, we have

d(O, us(z)) < M.

By Proposition 3.4.5, each parachute map us satisﬁes

1 " # 2 s d(O, us(z)) ≤ B · sup e (z) . z∈γ1

85 By Proposition 3.4.6, for each z ∈ γ1, we have

es(z) = 2|φs(z)|.

By Proposition 3.4.2, for r large enough, for all s > r, there exists a positive constant

M < ∞ such that, for all z ∈ γ1, we have

|φs(z)| < M.

Combining these, we arrive at our desired conclusion.

3.4.5 The Energy Estimate: Parachute maps tied to Scherk map

With the Bounded Core Lemma 3.4.4 in hand, we now derive an estimate on the discrepancy E(w, Ωr,s) − E(us, Ωr,s) between the energies of the parachute map us and the Scherk map w over their common domains.

Lemma 3.4.7. (The Energy Estimate) There is a function F(r) such that, when r is large enough,for

all s > r, the following inequality holds:

E(Ωr,s, w) − E(Ωr,s, us) < F(r)

Proof. We can estimate the discrepancy between the energies of w|Ωr,s and us|Ωr,s by the

2 energy of a harmonic map of a disc fs : Ωs → H . To do this, consider the unique harmonic

2 map ps : Ωr → H which minimizes energy in its homotopy class of maps with boundary

value ps|γr ≡ us|γr .

fs|Ωr = ps

fs|Ωr,s = us.

The map fs is such that E(Ωs, w) ≤ E(Ωs, fs), since w is actually smooth and energy minimizing in all of Ωr. Since we can decompose these energies over the domain Ωs = Ωr ∪ Ωr,s

86 as

E(Ωs, w) = E(Ωr, w) + E(Ωr,s, w)

E(Ωs, fs) = E(Ωr, ps) + E(Ωr,s, us)

their difference on Ωr,s can be bounded as

E(Ωr,s, w) − E(Ωr,s, us) ≤ E(Ωr, ps) − L,

where the constant L = L(k, r) is simply the energy of the Scherk map on the disk Ωr:

L(k, r) := (Ωr, w). Hence, it sufﬁces to exhibit a bound on E(Ωr, ps) in terms of r alone. To do this, we will use the Bounded Core Lemma 3.4.4 in addition to the iso-energy inequality

(Theorem 2.4.6) and Cheng’s interior gradient estimate (Theorem 2.4.5). We apply the iso- energy inequality (Theorem 2.4.6) ﬁrst. Since Ωr is a ball, the iso-energy inequality allows us to conclude

E(Ωr, ps) ≤ E(γr, ps|∂∆r) for which we know the right-hand side is uniformly bounded. Recalling that

E(Ωr,s, w) − E(Ωr,s, us) ≤ E(∆r, ps) − Kr,

we also arrive at E(Ωr,s, w) − E(Ωr,s, us) being uniformly bounded in s for all s > r. Let us now specialize Cheng’s interior gradient estimate (Theorem 2.4.5) to our situation. We use

2 M = Ωr,s and N = H , so that we can take K = 0. A choice for the points x0, y0 and the constants a, b will be chosen momentarily. We are guided by the following application of the Bounded Core Lemma.

Fix r for which there is a sequence of indices sj % ∞ as in the Bounded Core Lemma

3.4.4. It follows that

d(O, usj (z)) ≤ M

87 for all sj. By the maximum principle for the (subharmonic) distance function between the harmonic Scherk map w and the harmonic parachute map us, we have that

max {dH2 (us(p), w(p))} ≤ max {dH2 (us(p), w(p))} p∈Ωr,s p∈γr∪γs

≤ max{dH2 (us(p), w(p))} p∈γr for all s > r. The second inequality follows from the pointwise agreement of us with w on

γs. Applying the triangle inequality, we obtain

max {dH2 (us(p), w(p))} ≤ max{dH2 (us(p), O)} + max{dH2 (w(p), O)} p∈Ωr,s p∈γr p∈γr

≤ M + max{dH2 (w(p), O)} p∈γr

≤ M + P where P = P(k) is a constant deﬁned as

P(k) := max{dH2 (w(p), O)} < ∞. p∈γr

2 Fix a constant a > r, choose x0 ∈ γr ﬁx any point y0 ∈ H at a distance R of at least M + P from the set w(∆r). Then there exist constants b > 1 and β > 0 are some independent of sj, all of which are ﬁnite (although their admissible values are dependent on k and the map w on ∆r)). Choosing such b and β, Cheng’s Lemma 2.4.5 states that

b2(b2 − ρ2 ◦ u )2 |∇u (x )|2 ≤ c s s 0 m a2β

Because y0 is chosen to be a distance R from w(Ωr), for each p ∈ Ωr,s, we have:

s s ρ ◦ u (p) ≡ dH2 (u (p), y0)

s ≤ dH2 (u (p), w(p)) + dH2 (w(p), y0)

≤ (M + P) + R

88 2 2 Hence it follows that ρ ◦ us(x0) ≤ (M + P + R) and that

2 2 2 2 b − ρ ◦ us ≤ b + (M + P + R) .

2 This shows that |∇us(x0)| is uniformly bounded for all x0 ∈ γr, for all sj.

Finally, observe that ∇us has zero ∂n component along γr, so that

|∇us|(x0) = |∇us(∂θ)|(x0)

for all x0 ∈ γr. Since ps|γr ≡ us|γr is the parameterized Dirichlet boundary condition for

ps, it is thus true that

|∇ps(∂θ)|(x0) = |∇us|(x0).

