The Measurable Riemann Mapping Theorem

CHAPTER 1

1.1. Conformal structures on Riemann surfaces

Throughout, “smooth” will always mean C 1. All surfaces are assumed to be smooth, oriented and without boundary. All diffeomorphisms are assumed to be smooth and orientation-preserving. It will be convenient for our purposes to do local computations involving metrics in the complex-variable notation. Let X be a Riemann surface and z x iy be a holomorphic local coordinate on X. The pair .x; y/ can be thoughtD C of as local coordinates for the underlying smooth surface. In these coordinates, a smooth Riemannian metric has the local form E dx2 2F dx dy G dy2; C C where E;F;G are smooth functions of .x; y/ satisfying E > 0, G > 0 and EG F 2 > 0. The associated inner product on each tangent space is given by @ @ @ @ a b ; c d Eac F .ad bc/ Gbd @x C @y @x C @y D C C C Äc a b ; D d where the symmetric positive deﬁnite matrix ÄEF (1.1) D FG represents in the basis @ ; @ . In particular, the length of a tangent vector is f @x @y g given by

@ @ p a b Ea2 2F ab Gb2: @x C @y D C C

Deﬁne two local sections of the complexiﬁed cotangent bundle T X C by ˝ dz dx i dy WD C dz dx i dy: N WD 2 1 The Measurable Riemann Mapping Theorem

These form a basis for each complexiﬁed cotangent space. The local sections @ 1 Â @ @ Ã i @z WD 2 @x @y @ 1 Â @ @ Ã i @z WD 2 @x C @y N of the complexiﬁed tangent bundle TX C will form the dual basis at each point. ˝ The inner product on TX extends canonically to a Hermitian product on TX C. @ @ ˝ The matrix of this Hermitian product in the basis @z ; @z is given by f N g (1.2) P P ; O D where 1 Ä1 i (1.3) P : D 2 1 i It follows from (1.1), (1.2) and (1.3) that 1 Ä E GE G 2iF C : O D 4 E G 2iF E G C C Let us introduce the quantities 1 (1.4) .E G pEG F 2/1=2 WD 2 C C 1 E G 2iF (1.5) .E G 2iF / C : WD 4 2 C D E G pEG F 2 C C Note that !1=2 E G 2pEG F 2 (1.6) > 0 and C < 1: j j D E G 2pEG F 2 C C A straightforward computation then shows that " # 1 1 2 2 2 C j j N : O D 2 2 1 2 C j j Since the Hermitian product on TX C is given by ˝ @ @ @ @ ÄC A B ;C D AB N ; @z C @z @z C @z D O D N N N it follows that 2 @ @ 1 2 2 2 2 Á (1.7) A B .1 /. A B / 2 AB 2 A B : @z C @z D 2 C j j j j C j j C N C N N N 1.1 Conformal structures on Riemann surfaces 3

@ @ Now a tangent vector A B TX C is real (i.e., belongs to TX) if and @z C @z 2 ˝ only if B A. This simply followsN from D N @ @ @ @ a b .a ib/ .a ib/ : @x C @y D C @z C @z N Thus, for real tangent vectors, the formula (1.7) reduces to

@ @ A A A A : @z C N @z D j C Nj N The last expression suggests that if we are only concerned about lengths of real @ @ tangent vectors, the metric in the complex basis @z ; @z can be represented as f N g (1.8) .z/ dz .z/ dz ; D j C Nj with and deﬁned by (1.4) and (1.5). Let us see how the quantities and associated with transform under a holomorphic change of coordinates z w on X: 7! .z/ dz .z/ dz .w/ dw .w/ dw j C Nj D j C N j .w.z// w0.z/ dz .w.z//w0.z/ dz D j ˇ C Nj ˇ ˇ w .z/ ˇ ˇ 0 ˇ .w.z// w0.z/ ˇdz .w.z// dzˇ ; D j j ˇ C w0.z/ Nˇ from which we obtain

.z/ .w.z// w0.z/ D j j w .z/ .z/ .w.z// 0 D w0.z/ or simply .z/ dz .w/ dw j j D j j dz dw (1.9) .z/ N .w/ N : dz D dw dz It follows that .z/ dz and .z/ dzN are well-deﬁned forms on X. Note that z .z/ is a well-deﬁnedj j function on X, even though z .z/ is not. 7! j j 7! dz DEFINITION 1.1. .z/ dzN is called the Beltrami form associated with the metric . We say Dis a conformal metric if its Beltrami form is identically zero, in which case .z/ dz is a positive multiple of the Euclidean metric in every local coordinate.D j j 4 1 The Measurable Riemann Mapping Theorem

Conformal metrics exist on every Riemann surface. For example, the spherical 2 metric dz =.1 z / on C, the Euclidean metric dz on C, C and the tori j j C j j O j j C=ƒ, and the Poincare´ metric on hyperbolic Riemann surfaces are all conformal metrics.

