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Some Metric Properties of the Teichmüller Space of a Closed Set Some Metric Properties of the Teichmüller Space of a Closed Set in the Riemann Sphere

in the Riemann Sphere

Nishan Chatterjee

The Graduate Center, City University of New York

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Some Metric Properties of the Teichm¨ uller Space of a Closed Set

in the Riemann Sphere

by

Nishan Chatterjee

A dissertation submitted to the Graduate Faculty in Mathematics in partial fulfill- ment of the requirements for the degree of Doctor of Philosophy, The City University of New York.

2017

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2017 c

Nishan Chatterjee

All Rights Reserved

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This manuscript has been read and accepted for the Graduate Faculty in Mathe- matics in satisfaction of the dissertation requirements for the degree of Doctor of Philosophy.

(required signature)

Date Chair of Examining Committee

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Date Co-Chair of Examining Committee

(required signature)

Date Executive Officer

Ara Basmajian

Yunping Jiang

Linda Keen

Sudeb Mitra

Supervisory Committee

THE CITY UNIVERSITY OF NEW YORK

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Abstract

Some Metric Properties of the Teichm¨ uller Space of a Closed Set in the Riemann Sphere

by

Nishan Chatterjee

Advisors: Yunping Jiang and Sudeb Mitra

Associated to each closed subset E of the Riemann sphere b C, there is a con- tractible complex Banach manifold with a basepoint; this was first studied by G.

Lieb in his Cornell University doctoral dissertation [15]. We call this the Teich- m¨ uller space of the closed set E, denoted by T (E). Throughout this thesis, the blanket assumption will be that E is a closed set in b C and that 0, 1, ∞ belong to E.

The Teichm¨ uller space T (E) is intimately related with holomorphic motions of the closed set E.

In this thesis we study several metric and analytic properties of T (E).

For the Teichm¨ uller space of a Riemann surface, the principle of Teichm¨ uller contraction was introduced by Gardiner in the paper [9]. In [18] Mitra proved a δ-

form of Teichm¨ uller contraction for the generalized Teichm¨ uller space T (E). Our

first theorem in this thesis, is to extend Earle’s form of Teichm¨ uller contraction

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to the space T (E). This is Theorem A of this thesis. It improves and sharpens Mitra’s δ- form of Teichm¨ uller contraction for T (E).

We then study holomorphic isometries for T (E), thereby generalizing Theorem 5 of the paper of Earle, Kra and Krushka´l [7]. This is Theorem B of this thesis.

In his paper [6], Earle proved a version of Schwarz’s lemma for Teichm¨ uller space of a Riemann surface. Theorem C of this thesis extends Earle’s result to the generalized Teichm¨ uller space T (E).

Finally, we study complex geodesics and unique extremality for T (E), in order to generalize Theorem 6 of the paper of Earle-Kra-Krushka´l [7]. This is Theorem D of this thesis.

When E is finite there is a natural identification of T (E) with the classical Teich- m¨ uller space T eich(b C\E). For the more general case, when E is infinite, we consider an increasing sequence {E

n

} of finite subsets of E, such that 0, 1, ∞ always belong to E

n

and S

n

E

n

is dense in E.

It was proved by Mitra in [17] that T (E) is the metric inverse limit of the pointed metric spaces {T (E

n

)}. A basic technique in this thesis is to exploit this approximation. We summarize the main results of this thesis as follows. The precise definitions are given in Sections/Chapters indicated below. In what follows, let ∆ be the open unit disk in the complex plane, that is, ∆ := {z ∈ C : |z| < 1}.

Theorem A (Teichm¨ uller contraction in T (E)). Let µ ∈ M (C), P

E

(µ) = τ ∈ T (E)

and µ

0

be extremal in the E-equivalence class of µ; that is P

E

0

) = P

E

(µ) and

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0

k

≤ kµk

. Let k = kµk

and k

0

= kµ

0

k

, also

K = 1 + k

1 − k and K

0

= 1 + k

0

1 − k

0

, We define `

µ

as in §1.6

`

µ

(φ) = Z Z

C

µφdxdy , ∀φ ∈ A(E), ∀µ ∈ L

(C).

and

k`

µ

k

T (E)

= sup{|`

µ

(φ)|, φ ∈ A(E), kφk

1

= 1}.

Then

1 K

0

− 1

K ≤ 2

1 − k

2

(k − k`

µ

k

T (E)

) ≤ K − K

0

.

Theorem B (Holomorphic isometries from ∆ to T (E)). Let f : ∆ → T (E) be holomorphic, and let t

1

∈ ∆, if

1. d

T (E)

(f (t

1

), f (t

2

)) = ρ

(t

1

, t

2

) for some t

2

∈ ∆ \ {t

1

} or

2. kf

0

(t

1

)k

T (E)

= F

T (E)

(f (t

1

), f

0

(t

1

)) =

1

1−|t1|2

then f is a holomorphic isometry.

[The definitions of d

T (E)

and F

T (E)

are given in §1.6. The definition of holomor- phic isometry is given in Chapter 3.]

Theorem C (Schwarz’s lemma for T (E)). Let f : ∆ → T (E) be holomorphic and t ∈ ∆ \ {0} with f (0) = 0. If either of the inequalities

0

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2. k

0

(t) ≤ |t|

is strict, then both are strict and

ρ

 k

0

(t)

|t| , kf

0

(0)k

T (E)



≤ 2ρ

(0, t).

[The definition of ρ

is given in §1.3.]

Theorem D (Complex geodesics in T (E)). Let µ

0

∈ M (C), µ

0

6= 0 and µ

0

be ex- tremal in its E-equivalence class. Then the following four statements are equivalent:

1. The Beltrami coefficient µ

0

is uniquely extremal and |µ

0

| = kµ

0

k

a.e.

2. There exists only one geodesic segment joining P

E

(0) and P

E

0

).

3. There exists only one holomorphic isometry f : ∆ → T (E) such that f (0) = P

E

(0) and f (kµ

0

k

) = P

E

0

).

4. There exists only one holomorphic map g : ∆ → M (C) such that g(0) = 0 and P

E

(g(kµ

0

k

)) = P

E

0

).

[The definitions of complex geodesics and unique extremality are given in Chapter

5.]

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Acknowledgements

First of all, I would like to thank my advisors, Professor Yunping Jiang and Professor Sudeb Mitra, for their immense help during the course of my PhD. It was Professor Jiang who brought me to the complex analysis and dynamics seminar that opened up my views and outlook towards the subject. Throughout these six years he has always been available for help. He has shared his interests and ideas with us and had encouraged us greatly to be good mathematicians.

