The Journal of Geometric Analysis Volume 7, Number 4, 1997

Natural Extensions of Holomorphic Motions B y Zbigniew Slodkowski

ABSTRACT. We consider an arbitrary real analytic family Xz, z 9 -D, over the closed unit disc -D, of real analytic plane Jordan curves X z. l f Jeio , e iO ~ OD, is an arbitrary real-analytic family of orientation reversing homeomorphisms of C fixing X eio pointwise, we show that there is a unique holomorphic motion o f t extending the given motion of Jordan curves and consistent with the given family of involutions. If these generalized reflections are defined using the barycentric extension construction of Douady-Earle-Nag, then the resulting extension method for holomorphic motions of X is natural, that is Moebius-invariant and continuous with respect to variation of the given motion of Xo.

1. Introduction This paper deals with holomorphic motion of subsets of the Riemann sphere C -- C U {oo} over the unit disc D, that is the maps (z, w) > fz(w) : D x E > Csuchthat f0 = ide, fz : E >C are injections forallz e D, and for every w0 e E t h e m a p z > fz(WO) : D > Cisholomorphic. Holomorphic motions were introduced by Marie et al. [11], but implicitly they were already present in the work of Ahlfors and Bers [1]. They have found many applications, in particular to complex dynamics of the Riemann sphere and Teichmiiller theory, see [3, 6, 11, 14, 15, 16, 18, 19], where also background and further references can be found. A

Soon after introduction of holomorphic motions, it became evident, due to the work of Sullivan and Thurston [19] and Sullivan [18] that the questions of extensibility of holomorphic motions are crucial to the theory. After important partial extension results due to Sullivan and Thurston [19] and Bers and Royden [3], the author has shown, cf. [15, 16], that every holomorphic motion f as defined above can be extended to a holomorphic motion of the whole Riemann sphere, (z, w) > Fz(w) :

DxC

:,C.

In actual applications, it is frequently essential that the extended holomorphic motions be invariant with respect to an action of a group or semigroup. Although some equivariant extension theorems have been obtained, cf. [6, 15, 16], they are somewhat unsatisfactory (even though they

Math Subject Classifications. Primary 30F60; Secondary 32F15, 32D15. Key Words and Phrases. holomorphic motions, holomorphic families of domains, reflection principle, polynomial hulls, Fredholm theory, Toeplitz operators, Hankel operators, H~176 Acknowledgements and Notes. This research has been partially supported by an NSF grant. The result of this paper was communicated at the 3rd Analysis Colloquium, Bern, Switzerland, August, 1996. The author is grateful to Professor Dennis Sullivan for inspiring and informative conversations about the problem of natural extensions of holomorphic motions and its possible applications.

9 1997The Journalof GeometricAnalysis ISSN 1050-6926

638

Zbigniew Slodkowski

have found applications) due to the nonunique, essentially random, construction (based on so called "holomorphic axiom of choice") of the required extensions. In a word, they are not "natural." The problem of existence of natural extensions of holomorphic motions has been posed by Sullivan and Thurston [19]. A natural extension method should assign to each holomorphic motion its unique extension to a holomorphic motion of the whole Riemann sphere in a manner independent of the conformal change of coordinates in the base disc D and fiber-preserving biholomorphic change of coordinates in the space D x C. The aim of this paper and of its sequel is to address this question through a more general problem of parametrizing, by a natural side condition, all (or a reasonably large class of) holomorphic motions extending the given one. The following is the main result of this paper. Theoreml.1. L e t r 0 c-CbeaJordancurveandthemap(z, wo) > (Z, fz(WO)):-DxFo > D x C b e a h o m e o m o r p h i s m o n t o i t s r a n g e a n d ( z , wo) > fz(wo) beaholomorphicmotionofFo over D. Let 1-'z = fz(F0), C \ F z = Uz U Uz, z 9 -D, with Uz, Uz being Jordan domains varying continuously with z 9 D. Let jo : Ueio ~ Ueio, 0 9 [0, 2:r), be given orientation-reversing homeomorphisms preserving pointwise the common boundary F do = 0 Ueio = a U~o, and such that -

-

-

-

- - r

ajo r ,,I for w 9 Uo, 0 9 R. Assume further that the map ~A-e(w)l~ > law, (e iO, w)

> Jo(w) : U { e iO} x -ffeiO

>C

(1.1.1)

0

is real analytic (on some neighborhood of its domain). Then there is a unique real analytic homeomorphism

(z, w)

, (z, Fz(w)): D x Vo

> U {z} x Vz, Izl_
such that

(i)

-

-

D

(z, w) > Fz(w) : D x Uo > C is a holomorphic motion extending fz(W); for each wo 9 Uo there is a holomorphic mapping h : D > C such that h(z) 9 Uz, z9 jo(Feio(wo) ) = h(ei~ for 0 9 [0, zzr).

Briefly, specifying (a real analytic) family of involutions jo determines a unique holomorphic motion via condition (ii). If these involution are chosen in a conformally natural way as in Earle and Nag [7] (where they are defined in terms of barycentric extensions of Douady and Earle [5]), then the theorem determines a natural extension method. (There are some delicate points related to this which we discuss in Section 7.) To understand the role of condition (ii) consider a simple example. Let F0 = R U {c~} and let fz(W) = w be a trivial (constant) holomorphic motion. Then Uz = H + = {w e C : Im w > 0}, Yz* = H - = {w 9 C : Im w < 0}. The Earle-Nag involutions in this case are jo (w) = ~. Condition (ii) means that the holomorphic trajectory z > Fz (wo) has real boundary values and so is constant. Thus, Fz = iduz, for z 9 D. The main limitation of Theorem 1.1 is the assumption of real-analyticity of jo, which implies, in particular, that the bordered hypersurface {(z, w) : z 9 D, w 9 I'z} is real-analytic. In the sequel to this paper we will extend Theorem 1.1 relaxing considerably the regularity requirements on I"z, z 9 D. On the other hand, we will also present a counterexample that suggests that a natural extension method for holomorphic motions might be not possible without some assumptions of boundary regularity (at [z[ = 1).