It is clear from (1.8) that two smooth metrics and O have the same Beltrami forms if and only if they belong to the same conformal class, which means =O is a smooth positive function X R. Each conformal class is also called a smooth conformal structure on!X. On the other hand, given a Beltrami dz form .z/ dzN on X, we can pair it with an arbitrary conformal metric .z/ dz Dto construct the metric .z/ dz .z/ dz with the Beltrami formj.j The canonical conformal structureD jof XCis the oneNj represented by any conformal metric .z/ dz on X. It corresponds to the zero Beltrami form 0 which vanishes identicallyj j in every local coordinate on X.

COROLLARY 1.2. There is a one-to-one correspondence between smooth dz conformal structures on X and smooth Beltrami forms .z/ dzN which satisfy .z/ < 1 in every local coordinate z on X. D j j Here is a more geometric description for a Beltrami form as a ﬁeld of concentric ellipses on the tangent bundle of X. Fix a local coordinate z dz D x iy .x; y/ near a point p X and let .z/ N in this coordinate. C Š 2 D dz Consider the family of “circles” E.p/ v TpX v const: which depends only on and not on the choiceD of f the2 representativeW k k D metric.g If v @ @ @ @ D a b .a ib/ .a ib/ , then the “circles” v const: in TpX @x C @y D C @z C @z k k D correspond to the loci .a ib/ .aN ib/ const: in the real .a; b/-plane. i j C C i=2 j D Setting re and .a ib/e , we obtain the loci rN const: in the -plane,WD which is justWD theC family of concentric ellipses withj C the minorj D axis along the real direction and the major axis along the imaginary direction, and with the ratio of the major to minor axis equal to .1 r/=.1 r/ (compare Fig. 1.1). Transferring this family back to the .a; b/-plane,C it follows that E.p/ is a family of concentric ellipses in TpX with 1 angle of elevation of the minor axis arg D 2 1 ratio of the major to minor axis C j j: D 1 j j Note that in this geometric description, the zero Beltrami form 0 corresponds to the ﬁeld of round circles 2 a2 b2 const: in the .a; b/-plane. j j D C D 1.1 Conformal structures on Riemann surfaces 5

b E.p/

i=2 1=.1 r/ .a ib/e =2 D C a

1=.1 r/ C

FIGURE 1.1. Geometric interpretation of a Beltrami form as a ﬁeld of ellipses on the tangent bundle.

To summarize, we have at least three ways of thinking about Beltrami forms on a Riemann surface X: (i) as a . 1; 1/-tensor on X obeying the transformation rule (1.9); (ii) as a conformal structure on X; (iii) as a field of concentric ellipses on the real tangent bundle TX. Now let X and Y be Riemann surfaces and f X Y be a diffeomorphism. W ! Given a Beltrami form on Y , the pull-back f is defined as the Beltrami form of the pull-back metric f , where is any metric with the Beltrami form . It is easy to see that the definition is independent of the choice of . The pull- back operator on Beltrami forms has all the functorial properties of the similar operator on metrics. For example, if f X Y and g Y Z are given W ! W ! diffeomorphisms and is a Beltrami form on Z, then .g f / f .g/, or in short .g f / f g . ı D ı D ı Let us derive a local formula for the pull-back operator. Express f X Y locally as w f .z/, where z and w are local coordinates on X and YW , and! let D 6 1 The Measurable Riemann Mapping Theorem

.w/ dw .w/ dw be a metric on Y . Then D j C N j f .w.z// wz dz wz dz .w.z// .wz dz wz dz/ D j C N N C CN N N j .w.z// .wz .w.z// wz/ dz .wz .w.z// wz/ dz D j C N ˇ C N C Nˇj ˇ wz .w.z// wz ˇ .w.z// .wz .w.z// wz/ ˇdz N C dzˇ ; D C N ˇ C wz .w.z// wz Nˇ C N where we have used the fact that

wz wz and wz wz: N D N N N D It follows that Â Ã dw wz .w.z// wz dz (1.10) f .w/ N N C N : dw D wz .w.z// wz dz C N In particular, the pull-back f f 0 of the zero Beltrami form on Y is given by D

wz dz (1.11) f f .0/ N N : D D wz dz

The coefﬁcient f .z/ wz=wz is called the complex dilatation of f in local D N coordinates z and w. By the Cauchy-Riemann equations, f 0 if and only if f is a conformal isomorphism. D Now (1.10) can be written as

wz Â Ã f .z/ .w.z// dw wz dz (1.12) f .w/ N C N : wz dw D 1 f .z/.w.z// dz C wz Thus, at the level of coefﬁcients, the pull-back operator on Beltrami forms acts pointwise as the disk automorphism

. f / f wz (1.13) ı C ; where has modulus 1: 7! 1 f . f / D wz C ı

COROLLARY 1.3. The pull-back operator f acts pointwise on the coefﬁcients of Beltrami forms as an automorphism7! of the unit disk. When f is holomorphic, this automorphism reduces to a rotation about the origin. In particular, f X Y is a conformal isomorphism if and only if f 0 0. W ! D The central question now is Question. Given a Beltrami form on a Riemann surface X, does there exist a new complex structure on X with respect to which is the zero Beltrami form? 1.1 Conformal structures on Riemann surfaces 7