Professor Sudeb Mitra is also very insightful and considerate. He helped me work through all the minute details in my thesis, and he is the one who inspired my in- terest in complex analytic Teichm¨ uller theory. He also showed me various directions and projects the thesis might lead to.

I would also like to thank Professor Ara Basmajian and Professor Linda Keen for

being in my thesis committee. I have learned a great deal of geometry and analysis

from them.

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I would like to thank Professor Frederick Gardiner for arranging seminars at the Graduate Center that helped me a lot in understanding the subject.

I would also like to thank Professor Hiroshige Shiga, with whom I had several helpful and illuminating discussions.

I would like to thank some of my teachers, especially Professor Abhijit Champan- erkar, from the Graduate Center for helping me in my every need. Also, I would like to thank the teachers I met before coming to the Graduate Center. Especially, Professor Jørgen Andersen (Aarhus Universitet), Professor Mahan Mj(Tata Insti- tute of Fundamantal Research, Mumbai), Professor C. S. Aravinda(Tata Institute of Fundamental Research, Bangalore), Professor Ramesh Srikantan (Tata Institute of Fundamental Research, Bangalore) and Professor Sandeep Kunnath (Tata Institute of Fundamental Research, Bangalore).

I would like to thank Professor Saeed Zakeri for making me interested in com- plex analysis and Professor Jozef Dodziuk for helping me with any problem, be it mathematical or non-mathematical.

I would like to thank my colleagues Zhe Wang, Tao Chen, Shantanu Nandy, Marten

Fels and John Adamski from the Graduate Center. I would like to thank my friends

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Jaimalya Mukhopadhyay, Sagar Podder, Simantini Ghosh and Dhrubajyoti Gan- gopadhyay for providing me with much needed mental support. I will especially like to thank my friend Alice Kwon for always being at my side and discussing mathe- matics that helped me understand the subject in a better way.

I would like to thank my first two mathematics teachers, who inspired me to take mathematics as a career choice. One is my uncle Professor Kshitish Chattopadhyay (Burdwan University, Bardhaman, West Bengal, India) and the other is my cousin Professor Pralay Chattopadhyay (Institute of Mathematical Sciences, Chennai, In- dia).

I would also like to thank my family, my parents Subhas and Sikha Chatterjee, and my sisters Sucheta and Sanchita Chatterjee. Without them things would have been very difficult.

At the end, I would especially like to thank my wife Raya Debnath. Without her

motivation and constant support, nothing would have been possible.

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Dedicated to my uncle

Professor Kshitish Chattopadhyay.

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Contents

1 Introduction 1

1.1 Quasiconformal mappings . . . . 1

1.2 Infinite Dimensional Holomorphy . . . . 2

1.3 The Kobayashi metric . . . . 4

1.4 Teichm¨ uller space of a plane region . . . 17

1.5 Product Teichm¨ uller space . . . 20

1.6 Teichm¨ uller space of a closed set in the sphere . . . 23

2 Teichm¨ uller contraction on T (E) 32

3 Holomorphic isometries from ∆ to T (E) 43

4 Schwarz’s lemma for T (E) 51

5 Complex geodesics in T (E) 55

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Chapter 1 Introduction

1.1 Quasiconformal mappings

Throughout this dissertation we will use C for the complex plane, b C = C ∪ {∞} for the Riemann sphere, and ∆ for the open unit disk {z ∈ C : |z| < 1}.

Definition 1.1. A complex-valued function w = f (z) defined in a region V ⊂ C is called a quasiconformal mapping if it is a sense-preserving homeomorphism of V onto its image and its complex distributional derivatives

w

z

= 1 2

 ∂f

∂x − i ∂f

∂y



and w

z¯

= 1 2

 ∂f

∂x + i ∂f

∂y



are Lebesgue measurable locally square integrable functions on V that satisfy the inequality |w

z¯

| ≤ k|w

z

| almost everywhere in V , for some real number k with 0 ≤ k < 1.

If w = f (z) is a quasiconformal mapping defined on the region V then the

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function w

z

is known to be nonzero almost everywhere on V . Therefore the function

µ

f

= w

z¯

w

z

is well defined L

function on V , called the complex dilatation or the Beltrami coefficient of f . The L

norm of every Beltrami coefficient is less than one.

The positive number

K(f ) = 1 + kµ

f

k

1 − kµ

f

k

is called the dilatation of f . We say that f is K-quasiconformal if f is a quasicon- formal mapping and K(f ) ≤ K.

We call a homeomorphism of b C normalized if it fixes the points 0, 1, and ∞.

We will always denote by M (C) the open unit ball of the complex Banach space L

(C). Then, for each µ in M(C), there exists a unique normalized quasiconfor- mal homeomorphism of b C onto itself that has Beltrami coefficient µ (see [1]); this quasiconformal map will be denoted by w

µ

. Furthermore, we have the following fundamental theorem due to Ahlfors and Bers: for every fixed z ∈ C, the map µ 7→ w

µ

(z) of M (C) into C is holomorphic (see [2]).

1.2 Infinite Dimensional Holomorphy

Teichm¨ uller spaces are complex manifolds modeled on finite or infinite dimensional

complex Banach spaces. In this section we will discuss some basic facts about

holomorphic maps between Banach spaces. The standard reference is [4].

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Definition 1.2. Let E and F be two complex Banach spaces of finite or infinite dimensions. Let U be a nonempty open subset in E. A mapping f : U → F is holomorphic if and only if it is locally bounded and the complex Gˆ ateaux derivative f

0

(x)(λ) at x ∈ U in the direction λ ∈ E defined as:

f

0

(x)(λ) = lim

t→0

f (x + tλ) − f (x)

t ∈ F

exists (in the norm of F ) for t ∈ C and for each (x, λ) ∈ U × E.

If f is holomorphic, then f

0

(x)(λ) : E → F is a continuous C-linear map for each x ∈ U . This map is called the Fr´ echet derivative of f at x. We note the following

Proposition 1.3. A map f : U → F is holomorphic if and only if for each x in U there is a continuous complex linear map f

0

(x) : E → F such that

||f (x + y) − f (x) − f

0

(x)(y)||

F

||y||

E

→ 0 (1.2.1)

as y → 0 in E.