2 As we have shown that |∇us(x0)| is uniformly bounded for all x0 ∈ γr, for all sj, it follows

that E(∂Ωr, ps|∂Ωr ), and hence E(Ωr, ps) is bounded independently of s.

89 Chapter 4

Future directions

Theorem 1.1.1 provides some examples of handle crushing harmonic maps h : Σg → Pk from punctured Riemann surfaces Σg to some ideal polygons Pk in the hyperbolic plane.

There are some immediate natural questions to ask.

Generalizing the target somewhat,

Question 4.0.8. Fixing a once-punctured Riemann surface Σg, does there exist a harmonic map

hP : Σg → P for any ideal polygon P in the hyperbolic plane?

Or, generalizing the domain somewhat,

Question 4.0.9. Do there exist harmonic maps from multiply punctured Riemann surfaces to the hyperbolic plane, whose images lie inside unions of ideal polygons?

We expect these maps from multiply punctured surfaces, if they exist, to have Hopf differentials which are rational polynomials. We hope to investigate these and the following questions concerning global and local properties for our harmonic maps.

90 4.1 The whole picture: handle crushers between compact surfaces

Theorem 1.1.1 describes a harmonic map locally around a handle collapse. We conjecture two procedures which suggest this local model is indeed the local model for handle crushing harmonic maps between compact surfaces, in that it approximates well a neighborhood of the handle collapsing.

One process considers a sequence of handle crushing harmonic maps ht : (Σg, σt) →

(Σh

∗ ∗ ∗ ∗ map h : (Σg, σ) → (Σh, ρ) from a noded surface (Σg, σ) = (Σh, σ) ∨ (Σg−h, σ), we expect

∗ to recover (away from the node point) a harmonic diffeomorphism h : (Σh, σ) → (Σh, ρ)

∗ on one part and a handle crushing map h : (Σg−h, σ) → (Σh, ρ) (of the topology we have

produced) on the other part. Algebraically, we describe this as a decomposition of the

domain on which the limiting harmonic map induces an isomorphism or maps trivially

on fundamental groups.

The second process suggests our non-compact handle crushing map can be built up as

part of an almost harmonic map, and a singular perturbation argument (or more specif-

ically, a bridge principle) can be employed to correct it to become harmonic. We expect

the correction to be small, so that our model is a close approximate on the handle crush-

ing component. This correction is expected to be small by relating the data of the Hopf

differentials near the bridged components.

4.1.1 Limit of compact maps to a diffeomorphism and a handle crushing model

Let us ﬁrst describe an example of a family of holomorphic quadratic differentials Φt on a genus two surface which limits to a holomorphic quadratic differential Φ on a noded surface of genus two (which is topologically a union of two punctured genus one surfaces away from the node point). This example is borrowed from sec A.4 of [McM89].

91 Fix a torus with holomorphic quadratic differential dz2. Cut open two segments of its horizontal foliation, each of length L and away from the singularities, and glue in a

Euclidean cylinder of height H and circumference t := 2L with its foliation by circles. This results in a holomorphic quadratic differential Φt on a genus two surface with four zeroes at the endpoints of the slits. Suppose the slits are the sides of a square and let t → 0 while

H L is ﬁxed. Fixing a basepoint, the limit quadratic differential Φ lives on the punctured torus with a fourth order pole.

Let us adopt this notion of a limiting pair (Σg, Φt) of Riemann surface Σg with holomorphic quadratic differential Φt in the following construction. Let ht : (Σg, σt) → (Σg−1, ρ) be a sequence of handle crushing harmonic maps in which σt is an unbounded ray in the

Teichmuller¨ metric on Tg. Suppose ht induces the same homomorphism on fundamental groups for all t. Note that a single handle is being crushed for each t. Fixing a basepoint in

h a neighborhood of the crushed handle, we consider the limit of (Σg, Φ t ), and ask:

Question 4.1.1. Can this process be carried out to produce our handle crushing map of a square punctured torus onto an ideal square from Corollary 1.1.3?

4.1.2 Bridging harmonic diffeomorphisms with handle crushers

A bridge principle would be effective in showing that the harmonic non-compact handle crushing model from Theorem 1.1.1 closely describes the harmonic handle crushing near a neighborhood of a handle crushing map between compact surfaces. Let us describe our reasoning for expecting this.

Let (Σh, σ) be a compact Riemann surface, and choose a holomorphic quadratic differential Φ which has a zero p ∈ Σh of order k − 2. Consider the one-parameter family of holomorphic quadratic differentials

Φt := tΦ obtained by scaling Φ. For each value of t, there exists a unique hyperbolic metric ρt on Σh

92 for which there exists a unique harmonic diffeomorphism

ut ≡ uρt : (Σh, σ) → (Σh, ρt)

homotopic to the identity idσ,ρ : (Σh, σ) → (Σh, ρt), by work of Wolf [Wol89].

Observe that there is a sequence of Φt-disks Dt centered at p of increasing and diverg-

ing radii rt = 2diam(Dt). This produces a sequence of harmonic maps

ut : Dt → H2

with Hopf differentials Φ(ut) = tΦ (by construction). Hence, upon rescaling, we have a

sequence of harmonic maps: √ vt : t · Dt → H2 with Hopf differentials Φ(vt) = Φ. This implies that vt converges to a harmonic map

v : C → H2 with a Hopf differential Φv with a zero of order k − 2 at the origin. We view

(C, Φv) as the punctured sphere with a Hopf differential having a pole of order k + 2.

