CHAPTER HI
4. Space of Schwarzian Derivatives
112 III. Universal Teichmtiller Space
z
= x + ly, is finite. The space Q has a natural linear structure over the
complex numbers.Furthermore, Q is complete. For if is a Cauchy sequence in Q, then
= is a Cauthy sequence in Since is complete, there is a such that converges to in Here can be taken to be continuous. Then q,, converges locally uniformly to q = because is
locally bounded. It follows that p Il holomorphic, — H —,0,
and p
Q.We conclude that Q is a Banach space. Its points are Schwarzian derivatives:
By Theorem 11.1.1 every function p e Q is the Schwarzian derivative of a function f meromorphic in H.
Going back to the functions f,,, we write sp = By Theorem 11.1.3,
II ii 6. Therefore, [pJ —' maps T into Q. The mapping is well defined, for if v is equivalent to wehave = and hence s,, =
4.2. Comparison of Distances
We shall prove that the mapping [p3 -+ is a homeomorphism of T onto its image in Q. To this end we shall compare the a-distance of two given points
[p] and [v] of T to the distance of their images s,, and
in Q. We write(P2)= II(p1 — (I'I II for points of Q.
In the special case v = 0, estimates in both directions can be obtained
directly from our previous results. If = 0, then also = 0, and q(s,1, 0) =ii. Moreover, = By Theorem 11.3.2, This
holds no matter how p is chosen from the equivalence class. Consequently,
6fl([p],0).
(4.1)We remark that is equal to the distance of from H (cf.
the remark at the end of 1.5).
In order to get an inequality in the opposite,, direction, we choose an
arbitrary eQ such that II ii <2. By Theorems 11.1.1 and 11.5.1, there is a normalized quasiconformal mapping f of the plane which is conformal in H, for which S11K = (P,and whose complex dilatation p in the lower half-plane is obtained from the formula = —2y2q,(z), z H. For this mapping f =we have ii = 11/2. Hence,
(4.2)
We assumed that 0) < 2. But since all a-distances are <1, (4.2) holds trivially if
2.
We shall now generalize (4.1) and (4.2) for arbitrary points [p] and [v]. We start with the transformation rule
liSp — SVIIH= IS1 (4.3)
for the Schwarzian derivatives (formula (1.10) in 11.1.3). We write = f,(H) and apply Theorem 11.3.2 to the conformal mapping w = in the
4. of Derivatives 113
quasidisc .4,. It follows that
— oo(AJII
L
here o0(A,) = 6+ ô(Á,,) is the outer radius of univalence of A,. In view of (4.3), this yields the estimate
[v]), (4.4)
which is the desired generalization of(4.1). Since a0(A,) .12, we could use as the coefficient on the right-hand side the absolute constant 12. On the other hand, a0(A,) can be replaced by where AM fP(").
It is more difficult to generalize the inequality We choose v from its equivalence class so that f, has the smallest possible maximal dilatation K,..
After this, we consider Schwarzian derivatives whichare so close to s, that s is the constant of Theorem 11.4.1. Then, by formula (4.3),
HSWIIA. <s(K,).
We know that w has a quasiconformal extension, namely, f_of,'. However, we prefer to extend w by utilizing Theorem 11.4.1, which makes it possible to estimate the complex dilatation. By that theorem, w has a quasiconformal extension to the plane such that the complex dilatation K ofthe extended
mapping satisfies the inequality
£(K,,) —
45
If
the extended w is also denoted by w, then = wof,
isa quasiconformal extension of to the lower half-plane. Thus A. is equivalent to p. Because w = o ',we have, therefore,=
fJ([p],[vJ).Combining this with (4.5) we finally arrive at the inequality
e(K,)/3({uJ,[v]),
(4.6)valid also if e(K,). This contains (4.2) as a special case: If v = 0,
then K, = I and E(K,) = 2.Since the roles of p and vcan be interchanged, we can replace in (4.6) by e(K,fl.