Remark 1.4. The map f : U → F has a real Fr´ echet derivative at each point x ∈ U . If f

0

(x) : E → F is a continuous, real linear map that satisfies (1.2.1)

Definition 1.5. A complex Banach manifold modeled on a complex Banach space E is a topological (Hausdorff) space X with an open covering U

i

such that:

(i) For each U

i

there is a homeomorphism ϕ : U

i

→ ϕ

i

(U

i

) ⊂ E, where ϕ

i

(U

i

) is

an open subset of E, and

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(ii) The transition homeomorphism

ϕ

j

◦ ϕ

−1i

: ϕ

i

(U

i

∩ U

j

) → ϕ

j

(U

i

∩ U

j

)

is a holomorphic isomorphism, for each pair of indices i, j.

The following concept will be crucial throughout this dissertation.

Definition 1.6. A holomorphic map f between two complex Banach manifolds X and Y is a holomorphic split submersion if it is an open mapping and has holomorphic local sections. This means for any given x in X there is a holomorphic right inverse to f defined in some neighborhood of f (x) and mapping f (x) to x.

1.3 The Kobayashi metric

Recall that the Poincar´ e metric on ∆ is given by

ρ

(z

1

, z

2

) = tanh

−1

z

1

− z

2

1 − ¯ z

1

z

2

for z

1

and z

2

in ∆.

Let X be a connected complex manifold and let O(∆, X) be the set of holomor- phic maps from ∆ into X. The Kobayashi function δ

X

: X × X → [0, ∞] is defined for all x and y in X by

δ

X

(x, y) = inf{ρ

(0, t) : f (0) = x and f (t) = y for some f ∈ O(∆, X)} (1.3.1)

provided the set of maps described above is nonempty, and +∞ otherwise. It easily

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and f : X → Y is a holomorphic map, then

δ

Y

(f (x

1

), f (x

2

)) ≤ δ

X

(x

1

, x

2

) for all x

1

, x

2

∈ X.

Definition 1.7. The Kobayashi (pseudo)metric ρ

X

is defined to be the largest (pseudo)metric on X such that

ρ

X

(x, y) ≤ δ

X

(x, y) for all x, y ∈ X.

If δ

X

is a metric, then ρ

X

and δ

X

are equal. In any case, if X and Y are connected complex manifolds and f : X → Y is a holomorphic map, we have

ρ

Y

(f (x

1

), f (x

2

)) ≤ ρ

X

(x

1

, x

2

) for all x

1

, x

2

∈ X. (1.3.2)

Equality holds in Equation (1.3.2) if f is biholomorphic.

Recall the open unit ball M (C) of the complex Banach space L

(C).

Proposition 1.8. The Kobayashi metric on M (C) is given by:

ρ

M

(µ, ν) = tanh

−1

µ − ν 1 − ¯ µν

for all µ, ν in M (C). The infinitesimal Kobayashi metric on M(C) is given by:

K

M

(µ, λ) =

λ 1 − |µ|

2

for µ in M (C) and λ in L

(C).

See Proposition 7.25 in [8].

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We need the following inequality of Lindel¨ of; see [16] and also [11]. For the reader’s convenience, we give a self-contained proof.

Before the proof however, we proceed with a few preliminaries. To begin, for z, a ∈ ∆ define the pseudo-hyperbolic metric on ∆ to be

d

(z, a) = |φ

a

(z)| =

z − a 1 − ¯ az

.

Observe that if f : ∆ → ∆ is holomorphic then by the Schwarz-Pick Theorem

d

(f (z), f (a)) =

f (z) − f (a) 1 − f (a)f (z)

z − a 1 − ¯ az

= d

(z, a) and if f ∈ Aut(∆) then equality holds, i.e.

d

(f (z), f (a)) = d

(z, a) for all z, a ∈ ∆.

Lemma 1.9. d

is a metric

Proof. Let z, a ∈ ∆. For ease of notation, we will omit the subscript ∆ and use d to denote d

. That d(z, a) ≥ 0 is clear, as is the fact that d(z, a) = 0 if and only if z = a.

To see that d(z, a) = d(a, z) simply observe that

d(z, a) = |φ

a

(z)| =

z − a 1 − ¯ az

and d(a, z) = |φ

z

(a)| =

a − z 1 − ¯ za

. (1.3.3)

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But

|1 − ¯ az|

2

= (1 − ¯ az)(1 − a¯ z)

= (1 − ¯ za)(1 − z¯ a)

= |1 − ¯ za|

2

and so |1 − ¯ az| = |1 − ¯ za|. This implies, referring back to (1.3.3), that d(z, a) = d(a, z).

It remains to show the triangle inequality, and we claim it suffices to prove

d(t

1

, t

2

) ≤ |t

1

| + |t

2

|, for all t

1

, t

2

∈ ∆. (1.3.4)

Indeed for z, w, a ∈ ∆, the triangle inequality d(z, w) ≤ d(z, a) + d(a, w) holds if and only if d(φ

a

(z), φ

a

(w)) ≤ d(φ

a

(z), 0) + d(0, φ

a

(w)) since φ

a

∈ Aut(∆) and φ

a

(a) = 0. Letting t

1

= φ

a

(z) and t

2

= φ

a

(w), this becomes d(t

1

, t

2

) ≤ d(t

1

, 0) + d(0, t

2

) which is precisely (1.3.4) since d(z, 0) = |z| for any z ∈ ∆.

Thus our goal is to prove (1.3.4), but in fact we aim to prove a much stronger result, namely:

Lemma 1.10. For any t

1

, t

2

∈ ∆,

d(t

1

, t

2

) ≤ |t

1

| + |t

2

| 1 + |t

1

||t

2

|.

This of course implies (1.3.4) since 1/(1 + |t

1

||t

2

|) ≤ 1. To prove Lemma 1.10,

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we begin by observing that

1 − d(t

1

, t

2

)

2

= 1 −

t

1

− t

2

1 − t

1

¯ t

2

2

= (1 − t

1

¯ t

2

)(1 − ¯ t

1

t

2

) − (t

1

− t

2

)(¯ t

1

− ¯ t

2

)

|1 − t

1

t ¯

2

|

2

= (1 − |t

1

|

2

)(1 − |t

2

|

2

)

|1 − t

1

¯ t

2

|

2

≥ (1 − |t

1

|

2

)(1 − |t

2

|

2

)

(1 + |t

1

||t

2

|)

2

, (1.3.5)

where the last line follows from the triangle inequality. We also compute the follow- ing:

d(|t

1

|, −|t

2

|) =

|t

1

| − (−|t

2

|) 1 − |t

1

|(−|t

2

|)

=

|t

1

| + |t

2

| 1 + |t

1

||t

2

|

= |t

1

| + |t

2

| 1 + |t

1

||t

2

| and so

1 − d(|t

1

|, −|t

2

|)

2

= 1 −  |t

1

| + |t

2

| 1 + |t

1

||t

2

|



2

= (1 − |t

1

|

2

)(1 − |t

2

|

2

) (1 + |t

1

||t

2

|)

2

. Comparing the previous line with (1.3.5) we see that

1 − d(t

1

, t

2

)

2

≥ 1 − d(|t

1

|, −|t

2

|)

2

.