The foliation near the pole of Φv is similar to the foliation near the pole of a meromor-

∗ phic quadratic differential on a punctured Riemann surface Σg−h of positive genus. If k is

∗ even, Theorem 1.1.1 states there is a harmonic map w : Σg−h → Pk.

t Now, consider the harmonic diffeomorphisms u : (Σh, σ) → (Σh, ρt) above and the

∗ ut handle crushing harmonic map w : Σg−h → Pk. Near the zero of order k − 2 of Φ and

near the pole of Φw, the (unmeasured) foliations look more similar as t % ∞. Guided

by these similar foliations, we can cut appropriately and bridge these maps together to

produce an almost everywhere harmonic map. We ask:

t Question 4.1.2. Can we bridge the harmonic map u : (Σh, σ) → (Σh, ρt) (for some t) with the

∗ harmonic handle crushing map w : Σg−h → Pk to produce a harmonic map f : Σg → Σh between

compact Riemann surfaces?

If this is possible, the Hopf differential data and the diffeo-topological singularities of

the map f can be studied through our handle crushing models.

93 4.2 Minimal surfaces in H2 × R

We have seen in Corollary 1.1.3 that there are three possibilities for the Hopf differential of a handle crushing harmonic map h : Σ1 → P4 from the square punctured torus to the ideal square. If the divisor of Φh is given by arrangement (a), then we can perform a minimal suspension of h to produce a minimal surface. With the numerical evidence suggested by a minimal punctured torus with boundary in H2 × R (see Figure 1.2), we expect this to be the case.

h Conjecture 4.2.1. The Hopf differential Φ of the handle crushing harmonic map h : Σ1 → P4 is the square of a one-form.

This leads us to ask:

2 Question 4.2.2. Can this handle crushing harmonic maps h : Σg → H produce embedded Chen-

Gackstatter type minimal surfaces in H2 × R?

This would provide the ﬁrst example of an embedded minimal surface in H2 × R of this topological type and with this asymptotic data. Embeddedness could be deduced by verifying a sufﬁciency condition for embeddedness proved in [CG03]:

n Theorem 4.2.3. Suppose that Γ is a Jordan curve in M with total curvature Ctot(Γ) satisfying

Z 2 Ctot(Γ) := ||~κ||H2×R ds ≤ 4π + κ · inf Area(p××Γ) Γ p∈M where p××Γ denotes the geodesic cone over Γ with vertex p. Then, any minimal surface Σ ⊂ M with ∂Σ = Γ is embedded.

Again, numerical evidence suggests that a genus one Chen-Gackstatter type minimal surface exists and is embedded in H2 × R. It also seems likely that higher genus handle crushing harmonic maps h : Σg → P4 can be suspended to produce higher genus Chen-

Gackstatter type minimal surfaces in H2 × R.

94 4.3 Deforming handle crushers

Can the handle crushing harmonic maps from Theorem 1.1.1 by deformed? We discuss two reasonable methods to classify their deformations.

4.3.1 Deformations changing the image polygon

Can we perturb our handle crushing harmonic map image h(Σg) = Pk?

For example, consider the one-parameter family of translations ωs along an axis γ of a hyperbolic isometry of H2. Post-composition of any handle crushing harmonic map

2 h : Σg → Pk from Theorem 1.1.1 by ωs produces a map ωs ◦ h : Σg → H into a different ideal polygon, ωs(Pk). Note that the Hopf differential for each of these translated maps

ωs ◦ h is the same as that of h.

However, these simple deformations obtained by post-composition by an isometry of

H2 do not change the set of cross-ratios formed by all the order 4 subsets of the boundary

points. We suspect that a change in the set of cross-ratios of boundary points is correlated

to a change of the Hopf differential.

Question 4.3.1. Can we deform the harmonic map h : (Σg, σ) → Pk along a one-parameter family

2 of harmonic maps ht : (Σg, σ) → H so that h0 = h, and such that ht(Σg) and h(Σg) are not isometric for each t? Note that the complex structure σ on Σg is ﬁxed. If so, how do the meromorphic

Hopf differentials Φht and Φh compare? For example, is it true that Φht 6= Φh for t 6= 0?

It is reasonable to suspect that, if the cross ratios are not the same, then the Hopf differentials will differ.

4.3.2 Deformations preserving the image set

Can we deform our handle crushing harmonic map so as to preserve its image?

95 Note that, on the complex plane C, any polynomial holomorphic quadratic differential

2k+1 k+1 k 2 Φ = (a2k+1z + ··· + ak+1z + ckz + ··· + c1z + c0)dz can be realized as the Hopf differential of a Scherk map w : C → H2, which is a harmonic diffeomorphism, i.e., Φw ≡ Φ. This follows from Theorem 2.4.1. Furthermore, the image w(C) is the interior of an ideal polygon with 2k + 3 vertices.