4.3. Imbedding of the Universal Teichmüller Space
The estimates (4.4) and (4.6) show that the and q-metrics are topologically equivalent. Thus a new important model is obtained for the topological space (T,t).
14 III. Universal Teichmüller Space
Theorem 4.1. Themapping
[/4]
a homeomorphism of the universal Teichinüller space onto its image in Q.
We noted already in that (47) is well defined in 7 If [p3and [v]
ave the same image, it follows from the normalization that =f,IH,i.e., and v are equivalent. Hence (4.7) is injective. Inequality (4.4) shows that i.7) is continuous, and (4.6) that its inverse is continuous.
0
In 4.5 we shall show that the image of T under the homeomorphism (4.7) open in Q.(Wehave almost proved this in establishing (4.6).) The mapping
which will later be considered in connection with an arbitrary Teich- nüller space, is called the Bers imbedding of TeichmUller space.
By Theorem 4.1, the convergence Sfl,. in Qimpliesthat —' [it] in Hence, by Lemma 2.2 and the remark following Theorem 2.3, —*
uniformly in H.
Anticipating developments in Chapter V, we denote the image of T under (4.7) by T(1). When there is no fear of confusion, we often identify T(1) with universal Teichmüller space. Like X, the space T(1) is simpler than T in that its points are functions and not equivalence classes of functions.
We can also define T(l) =
{Sf(fis conformal in H and has a quasicon- formal extension to the plane}. For such an f is equal to a normalized
mapping modulo a Möbius transformation, which does not change the Schwarzian derivative.By Theorem 11.5.1, the set
T(l) contains the open ball
B(O, 2)<2). In this ball, the inverse of the mapping (4.7) can be
described explicitly:
—÷[p], — —2y2q(z).
The space (T(l), q) is not complete, even though it is homeomorphic to the complete spaces (T,r), (T,fl) and (X,p). In order to prove this, it is sufficient to find an Sf EQ\T(1)and functions S1e T(1), n = 1, 2,
...,
such that Sf —'S1in Q. Then (S1) is a Cauchy sequence in T(l) but its limit is not in T(l). An example is provided by the functions z -+ f(z) = logz, z = in H,
which we considered, for another purpose, in 11.1.4. Since S1,,(z) = (1 —
we have = 2(1 — 1/n2) <2. By Theorem 11.5.1, the has a quasiconformal extension to the plane. Hence S1e T(1). In 11.1.4 we saw already that
— S,Ij = 2/n2 -+0.
But since z —*logz does not even have a homeomorphic extension, Sf is not in T(l).
4. Space at Schwanian Derivatives 115
4.4. Schwarzian Derivatives of Univalent Functions
Let us define the set
u =
univalentin H).Then trivially T(1) U, and by Theorem 11.1.3, U c Q. More precisely, we conclude from Theorem IL 1.3 that U is contained in the closure of the ball
B(O, 6) {4 QI 114' II <6}.On the other hand, it follows from Theorem 11.5.2 that U contains the closure of B(O, 2).
Let f1(w) = w+ 1/w,f2(w) = w— l/w. Ifh is a conformal mapping of the upper half-plane onto {wlfwI> 1), then S1101,, 5,20,16 U. From the calcula- tions in 11.2.6 it follows that — S120,.II = 12.
