Models of the Universal Teichmüller Space

CHAPTER HI

1. Models of the Universal Teichmüller Space

1.1. Equivalent Quasiconformal Mappings

Let us consider the family of all quasiconformal mappings of a fixed domain in the plane. In this section we assume that this domain is the upper half- plane H. We wish to introduce additional structure to this family and begin by regarding two mappings as equivalent if they differ by a conformal map- ping. In view of the Riemann mapping theorem, we may then restrict our- selves to self-mappings of H and require that they are normalized so as to keep fixed the three boundary points 0, 1 and Wedenote by F the family of such normalized mappings. (Recall: every element' of F can be extended to a homeomorphic self-mapping of the closure of H. It is actually the extended mappings to which the normalization requirements apply.)

By the existence and uniqueness theorems for Beltrami'equations (Theo- rems 1.4.4 and 1.4.2), there is a one—one correspondence between F and the open unit ball B of the Banach space which consists of all on H.

A more interesting space is obtained if we introduce a weaker equivalence relation.

Defánition. Two mappings of the family F are equivalent if they agree on the real axis. The complex dilatations of equivalent mappings are also said to be equivalent. The set of the equivalence classes is the universal Teichmüller space

We thus have two models for T: Its points are classes of equivalent map- pings i'ithefamily F or of equivalent functions on the ball B.

A third model is obtained in terms of quasisymmetric functions. We recall that a quasisymmetric function is said to be normalized itit fixes the points 0 and 1. Let X denote the class of all normalized quasisymmetric functions. If

98 III. Universal TeicbmUller Space

[f

^J is the point of T represented by the mapping fe F, then

(1.1) is a bijective mapping of T onto X. For it is clear from the definition of T that (1.1) is well defined and injective. By Theorem 1.5.1, it maps T into X, and by Theorem 1.5.2, it is surjective. It follows that we can rephrase thedefinition of the universal Teichinüller space:T is the setofallnormalized quasisymmetric functions.

This observation allows an important conclusion:

Theorem 1.1. Every pointoftheuniversal Teichmüllerspace can be represented by a real analytic quasiconformal mapping fEF or by a real analytic complex dilatation pe B.

PROOF. The result follows immediately from Theorem 1.5.3. (For a complete

proof, see 11.5.2.)

0

We shall see in V.3.2 that the universal Teichniüller space contains as a subset the Teichmüller space of any Riemann surface which allows a half- plane as its universal covering surface. It was Bers [7,8] who recognized the importance of this largest and, in many way's, simplest Teichmüller space and gave it the name universal.

J.2.

1ff belongs to F, then so does its inverse f'; along with f and g in F, the

compositionfog is also in F. The family F can thus be regarded as a group.

From the definition of the universal Teichmüller space it follows that T

inherits this group structure: T is thequotientof thegroup

F of

all normalized quasiconformal self-mappings of the upper half-planeby thenormal subgroup of mappings equivalent to the identity.

1ff

g e F, the rule

(f]o[g]= [fog]

(1.2)

defmes the group operation in T The neutral element, i.e., the point of T

determined by the identity mapping (or by the complex dilatation which is identically zero) is called the origin of T and denoted by 0.

Normalized quasisymmetric functions also form a group under composi- tion. This follows from Theorems 1.5.1 and 1.5.2 or from an easy elementary computation. We see that the mapping (1.1) is an isomorphism between the

groups T and X.

Let us consider, a moment, quasiconformal self-mappings of II which are not necessarily normalized. If 11 and 12 are two such mappings, we still say that is

to f21f is the identity. Let JEF be a

1. Models of the Universal Teichmülier Space 99

normalized and g an arbitrary quasiconformal self-mapping of H. We choose a Möbius transformation h, mapping H onto itself, such that

eF,

and defme w = ^w151: T —, Tby the formula

co([f]) =

Then w, which depends only on the equivalence class [g], is a well defined bijection of T onto itself. Further, [g] =

⁰

implies that w is the

identity and

=

[h1 co[g,]01921([f]).

We conclude that when g runs through all quasiconformal self-mappings of H, the transformations T —, T form a group. It is called the universal modular group M.