It has been observed that, by fixing the ai coefficients and varying the ci coefficients of

Φ along a one parameter family, say

t 2k+1 k+1 k 2 Φ = a2k+1z + ··· + ak+1z + ck(t)z + ··· + c1(t)z + c0(t) dz , we obtain the corresponding family of Scherk maps wt. For each t, the ideal polygon

t w (C) = P2k+3 = w(C) coincides with the initial ideal polygon with 2k + 3 sides. As an analogous example for the case of an even degree polynomial, Au-Wan proved that any Scherk maps w : C → H2 with a Hopf differential z2m − (a + ib)zm−1 dz2, for a, b ∈ R, has image w(C) lying inside the regular ideal polygon Pm+2, regardless of a and b. [AW05].

We can ask whether the analogy occurs for higher genus domains:

Question 4.3.2. Suppose h : (Σg, σ) → Pk and v : (Σg, σ) → Pk are two handle crushing harmonic maps with the same punctured Riemann surface domain (Σg, σ) and with the same image

h v set h(Σg) = Pk = v(Σg). How do Φ and Φ compare on Σg?

By Theorem 1.1.2, such maps h and v must be quite different as they approach the ideal polygon boundary.

4.4 Combinatorial data from Hopf differential

Here we collect some questions of a combinatorial nature.

96 Fact 4.4.1. The space of ideal k-sided polygons of H2, modulo isometries, is of real dimension

2k − 1. They are parameterized by 2k − 1 real numbers called cross-ratios.

Proof. Let P be an ideal k-sided polygon. Then there exist boundary points points {z1, z2,..., zk} ⊂

∂H2 for which

P = convex hull ({z1, z2,..., zk}) .

The map

{z1, z2,..., zk} 7→ CH ({z1, z2,..., zk})

is a bijection between cyclicly ordered sets and ideal k-sided polygons. The k complex

numbers

zi−1 − zi zi+2 − zi+2 Ci ≡ [zi−1, zi; zi+1, zi+2] := zi−1 − zi+1 zi − zi+2

called the shape parameters determine. These complex shape parameters

Geometrically, the shape number Ci is deﬁned by this: A unique Mobius transforma-

2 2 2 tion φi : H → H takes the ordered triple zi−1, zi, zi+1 to 1, ∞, 0. Then φ(zi+2) ∈ ∂H is

exactly the value of Ci.

Question 4.4.2. Are the locations and multiplicities of zeroes of Hopf differentials stable under the

deformations described in Questions 4.3.1 and 4.3.2?

In particular, are the sum of orders of zeros in each connected component of the J < 0

and J > 0 sets constant along deformations?

Question 4.4.3. Fix a punctured Riemann surface Σg. Which Hopf differentials are realizable as

the Hopf differential of a harmonic map to an ideal polygon in H2?

Because of the numerical evidence for the Chen-Gackstatter type minimal surface in

H2 × R, we expect the Hopf differential of Corollary 1.1.3 to be in conﬁguration (a) in.

Recall that the Hopf differential in that conﬁguration has a zero of order 2 in the J < 0

subset of Σ1. So, we expect:

97 Conjecture 4.4.4. The Hopf differentials of our handle-crushing harmonic maps have zeros of order

≥ 2 in each region on which orientation is reversed.

This would also seem to suggest:

Conjecture 4.4.5. The space of holomorphic quadratic differentials realizable as the Hopf differential of a handle crushing harmonic map satisﬁes this constraint, and its dimension can be calculated in this way.

This makes sense at least when the J < 0 region is very large in the Hopf differential metric. For, the harmonic map is a diffeomorphism onto its image from the J < 0

region, and the image tends towards a region bound by cusps and geodesic segments (see

Appendix A.2 for description of geometry of image).

4.5 Germs of harmonic maps

What are the germs of our harmonic maps near their singular (Jacobian vanishing) sets?

Loosely speaking, the Hartman-Wintner Lemma [HW53] states that the local expan-

sion for any harmonic map between surfaces can be given coordinate-wise by the real and

imaginary parts of holomorphic functions with zeros of order k and l, respectively. More speciﬁcally, the theorem reads:

Theorem 4.5.1. (Hartman-Wintner Lemma) Let D ⊂ R2 be an open disk with center (x, y) =

(0, 0). Suppose g11, g12, and g22 are C1(D) such that g11(0, 0) = g12(0, 0) = g22(0, 0) = 0. Let

H(x, y, u, p, q) be a continuous function of its ﬁve real variables such that, for every e > 0,M > 0,

there exists a constant K = K(e, M) for which

|H(x, y, u, p, q)| ≤ K(|u| + |p| + |q|)

whenever x2 + y2 < e2, |u| ≤ M, |p| ≤ M, and |q| ≤ M. Let u(x, y) be a C2 solution of

∂2u ∂2u ∂2u ∂u ∂u g11 + 2g12 + g22 + H x, y, u, , = 0 ∂x2 ∂x∂y ∂y2 ∂x ∂y

98 with u(0, 0) = 0. Then:

p (a)u = u(x, y) is not o(ρn) for any n ∈ N, where ρ = x2 + y2, unless u ≡ 0.

(b) either u ≡ 0, or else by writing z = x + iy, u admits the expansion

l l u = R(alz ) + o(ρ )

where l ∈ N and al ∈ C \{0}. Furthermore, the partial derivatives admit the expansions

l−1 l−1 ux = R(lalz ) + o(ρ )

l−1 l−1 uy = −I(lalz ) + o(ρ )

A natural question to ask is thus:

∗ Question 4.5.2. At the singular sets of the handle crushing harmonic maps h : Σg → Pk from

Theorem 1.1.1, what are the possible degrees k and l in the local expansion provided by the Hartman-

Wintner Lemma?