We conclude that the
diameter of U is 12.The set U is closed in Q. For suppose that
eU, n =
1, 2,...,
andthat S, converges to S1 in Q. We show that f is univalent.We are free to compose the functions with arbitrary Möbius transforma- tions. There is no loss of generality, therefore, in assuming that every f,, fixes the same three points a1, a2, a3 in H. By Theorem 1.2.1, the family is then normal. Consequently, (fe) contains a which is locally uniformly convergent in H. By renumbering the functions we may assume that (fe) itself
has this property. The limit g = fixes a1, 02 and a3, and is there-
fore univalent in H. At every point z EH we have urn=
S9(z)and also urn S1(z) = Sf(z). Hence, I differs from g by a Möbius transformation, and so f is univalent.Since U is closed and T(l) U, the closure of T(1) is contained in U. If U, we can always find functions with T(1) such that
f(z) =
locally uniformly in H. (4.8)An approximating sequence with T(I) is obtained as follows. Set
z + i/n
1 — n = 2,3
•Then maps
H onto the disc =
{wlIw —iI <((n — 1)/(n + 1))Iw + it), whose closure lies in H. As n —* the discs exhaust H, and g,,(z) -+locally
uniformly. Hence, by setting J, =
fog,,, weobtain a sequence of
functions for which (4.8) is true. The property T(1) follows from the fact that f,,(R) =f(aD,,)is a quasicircie (cf. the remark in 1.6.1).If (4.8) holds, the derivatives off, converge to the derivatives off. Hence, S1(z) = urnS1(z)
fl-4
locally uniformly in H. However, as we showed in 11.1.4, it does not neces- sarily follow that —'
in Q. We cannot conclude, therefore, that the
closure of T(1) coincides with U. A counterexample such as the one in 11.1.4 does not disprove this either, but actually the closure of T(1) is not the whole of U. This will be explained in 4.6.116 III. Universal TeichmUller Space
4.5. Univalent Functions and the Universal TeichmUller Space
From Theorems 11.4.1 and 11.4.2 we obtain a remarkable connectibn between the sets T(l) and U (Gehring [2]).
Theorem 4.2. The set T( 1) is the interior of U.
PROOF. We prove first that T(1) is an open subset of Q. Fix an arbitrary point S1 of T(l). For ShEQ we write g =
hof1,
and conclude that g is meromor- phic in the quasidisc f(H). By Theorem 11.4.1, there exists a positive constant e such that if II IIf(H) <e,then g is univalent in f(II) and has a quasicon-
formal extension to the plane. Now choose SheQ such that II Sh —S,HR <c.Then
IISgIIj1ø> ItS,, — SIliii<
Because h = gof,we conclude that S,,E T(1). It follows that T(l) is àpen.
Since T(1) U, the proof will be complete if we show that mt U c T(1).
Choose a point mt U. We then have an £ > 0 such that
Let g be an arbitrary meromorphic function in the domain f(H), with the
property S5 II((H)If
h = g of,then—
=
HSSIIJ(u)8.
It follows that S,,e V c U, i.e., h is univalent in H. But then g = hof'
isunivalent in f(H). What we have proved is that f(H) is an e-Schwarzian domain. Hence, by Theorem 11.4.2, the domain f(H) is a quasidisc. We
conclude that S1e T(l) (cf. Lemma 1.6.2, statement 30) aswe wished to
show.
0
The result that T(1) is open in Q was first proved by Ahlfors [4].
4.6. Closure of the Universal Teichmüller Space
For a long time it was a famous open problem, raised by Bers, whether the closure of T( 1), which is contained in U, actually agrees with U. In 1978, Gehring [3] showed that the answer to this question is in the negative. He constructed a counterexample with the help of the simply connected domain G which is the complement of the curve
y = {z = t < co) u {0},
where a > 0 is small (Fig. 6). This 6 is not a Jordan domain, but more than
that, at the origin its boundary is so rigid that G possesses the, following
4. SpaceofSchwarzian Derivatives 117
Figure 6. not in the closure of T(1).
property: There a positive constant e such that if f is a conformal mapping
of G and uS1 II e, then f(G) is not a Jordan domain.
For the proof we refer to Gehring [3]. With the aid of this result the
negative answer to the question of Bers is readily established.Theorem 4.3. The closure of T(1) isapropersubset of U.
PROOF. Let G be the domain delined above and c > 0 the associated constant.
If h is a conformal mapping of the upper half-plane onto G, we prove that S1, does not lie in the closure of T(l).
Consider an arbitrary point S,,, of the neighborhood
h'
—S1,II,,For
f
w o h' we then have H hG = II — S1,hi H Therefore, eitherf isnot univalent or f is univalent but f(G) =
w(H)is not a Jordan domain. It
follows that is not in T(1).Recently, Theorem 4.3 has been strengthened: There, exists a confonnal mapping h: H G,