639

Natural Extensions of Holomorphic Motions

2. Preliminaries We will modify slightly the formulation of Theorem 1.1 to make it more convenient to prove and to strengthen it. The assumption of real-analyticity of the map (1.1.1) implies that the toms {(e iO, w) : w E l-'e/0 } is real-analytic. This in turn implies, by arguments similar to those used in [15, Proof of L e m m a 1.3] that the holomorphic motion fz defining F z can be extended to a holomorphic motion (z, w) > fz(W) : D(0, 1 + E) x P0 > C, E > 0. To simplify the exposition we will simply include this into our assumptions. In general, we will call {Uz}ze-B, D = { Izl < 1 ], a holomorphic family of Jordan domains over D, if U0 is a Jordan domain and there is a holomorphic motion (z, w) > az(W) : D(0, 1 + E) > C, with e > 0 dependent on {Uz}ze-B, such that Uz = az(Uo), z e D. Denote by Fz the quasicircles OUz, so that 1-'z = az(l-'0). In regard to future use, we introduce the following definition in greater generality than required by this paper. m

Let{Uz}zc~beaholomorphicfamilyofJordandomaJnsoverDandjo

Definition2.1.

B

" Ueio

>

C\Ueio, 0 ~ [0, 2~r], j0 = j2~, a continuous family of orientation reversing homeomorphisms such that jo IF0 = idro. Then (f, f * ) is called a canonical pair of selections relative to {Uz, jo} if (i) (ii) (iii)

f, f * : D

> C are holomorphic (more precisely meromorphic);

f(z) e Uz, f*(z) e Uz , z e D; Cl(f, f * ; e i~ C gr(jo) = {(e g~ jo(w)) : w E -Ueio},

where the joint cluster of f, f * at e i0 is defined as C1 (f, f * , e iO) : = [(W, V) E C x C : 3Zn ~

e iO, f ( Z n ) ~

11), f * ( z n )

--+ v } .

Function f itself is called a canonical selection if f * with required properties exists. It is not convenient to deal with meromorphic canonical selections f, f * . Choosing (by the holomorphic axiom of choice, cf. [14]) a meromorphic function ~. : D(0, 1 + E) > C such that L(z) ~ Uz, for every [zl < 1 + E, and applying biholomorphic coordinate change (z, w) > (z, (w - Z(z) -1) : D x C > D x C, we can assume without loss of generality that Uz C C, z ~ D. To avoid dealing with meromorphic functions f * we will consider an analogous situation in which the role of unbounded domains U z is played by bounded ones. To this end we choose any analytic function X : D(0, 1 + ~')

> C, Et > 0, such that

f(z) ~ Uz, Iz[ _< 1, consider the biholomorphic mapping (Z,W)

> (z,v)=

(1)

Z , - w - x (z)

:DxC

> DxC

and define V C D x C to be the image of D x C \ U under the above map, i.e.,

Vz=

{v--

'

ee:weC\Uc

w - x (z)

: = ~.J {z} x ~ z , Izl_
Vo = Veio ,

}

-, zeD,

640

Zbigniew Slodkowski

and To : U o

> Vo is defined by TO(W) = (jo(w)

--

X(Z)) -1, W E Uo-

(2.2)

P r o p o s i t i o n 2.3. Under the above assumptions and constructions, the domains Vz, z ~ D and maps To : Uo ~ Vo satisfy the following properties: (i)

{Vz}ze ~ is a holomorphic family o f quasidiscs and Z z = OVz, z ~ D, is a holomorphic family o f quasicircles over D. (i?) To : U o ) V o is a homeomorphism onto V o , 0 ~ R. (iii) To(Fo ) = Zo and To(Fo ), considered with orientation induced by that o f Fo has opposite orientation to that o l e o , provided both Fo and Zo have the same orientation (e.g., positive) relative to the point at c~. (iv) (e iO, w) ) (e iO, Tow) : U{e iO} • -fro ) U{e i~ } x v o is a homeomorphism which is o o 8To real-analytic on the set[_J{e i~ • -uo with I-~(w)l > 07"0 0w,( W - ,~ 9 0

(v)

(vi)

For every holomo phic function a : -B C such that graph (a) = { ( z , a ( z ) ) : z -B} C F (F is foliated by such graphs) there is a holomorphic function b : D ) C such that To(a(ei~ = b(ei~ graph Co) C Z, and every analytic disc is Z can be obtained in this way. Whenever (f, f * ) is a canonical pair o f selections relative to (Uz, Jo)z,O and i f g(z) = ( f * (z) - X(z)) -1, z ~ -D, then g : D ) C is holomorphic, g(z) ~ Vz, z ~ D ,

(2.3.1)

C l ( f , g; e i~ C graph (To),

(2.3.2)

and for every O ~ IR.