Which implies

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d(t

1

, t

2

) ≤ d(|t

1

|, −|t

2

|) = |t

1

| + |t

2

| 1 + |t

1

||t

2

| . This proves Lemma 1.10 and hence Lemma 1.9 as well.

Our next goal is to prove Lindel¨ of’s Inequality as stated previously, but first we derive one more inequality concerning the metric d.

Claim: For all z, a ∈ ∆, the pseudo-hyperbolic metric d satisfies

d(|z|, |a|) ≤ d(z, a) ≤ d(|z|, −|a|).

Proof. We need only prove the first inequality since the second was verified in the proof of Lemma 1.10. To this end, note that

d(|z|, |a|) =

|z| − |a|

1 − |z||a|

= ||z| − |a||

1 − |z||a|

and so

1 − d(|z|, |a|)

2

= 1 − ||z| − |a||

2

(1 − |z||a|)

2

= 1 − (|z| − |a|)(|¯ z| − |¯ a|) (1 − |z||a|)

2

= 1 − (|z| − |a|)(|z| − |a|) (1 − |z||a|)

2

= (1 − |z|

2

)(1 − |a|

2

)

(1 − |z||a|)

2

. (1.3.6)

But the above is less than or equal to 1 − d(z, a)

2

. Indeed, consider the following:

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1 − d(z, a)

2

= (1 − |z|

2

)(1 − |a|

2

)

|1 − ¯ az|

2

≤ (1 − |z|

2

)(1 − |a|

2

)

(1 − |z||a|)

2

= 1 − d(|z|, |a|)

2

.

The leftmost equality was derived in the proof of Lemma 1.10, and the inequality follows since |1 − ¯ az| ≥ ||1| − |a¯ z|| = |1 − |a||z||. Of course the rightmost equality is (1.3.6). From this we conclude d(|z|, |a|) ≤ d(z, a) as desired.

Finally we are ready to derive Lindel¨ of’s inequality.

Proposition 1.11 (Lindel¨ of). If f : ∆ → ∆ is holomorphic then

|f (0)| − |z|

1 − |f (0)||z| ≤ |f (z)| ≤ |f (0)| + |z|

1 + |f (0)||z| for all z ∈ ∆.

Proof. We prove the rightmost inequality first. Let z ∈ ∆. By our claim above we know that

d(|f (z)|, |f (0)|) ≤ d(f (z), f (0)) ≤ d(|z|, 0).

Which by definition of d implies

|f (z)| − |f (0)|

1 − |f (z)||f (0)|

≤ |z| (1.3.7)

and so ||f (z)| − |f (0)|| ≤ |z|(1 − |f (z)||f (0)|). Thus

|f (z)| ≤ ||f (z)| − |f (0|| + |f (0)|

≤ |z|(1 − |f (z)||f (0)|) + |f (0)|.

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Rearranging terms we find that |f (z)|(1 + |z||f (0)|) ≤ |f (0)| + |z| and so

|f (z)| ≤ |f (0)| + |z|

1 + |f (0)||z|

as desired. For the lower bound on |f (z)| observe that (1.3.7) implies

|f (0)| ≤ ||f (0)| − |f (z)|| + |f (z)|

≤ |z|(1 − |f (z)||f (0)|) + |f (z)|

and hence, again with some rearranging, |f (0)| + |z||f (z)||f (0)| ≤ |f (z)| + |z| and thus

|f (0)| − |z|

1 − |f (0)||z| ≤ |f (z)|.

The following theorem appears in Earle’s paper [6] as Theorem 3. This result will be crucial in our thesis. For the reader’s convenience, we give a complete proof.

Theorem 1.12 (Earle [6]). Let V be a complex Banach space and g : ∆ → V be a holomorphic map with g(0) = 0 and kg(t)k ≤ 1, ∀t ∈ ∆. Fix t ∈ ∆ \ {0}. If either of the inequalities

1. kg

0

(0)k ≤ 1

2. kg(t)k ≤ |t|

is strict then both are strict and

ρ

 kg(t)k

|t| , kg

0

(0)k 

≤ ρ

(0, t).

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Proof. We observe that Lindel¨ of’s inequality is valid with the assumption f : ∆ →

∆, since if for any t

0

∈ ∆, |f (t

0

)| = 1, by maximum modulus principle f (t) = f (t

0

), ∀t ∈ ∆ and the inequality reduces to

1 − |t|

1 − |t| ≤ |f (t)| ≤ 1 + |t|

1 + |t| ,

which implies |f (t)| = 1 and hence the inequality is trivially true.

So, without loss of generality we can say that if f is holomorphic on ∆ and

|f (t)| ≤ 1 for all t ∈ ∆ then ∀t ∈ ∆ the folowing inequality is true

|f (0)| − |t|

1 − |t||f (0)| ≤ |f (t)| ≤ |f (0)| + |t|

1 + |t||f (0)| . (1.3.8) Let ` : V → C be a linear functional on V with k`k = 1, where

k`k = sup n |`(x)|

kxk , x ∈ V, kxk 6= 0} = sup{|`(x)|, x ∈ V, kxk = 1 o .

Consider the holomorphic map g : ∆ → V , we already know that kg(t)k ≤ 1 and hence from the definition of k`k we conclude that |`(g(t))| ≤ 1, So the function h : ∆ → ∆ defined as h(t) = `(g(t)) is holomorphic, since, g is holomorphic and

` is linear. Moreover h(0) = 0, so applying Schwarz’s lemma we see that |h(t)| =

|`(g(t))| ≤ |t|. Hence

|`(g(t))||t|

≤ 1, ∀t ∈ ∆

Also since ` is linear, we see that h

0

(0) = `(g

0

(0)) and hence again by Schwarz’s

lemma and the fact k`k = 1 we see that |h

0

(0)| = |`(g

0

(0)| ≤ 1.