If such a complex structure exists, we say that it is compatible with and we call integrable. The surface X with the complex structure compatible with is denoted by X. Here is an equivalent question Question. Is there a Riemann surface Y and a diffeomorphism f X Y such W ! that f 0 ? D Any such f is said to integrate . Geometrically, the condition means that the derivative Df pulls back the ﬁeld of round circles on Tf .p/Y (generated by 0) to the given ﬁeld of ellipses on TpX generated by . To see the equivalence of the two formulations, suppose f X Y integrates and let X be X equipped with the pull-back of the complexW ! structure of Y under f (whose local coordinates are the composition of f with the local coordinates of Y ). Conversely, if X exists, the identity map X X clearly integrates . ! If there are two diffeomorphisms f X Y and g X Z that integrate 1 W ! W ! in the above sense, then .g f /0 0, which by Corollary 1.3 means 1 ı D g f Y Z is a conformal isomorphism. This shows that the Riemann surfaceı YW in the! integrability question is unique up to biholomorphism. By (1.11) the integrability of Beltrami forms can be easily expressed in local coordinates. If f X Y integrates , then f f 0 . Writing f W ! D D locally as w f .z/, this translates into f .z/ wz=wz .z/, or D D N D (1.14) wz wz: N D This partial differential equation is called the Beltrami equation. When D 0 0, it reduces to the classical Cauchy-Riemann equation wz 0. There is alongD history of attempts to solve the Beltrami equation under variousN D regularity conditions on . The most basic result is the following

THEOREM 1.4 (Gauss). Suppose is a smooth complex-valued function defined in the unit disk D which satisfies .z/ < 1 at every z D. Then j j 2 there exists a diffeomorphism f D Š f .D/ C such that fz fz in D. W ! N D Note that either f .D/ C or f .D/ is conformally isomorphic to D by the Riemann Mapping Theorem.D Thus, after post-composing with a biholomorphism if necessary, we can assume that f is a diffeomorphism D D or D C; in ! ! other words, either D C or D D. Š Š EXAMPLE 1.5. Let .z/ k in D where 0 k < 1 is a constant. The affine map D Ä 2 2 2 x y f D U .x; y/ R < 1 W ! WD 2 W .1 k/2 C .1 k/2 C 8 1 The Measurable Riemann Mapping Theorem

deﬁned by w f .z/ z kz integrates since clearly wz k wz. Post-composing w with a D D C N N D biholomorphism U D given by the Riemann Mapping Theorem, we obtain a diffeomorphism ! D D which integrates . In particular, D D. ! Š

2 EXAMPLE 1.6. Let .z/ z in D. The diffeomorphism f D C deﬁned by w f .z/ z=.1 z 2/ integrates sinceD W ! D D j j z2 2 2 wz .1 z / 2 N j j z : wz D 1 D .1 z 2/2 j j It follows that D C. Š The key difference between the above examples is that < 1 in the k k1 former while 1 in the latter. It turns out that .D; / is always isomorphic k k1 D to D if < 1. This will be a corollary of the measurable Riemann mapping theorem.k k1

COROLLARY 1.7. Every smooth Beltrami form on a Riemann surface is integrable.

PROOF. Let be a smooth Beltrami form on a Riemann surface X. Cover X with countably many charts Ui with local coordinates zi Ui D. Let i be the 1 W ! pull-back of restriction Ui under zi . The Beltrami forms i on D are integrable j by Theorem 1.4, so there are diffeomorphisms fi D Vi fi .D/ C such W ! WD that f 0 i . Now equip X with a new complex structure in which the i D gi fi zi Ui Vi are the local coordinates. Note that the change of WD ı W ! coordinates are indeed holomorphic since if Ui Uj , then \ ¤ ; 1 1 1 .gj g /0 Œ.f / .z / z f 0 ı i D i ı i ı j ı j 1 1 1 1 Œ.f / .z / z j Œ.f / .z / D i ı i ı j D i ı i 1 .f /i 0: D i D Evidently, the Beltrami form in this new complex structure is identically zero 1 since in Ui it is seen as .g / 0. i D 1.2. Quasiconformal maps In many applications, one is bound to consider conformal structures on Riemann surfaces which are only measurable. The integrability question for such conformal structures still makes sense, but maps that would integrate such structures can no longer be smooth. Simple examples show that measurable 1.2 Quasiconformal maps 9 conformal structures are not generally integrable. However, with an extra boundedness assumption, they are integrable and the maps which integrate them are homeomorphisms which enjoy some degree of regularity. This leads to the idea of quasiconformal homeomorphisms between Riemann surfaces. In the category of quasiconformal maps rigidity and flexibility coexist, and that makes them an extremely powerful tool. Roughly speaking, quasiconformal maps are a.e. differentiable homeomorphisms with bounded small-scale geometry. More specifically, a quasiconformal map f U V between planar domains has three essential features: (i) it is an orientation-preservingW ! homeomorphism; (ii) it is differentiable at almost every p U and the derivative Df.p/ R2 R2 is non-singular; (iii) Df.p/ pulls back2 round circles to ellipses whoseW eccentricity! is bounded independent of p. This is in essence what quasiconformality means, except that the a.e. existence of the derivative should be replaced by a stronger condition called “ACL” (see Definition 1.8 below) or, equivalently, by f having locally integrable partial derivatives in the sense of distributions. Although a bit technical at first, this is just the right degree of regularity one needs to exclude pathologies and have a well-behaved class of maps. It will be convenient to begin the discussion with quasiconformal maps between planar domains and consider the case of Riemann surfaces later.