The proof of this proposition follows from inspection of the definitions. Definition 2.4. We say that functions f, g ~ H ~176 (D) form a canonical pair of selections relative to {Uz, Vz, To : z ~ -D, 0 ~ [0, 2zr) }, where Uz, Vz, To satisfy condition (i) through (v) of the above proposition if f , g satisfy conditions (2.3.1) and (2.3.2) above. In view of the preceding discussion our main result, Theorem 1.1, is a direct consequence of the following, in fact more general, fact. T h e o r e m 2.5. Let {Uz , Vz }z~O be holomolphic families (over D) o f real-anaiytic Jordan domains i n C a n d T o : Uo ) Vo,O ~ [0, (2zr)], T0 = T2zr, beareal-anaiyticfamilyoforientation-reversing homeomorphisms satisfying conditions (ii) through (v) o f Proposition 2.3. Then all functions f H~176 D ) such that (f, g) is a canonical pair o f selections for some g are real analytic on D and their graphs form a foliation oftAzeO{z } • Uz. This foliation defines a holomorphic motion

(z, w)

> (z, Fz(w)) : o • -Uo

, C,

w h i c h e x t e n d s t o a h o m e o m o r p h i s m D • Uo ) U, suchthatFz(Uo) = Uz, Fz(l-'0) = Fz (andfor w ~ Uo, t h e m a p z ) Fz(w ) i s o n e o f t h e c a n o n i c a l s e l e c t i o n s f ) .

The proof of this theorem will be given in Sections 3 through 6.

641

Natural Extensions of Holomorphic Motions

3. A uniform estimate We will prove Theorem 2.5 by the continuity method. To propagate the local argument, we need a uniform estimate which will be established here with help of the reflection principle.

Lemma 3.1.

Under assumptions o f the above Theorem 2.5, let .~ C H~176 ( D ) x H~176 ( D ) denote the set o f all canonical pairs o f selections relative to { Uz , Vz , T }. Then there are E > 0 and M < +oo, such that every ( f , g) E ~ extends to a holomorphis function ( f , ~ in D(0, 1 + E) such that

II.~llzoo(oo+E)), Ilffllzoo(O(l§

(3.1.1)

~ C0,

for some constant Co.

The proof of the lemma is based on the following two facts. T h e o r e m 3.2. [4, 2]. Let W C C n be an open set and M C W be a relatively closed real-analytic submanifold o f W. A s s u m e that M is totally real in W (i.e., Ta(M) M iTa(M) = (0) for every a E M). Then there is an open neighborhood W1 o f M in W such that every complex sub-variety S in W o f ]3ure dimension 1 (not necessarily closed), such that~\ S C M has an extension to a complex variety S ~ S in W such that S fq W1 is relatively closed, S N M = -S\ S.

Proposition 3.3.

Let X C 8 D x C n be a compact set. A s s u m e that for every 0 E R, the fiber = Xo). Let Y = the polynomial hull X o f X. Then Y M (8D x C n) = X. X 0 = {113 E C n : (e iO , 113) E X } is polynomially convex in c n ( x o

The proof is essentially folklore; an easy argument can be adapted from [14, The proof of Lemma 5]. P r o o f of L e m m a 3.1.

The map (iv), Proposition 2.3 can be extended to a real-analytic map (e iO, w)

' (e iO, To (w))

defined in some neighborhood of the set {(e i~ w); w E U0 },with ~ Because of the latter condition, the graph

~ 0 in this neighborhood.

M := {(e i~ w, T o ( w ) ) : w E neighborhood ofV-0, 0 E R} is a totally real, real-analytic submanifold of C 3. Choose an open neighborhood W of

w Uz,

V Vz}

in C 3, so that M is relatively closed in W. Let W1 be an open neighborhood of M in W satisfying conclusions of Theorem 3.2. Denote X :={(e iO, w, Tow) : 0 E R, w E -Uo} 9

Fix (f, g) E )r and let S be the graph of (f, g). Then S C W, S is a complex one-dimensional submanifold in W; S\S C X C M. By Theorem 3.2 there is a complex variety S 3 S such that S fl W1 is relatively closed. However, it does not follow immediately that S is the graph of a function. To obtain the required uniform

642

~isniew Slodkowski

estimates for )Tand ff we have to actually utilize some elements of the proof of Cirka's theorem in [4, Sec. 1]. As it is well known, cf. [4], a totally real manifold, e.g., M, admits locally antibiholomorphic involutions, i.e., maps ~).a : W a > W a, a ~ M, a ~ W a such that ~ra IW a fq M = id, 1~ra O ~r a = idw a , and Ca is biholomorphic. Since Cas are germ-wise unique, they can be patched together to form an antibiholomorphic involution r : W1 > W1, W1 ~ M. (Here W1 can be the W1 from Theorem 3.2, somewhat smaller if needed.) In the case of our M, the involution r has more special structure and can be obtained as follows. Fix 0 ~ ~, and consider the totally real, real-analytic submanifold Mo = {(w, To(w)) : w ~ neighborhood of (U0) } of C 2. By the above outline there is an antibiholomorphic involution (z, w) > aPo(z, w)ofaneighborhoodofMo onto itself, (r or = id),preserving M0 pointwise. In particular aP0(z, w) = q~0(z, w), where (z, w) > q~0(z, w) is biholomorphic on a neighborhood of Mo.

Assertion 3.4.

The map (e iO, z, w)

> ~0 (z, w) is real analytic on a neighborhood o f M =

U { e iO} x MO i n S 1 x C 2. It has (a germ-wise unique) extension to a holomorphic map, (z, w, v) o

>

C~z(W, v) holomorphic in some neighborhood o f M in C 3. The map

is an antibiholomorphic involution r o f some neighborhood W1 o f M, tixing M pointwise.

The proof of this assertion, part exercise and part folklore, is omitted. We may assume (by shrinking) that W1 satisfies the conclusion of Theorem 3.2. To finish the proof, choose a compact neighborhood N of X in W1, in such a way that r (N) = N, i.e., X C Int N C N = N C W1. Let Y = X, the polynomial hull of X.