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Now consider the function f defined as follows

f (t) =

(

`(g(t))

t

if t 6= 0

`(g

0

(0)) if t = 0.

From our discussion above we see that |f (t)| ≤ 1 and since

f (0) = `(g

0

(0)) = lim

t→0

`(g(t)) t , f is holomorphic.

So we see that f : ∆ → ∆ is holomorphic and hence we get

|f (0)| − |t|

1 − |t||f (0)| ≤ |f (t)| ≤ |f (0)| + |t|

1 + |t||f (0)| , ∀t ∈ ∆.

Hence, since our only assumption on ` was k`k = 1, we conclude ∀` : V → C, linear, with k`k = 1 the following is true for all t ∈ ∆,

|`(g

0

(0))| − |t|

1 − |t||`(g

0

(0))| ≤

`(g(t)) t

≤ |`(g

0

(0))| + |t|

1 + |t||`(g

0

(0))| . (1.3.9) By Hahn Banach theorem we can choose `

1

and `

2

, linear functionals on V with k`

1

k = k`

2

k = 1 such that `

1

(g

0

(0)) = kg

0

(0)k and `

2

(g(t)) = kg(t)k.

Since (1.3.9) is valid for all `, linear functionals with k`k = 1, it’s true for `

1

and

`

2

as well.

Notice first |`

1

(g(t))| ≤ kg(t)k since k`

1

k = 1 and hence

|`1(g(t))||t|

kg(t)k|t|

,

∀t ∈ ∆ \ {0}.

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Also, observe

|`

2

(g

0

(0))| + |t|

1 + |t||`

2

(g

0

(0))| ≤ kg

0

(0)k + |t|

1 + |t|kg

0

(0)k (1.3.10)

⇔ (|`

2

(g

0

(0))| + |t|)(1 + |t|kg

0

(0)k)

≤ (kg

0

(0)k + |t|)(1 + |t||`

2

(g

0

(0))|)

⇔ |`

2

(g

0

(0))| + |t||`

2

(g

0

(0))|kg

0

(0)k + |t| + |t|

2

kg

0

(0)k

≤ kg

0

(0)k + |t||`

2

(g

0

(0))|kg

0

(0)k + |t| + |t|

2

|`

2

(g

0

(0))|

⇔ |`

2

(g

0

(0))| + |t|

2

kg

0

(0)k ≤ kg

0

(0)k + |t|

2

|`

2

(g

0

(0))|

⇔ |`

2

(g

0

(0))|(1 − |t|

2

) ≤ kg

0

(0)k(1 − |t|

2

)

⇔ |`

2

(g

0

(0))| ≤ kg

0

(0)k.

Since k`

2

k = 1 we already know |`

2

(g

0

(0))| ≤ kg

0

(0)k and so we conclude (1.3.10) is true as well.

Now fix a t in ∆ \ {0}.

Applying (1.3.9) on `

1

we obtain kg

0

(0)k − |t|

1 − |t|kg

0

(0)k = |`

1

(g

0

(0))| − |t|

1 − |t||`

1

(g

0

(0))| ≤ |`

1

(g(t))|

|t| ≤ kg(t)k

|t| . (1.3.11)

Applying (1.3.9) and (1.3.10) on `

2

we obtain kg(t)k

|t| = |`

2

(g(t))|

|t| ≤ |`

2

(g

0

(0))| + |t|

1 + |t||`

2

(g

0

(0))| ≤ kg

0

(0)k + |t|

1 + |t|kg

0

(0)k . (1.3.12)

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Combining (1.3.11) and (1.3.12), we get kg

0

(0)k − |t|

1 − |t|kg

0

(0)k ≤ kg(t)k

|t| ≤ kg

0

(0)k + |t|

1 + |t|kg

0

(0)k . (1.3.13) Now if kg

0

(0)k = 1, then left of (1.3.13) tells us

1 = 1 − |t|

1 − |t| ≤ kg(t)k

|t| .

Hence kg(t)k ≥ |t|, so contrapositively kg(t)k < |t| ⇒ kg

0

(0)k < 1 and if kg(t)k ≥ |t|

then of (1.3.13) gives us

kg

0

(0)k + |t|

1 + |t|kg

0

(0)k ≥ 1

⇔ kg

0

(0)k + |t| ≥ 1 + |t|kg

0

(0)k

⇔ kg

0

(0)k(1 − |t|) ≥ (1 − |t|)

⇔ kg

0

(0)k ≥ 1.

Hence contrapositively kg

0

(0)k < 1 ⇒ kg(t)k < |t|. Hence we get

kg

0

(0)k < 1 ⇔ kg(t)k < |t|.

For a fixed t, from the left of (1.3.13) we have kg

0

(0)k − |t|

1 − |t|kg

0

(0)k ≤ kg(t)k

|t|

⇔ kg

0

(0)k − |t| ≤ kg(t)k

|t| − kg(t)k

|t| |t|kg

0

(0)k

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⇔ −|t|(1 − kg(t)k

|t| kg

0

(0)k) ≤ kg(t)k

|t| − kg

0

(0)k

⇔ −|t| ≤

kg(t)k

|t|

− kg

0

(0)k

1 −

kg(t)k|t|

kg

0

(0)k . (1.3.14)

From the right hand side of (1.3.13) we have kg(t)k

|t| ≤ kg

0

(0)k + |t|

1 + |t|kg

0

(0)k

⇔ kg(t)k

|t| + kg(t)k

|t| |t|kg

0

(0)k ≤ kg

0

(0)k + |t|

⇔ kg(t)k

|t| − kg

0

(0)k ≤ |t|(1 − kg(t)k

|t| kg

0

(0)k)

kg(t)k

|t|

− kg

0

(0)k

1 −

kg(t)k|t|

kg

0

(0)k ≤ |t|. (1.3.15)

Combining (1.3.14) and (1.3.15) we get

kg(t)k

|t|

− kg

0

(0)k 1 −

kg(t)k|t|

kg

0

(0)k

≤ |t|. (1.3.16)

Hence from (1.3.16) we get

tanh ρ

 kg(t)k

|t| , kg

0

(0)k 

≤ tanh ρ

(0, t). (1.3.17)

Since as tanh is a monotonically increasing bijection we finally get

ρ

 kg(t)k

|t| , kg

0

(0)k



≤ ρ

(0, t).