DEFINITION 1.8. Let U and V be non-empty open sets in C. An orientation- preserving homeomorphism f U V is called quasiconformal if W ! (i) f is absolutely continuous on lines (ACL). This means that for every closed rectangle Œa; b Œc; d U , the restriction x f .x iy/ is absolutely continuous on Œa; bfor a.e. y Œc; d, and7! the restrictionC y f .x iy/ is absolutely continuous on2Œc; d for a.e. x Œa; b. 7! C 2 (ii) the partial derivatives of f (which by (i) exist a.e.) satisfy

fz k fz a.e. in U; j N j Ä j j for some constant 0 k < 1. Ä To make the attribute more quantitative, we say that f is K-quasiconformal, where K .1 k/=.1 k/ 1. WD C EXAMPLE 1.9. Let K 1 and deﬁne f C C by W ! ( x iKy if y 0 f .x iy/ C : C WD x iy if y < 0: C 10 1 The Measurable Riemann Mapping Theorem

Then f is ACL, with ( ( .1 K/=2 if y > 0 .1 K/=2 if y > 0 fz.x iy/ C and fz.x iy/ C D 1 if y < 0 N C D 0 if y < 0

(note that the partial derivatives do not exist along the real axis unless K 1). Thus, fz=fz .K 1/=.K 1/ and f is K-quasiconformal. D j N j Ä C

EXAMPLE 1.10. Let 0 k < 1 and deﬁne f C C by Ä W ! ( z kz if z 1 f .z/ C N j j Ä WD z k=z if z > 1: C j j Then f is easily seen to be ACL, with ( ( 1 if z < 1 k if z < 1 fz.z/ j j and fz.z/ j j D 1 k=z2 if z > 1 N D 0 if z > 1 j j j j (again, the partial derivatives do not exist along the unit circle unless k 0). It follows that D fz=fz k, so f is K-quasiconformal with K .1 k/=.1 k/. j N j Ä D C Below we list some basic properties of quasiconformal maps. For a complete treatment and the proofs, we refer the reader to [A] or [LV]. Suppose f U V is K-quasiconformal. Then W ! (QC1) f is differentiable almost everywhere in U , that is, for a.e. p U , 2 f .p z/ f .p/ fz.p/ z fz.p/ z ".z/; C D C C N N C where ".z/=z 0 as z 0. ! ! (QC2) The Jacobian 2 2 Jac.f / fz fz D j j j N j is positive a.e. and locally integrable in U , and the formula Z Jac.f / dx dy area.f .E// E D holds for every measurable set E U . In particular, f maps sets of area zero to sets of area zero.

(QC3) The partial derivatives fz and fz are locally square-integrable in U . In N fact, since fz k fz for k .K 1/=.K 1/, the deﬁnition of Jac.f / showsj N thatj Ä j j D C 2 2 1 2 k fz Jac.f / and fz Jac.f /: j j Ä 1 k2 j N j Ä 1 k2 1.2 Quasiconformal maps 11

(QC4) The partial derivatives fz and fz are the distributional derivatives also, that is, N Z Z Z Z fz ' f 'z and fz ' f 'z U D U U N D U N for every compactly supported smooth test function ' U C. W ! (QC5) For every topological annulus A U , 1 K mod.A/ mod.f .A// K mod.A/: Ä Ä Surprisingly, this geometric condition is in fact equivalent to K- quasiconformality. 1 (QC6) The inverse f V U is K-quasiconformal. This follows at once from (QC5). W ! (QC7) If g V W is K -quasiconformal, the composition g f U W W ! 0 ı W ! is KK0-quasiconformal. This is also immediate from (QC5).

EXAMPLE 1.11. The unit disk D and the complex plane C are not quasiconformally homeomorphic: If there were a quasiconformal map f D C, then by (QC5) the annulus f . z 1=2 < z < 1 / would have a ﬁnite modulus. ButW this! annulus contains the punctured diskf zW z >j r j for ag large r, whose modulus is inﬁnite. f W j j g The following result will be frequently used:

THEOREM 1.12 (Weyl’s Lemma). A 1-quasiconformal map is conformal.