Assertion 3.5. The fiber Xo = {(w, v) e C2; (e i~ w, v) e X} = {(w, Tow); w ~ -Uo} is polynomially convex in C 2. In case Uo is the unit circle, this is proven in [8]. The case when Uo is a Jordan domain with real-analytic boundary can be reduced to the case of the disc as follows. Let h : Uo ~ D be a homeomorphism such that h lUo is the Riemann mapping. Suppose Xo # Xo and let Z = ~"0\Xo. Then Z \ Z C Xo and Z has the local maximum property in the sense that there is no plurisubharmonic function ~t defined in a neighborhood of any x0 e Z such that r = 0 > ~(x) for x ~ Z\{0}, Ix - x01 < E, E > 0 (by the Rossi's local maximum modulus principle). Let Z* = {(h(w), v) : (w, v) e Z}. Then Z* has local maximum property and Z*\Z* C graph(T0 o h - l ) , and so Z* is contained in the polynomial hull of graph(T0 o h - l ) , To o h -1 : D > C. But the latter set is polynomially convex by [8]; hence, Z* = 0 and Z = 0, which proves Assertion 3.5. By Proposition 3.3, Y N (S 1 x C 2) = X. Hence (in view of the compactness of Y), there is r0 < 1 such that Y fq {(z, w, v) : v0 < Izl

_< 1} C

N.

(3.6)

Consider now an arbitrary canonical pair of selections (f, g) ~ 7 , and S = {(z, f ( z ) , g(z)) :

643

Natural Extensions of Holomorphic Motions

z ~ D}, thenS C Y, (for S \ S C X), andby (3.6) the set

Sr = SA{(z, w, v) : r < Izl _ 1} = {(z, f ( z ) , g ( z ) ) : r < Izl < 1} is contained in N. Since r preserves N, ~P(Sr) is contained in N. Now

~t(Sr)

=

,

=

Oz(f(z), g(z))

)

: v < [Z] < 1

}

{(~,4~l/~(f(1/~),g(1/~))):l
~z (w, v) is holomorphic in (z, w, v) we obtain that there are holomorphic functions fl, gl in {1 < ]~[ _< 1 } such that ,(sr)

=

{<

1}

~, A ( ~ ) , g, (r

: 1 < I~1 5_ -/-

9

We know already that f, g (and so fl, gl) must be holomorphic on OD as well. Since lp ]M = id,

f (e iO) = f l (ei~ g(e iO) = gl (ei~ Thus, 27(z) ~(Z)

= =

[ f (z)

r <_ Izl ___1

/ fl(z)

1 < Izl_ r

[ g(z)

r <_ I z l _ 1

[ gl(z)

1 < Izl _< T

1

1

are holomorphic on D(0, 1 + E). Since (3.1.1) graph (27, ~ C N, the required uniform estimate holds with Co = max{Iwl + Ivl : (z, v, w) ~ N}. []

4. Application of angular derivative The following lemma will be crucial in proving the uniqueness of canonical foliation postulated in Theorem 2.5.

Lemma 4.1.

Under assumptions of Theorem 2.5, whenever (f, g) is a canonical pair of selections such that f (e i~176~ r'oo for someOo ~ R, then f ( z ) E Fz, g(z) ~ Ez, z ~ D . (Note that f ( e i~176is defined because of Lemma 3.1.) m

Proof. Let a : D -+ C and b : D ~ C be the (unique) holomorphic functions such that gr(a) gr(b)

c C

F, a(e iO~ = f ( e iO~ , ~ , b(e iO~ = g(e iOo) .

(cf. Section 2, Proposition 2.3 (v)). We have to show that f = a, g -- b.

Assertion 4.2.

We can assume without loss of generality (by making appropriate biholomorphic coordinate changes) that I

O0=0, g(1)=l,

f(1)=l,

l~Fz,

1 C Ez,

Vz C D

-

-

Uz C D

forz~D for z E D ,

644

Zbigniew Slodkowski

and To(1) = 1, r o ( f (O)) = g(O), forO =- R. Applying the rotation (z, w) --~ (e-i~176w) we reduce the situation to the case 00 = 0. Since F0 are real-analytic regular (simple) curves, there is R0 > 0 such that for every 0 ~ R, w E F0, there is a closed disc of radius R0, passing through w and contained in U~. Denote by v(O, w), for w E F0, the unit outward normal vector to F0 at w. Then v(O, w) is real-analytic on U{e i~ } x F0; in particular, 0 e iO

>

v(O, a(ei~

: OD

> OD C C

is a real analytic function of index O. It is well known that there is a positive real-analytic function r : OD > (0, § such that for e > O, theproducter(O)v(O,a(ei~ is the boundary values of a nonvanishing holomorphic function

c:D

> C, c(z) ~ 0 ;

(take - logr(0) = the Hilbert transform of the continuous branch of Arg v(O, a(ei~ E > 0 small enough we can assume that ltcllLoo < R0.

Choosing

A s s e r t i o n 4.3.

{w : Iw - (a(z) -b c(z))l < Ic(z)l} n Uz = 0, z ~ -B . Postponing for awhile the proof of Assertion 4.3, we conclude now the proof of the lemma. The fractional-linear transformation

w

> wl = c ( z ) / ( a ( z ) + c(z) - w) : C • C

maps the disc D(a(z) x c(z), Ic(z)l) onto {z ~ C : Izl > Uz under the map (4.4) by Uz. Then Coy Assertion 4.3) Uz o o ( 0 , 1 ) ;

(4.4)

1} and a(z) onto 1. Denote the image of

I ~ 0 U z, z ~ O .