To visualize the behavior of a harmonic map near a cusp, we can construct concrete local models for self maps of the Euclidean disk using the Poisson kernel for harmonic functions. We include an example of a harmonic map from the disk to itself obtained coordinate-wise through the Poisson integral formula.

iθ 2 Let e , for θ ∈ [0, 2π), parameterize the boundary γ1 of the Euclidean disk Ω1 ⊂ R .

Consider the Dirichlet boundary condition    3 ≤ < 2π  2 θ 0 θ 3   θ 7→ 1 sin(3θ) 2π ≤ θ < 2 2π  2 3 3    3 2π ≤ <  2 θ 2 3 θ 2π where we have identiﬁed θ with eiθ. Observe that this boundary condition can be parameterized coordinate-wise, say by ( f (θ), g(θ)).

99 Thus, we can ﬁnd the unique harmonic map h : Ω → Ω satisfying this boundary constraints by producing the unique harmonic functions u(z) and v(z) with boundary values

f (θ) and g(θ), respectively, and forming h := (u, v) component-wise. In Figure 4.1, we dis-

play the resulting harmonic map h; sub-ﬁgure (a) depicts the images of concentric circles

γr (for various values of r ∈ (0, 1)) about the origin, sub-ﬁgure (c) depicts the images of

rays (0, 1) · eiθ (for various values of θ ∈ [0, 2π)) emanating from the origin, and sub-ﬁgure

(b) super-imposes these.

1.0

0.5 0.5 0.5

-1.0 -0.5 0.5 1.0 -0.5 0.5 -0.5 0.5

-0.5 -0.5 -0.5

-1.0

(a) Images of circles. (b) Super-imposed images of cir- (c) Images of rays.

cles and rays.

Figure 4.1: Harmonic self map of unit disk exhibiting a cusp.

Such harmonic self-maps of the disk Ω1 seem very ﬂexible, but nonetheless, we expect the degrees k and l of our handle crushing harmonic maps to be related to the order of the

zeros of their Hopf differentials in their orientation reversing subsets (see arrangement (a)

of the Hopf differential in Corollary 1.1.3).

Furthermore, observe that the leading order terms Re(zk) and Im(zl) described in the

Hartman-Wintner Lemma are harmonic homogeneous polynomials. We have asked only

about k and l - essentially only about the highest order terms. Yet, we can think of har-

monic germs as perturbations of harmonic polynomials with lower order terms involving

harmonic polynomials. We include a brief discussion of harmonic polynomials before ask-

100 ing about these lower order terms.

4.5.1 Space of harmonic polynomials

Let Vk denote the real vector space of homogeneous polynomials in the variables x and y.

Note that each Vk is a ﬁnite dimensional subspace of R[x, y]. The Laplacian ∆ is a linear operator which satisﬁes

∆ : Vn → Vn−2

2 2 ∆( f ) = (∂x + ∂y) f 1 1 = ∂ (r∂ ) + ∂2 r r r r2 θ

a b since ∆ is a R−linear and, for x y ∈ Vn, we have

a b a−2 b a b−2 ∆(x y ) = a · (a − 1)x y + b · (b − 1)x y ∈ Vn−2. p 2 2 −1 y Employing polar coordinates (r, θ) = x + y , tan x , note that any homogeneous polynomial of degree n satisﬁes

f (xn, yn) = f (rncos(nθ), rnsin(nθ))

= rn f (cos(nθ), sin(θ)).

Let the subspace of harmonic polynomials of degree n be denoted

Hn := ker(∆) ⊂ V0 ⊕ · · · ⊕ Vn.

Consider an element f ∈ Hn, and deﬁne F(θ) := f (cos(nθ), sin(nθ)). Then F satisﬁes the

second-order ordinary differential equation

F00(θ) + n2F(θ) = 0.

It is straightforward to check that

n Rn(x, y) := R((x + iy) )

n In(x, y) := I((x + iy) )

101 are homogeneous polynomials of degree n and form a linearly independent set of solutions to this ordinary differential equation. Hence, Rn and In span Hn. Thus, the space of

∞ harmonic polynomials is H := ⊕i = 0 Hi.

We now ask whether we can get any more information about the lower order terms from the Hartman-Wintner Lemma:

Question 4.5.3. Which harmonic polynomials describe the germs of our handle-crushing harmonic maps? (This is an extension of the question above, asking about the lower order terms which may not appear in the R zk or I zl terms.) Can this data be recovered from the Hopf differential?

Although this seems like a local question, the Hopf differential for harmonic maps between compact surfaces are global objects. Recall that two harmonic maps from a ﬁxed genus g surface can have the same Hopf differential, even though one may be a diffeomorphism onto a genus g surface and the other may be a handle crushing map to a lower genus h < g surface. We essentially ask whether the Hopf differential detects restrictions on the realizable germs for a harmonic map.

102 Chapter 5

Tangential questions

In this chapter, we present some problems related to handle crushing harmonic maps from surfaces. These problems motivate the future directions we outlined in Chapter 4.

5.1 Surface group representation domination by Fuchsian repre-

sentations

Consider a handle crushing harmonic map between compact genus g and genus h < g surfaces. This provides a concrete example a representation of π1(Σg) into PSL(2, R) which is discrete but not faithful, obtained by the composition of the handle crushing map with the Fuchsian representation of the target genus h surface deﬁning its hyperbolic structure.