Note that 1-quasiconformality of f means that the Cauchy-Riemann equation fz 0 holds a.e., and to conclude that f is conformal in this case, it is essential thatN D we assume it is ACL.

EXAMPLE 1.13. Let Œ0; 1 Œ0; 1 be any continuous non-decreasing function, with .0/ 0, .1/ 1, and W.x/ 0!a.e. (the graph of such is called a devil’s staircase). Extend D D 0 D to a map R R by setting .x n/ .x/ n for n Z. Deﬁne f C C by ! C D C 2 W ! f .x iy/ x i.y .x//: C WD C C Then f is a homeomorphism which satisﬁes the Cauchy-Riemann equation fz 0 a.e. in N D C. However, f is not conformal. This does not contradict Weyl’s Lemma since fails to be absolutely continuous, so f is not ACL, hence not quasiconformal.

The standard chain rule formulas applies to compositions of quasiconformal maps. Suppose f U V and g V W are quasiconformal. By (QC7), W ! W ! 12 1 The Measurable Riemann Mapping Theorem

g f U W is quasiconformal. By (QC1), there are sets of measure zero Aı UW and!B V away from which f and g are differentiable. Furthermore, 1 by (QC2) and (QC6), f .B/ has measure zero. Thus, g f is differentiable 1 ı outside the measure zero set A f .B/ and the following chain rule formulas hold: [

.g f /z .gz f / fz .gz f / fz ı D ı C N ı N .g f /z .gz f / fz .gz f / fz ı N D ı N C N ı NN 2 2 If f U V is quasiconformal, (QC2) says that Jac.f / fz fz > 0 W ! D j j j N j a.e., so the partial derivative fz must be non-zero a.e. in U . Inspired by the smooth case treated before, we call the measurable function

f fz=fz WD N the complex dilatation of f . The measurable function

1 f fz fz Kf C j j j j C j N j WD 1 f D fz fz j j j j j N j is called the real dilatation of f . Thus f is K-quasiconformal precisely when Kf K or f k .K 1/=.K 1/. k k1 Ä k k1 Ä WD C The chain rule gives the following formula for the complex dilatation of the composition of quasiconformal maps:

.gz f / fz .gz f / fz g f ı N C N ı NN ı D .gz f / fz .gz f / fz ı C N ı N fz .g f / fz N C ı D fz .g f / fz C ı N fz f .g f / C ı fz D fz 1 f .f g/ C ı fz which, for good reason, is similar to the local formula for the pull-back operator in (1.13). Here is a list of a few deeper properties of quasiconformal mappings:

(QC8) (Mori) If f D D is K-quasiconformal with f .0/ 0, then W ! D 1=K f .z/ f .w/ 16 z w .z; w D/: j j Ä j j 2 As a corollary, it follows that quasiconformal maps are locally Holder:¨ If f U V is K-quasiconformal, for every compact set E U there W ! 1.2 Quasiconformal maps 13

exists a constant C C.E; K/ > 0 such that D f .z/ f .w/ C z w 1=K .z; w E/: j j Ä j j 2 (QC9) The family of all K-quasiconformal maps deﬁned in a domain U which ﬁx three marked points in U is compact. As a result, for any sequence fn of K-quasiconformal maps in U there is a sequence n of Mobius¨ f g f g maps such that n fn converges locally uniformly in U to a K- quasiconformal map.f ı g (QC10) (Astala) If f D D is K-quasiconformal with f .0/ 0, then W ! D area.f .E// C area.E/ Ä for every measurable set E D. Here C C.K/ > 0. As a corollary, D p if f U V is K-quasiconformal, then Jac.f / Lloc.U / for every 1 < pW

f integrates f f 0 fz fz a.e. ” D D ” N D We say that a measurable conformal structure, or its associated Beltrami form has bounded dilatation if sup .z/ < 1: k k1 D z X j j 2 14 1 The Measurable Riemann Mapping Theorem

Thus, f X Y is quasiconformal if and only if the Beltrami form f W ! D f .0/ has bounded dilatation. The following theorem was proved by Morrey-Ahlfors-Bers:

THEOREM 1.14 (The measurable Riemann mapping theorem). Let be a measurable Beltrami form on the Riemann sphere C with k < 1. Then O k k1 D (i) There exists a unique quasiconformal homeomorphism f f Œ C D W O ! C which ﬁxes 0; 1; and solves the Beltrami equation f . O 1 D (ii) f is a C 1 (resp. real analytic) diffeomorphism if is C 1 (resp. real analytic). (iii) The normalized solution f Œt, given by part (i), depends holomorphically on the complex parameter t D.0; 1=k/. 2 COROLLARY 1.15 (MRMT, the disk case). Let be a measurable complex- valued function on the unit disk D with < 1. Then there exists a k k1 quasiconformal homeomorphism f D D which satisﬁes the Beltrami W ! equation fz fz almost everywhere. Moreover, f is unique up to post- N D composition with a conformal automorphism of D.