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1.4 Teichm¨ uller space of a plane region

Let Ω be a plane region whose complement C \ Ω contains at least two points.

Two quasiconformal mappings f and g with domain Ω are said to be in the same Teichm¨ uller class if and only if there is a conformal map h of f (Ω) onto g(Ω) such that the self-mapping g

−1

◦ h ◦ f of Ω is isotopic to the identity rel the boundary of Ω. (This means that g

−1

◦ h ◦ f extends to a homeomorphism of the closure of Ω onto itself that is isotopic to the identity by an isotopy that fixes the boundary pointwise.) The Teichm¨ uller space T eich(Ω) is the set of all Teichm¨ uller classes of quasiconformal mappings with domain Ω.

The Teichm¨ uller class of f depends only on its Beltrami coefficient, which is a function µ in the open unit ball M (Ω) of the complex Banach space L

(Ω). The canonical projection Φ : M (Ω) → T eich(Ω) maps µ to the Teichm¨ uller class of any quasiconformal map whose domain is Ω and whose Beltrami coefficient is µ. The basepoints of M (Ω) and T (Ω) are 0 and Φ(0) respectively.

Teichm¨ uller metric. The Teichm¨ uller metric on M (Ω) is defined as:

ρ

M

(µ, ν) = tanh

−1

µ − ν 1 − µν

(1.4.1)

for any µ and ν in M (Ω). This Teichm¨ uller metric ρ

M

provides the same topology to M (Ω) as does the Banach norm topology of L

(Ω). The Teichm¨ uller metric on T eich(Ω) is the quotient metric:

d

T

(s, t) = inf{ρ

M

(µ, ν) : µ and ν in M (Ω), Φ(µ) = s, and Φ(ν) = t} (1.4.2)

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for all s and t in T eich(Ω). The infimum is always attained (see [1]) and we can choose µ in Φ

−1

(s) arbitrarily so that we have

d

T

(Φ(µ), t) = min{d

M

(µ, ν) : ν ∈ M (Ω) and Φ(ν) = t}

for all µ in M (Ω) and t in T eich(Ω). It follows easily that the canonical projection Φ is an open continuous map with respect to the Teichm¨ uller metric.

A fundamental fact in the theory of Teichm¨ uller spaces is the following:

Theorem 1.13. There is a unique complex Banach manifold structure on T eich(Ω) such that Φ : M (Ω) → T eich(Ω) is a holomorphic split submersion, that is Φ is surjective, holomorphic, its derivative is surjective and Φ has local holomorphic sections.

See, for example, [13] or [19].

The tangent space at the basepoint. We identify L

(Ω) with the dual space of L

1

(Ω) in the obvious way, by sending µ in L

(Ω) to the bounded linear functional

`

µ

(f ) = Z Z

µ(z)f (z) dx dy, f ∈ L

1

(Ω).

The space A(Ω) of integrable holomorphic functions on Ω is a closed subspace of L

1

(Ω), and hence a (complex) Banach space. The orthogonal complement of A(Ω) is the set:

A(Ω)

= {µ ∈ L

(Ω) : `

µ

(φ) = 0 for all φ in A(Ω)}.

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Proposition 1.14 (Teichm¨ uller’s Lemma). ker(Φ

0

(0)) = A(Ω)

.

For the proof we refer to [19].

By a well-known principle (see [20]), we can identify L

(Ω)/A(Ω)

with the dual space of A(Ω). Hence Teichm¨ uller’s lemma implies the following

Corollary 1.15. The tangent space to T eich(Ω) at its basepoint is naturally iso- morphic to the Banach dual A

(Ω) of A(Ω).

Teichm¨ uller contraction:

For each tangent vector v to T eich(Ω) at Φ(0), there exists some µ in L

(Ω) such that v = Φ

0

(0)µ. The natural isomorphism sends v to the linear functional ϕ 7→ `

µ

(ϕ) on A(Ω).

We write

k`

µ

k

T

= sup

kφk=1

n

Z Z

C

µφdxdy

, φ ∈ A(Ω) o

It is therefore clear that k`

µ

k

T

≤ kµk

.

Let µ

0

in M (Ω) be extremal in its Teichm¨ uller class; which means Φ(µ

0

) = Φ(µ) and kµ

0

k

≤ kµk

. Let k

0

= kµk

, and k = kµk

; also, let

K

0

= 1 + k

0

1 − k

0

and K = 1 + k 1 − k .

In [6], Earle proved the sharp form of the principle of Teichm¨ uller contraction:

Theorem 1.16 (Earle). Let µ be in M (Ω) with kµk

= k < 1, and let K and K

0

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be defined as above. Then 1 K

0

− 1

K ≤ 2

1 − k

2

(k − k`

µ

k

T

) ≤ K − K

0

.

Changing the basepoint. Let h be a quasiconformal map whose domain is Ω and whose image is a plane region e Ω. Let g be any quasiconformal map whose domain is e Ω. By definition, the allowable map h

from T eich(e Ω) to T eich(Ω) maps the Teichm¨ uller class of g to the Teichm¨ uller class of g ◦ h.

Proposition 1.17. The allowable map h

: T eich(e Ω) → T eich(Ω) is biholomorphic.

If µ is the Beltrami coefficient of h, then h

maps the basepoint of T eich(e Ω) to the point Φ(µ) in T eich(Ω).

See [19] for the details.

We conclude this section with the following important result due to Royden when Ω is a finite analytic type Riemann surface and Gardiner when Ω is an infnite analytic type Riemann surface. A simpler proof was given by Earle, Kra and Krushka´l in [7].

Proposition 1.18. The Teichm¨ uller metric on T eich(Ω) is the same as its Kobayashi metric.

Proof. See [7].

1.5 Product Teichm¨ uller space

Let {X

n

}

n=1

be a countable collection of nonempty plane regions, none of which

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space of X, which we denote by T (X); we will also discuss some of its properties.

For the details we refer the reader to [8] or [15].

Teichm¨ uller space of X. For each n ≥ 1 let T eich(X

n

) be the Teichm¨ uller space of the region X

n

, let 0

n

be the basepoint of T eich(X

n

), and let d

Tn

be the Teichm¨ uller metric on T eich(X

n

). By definition, the Teichm¨ uller space T eich(X) is the set of sequences t = {t

n

}

n=1

such that t

n

belongs to T eich(X

n

) for each n and

sup{d

Tn

(0

n

, t

n

) : n ≥ 1} < ∞.