Tholozan-Deroin prove that this representation is strictly dominated by the Fuchsian rep-

resentation [TD13].

Question 5.1.1. Can we parameterize these dominated representations by their Hopf differentials?

103 5.2 Discrete, non-faithful representations and branched hyper-

bolic structures

Example 8 from [Tan94] provides an example of a discrete, non-faithful representation ρ of a genus 3 surface group into Γ < PSL(2, R) whose image Γ is isomorphic to a genus

2 surface group. The prescribed holonomy ρ cannot arise from a branched hyperbolic structure (proven in Proposition 3 of [Tan94]), but can be geometrically described by the harmonic map constructed above.

Tan asks (stated in our notation) [Tan94]: For (2 − 2g) ≤ k ≤ 1, consider the component

−1 −1 e (k) of Hom(πg, PSL(2, R)). Is the subset e (k) consisting of all the representations that

−1 occur as the holonomy of branched hyperbolic structures on the surface Σg dense in e (k)?

Fixing symplectic bases for Σ1, Σ2,..., Σg, observe that any harmonic map h : Σg →

Σk+1 between hyperbolic surfaces (for (k + 1) < g) gives rise to a representation ρh ∈

−1 e (k). By a result of Eells-Lemaire in [EL81], the harmonic map (and hence ρh) varies

continuously with respect to the variation of the metric on Σk+1.

Question 5.2.1. Is this Hopf diffferential unique?

If we have uniqueness of the Hopf differential, then this gives us a 6k − 6 dimensional

subspace of e−1(k). Hitchin showed that e−1(k)/PSL(2, R) is a 6g − 6 real dimensional

vector space, diffeomorphic to a complex vector bundle of rank (g − 1 + k) over the sym-

2g−2−k metric product S Σg (Theorem 10.8 of [Hit87]).

104 Appendix A

Riemannian geometry computations

A.1 Concerning Hopf differentials

Suppose (Σ, σ) is a Riemann surface and (N, ρ) is a smooth Riemannian manifold. Let

u ∈ C2(Σ, N) be a map between them. Then, using the complex coordinate z on Σ, the

Hopf differential Φ = φ(z)dz2 of u is the quadratic differential deﬁned (locally) by the equation

∗ (2,0) 2 (u (ρ)) = ρ(u(z))uzuzdz

=: φ(z)dz2

assertions relating the holomorphicity of Φ and the harmonicity of u.

Proposition A.1.1. Suppose u ∈ C1(Σ, N). If u is harmonic, then its Hopf differential Φu is holomorphic.

Proof. In local coordinates, holomorphicity of Φ is deﬁned by the vanishing of

∂ ∇ φ(z)dz2 ≡ φ(z) dz2 + φ(z) ∇ dz2 . ∂z ∂z ∂z

This is equivalent to the vanishing of

∂ (ρ(u(z))u u ) , ∂z z z

105 2 since ∇∂z dz = 0. We proceed by computing:

∂ (ρ(u(z))u u ) = ((ρ u + ρ u ) u u ) + ρ (u u + u u ) ∂z z z u z u z z z zz z z zz

= (ρuuzuz + ρuzz) uz + (ρuuzuz + ρuzz) uz

u u =ρ τ uz + τ uz

Since u is harmonic, we have τu ≡ 0 and τu ≡ 0.

Proposition A.1.2. Suppose that u : Σ → N is either C2 or a C1-diffeomorphism. If its Hopf differential Φu is holomorphic, then u is harmonic.

Proof. If Φu is holomorphic, then, as above,

∂ 0 = (ρ(u(s))u u ) = τuu + τuu . ∂z z z z z

u u 2 2 Hence, τ must vanish at least where J = |uz| − |uz| and ρ ◦ u do not vanish. In particular, this implies that τu vanishes almost everywhere. If u ∈ C2, then τ must actually vanish everywhere. Hence, u is harmonic.

1 u 2 2 If u ∈ C is a diffeomorphism, then J = |uz| − |uz| does not vanish. Hence, |uz| 6=

|uz| holds everywhere and, in particular, the complex numbers uz and uz do not have the same modulus. Note that τu and τu have the same modulus. Furthermore, the complex

u u u numbers τ uz and τ uz can only sum to be 0 if their moduli are equal. Therefore τ must vanish, and u is harmonic.

Harmonicity is a weaker notion than conformality. We also have the following characterization of conformal maps through their Hopf differentials:

Proposition A.1.3. Suppose u ∈ C1(Σ, N). Then Φu = 0 if and only if u is conformal or anti-

conformal.

Proof. Vanishing of the Hopf differential Φ yields

ρ(u(z))uzuz = 0.

106 1 Hence, either uz = 0 or uz = 0 almost everywhere. Since u is C , this implies that in fact either uz = 0 (and u is conformal) or uz = 0 (and u is anti-conformal) everywhere.

Note that there is a counter-example for the correlation between harmonicity of u and

holomorphicity of Φ when u is not C1: there exists a harmonic map from the torus C/Z2 →

CP1 which is not harmonic, yet whose Hopf differential is not harmonic [Jos91].