PROOF. Extend to C by setting .z/ 0 for z > 1 and apply O D j j Theorem 1.14. The restriction to D of the normalized solution given by that theorem will provide a quasiconformal solution h D h.D/ of the Beltrami W ! equation. We have h.D/ C since C and D are not quasiconformally homeomorphic, so by the Riemann¤ mapping theorem there is a conformal map g h.D/ D. The composition f g h will then be the required map. W ! D ı (Alternatively, extend to C symmetrically by setting .z/ .1=z/ for O D N z > 1, ﬁnd the similar map h and use symmetry to show that h.D/ D j j D already.)

The local case of the theorem easily yields the general version:

COROLLARY 1.16 (MRMT, the Riemann surface case). Let be a measurable Beltrami form with bounded dilatation on a Riemann surface X. Then there exists a Riemann surface Y and a quasiconformal homeomorphism f X Y such W ! that f f .0/ . If g X Z is another such homeomorphism, the map g fD 1 Y DZ is a biholomorphism.W ! ı W ! 1.2 Quasiconformal maps 15

Problems. (1) Let f U V be quasiconformal. Show that W ! fz f 1 f f : ı D fz (2) Suppose is a Beltrami form on a Riemann surface X that is locally of bounded dilatation in the sense that sup .z/ < 1 z K j j 2 for every compact set K X. Show that is integrable in the following sense: There is a Riemann surface Y and a homeomorphism f X Y such that W ! (i) each p X has a neighborhood Up for which f U is quasiconformal; (ii) 2 j p f 0 . D (3) Let be a smooth Beltrami form in D which is rotationally symmetric so that depends only on r z . Show that the Riemann surface D is conformally j j D j j isomorphic to D or C according as Z a 1 dr lim C j j a 1 1 1 r ! 2 j j is ﬁnite or inﬁnite.

(4) Study compactness properties of QS0.M /. R x (5) Every homeomorphism h R R can be written as h.x/ h.0/ 0 d for some positive measure W without! atoms. Show that h is quasisymmetricD C iff is a doubling measure. Research problem.

(1) Let be a smooth Beltrami form in D. Develop a method for deciding whether the Riemann surface D is conformally isomorphic to D or C.

CHAPTER 2

Some Applications of Quasiconformal Maps

2.1. The diffeomorphism group of the 2-sphere We begin our application tour by proving a classical result on the homotopy type of the diffeomorphism group of the 2-sphere [S]:

2 THEOREM 2.1 (Smale). The group DiffC.S / of smooth orientation-preserving diffeomorphisms of the 2-sphere has a strong deformation retraction onto the rotation subgroup SO.3/.

Recall that a space X has a strong deformation retraction onto a subspace Y if there is a continuous map F X Œ0; 1 X such that W ! F .x; 0/ x and F .x; 1/ Y for all x X; F .y; t/ D y for all y Y2and t Œ0; 12. D 2 2 It is easily seen that in this case X and Y have the same homotopy type; in fact, the inclusion Y, X is a homotopy equivalence. Thus, Smale’s theorem shows 2 ! that DiffC.S / has the homotopy type of a ﬁnite cell-complex. This is also true 3 for DiffC.S / (Hatcher) and is known to be false for all dimensions 7. Let us identify 2 with the Riemann sphere via, say, the stereographic S CO projection. The idea of the proof consists of ﬁrst constructing a strong deformation retraction from Diff . / onto the Mobius¨ group Aut. / and then follow it by C CO CO another from Aut. / onto SO.3/. CO

PROOF. Let ' Diff . / and , so < 1. Let ˆ Aut. / be C CO ' CO the unique Mobius¨ 2 map which agreesD with ' onk thek1 set 0; 1; . Consider2 the f 1g Œt smooth Beltrami forms t on C for 0 t 1, and set 't ˆ ' (here Œt O Ä Ä WD ı ' C C is the unique solution of the Beltrami equation wz .t/ wz W O ! O N D which ﬁxes 0; 1; ). Then t 't is a continuous path in DiffC.C/ that connects 1 7! O '0 ˆ to '1 '. The map F DiffC.C/ Œ0; 1 DiffC.C/ deﬁned by D D W O ! O F .'; t/ '1 t is a strong deformation retraction onto Aut.C/. WD O 18 2 Some Applications of Quasiconformal Maps

Next we construct a strong deformation retraction G Aut.C/ Œ0; 1 W O ! Aut.C/ onto the rotation subgroup SO.3/. The map H DiffC.C/ Œ0; 1 O W O ! Diff . / deﬁned by C CO ( F .'; 2t/ t Œ0; 1=2 H.'; t/ 2 WD G.F .'; 1/; 2t 1/ t Œ1=2; 1 2 will then be a strong deformation retraction of Diff . / onto SO.3/. C CO