Sincethemap


> (z,

= (z, c(z)l(a(z) + c ( z ) -

is biholomorphic on a neighborhood of D • C in C x C, it is clear that {Uz}ze-~, is a holomorphic family of quasicircles satisfying all assumptions of Proposition 2.3. Let finally f * ( z ) ----c(z)l(a(z) + c(z) - f ( z ) ) . Then f * ( z ) ~ U z, f*(1) = 1. The same work applied to {Vz } yields a biholomorphic coordinate change

(z,v)

> (Z, V l ) : D x C

> DxC,

domains Vz* and function g*(z), with identical properties. Now define orientation reversing homeomorphism To* : ~00 > V~0 by composing the maps v > vl, To, and the inverse to w > Wl, in the natural order. It is clear that U z, Vz, To*, f * , g* satisfy all the assumptions in the lemma (in particular To*(f*(O)) = g*(O)) as well Assertion 4.2. We conclude now the proof of the lemma in the setting of Assertion 4.2.

NaturalExtensionsof HolomorphicMotions

645

Since f : D > D (g : D > D) and f ( 1 ) ----- 1 (g(1) = 1) and f , g ~ C ( I ) ( D ) , the classical results of Julia on the angular derivative, cf. [10, Ch. I, Exercise 7], imply that either f ( z ) --- 1, or f ( 1 ) 6 (0, + o o ) (and either g(z) -------1, or gf(1) 6 (0, + ~ ) ) . If f ( z ) -- 1 or equivalently g(z) ~ 1, we are done. (Note To(f(O)) = g(O), T0(1) = 1.) Assume that f ' ( 1 ) > 0, g'(1) > 0. Differentiating with respect to 0 the equation Tof(O) = g(O), we obtain, at (0, w) = (0, 1),

aTo

]

aTo T + - ~ - ( ) 9 [ - i F ( - - ~ + a-~T~( 1 ) [ i f ' ( 1 ) ] = ig'(1) .

ao (1) ]

0=0

(4.5)

01/3

Since To(l) = 1, 0 6 R, we get ~0~ (1) = 0. Since To(I'o) = Eo, DTo((TI['O)) = (TI(EO)). As Uo C D and l ~ t o , the tangent line to Fo at 1 is {1 + iy : y E R} and the tangent space is Tt(Fo) = {iy : y E R}. Similarly T1 (~o) = {i 0 : O 6 R}. The differential D To must be of the form (DTo)(w) = ot~+/3w, or,/3 6 C, w E C. Since it maps {iy : y E R} onto {ia : r ~ R}, -o~ + / 3 must be real, and since To reverses orientation, -or + / 3 < 0 . (4.6) Substituting this information to (4.5) we get (remembering that f1(1) = f ' ( 1 ) ) (--or + / 3 ) i f ' ( 1 ) = i g ' ( 1 ) . As f ' ( 1 ) , g'(1) > 0 and -or + 13 < 0, this is a contradiction. We will complete now the proof by justifying Assertion 4.3. Let (z, w) --~ fz(W) : D(0, 1 + 6) x U0 > C be a holomorphic motion such that fz(Uo) = Uz. Let f w : D(0, 1) > C be the analytic function defined by fW(z) = fz(W), Izl _< 1. Denote aU(z) = a ( z ) + c(z) -/xc(z), z E D, for I1zl < 1. Then, by our construction

f w _ atZ : S 1

> C\{0},

w ~ U---0, I~1 > 1,

and so, by continuity i n d ( f w - a tz, 0) = const. By construction, fwo = a, where w0 = a(0), has graph disjoint with that of a g, I~1 < 1, and so i n d ( f w~ - a g, 0) = 0 for I/zl < 1. Hence, i n d ( f w - a Iz, 0) = 0 for w ~ U0, I/zl < 1, that is g r ( f w) n gr(a u) = 0 for w ~ U0, I/zl < 1, which proves Assertion 4.3. []

5. Application of singular integrals Notation.

F o r 0 < ot < 1, denoted by Cg the Banach space of all functions f : a D > Cwhich are H61der continuous of order or; let A(aD) = the set of all continuous functions on a D whose 1 f2Jr f(eiO)dO = 0}. Poissonintegralis analytic in D, denote A ~ = C~ OA ( OD ), A~ = {f ~ A a : ~-~

[] L e m m a 5.1. Let or,/3 ~ C a. Assume that ot(0) ~ 0 for every 0 and that the winding number of 0t : S 1 ---->C\{0} is O. Then m

f

, o t f + / 3 f + Ag : A ~

, Ca/Ag

(5.1.1)

646

Zbigniew Slodkowski

is a bounded Fredholm operator o f index zero.

P r o o f . Since we could not find precise references for this undoubtedly well-known fact, we give a few comments to reduce it to standard results. Following Pr6ssdorf [13, Section 3.4] let (sf)(z)

=

1 ~!

--

f

dr,

-~

z

=

1

Iz[=l

in the sense of principal value. By Privalov's theorem S : C a > C a is a bounded invertible operator with S 2 = I and P ---- 1 ( I + S), Q = 1 ( I - S) are projections in C a with P ( C a) = A a, Q ( C a) = Ag, P Q = Q P = O. Let A, B : C a > C a be the operators of multiplication by ~, t , respectively. By [13, Corollary 4.4] the commutators

[B, P], [A, P] : C a

> C a are compact operators.

Consequently Q B P = Q[ B, P] is compact.

Assertion 5.2.

The operator f ---> [flf ] : A a ---> Ca / Ag is compact.