We now return to normalized quasiconformal self-mappings of H and

consider the subgroup M, of the universal modular group consisting of trans- formations with g e F. Then

=

i.e.,elements of are right translations of the group T

In 2.1 we shall introduce a metric into T and prove that every is an isometry with respect to this metric.

The group M, of right translations is transitive: If

[fr] and P2 = [f2]

are given points of there is an

E such that P2 = This is clearly the case if g = of1..

1.3. Normalized Conformal Mappings

The mappings belonging to F can be continued quasiconformally to the

plane by reflection in the real axis. However, such an extension does not give new insight into the properties of T It was a fundamentai observation of Bers [4] that one should extend, not the mappings of F but rather their complex

dilatations, in such a way that the corresponding extended mappings are

conformal in the lower half-plane. The machinery developed in Chapter II can then be applied to the study of T

Let B and f'4 be the mapping of F withcomplex dilatation i. We extend

,u to the lower half-plane H' by giving it' there the value 0. Let

be the quasiconforinal mapping of the plane which 5*ce 0,1, co and whose complex dilatation agrees with the extended Then is conformal.

Theorem 1.2. The complex dilatations and v are equivalent and only the conformal mappings f,LIH' and H' coincide.

PROOF. Suppose first that

H' f,IH'. The mappings j a

and

are both conformal in the upper half-plane H, which they map

100 ^III. TeichmSller Space

ontothe same quasidisc. Because they fix 0, 1, oo,it follows that they agree in H, and hence also on the real axis R. Since f,, =

f, on R, we condude that

=

on R, i.e.,p and v are equivalent.

Assume, conversely, that f" =1' on R. We define a mapping w of the plane by the requirements w =

in f,(H' '.iR), and w = f,,^o

of' of,1 in

_f,(H). From the hypothesis

f' on R it follows

that w is a homeomorphisin of the plane. In addition, is conformaL

But so is also wjf,(H), because f,,o(fT'

^and

f'of,1

arc conformal. Since f,(R) is a quasicircie, we infer from Lemma 1.6.1 that w is a Möbius trans- formation. Owing to the normalization, w is the identity mapping and so

0

Let F* be the family of all quasiconformal mappings of the plane which fix the points 0,1, and arc conformal in the lower half-plane. Two mappings f,, and f, of F* are said to be equivalent if they agree in the lower half-plane.

Theorem 1.1 says that every equivalence class [f"] contains real analytic mappings. It follows that each class [1,3 has representatives which arc real analytic in the upper half-plane H. For in H,

where

f,1o(fT'

^isconformal. Therefore, f,,IH is real analytic is.

In particular, there arc mappings f,, which are conformal in H' and real analytic in H but which, nonetheless, are very irregular on the real axis R.

We recall that the Hausdorif dimension of the image cur e ban be arbi- trarily close to 2 (cf. 1.6.1).

By Theorem 1.2 the space T can be regarded as the set of the equivalence classes [1,3. Or more explicitly: The universal Teichmüller space is theset of

thenormalizedconformal mappings

1.4. Sewing Problem

The characterization of T by means of the conformal mappings leads to far-reaching conclusions. Section 4, in particular, will be devoted to consi- derations emanating from this model of

Anticipating a need in section 1.5, we

give a solution to the following

sewing problem:

Let h be a strictly increasing continuous function on the

real axis, growing from — to + cc. Find conformal mappings 11 and f2 of the upper and lower half-plane, respectively, onto complementary Jordan domains such that

fr1ofz=h

on the real axis. We call the pair normalized if and 12 both fix 0, 1 and cc.

Depending on h, the sewing problem, to which many questions in complex

1. Models o( the Univeiial Teichmüikr Space 101

analysis seem to lead, need not have any solution or it may have infinitely many pairs of solutions. However, if h is a normalized quasisymmetric func- tion, the existence of a unique nonnalized solution can be easily established by aid of the quasiconformal mappings f and It is in this form that the result will be used for studying the universal Teichmülier space.

Lemma 1.1. Leth be a normalized quasisymmétric function. Then the sewing problem has aunique normalized pair ofsolutions.

PROOF. Given a function he X, there is a mapping f' e F such that f"I ft = h.

Then

is a solution of the sewing problem. This can be verified immediately.