The basepoint of T eich(X) is the sequence 0 = {0

n

} whose nth term is the basepoint of T eich(X

n

).

Teichm¨ uller metric on T eich(X). The Teichm¨ uller metric on T eich(X) is defined by:

d

T

(s, t) = sup{d

Tn

(s

n

, t

n

) : n ≥ 1}.

Let L

(X) be the complex Banach space of sequences µ = {µ

n

} such that µ

n

belongs to L

(X

n

) for each n and the norm kµk

= sup{kµ

n

k

: n ≥ 1} is finite.

Let M (X) be the open unit ball of L

(X); note that if µ belongs to M (X) then µ

n

belongs to M (X

n

) for all n ≥ 1 (but the converse statement is false). As before, the Teichm¨ uller metric on M (X) is defined by the formula:

ρ

M

(µ, ν) = tanh

−1

µ − ν 1 − µν

, µ and ν in M (X), (1.5.1)

and it again induces on M (X) the same topology that M (X) inherits from L

(X).

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For each n ≥ 1 let Φ

n

be the canonical projection from M (X

n

) to T eich(X

n

).

For µ in M (X) let Φ(µ) be the sequence {Φ

n

n

)}. It is easy to see that Φ(µ) belongs to T eich(X) and that the map Φ from M (X) to T eich(X) is surjective. We call Φ the canonical projection of M (X) onto T eich(X). We have

d

T

(s, t) = inf{ρ

M

(µ, ν) : µ and ν in M (X), Φ(µ) = s, and Φ(ν) = t} (1.5.2)

for all s and t in T eich(X). Once again we have

d

T

(Φ(µ), t) = min{ρ

M

(µ, ν) : ν ∈ M (X) and Φ(ν) = t}

for all µ in M (X) and t in T eich(X). Therefore, the topologies on M (X) and T eich(X) determined by their Teichm¨ uller metrics make Φ an open continuous map.

The following result was proved in [8] and [15].

Theorem 1.19. There is a unique complex Banach manifold structure on T eich(X) the map Φ : M (X) → T eich(X) is a holomorphic split submersion.

Corollary 1.20. For each n ≥ 1 the map t 7→ t

n

from T eich(X) to T eich(X

n

) is a holomorphic split submersion.

Proof. This is true since the projections Φ : M (X) → T eich(X), Φ

n

: M (X

n

) → T eich(X

n

) and µ 7→ µ

n

from M (X) to M (X

n

) are holomorphic split submersions.

The tangent space at the basepoint. Let L

1

(X) be the complex Banach

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norm kf k

1

= P

n=1

kf

n

k

1

is finite. We can identify L

(X) with the dual space of L

1

(X) by sending µ in L

(X) to the linear functional

`

µ

(f ) =

X

n=1

Z Z

Xn

µ

n

(z)f

n

(z) dx dy, f ∈ L

1

(X).

By definition, A(X) is the closed subspace of L

1

(X) consisting of the ϕ such that ϕ

n

belongs to A(X

n

) (i.e. is a holomorphic function) for each n. The orthogonal complement of A(X) in L

(X) is by definition the set

A(X)

= {µ ∈ L

(X) : `

µ

(ϕ) = 0 for all ϕ in A(X)}.

Proposition 1.21 (Generalized Teichm¨ uller Lemma). ker(Φ

0

(0)) = A(X)

.

Proof. It is easy to verify that µ ∈ ker(Φ

0

(0)) if and only if µ

n

∈ ker(Φ

0n

(0)) for each n and also that µ ∈ A(X)

if and only if µ

n

∈ A(X

n

)

for each n. Therefore the proposition follows immediately from Proposition 1.14.

Corollary 1.22. The tangent space to T eich(X) at its basepoint is naturally iso- morphic to the dual space A(X)

of A(X).

The natural isomorphism sends the tangent vector v = Φ

0

(0)µ to the linear functional ϕ 7→ `

µ

(ϕ) in A(X)

.

1.6 Teichm¨ uller space of a closed set in the sphere

In this section we define the Teichm¨ uller space T (E) of a closed subset E in ˆ C and

discuss some of its properties. Remember our blanket assumption that 0,1, and ∞

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belong to E.

Definition 1.23. The normalized quasiconformal self-mappings f and g of b C are said to be E-equivalent if and only if f

−1

◦ g is isotopic to the identity rel E. The Teichm¨ uller space T (E) is the set of E-equivalence classes of normalized quasicon- formal self-mappings of b C. The basepoint of T (E) is the E-equivalence class of the identity map.

As in §1.1, for each µ ∈ M (C) we denote by w

µ

the unique normalized quasicon- formal homeomorphism of b C onto itself that has Beltrami coefficient µ. The quotient map P

E

of M (C) onto T (E) is defined by setting P

E

(µ) equal to the E-equivalence class of w

µ

. This is surjective and basepoint preserving. (We will use the same notation 0 for the basepoint in M (C) and the basepoint in T (E).)

The product space T eich(E

c

) × M (E). We will use the symbol E

c

for the complement b C \ E of E in b C. If E

c

is not empty, it is the union of its connected components X

n

, each of which has a Teichm¨ uller space T eich(X

n

). If the number of components is finite, T (E

c

) is by definition the cartesian product of the spaces T eich(X

n

). If there are infinitely many components, then E

c

is the disjoint union of the countably many X

n

, and T eich(E

c

) is the product Teichm¨ uller space defined in

§1.5. In all these cases T eich(E

c

) is a complex Banach manifold. Recall that M (E)

is the open unit ball in L

(E); the product space T eich(E

c

) × M (E) is a complex

Banach manifold.

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T eich(E

c

) × M (E) is defined by the formula

P

E

(µ) = (Φ(µ

Ec

), µ

E

) for all µ in M (C). (1.6.1)

In the above formula, µ

Ec

and µ

E

are the restrictions of µ to E

c

and E respec- tively and Φ : M (E

c

) → T eich(E

c

) is the canonical projection defined in §1.5. If E

c

is empty, P

E

is the identity map of M (C) onto itself.

Proposition 1.25 (Lieb). For all µ and ν in M (C) we have P

E

(µ) = P

E

(ν) if and only if P

E

(µ) = P

E

(ν). Thus, there is a well defined bijection θ : T (E) → T (E

c

) × M (E) such that θ ◦ P

E

= P

E

, and T (E) has a unique complex manifold structure such that P

E

is a holomorphic split submersion and the map θ is biholomorphic.