LEMMA 2.2. Every Mobius¨ map ' Aut.C/ can be decomposed uniquely as 2 O ' ˛ ˇ; D ı ı where SO.3/, ˛.z/ rz for some r > 0, and ˇ.z/ z for some C. 2 D D C 2 Assuming this lemma for a moment, it is easy to construct G: Given ' Aut. /, CO take the decomposition ' ˛ ˇ as in the lemma, and define 2 D ı ı G.'; t/ ˛t ˇt ; WD ı ı 1 t where ˛t .z/ r z and ˇt .z/ z .1 t/. This proves Theorem 2.1. WD WD C PROOFOF LEMMA 2.2: The decomposition is immediate when '. / , for 1 D 1 in this case '.z/ az b for some a; b C with a 0, and we can take D C 2 ¤ a b .z/ z; ˛.z/ a z; and ˇ.z/ z : D a D j j D C a j j For the general case, first post-compose ' with an element of SO.3/ to keep fixed, and then decompose the resulting map as above. 1 To show uniqueness, suppose ' has two such decompositions, say

' 1 ˛1 ˇ1 2 ˛2 ˇ2; D ı ı D ı ı where ˛i .z/ ri z and ˇi z i . The map D D C 1 1 r2 2 1 ˛2 ˇ2 .˛1 ˇ1/ z z r2.2 1/ ı D ı ı ı W 7! r1 C is in SO.3/ and afﬁne, so it can only be a rotation around the origin. This implies ˇ ˇ ˇr2 ˇ ˇ ˇ 1 and r2.2 1/ 0; ˇr1 ˇ D D

which gives r1 r2 and 1 2. 2 D D 2.2 Quasiconformal conjugacy classes of rational maps 19

2.2. Quasiconformal conjugacy classes of rational maps Two rational maps f; g of the Riemann sphere are quasiconformally conjugate if there is a quasiconformal map ' C C which satisﬁes W O ! O ' f g ': ı D ı This is clearly an equivalence relation, and each equivalence class under this relation is called a quasiconformal conjugacy class. Suppose f; g are quasiconformally conjugate by ' as above. Then the Beltrami form ' is f -invariant. This is simply because g is holomorphic: D f f .'0/ .' f /0 .g '/0 D D ı D ı '.g0/ '0 : D D D Conversely, suppose is an f -invariant Beltrami differential with < 1. Then a similar argument shows that the branched covering g 'Œ fk .'k1Œ/ 1 D ı ı preserves 0, so it is locally 1-quasiconformal away from the branch points, hence is a rational map. Thus, there is a correspondence between f -invariant Beltrami forms of bounded dilatation and rational maps which are quasiconformally conjugate to f (the correspondence need not be one-to-one). As a basic dynamical application of the preceding remark, we show

THEOREM 2.3. The quasiconformal conjugacy class of a rational map f is always path-connected.

Of course this conjugacy class could reduce to a point, in which case f is called quasiconformally rigid. For example, in the family of normalized 2 quadratic polynomials fc.z/ z c c C, the map fc is quasiconformally rigid if and only if c belongsf to theD boundaryC g 2 of the Mandelbrot set or is the center of the hyperbolic component.

PROOF. Suppose ' is a quasiconformal conjugacy between f and another rational map g and set '. Then, for every 0 t 1, D Ä Ä f .t/ tf t: D D Here we have used the fact that the pull-back operator f acts as a rotation about Œt the origin and hence is linear. As before, set 't ˆ ' where ˆ is the unique D ı 1 Mobius¨ map which agrees with ' on 0; 1; . Then t gt 't f .'t / f 1g 7! WD ı ı is a path within the quasiconformal conjugacy class of f which connects g0 D 20 2 Some Applications of Quasiconformal Maps

1 1 ˆ f ˆ to g1 g. Joining this path to t ˆt f ˆ in which t ˆt ı ı D 7! ı ı t 7! is a path in Aut. / which connects id to ˆ, we obtain a path from f to g. CO

2.3. Local fixed-point theory Let f .z/ z O.z2/ be the germ of a holomorphic map in the plane D C fixing the origin. The multiplier f 0.0/ is clearly invariant under smooth conjugacies. On the other hand, z D2z is topologically (even quasiconformally) conjugate to z 3z. 7! 7! A remarkable theorem of Naishul asserts that when the origin is an indifferent fixed point in the sense that 1, then the multiplier is invariant under topological conjugacies. Herej wej D prove a weaker version of this result by using the measurable Riemann mapping theorem.