Suppose that the pair is also a normalized solution. Then g2IR oh ofUIR. Hence, the mapping w which agrees with

of's in H u ft

and with g2 in H' is a homeomorphism of the plane. OfT the real axis it is quasiconformal. By Lemma 1.6.1, w is quasiconformal everywhere. Since w has the same complex dilatation as and both mappings fix 0, 1 and CX), it follows from the uniqueness theorem (Theorem 1.4.2) that w = Compari- son of the definitions of w, and 12 then shows that g1 = f1, g2 = 12.

0

Note that 11 and f2 map the onto quasidiscs. Lemma 1.1 is due to Pfluger [21; in [LV], p. 92, it was proved without the use of the existence theorem for Beltrami equations.

1.5. Normalized Quasidiscs

We shall now express in geometric terms the fact that points of the universal TeichmUller space can be represented by the conformal mappings We call a quasidisc normalized if its boundary passes through the points 0, 1, and is so oriented that the direction from 0 to 1 to IX to 0 is negative with respect to the domain. Let A denote the class of all normalized quasidiscs.

1ff E^Fe,then

[f]—*f(H')

(1.3)

is a bijective mapping of Tonto We first conclude from Theorem 1.2 that (1.3) is well defined. If f(H') = g(H'), then of is a conformal self-mapping of H' fixing 0, 1, ^Hence

fIH' gIH', i.e., [f) =

[g], and it follows that (1.3) is Finally, by Lemma 1.6.2, every quasidisc is the image of H' under a quasiconformal mapping of the plane which is conformal in H'. Since the required normalization is achieved by use of a suitable Möbius transfor- mation, we conclude that (1.3) is suijective.

IlL Universal

Thebijection (1.3) provides one more model for the universal Teichmüfler space: Tis thecollection of all normalized quasidiscs.

Using Lemma LI we obtain a connection between nonnalized quasidiscs and the group structure of T(Gardiner [1]).

Theorem 13. Tho points [f"], [f"]

Tare inverse elements of the group Tjf

and only the and f,(II') aremirrorimages withrespecttothe realaxis.

PROOF. Assume first that and [fVJ _are

inverse; we can then take

=

(f'Y'.

^Let be the quasiconformal mapping of the plane which fixes

the points 0, 1,

and whose complex dilatation

vanishes in H and

equals at almost all points zeH'. We write 9i = anddenote by g2 the unique mapping of H' onto f,4.(H') which keeps 0, 1,cç fixed.

Then 9i ^and92 are normalized conformal mappings of the upper and lower half-planes, respectively, onto complementary quasidiscs.

In order to study ^og2on the real axis R, we continue fP by reflection in and use the same notation f" for the extended mapping. Then

inH', because both sides are normalized quasiconformal self-mappings of H' with the same complex dilatation. Hence, on R

og2 = (fMYl = 7.

Now set

f1 =

Ii

⁼

Then

and f2 are also normalized conformal mappings of the upper and lower half-planes onto complementary quasidiscs. On the real axis,

of2

= 7

= ^{og2. We}conclude from Lemma 1.1 that g1 = g2= 12.

From the definition of it follows that

=

one way to verilS'

this is to compute thr

derivatives. Since f,(H) =f1(H) = g1(H) =

we obtain

f,(H') =

=

and the first part of the theorem has been proved.

After this the converse is easily established. Suppose that f,6(H) f,(H').

By what was just proved, there is a quasiconformal mapping where is determined by fA = (fm', such

that

_=fA(H'). Since the mapping (1.3) is injective, we conclude that A is equivalent to v. It follows that ff") and

[7) are inverse elements of

0

In 11.2.1 we defined the distance ö(f(H')) = i),,. between the domains

f(h")

andH'. This notion was generalized in 11.2.7 to apply to two arbitrary domains cotiformally equivalent to discs. When the domains are normalized

2. Metric of the Universal Teichmüller Space 103

quasidiscs, we can give a new definition: if J, ^{g J. then} tween f(H') arid g(H') is defined by

q(f(H'), q(H')) = ^{-— 'I}

It

is easy to cheek that q, so defined, is a metric on

By use of the bijection (1.3), the metric can be transferred to T This q-metric will be in detail in section 4. Before that in section 2, a different metric will be defined for Tin a more direct manner.