Indeed, let Jr : C a / A ~ > C a / A a be the natural surjection. Then for f ~ A a : rr(flf) = A B P f + A a ~ C a / A a. Thus, f ---> rr([flf]) : A a ---> C a / A s is compact. Since Jr is Fredholm f > [fly] : A a ---> C a / A g is compact. To complete the proof of Lemma 5.1, it suffices to show the following. m

AssertionS.3. f > [~f]: A s > Ca/Ag isFre~olmof&dexO(sincethisoperatordiffers just by conjugation from the first term o f (5.1.1)). In natural identification of C S / A~ with A s this operator becomes C = P A P : A s > A s. SinceA -1 : C a > C a exists and is bounded, the operator C1 = P A - 1 P : A a > Aa isbounded and C1C -- I = P A - I [ P , A]Po Since [P, A] is compact, C is invertible modulo the ideal of compact operators and so is Fredholm. Finally, the symbol ot(e iO) E C\{0} has a continuous (in fact C a) branch of logarithm, the operator C can be continuously deformed to the identity through Fredholm operators, and so has index 0. [ ]

6. Existence of the canonical foliation Proof of Theorem 2.5. For the purpose of the proof, we will consider the map (0, w) ---> Tow to be defined and real analytic in some neighborhood U { e i~ } x U~ of U { e i~ } x -fro, assuming also 0

0

that ~OTo ~ 0 throughout this neighborhood. Denote W ' = { f ~ C a : f ( e i~ ~ U~, 0 ~ R} and let ( T f ) ( e i~ ---- T o f ( e i~ for f ~ W'. Clearly T : W t > C a is a continuously differentiable nonlinear operator (in fact real analytic) defined in the open set W'. Consider the operator f

, Tf=Tf+A~:W

> Ca/A~, f~W=W'AA

u.

647

Natural Extensions of Holomorphic Motions m

It is clear that a function f 9 A a, such that f(O) 9 Uo, 0 9 R, is a canonical solution if and only if T f = [k] = k + Ag 9 C a / A g , for some constant k 9 C. Recall that by L e m m a 3.1 every canonical solution f is real analytic on OD, and in particular belongs to A a.

Assertion 6.1. I f f 9 A a is real-analytic, then the differentied o f T at f , ( D T ) ( f ) C a/A~, is an invertible operator.

: Aa

Clearly

( D T ) ( f ) ( A f ) ( e i~ = ot(O)(Af)(O) + fl(O)(Af)(O) + A~ where or(0) = -aTo OTo ~ ( f ( 0 ) ) , fl(0) = -u~(f(O)). Since or, fl 9 C a (in fact they are real analytic), (DT)(f) : A a > C a / A g is a Fredholm transformation of index 0 by L e m m a 5.1. To prove its invertibility, it suffices to show that k e r ( D T ) ( f ) = (0). Suppose, by contradiction, that there is A f ~ 0; A f 9 A a, such that

(DT)f(Af)

= O,

i.e., or(A f ) + f l ( A f ) = Ag 9 Ag. It is presumably a well-known fact that A f , Ago must be real-analytic on OD. We can observe that this follows also from Theorem 3.2, applied to the variety A = {(z, ( A f ) ( z ) , (Ag)(z)) : z 9 D} C C 3 and to the submanifold

M ={(e iO, w, v) : ot(O)N+ fl(O)w = v, w, v 9 C} , D

for M is totally real (c~(0) ~ 0 for all 0) and A \ A C M. Let ,4 = exp[log or( 9 ) + / l o g o~( 9 )] where ~ denotes the conjugate function, normalized to vanish at 0. By Privalov's theorem ,4 9 AU; moreover, ,4 : OD > C\{0} and ind(,4, 0) = 0. Let u(O) = a(0)/,4(0). Then 1,4(ei~ = la(ei~ and so u 9 C a, u : S 1 > S 1 and ind(u, 0) = 0. Only the last property requires an explanation. It is enough to observe that ind(ot, 0) = 0. To this end consider a homotopy ft, 0 < t < 1, such that ft 9 A a, ft(eiO) 9 -Uo, 0 9 R, t 9 [0, 1], and f0 = f and g r ( f l ) C F. Let ott(e i0) = "~.Jt~.OTo t r teiO~jj. Then ott: S 1 > C\{0}, o t t 9 C a, t > ott is continuous and o~0 = or. Thus, ind(ott, 0) = const and so ind(ot, 0) = ind(Otl, 0). We will observe now that ind(oq, 0) = 0 by utilizing portions of the proof of L e m m a 4.1, in particular Assertion 4.2. Due to suitable coordinate changes considered there, one can assume without loss of generality that f l (z) -- 1 9 OUz, T0(1) = 1, and the tangent lines to FeiO, ZeiO at 1 are {1 + iy : y 9 R}. Then Oll(e i0) = -OTo ~ ( 1 ) 9 ( - o o , 0) for 0 9 R; hence, ind(ot, 0) = 0. Let A f = I F be the inner-outer factorization of A f . Since A f is real-analytic, I is a finite Blaschke product and F is an outer function, also real analytic, with at most finitely many zeros, all N

on the boundary. Represent F(z) = P(z)Fo(z) where P(z) = I-I (z - ei~ j=l

Then

.~ Fo(e i~ Ag u(O)I (e i~ e ( e ' ~ ) ~ + (MA)I P = 9 n~ . AFo ro[e .~)

(6.2)

Since P(e iOj) = 0, i = 1 . . . . . N, we have that ILHS(z)I < C o n s t l z - e i~ z 9 D , i = 1. . . . . N, and so { A g / ( ` 4 F o P ) } 9 H ~ (the class of bounded holomorphic functions vanishing at

648

ZbigniewSlodkowski N iO) = ei~176 -iNO, where Oo = NTr + ~ Oj, for e iO ~= ei~ j = 1. . . . . N

0). Since p ( e i ~

j=l

(cf. Nikolski [12] we obtain, dividing (6.2) by P, that

b(O) + ( f l / A ) I ~ H ~

(6.3)

where

h(O) = [ei~176176176 By the above consideration h : S 1

'~) jl[e-iU~176

"

> S 1 is a continuous function of constant modulus one, with

nonnegative index. On the other hand, o"~ ~u~, la(eiO)l < 1 by the antiquasiconformality of To, uniformly in 0. Since IA(0)I = la(0)l, 0 ~ R, we have II~/AII~ < 1. In other words dist(h, H ~ ) < 1, which is impossible by [10, Ch. IV, Ex. 4.6]. This completes the proof of Assertion 6.1.