A.2 Geometry of harmonic mappings via their Hopf differentials

Proposition A.2.1. Let u : (Σ, σ) → (N, ρ) be a harmonic map. In the natural coordinate system

ζ = ξ + iη in which Φu = dζ2, valid away from the zeros of Φu, we have the identity

σ · e σ · e u∗(ρ(u)du du) = + 2 dξ2 + − 2 dη2. |Φu| |Φu|

∂ζ ∂ζ Proof. This is a computation using the local change of variables dζ = ∂z dz + ∂z dz. Since √ √ ∂ζ u ∂ζ u ∂z = Φ and ∂z = 0 by holomorphicity of Φ , the local change of variables reduces to √ √ dζ = Φudz (and consequently dζ = Φudz). We compute:

u∗(ρ(u) · du du) = φudz2 + σ · e · dz dz + φudz2 ! !2 1 2 1 1 1 = φu · √ dζ + σ · e √ dζ √ dζ + φu √ dζ φu φu φu φu

σ · e 2 = dζ2 + dζ dζ + dζ |Φu| σ · e = (dξ + i · dη)2 + (dξ + i · dη) (dξ − i · dη) + (dξ − i · dη)2 |Φu| σ · e σ · e = + 2 dξ2 + − 2 dη2 |Φu| |Φu|

This expression is valid away from the zeros of Φu. Furthermore, by smoothness of the harmonic map and the continuity of the target metric, the pullback metric can be determined through this expression at the zeroes of Φu by taking limits.

107 Proposition A.2.2. Let u : (Σ, σ) → (N, ρ) be a harmonic map. The image of the horizontal leaf

γ(t) = (t, η0) has geodesic curvature

1 1 σ · e 2 ∂ σ · e κ (t) = − − 2 · log + 2 . γ 2 |Φu| ∂η |Φu|

Proof. The geodesic curvature is computed with respect to the pull-back metric described

above, which is conveniently expressed in isothermal coordinates. This metric is of the

form Edu2 + 2Fdu dv + Gdv2 with the coefﬁcients given by

σ · e σ · e E = + 2 F = 0 G = − 2 |Φu| |Φu|

In general, a curve parameterized by (u(t), v(t)) in the (ξ, η) coordinate system has geodesic

curvature computed by the formula √ EG − F κ(t) = · M, [E(u0)2 + 2Fu0v0 + G(v0)2]3/2 where M involves the Christoffel symbols, u, and v, and is given by

2 0 3 00 0 2 1 0 2 0 M := − Γ11(u ) + u v − (2Γ12 − Γ11)(u ) v

1 2 0 0 2 0 00 1 0 3 + (2Γ12 − Γ22)u (v ) − u v + Γ22(v ) .

k Furthermore, the Christoffel symbols Γij can be expressed 1 E 1 E Γ1 = u Γ2 = − v 11 2 E 11 2 G 1 E 1 G Γ1 = Γ1 = v Γ2 = Γ2 = u 12 21 2 E 12 21 2 G 1 G 1 G Γ1 = − u Γ2 = v 22 2 E 22 2 G

Thus, the image of the curve γ(t) = (t, η0) lying on a horizontal leaf has geodesic curvature √ EG 2 κγ(t) = · −Γ11 [E]3/2 √ 1 E = GE−1 v 2 G ∂ σe + r σe 1 1 ∂η |Φu| 2 = − · · · u 2 σe |Φ | | u| + 2 2 σe − Φ |Φu| 2 since u0 = 1, u00 = 0 and v0 = 0, v00 = 0.

108 Appendix B

Holomorphic quadratic differential structures

B.1 Measured foliations

To any holomorphic quadratic differential Φ on a simply connected domain D ⊂ Σ of a

Riemann surface Σ, there is an associated pair of mutually orthogonal singular foliations, the horizontal foliation FH and the vertical foliation FV . In a local coordinate z in which

2 Φ = φ(z)dz , we ﬁx a point z0 which is not a zero of φ(z), and form the C valued integral

Z z √ Z z q ζ(z) := Φ = φ(z)dz. z0 z0

Since φ is holomorphic, this deﬁnes a locally injective, conformal map of D to C which in essence provides a local C-chart on Σ.

Pulling back the horizontal lines (lines parallel to the x-axis), we obtain the horizontal

trajectories for the horizontal foliation. Analytically, a horizontal trajectory is described by

λ the set of curves γH(x) for which ζ(z) = x + iλ has a constant imaginary part. Similarly,

λ we deﬁne the vertical trajectories as the set of curves γV (y) for which ζ(z) = λ + iy has

constant real part. The horizontal and vertical trajectories deﬁne leaves of the horizontal

109 (a) FH (b) FV

Figure B.1: Foliations for Φu = z2dz2 on C

and vertical foliations FH and FV , respectively. Observe that the foliations are globally deﬁned because the leaves do not depend on the choice of a sign of the square root.

The foliations are mutually orthogonal because ζ(z) is a conformal map (by holomorphicity of Φ). The zeros of φ(z) form the singularities of the foliations. We call a leaf singular if it meets a zero of Φ. Generally, at a zero of order k ≥ 1 of φ, there are (k + 2) singular leaves emanating at equal angles from the zero, for each of the horizontal and vertical foliations.

For example, if Ω = C and Φu = z2dz2, then

Z ∗ (x2 − y2 + C) + i(2xy + D) = z dz. z0

The horizontal and vertical foliations FH and FV are depicted in Figure B.1. For concrete- ness, we describe the foliations:

Horizontal foliation FH = Level sets of imaginary part xy

2 2 Vertical foliation FV = Level sets of real part x − y

These orthogonal foliations are in fact measured foliations, with measures:

µH := |dIζ| on FH

µV := |dRζ| on FV

110 Note that the measured foliations are also globally deﬁned on the entire surface Σ, since the derivative applied in deﬁning the measures negates the constant of integration coming from a choice of a basepoint for the path integral.