See [8] or [15] for a complete proof. This means that there is a canonical biholo- morphism between T (E) and T eich(E

c

) × M (E).

The space T (E) is contractible.

Proposition 1.26. There is a basepoint preserving continuous map s from T (E) to M (C) such that P

E

◦ s is the identity map on T (E).

See Proposition 7.22 in [8]

Corollary 1.27. The space T (E) is contractible.

Proof. Since M (C) is contractible, it follows that T (E) is contractible.

The tangent space at the basepoint. Let A(E) be the closed subspace of

L

1

(C) consisting of the functions f in L

1

(C) whose restriction to E

c

is holomorphic.

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We identify L

(C) with the dual space of L

1

(C) in the usual way. Set

A(E)

= {µ ∈ L

(C) : `

µ

(f ) = Z Z

C

µ(z)f (z) dx dy = 0 for all f in A(E)}.

Proposition 1.28 (Teichm¨ uller’s lemma for T (E)). ker(P

E0

(0)) = A(E)

.

See Proposition 7.18 in [8].

Corollary 1.29. The tangent space to T (E) at its basepoint is naturally isomorphic to A(E)

.

The natural isomorphism sends the tangent vector P

E0

(0)µ to the linear functional f 7→ `

µ

(f ) on A(E).

Changing the basepoint. Let h be a normalized quasiconformal self-mapping of b C, and let e E = h(E). By definition, the allowable map h

from T ( e E) to T (E) maps the e E-equivalence class of g to the E-equivalence class of g ◦ h for every normalized quasiconformal self-mapping g of b C.

Proposition 1.30. The allowable map h

: T ( e E) → T (E) is biholomorphic. If µ is the Beltrami coefficient of h, then h

maps the basepoint of T ( e E) to the point P

E

(µ) in T (E).

Proof. As in Proposition 1.17, let e h be the map that sends µ in M (C) to the Beltrami

coefficient of w

µ

◦ h. Again P

Ee

and P

E

are holomorphic split submersions, and the

standard computation shows that e h is a biholomorphic map.

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Forgetful maps. If E is a subset of the closed set e E and µ is in M (C), then the E-equivalence class of w e

µ

is contained in the E-equivalence class of w

µ

. Therefore, there is a well-defined forgetful map p

E,Ee

from T ( e E) to T (E) such that P

E

= p

E,Ee

◦ P

Ee

.

Proposition 1.31. The forgetful map p

E,Ee

is a basepoint preserving holomorphic split submersion.

Proof. Since P

E

= p

E,Ee

◦ P

Ee

and P

E

and P

Ee

are holomorphic split submersions, so is p

E,Ee

.

The following proposition will be very crucial in our thesis.

Proposition 1.32. Let f be any holomorphic map of ∆ into T (E) and let µ be any point in M (C) such that P

E

(µ) = f (0). There is a holomorphic map b f from ∆ to M (C) such that b f (0) = µ and P

E

◦ b f = f .

See Proposition 7.27 in [8]. For a different approach we also refer to the papers [14] and [3].

The Kobayashi and Teichm¨ uller metrics on T (E). By definition, the Teichm¨ uller metric d

T (E)

on T (E) is given by

d

T (E)

(P

E

(µ), t) = inf{ρ

M

(µ, ν) : ν ∈ M (C) and P

E

(ν) = t}

for all µ in M (C) and t in T (E).

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The infinitesimal Teichm¨ uller metric F

T (E)

is defined on the tangent bundle of T (E) by the formula

F

T (E)

(P

E

(µ), v) = inf{K

M

(µ, λ) : λ ∈ L

(C) and P

E0

(µ)λ = v},

for any µ in M (C) and tangent vector v to T (E) at the point P

E

(µ).

Proposition 1.33. The Teichm¨ uller and Kobayashi metrics on T (E) are equal, and the infinitesimal Teichm¨ uller and Kobayashi metrics are also equal.

See Proposition 7.30 in [8].

The following proposition is obvious.

Proposition 1.34. If E is a subset of e E and p

E,Ee

: T ( e E) → T (E) is the forgetful map then

d

T (E)

(p

E,Ee

(s), p

E,Ee

(t)) ≤ d

T ( eE)

(s, t) for all s and t in T ( e E).

We defined a natural isomorphism mapping the tangent space to T (E) at its basepoint at its basepoint onto a Banach space A(E)

. That isomorphism is an isometry with respect to the infinitesimal Teichm¨ uller metric on the tangent space and the usual norm on A(E)

. Throughout this thesis we will denote this infinites- imal Teichm¨ uller norm by `

µ

; so `

µ

is the norm of the linear functional

`

µ

(φ) = Z Z

C

µφdxdy on A(E).

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Henceforth, we will denote this by

k`

µ

k

T (E)

= sup

kφk=1

n

Z Z

C

µφdxdy

, φ ∈ A(E) o .

It is clear that k`

µ

k

T (E)

≤ kµk

for µ in L

(C). We say that µ is infinitesimally extremal in its E-equivalence class, if k`

µ

k

T (E)

= kµk

.

A Beltrami coefficient µ in M (C) is called extremal in its E-equivalence class, if P

E

(µ) = P

E

(ν) and kµk

≤ kνk

. Equivalently, µ in M (C) is extremal in its E-equivalence class if d

T (E)

(0

T

, P

E

(µ)) = ρ

M

(0, µ).

When E is finite. Let E be a finite set (as usual, 0, 1, and ∞ belong to E).

Its complement E

c

= Ω is the Riemann sphere with punctures at the points of E.

We already saw after Proposition 1.25 that there exists a biholomorphism between T (E) and T eich(E

c

) × M (E). When E is finite, it follows that T (E) is canonically identified with T eich(b C \ E). For the reader’s convenience we give an independent proof given in Example 3.1 in [17].

Proposition 1.35. Let E be a finite subset of b C with {0, 1, ∞} ⊂ E. Its compli- ment Ω = b C \ E is a sphere with finitely many punctures and there is a natural identification of T (E) with the classical Teichm¨ uller space T eich(Ω).

Proof. We define a map θ : T (E) → T eich(Ω) by setting θ(P

E

(µ)) equal to the

Teichm¨ uller class of the restriction of w

µ

to Ω. Suppose the restrictions of w

µ

and

w

ν

to Ω are in the same Teichm¨ uller class. Then there exists a conformal map h of

w

µ

(Ω) onto w

ν

(Ω) such that (w

ν

)

−1

◦ h ◦ w

µ

is isotopic to the identity rel E. The

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