Assertion 6.4. Let fo E W C C a be a solution of the equation Tyo = [ko], ko a constant. Then there is a neighborhood V of fo in C a, and ~ > 0 such that for every k ~ D(ko, E) there is a unique function f such that f E V c C a, T f = [ k ] = k + A ~ ECa/A~. Since T : A a > Ca/A'~ is C (1) and (DT)(f0) is invertible, T must be a local homeomorphism in a neighborhood V of f0, which constitutes Assertion 6.4. (Note that we can apply Assertion 6.1 because YolOD is real analytic by Lemma 3.1.)

Assertion 6.5. IrE > 0 and V in Assertion 6.4 are small enough, then for every f , fl E V such thatT f = [k], T f l = [kd withk, kl ~ D(ko, E), either f =- fl o f f ( z ) # Yl(z) foreveryz ~ -D. Indeed, choosing V in Assertion 6.4 small enough we can assume that 1 > v =

{

OTo

0

OTo

}

sup II/~(-)/o~(-)11~ : a(0) = -~--~-(f()), fl(O) = ff-~w( f (O)) . Choose 8 > 0 such that v + ~ < 1. Since To's are differentiable, uniformly in 0, it holds:

To fl(O) = Toy(O) + (DTo)f(o)(Yl(O) - y(o)) + r(O, f ( 0 ) , f l ( 0 ) ) , with

lim

r(O, W, Wl)/lw - wl l = O, uniformly in w, wl, O and f ~ V. Hence, wecanshrink

W--Wl--->O

V so that

r(O, f(O), fl(0)) < 3If(0) -- fl(0)l,

for f, fl E V, 0 E R .

(Replacing ~ > 0 by a smaller one we can still assume that V, E satisfy Assertion 6.4). Writing now A f = fl -- f , (Ag)(e iO) = Tofl(e iO) - Tof(e iO) we obtain ot(0)Af + fl(O)Af + r(f, f l ) = Ag with A f , Ag 6 A a; in fact All real-analytic on D. Except for the presence of the remainder term r ( f , fl), the rest of the argument is the same as in the proof of Assertion 6.1. Suppose f ~ f l , i.e., A f ~ 0. We have to show that A f has no zeros in D. Let A, u, A f = I F = IPFo have the same meaning as in the proof of Assertion 6.1. A f has finitely many zeros and at least I or P is nontrivial. Dividing by F = PFo we obtain:

h(O) + (fl/.A)l + r(f, fa)/PFo ~ H ~ ,

Natural Extensionsof Holomorphic Motions

649

when h is defined by (6.3). Since II(/VA)I + r / P folloo < II/~/all~r +

m0axIr(0, f ( 0 ) ,

f l ( O ) ) / A f ( O ) l < v + 8 < 1,

we have that distoo(h, H ~176< 1. Since h : S 1 > S 1 is a continuous unimodular function of nonpositive index this implies, by [10, Ch. IV, Ex. 4.6], that h has index 0, i.e., A f = fl - f does not vanish in D. This confirms Assertion 6.5.

Assertion 6.6.

/ f T f 0 = [k0] 9 C~ / A~ and V ~ fo is small enough, the map f

> f ( 0 ) : { f 9 V : T f = [k],k 9 C}

>C

(6.6.1)

is a local homeomorphism onto an open neighborhood o f f0(0) 9 C. By Assertions 6.4 and 6.5, the map (6.6.1) is a homeomorphic embedding onto a subset, say Z 9 f0(0), a y Assertion 6.4 the map k > T - l ( [ k ] ) = f : D ( k o , E) > V is ahomeomorphic embedding and so the map k > f ( 0 ) : D(k0, E) > Z is a homeomorphism; hence, Z is a neighborhood of f0 (0). We can now conclude the proof of Theorem 2.5. Let ~b denote the set of canonical selections, i.e., q~ = { f : 3 g ( f , g ) 9 br} = { f 9 A u : T f = [const] and f ( z ) 9 U--z, z 9 (Recall that f ' s in Assertions 6.1 through 6.6 could have graphs in some neighborhood of U.) Denote zr(f) = f ( 0 ) . The set q~ is compact. We will use the following version of the Monodromy Theorem.

6.1.

(Monodromy Theorem)

Let ~b be a compact space and U0 a Jordan domain. Let 7r:4~

~ U0

be a continuous map such that m

(i) (ii) (iii)

forevery f 9 qbthereisaneighborhoodH of f inqbsuchthat~rlH : H is a local homeomorphism;

~ 7r(H) C Uo

with ~b0 = zr-l(0Uo), assume that 7rl~bo is one-to-one; for every f 9 q~\~bothere is a neighborhood H of f in ~b such that rrlU is one-to-one and zr (H) is an open subset of C. Then Jr : 4~ > U0 is a homeomorphism onto. m

The proof, which consists in showing that lifting property holds for every arc F in U0, is omitted. We observe only that conditions (i) through (iii) are fulfilled. Indeed (i) follows from Assertion 6.6, and (ii) is a basic property of holomorphic motions. As for (iii), by Lemma 4.1, if f 9 ~b\~b0, i.e., graph ( f ) ~ F, then graph ( f ) r F = 0, i.e., some neighborhood V of f in A a is contained in q~\q~0; the rest follows again from Assertion 6.6. Thus, the map f > f(O) : dp > Uo is a homeomorphism onto. It is, however, clear that point 0 9 D is not distinguished in any way in this argument and, in the same way, for each, z0 9 D, the map f > f ( z o ) : dp > Uzo is a homeomorphism onto.