B.2 Harmonic maps to R-trees

Associated naturally to each of the horizontal and vertical measured foliations FH and FV coming from a holomorphic quadratic differential Φ on Σ is a (singular) R-tree, TH and

TV . Furthermore, we can deﬁne a pair of projections to these R-trees,

πH :Σ = FH → TH

πV :Σ = FV → TV

which are harmonic maps in the sense of Gromov-Schoen [GS92]. Here, we abuse notation

and identify the surface Σ with its foliation decompositions:

λ Σ = FH = ∪ γH λ∈R

λ Σ = FV = ∪ γV λ∈R

We recall below the projection map constructions from [Wol98].

Consider the lift Φe of Φ to the universal cover Σe of Σ. Using the holomorphic quadratic

differential Φe on Σe, we can deﬁne the (lifted) horizontal and vertical foliations, FgH and FfV .

From these foliations, we deﬁne the identiﬁcation spaces TH and TV : a point of TH and TV

λ is an equivalence class of points belonging to the same horizontal leaf γfH ⊂ FgH or vertical

λ leaf γfV ⊂ FfV .

The TH and TV are called the leaf spaces of the horizontal and vertical foliations, respec-

tively. Lifting to the universal cover ensures that each leaf space is an R-tree. In particu-

lar, there cannot be any cycles in TH or TV because the cycle would have to come from a

cylinder of leaves on the domain, and so would be represented by an element of the ﬁrst

111 homology group of Σe; however, the ﬁrst homology group of Σe vanishes since it is simply

connected.

Away from the discrete set of zeros of Φe, the projections have the local form (deﬁned

on each leaf away from the singularities, if any)

λ πH : γH −→ R ⊂ TH Z z ζ(z) = s + iλ 7−→ s ≡ dRζ(z(s))ds z0

λ πV : γV −→ R ⊂ TV Z z ζ(z) = λ + it 7−→ t ≡ dIζ(z(s))ds, z0 where z0 ∈ Σ is an arbitrary point which is not a zero of Φe, and the path integral is taken along any path not running through a zero of Φe. Here, the targets R denote a real line embedded in the leaf spaces TH and TV .

112 Appendix C

Minimal suspensions in H2 × R

Here, we recall the suspension procedure [Wol07] of a harmonic map u : Ω → H2 from a domain Ω ⊂ C. This construction of minimal graphs in H2 × R is an application of the observation from [Wol98], and we will follow the notation from Appendix B. Moreover, all minimal graphs in H2 × R can be obtained this way.

Data from the harmonic map: Consider the Hopf differential Φu of the map u : Ω → H2.

u From the data of Φ , we obtain the horizontal foliation FH and its leaf space (TH, µH), √ R z u which is an R-tree with a metric deﬁned by the horizontal measure µH = dR Φ z0

(see Appendix B). Rescaling µH by a factor of 2, we can consider the rescaled leaf space space (TH, 2µH) with the local harmonic projection map

λ πH : γH −→ R ⊂ TH Z z ζ(z) = s + iλ 7−→ 2s ≡ 2 dRζ(z(s))ds. z0

The Hopf differential of this projection map πH is thus

h i πH 1 2 2 Φ = ||DπH (∂x)|| − DπH ∂y − 2i DπH (∂x) , DπH ∂y 4 = −1

113 in the natural coordinate system ζ = x + iy for which Φu = dζ2, since

DπH (∂x) = 0

DπH ∂y = 2∂s

Hence, the Hopf differential ΦπH in the z coordinate (away from the zeros of Φu) on Ω is simply ΦπH (z) = −Φu(z). By the unique continuation of the holomorphicity of ΦπH , we have that, in fact, ΦπH ≡ −Φu everywhere.

Deﬁning the height function: The real coordinate in H2 × R serves as a “height” for a minimal graph over H2. Observe that a surface Σ is minimal if it has a local parameterization (u, h) : Ω → H2 × R which is conformal and harmonic. Conformality of the map

(u, h) is equivalent to the vanishing of its Hopf differential Φ(u,h) = Φu + Φh, by the product structure of H2 × R (see Appendix A.1). Harmonicity of the map (u, h) is equivalent to the vanishing of τ(u,h) = (τu, τh), again by the product structure of H2 × R. In particular, this means that τh = 0, so that h : Ω → R is a harmonic function. Deﬁne the folding map

f : TH → R so that f (s) = f (t) whenever s = t. We call the composition

f ◦ πH : Σ = FH → TH → R the folded projection. This serves as the height function for the minimal graph.

For example, the folded projection for Φu = z2dz2 on C is given by the function

f ◦ πH : Σ −→ R

λ γH 3 s 7−→ s + C

x + iy 7−→ x2 − y2 + C

where C is the constant of integration from choosing an arbitrary base point z0 for path integration. This artifact C is related to the simple fact that a harmonic function plus a constant remains harmonic, which is equivalent to translating the minimal graph along the R axis (a Jacobi ﬁeld) in H2 × R.

114 Figure C.1: Suspension of u : C → H2 with Φu(z) = z2dz2.∗

The minimal suspension of the Scherk map u : C → H2 having Hopf differential

Φu(z) = −z2dz2 is depicted in Figure C.1.

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