650

Zbigniew Slodkowski

Finally, define the map (z, w) ~ Fz(w) : D x U0 ~ C as Fz(w) = f ( z ) , where f E is the unique function with f ( 0 ) = w. By compactness of ~ the map is continuous, and, by the above argument, Fzo : Uo ~ Uzo is a homeomorphism for every Iz01 < 1. If It I = 1, F~ 10U0 : OUo ~ OUt is a homeomorphism (because U {z} x OUz is fibered by canonical selections) and Izl_
F~ : U0

~ U~ is a local homeomorphism by Assertion 6.5; hence, it is a homeomorphism. Clearly z --~ Fz is a holomorphic motion over D satisfying all requirements of Theorem 2.5. []

7. Concluding remarks In order to obtain a natural extension method of holomorphic motions by using Theorem 1.1, one needs to assign to each holomorphic family of Jordan domains {Uz}z~-B a family of generalized reflections jo : Uo ) U~, 0 e [0, 2zr) in a conformally invariant, functorial, manner. This can be done by taking jo to be the conformally natural reflection in Jordan curve F0 defined by Earle and Nag [7] in terms of the barycentric extensions of [5]. Since it seems to be still unknown (although highly probable) that real-analytic homeomorphisms of the circle have realanalytic barycentric extensions, the desired natural holomorphic motion cannot be obtained as a direct application of Theorem 1.1. Instead, to show that the family of all canonical selections (which is an invariant object) defines a foliation of U {z} x Uz, we can approximate Earle-Nag's reflection jo, uniformaly in 0, by real zeD

analytic maps jn,o : UO ~ U~. One proves that the canonical foliations relative to jn.o (which exist by Theorem 1.1) converge to the family of all canonical selections relative to {j0}, and so the latter defines a foliation of U {z} x Uz. zED

This method of proof may be not the most economical in the case of real-analytic {F0 }0~[0,2~r), but it has the advantage of working in much greater generality, under mild regularity assumptions on quasicircles {F0 }. Since the techniques used to obtain these improvements are rather different from those of this paper, we postpone the details to Part H.

References [1] [2] [3] [4]

Ahlfors, L.V. and Bers, L. Riemann's mapping theorem for variable metrics, Annals Math., 72(2), 385-404, (1960). Alexander, H. Continuing 1-dimensional analytic sets, Math. Ann., 191, 143-144, (1971). Bers, L. and Royden, H.L. Holomorphic families of injections, Acta Math., 157, 259-286, (1986). C'trka,E.M. Regularity of boundaries of analytic sets, Math USSR-Sb, 117, 291-334, (1982), (Russian), Math USSR-Sb, 45, 291-336, (1983).

[5] Douady, A. and Earle, C.T. Conformally natural extensions of homeomorphisms of the circle, Acta Math., 157, 23-48, (1986). [6] Earle, C.T., Kra, I., and Krushkal, S.L. Holomorphic motions and TeichmiiUer spaces, to appear. [7] Earle, C.T. and Nag, S. ConformaUy natural reflections in Jordan curves with applications to Teichmiiller spaces, in Holomorphic Functions andModuli, II, Springer Verlag, New York, 1988, 179-194. [8] O'Farrell, A.G. and Preskenis, K.J. Approximation by polynomials in two diffeomorphisms, Bull. Am. Math. Soc., 10, 105-107, (1984). [9] Samelin, T.W., Garnett, T.B., Rubel, L.A., and Shields, A.L. On badly approximable functions, J. Approx. Theory, 17, 280-296, (1976). [10] Garnett, J.B. Bounded Analytic Functions, Academic Press, San Diego, 1981.

Natural Extensions of Holomorphic Motions

[ 11] [12] [13] [14] [15] [16] [17] [18] [19] [20]

651

Marie,R., Sad, E, and Sullivan, D. On the dynamics of rational maps, Ann. Sci. Ecole Norm. Sup., 16, 193-217, (1983). Nikolski, N.K. Treatise on the Shift Operator, Springer-Verlag,Berlin, 1986. Pr~Sssdorf,S. Some Classes of Singular Equations, North-Holland, Amsterdam, 1978. Slodkowski, Z. Polynomially convex hulls with convex sections and interpolation spaces, Proc. Amer. Math. Soc., 96, 255-260, (1986). Slodkowski, Z. Holomorphic motions and polynomial hulls, Proc. Amer. Math. Soc., 111, 347-355, (1991). Slodkowski, Z. Extensions of holomorphic motions, Annali Scuda Norm. Sup. Sev. IV, XXII, 185-210 (1995). Slodkowski, Z. Holomorphic motions commuting with semigroups, Studia Math., 119, 1-16 (1996). Sullivan, D. Quasiconformal homeomorphisms and dynamics, III: Topological conjugacy classes of analytic endomorphisms, preprint, 1985. Sullivan, D. and Thurston, W.E Extending holomorphic motions, Acta Math., 157, 243-257, (1986). Wermer,J. Banach Algebras and Several Complex Variables, Markham Publishing Company, Chicago, 1L, 1971.

Note added in proofs (Feb. 22, 1999): In the meantime, the real analyticity assumptions required above were substantially lowered, and m a t c h i n g regularity results for families of E a r l e - N a g reflections were obtained, cf. ([21]). In ([22]), the results of this paper were applied to study some p o l y n o m i a l hulls in C 3. [21] Slodkowski, Z. Orientation-reversing diffeomorphisms and holomorphic motions, preprint. [22] Slodkowski, Z. Polynomially convex hulls in C3 with totally real fibers, Complex Variables, 32, 321-330, (1997).

Received December 30, 1994 Department of Mathematics, University of Illinois at Chicago, 851 South Morgan Street, Chicago, IL 60607-7045 [email protected] Communicated by John Mather