THEOREM 2.4. Let f .z/ z O.z2/ and g.z/ z O.z2/ be quasiconformally conjugate nearD 0. IfC 1, then .D C j j D D PROOF. Let ' be a quasiconformal homeomorphism defined near 0 which satisfies '.0/ 0 and ' f g '. Consider the Beltrami form ' defined near theD origin, whichı D is clearlyı f -invariant. Let ı > 0 be smallD and extend to C by setting .z/ 0 for z > ı. Consider the Beltrami forms t D j j for t < 1= . Since f and f is linear, it follows that f .t/ j j k k1 Œt D 1 D t near 0. Let 't ' . Then gt 't f ' is a 1-quasiconformal D D ı ı t homeomorphism near the origin, hence holomorphic there. Moreover, t gt .z/ is holomorphic for each fixed z sufficiently close to 0 [Caution: This is7! true but 1 non-trivial, as the inverse 't may not depend holomorphically on t]. Writing 2 gt .z/ t z O.z /, it follows that t t is also holomorphic. But gt is D C 7! conjugate to f whose fixed point at z 0 is indifferent, so t 1 for all D j j D t D.0; 1 "/. The open mapping theorem now implies that t t is constant. 2 C 17! Since '0 id, we have g0 f so 0 . Similarly, '1 ' is conformal, so D D D ı g1 is holomorphically conjugate to g, so 1 . We conclude that . D D As another application, consider the problem of linearizing holomorphic maps near attracting or repelling fixed points. A classical theorem of Koenigs asserts that every holomorphic germ f .z/ z O.z2/ with 0; 1 is holomorphically linearizable. The classical proof,D forC < 1, consistsj j of ¤ showing n n j j that the sequence f ı .z/ n 1 converges uniformly in a neighborhood of the f g origin to a holomorphic map ˆ. It is then clear that ˆ0.0/ 1 and ˆ.f .z// ˆ.z/. Here we give a proof of this result by applying the measurableD RiemannD mapping theorem. 2.4 Quasiconformal surgery on rational maps 21

THEOREM 2.5 (Koenigs). If f .z/ z O.z2/ is a holomorphic germ with 0; 1, there exists a holomorphicD changeC of coordinate z ˆ.z/ defined nearj j ¤ the origin such that ˆ.0/ 0 and ˆ.f .z// ˆ.z/. 7! D D PROOF. Without losing generality assume < 1 (otherwise consider the local inverse of f ). Choose a disk U centered atj j0, small enough so that f .U / n is compactly contained in U . It follows by induction that f ı .U / is compactly n 1 n contained in f ı .U / for all n 1, and that diam f ı .U / 0 as n . Let L denote the linear contraction z z=2. Take a smooth! diffeomorphism! 1 7! A z C 1=2 z 1 U r f .U / subject only to the condition W D f 2 W Ä j j Ä g ! .L.z// f . .z// for z 1. Extend to a homeomorphism D U by D j j Dn n ! defining .0/ 0 and .Lı .z// f ı . .z// for all n 1 and all z A. Then is quasiconformalD and satisfiesD 2

.L.z// f . .z// for all z D: D 2 Now consider the Beltrami form on D. Extend to the entire plane by taking pull-backs under L. The resultingD Beltrami form (still denoted by ) is easily seen to be L-invariant and with bounded dilatation. The quasiconformal Œ Œ 1 map g ' L .' / C C preserves 0, hence is holomorphic. Since g.0/ D0, we mustı ı have g.z/W !z for some 0. D D ¤ 1 Set ˆ ' . Then ˆ is a 1-quasiconformal homeomorphism deﬁned in a neighborhoodD ı of the ﬁxed point 0. By Weyl’s Lemma, ˆ is holomorphic. Moreover, ˆ conjugates f to g near 0, so g .0/ f .0/ . D 0 D 0 D

2.4. Quasiconformal surgery on rational maps Here is an outline of the construction of Herman rings by surgery, following Shishikura. Suppose f is a rational map of degree d 2 with a ﬁxed Siegel disk of rotation number . Take another rational mapg of degree d 2 with a 0 ﬁxed Siegel disk 0 of rotation number . We will construct a rational map F , of degree d d 1, with a Herman ring of rotation number . The idea is to C 0 cut out invariant disks from and 0 and paste the resulting spheres-with-hole along their boundaries to obtain a sphere. There is an obvious action on this sphere coming from the action of f and g on the pieces. We apply the measurable Riemann mapping theorem to realize this action as a rational map.

More precisely, let Š D.0; 2/ and 0 Š D.0; 2/ be conformal isomorphisms which satisfyW ! W !

2i 2i .f .z// e .z/ and .g.z// e .z/: D D 22 2 Some Applications of Quasiconformal Maps

Consider the invariant Jordan curves

z .z/ 1 and 0 z 0 .z/ 1 : D f 2 W j j D g D f 2 W j j D g 1 The map h 0 deﬁned by h.z/ ..z// is a smooth orientation- reversing diffeomorphismW ! which satisﬁesD h.f .z// g.h.z// for all z . D 2 Extend h to a quasiconformal homeomorphism h with the following CO CO properties: W !

h maps int. / to ext. 0/ and ext. / to int. 0/. (Here “int” refers to the complementary component of the invariant curve which contains the center of the Siegel disk and “ext” refers to the other component.) 1 h is conformal in a neighborhood of C r . h .0//. O \ Define ( f .z/ if z ext. / F .z/ 2 [ Q D .h 1 g h/.z/ if z int. / ı ı 2 It is easy to check that F is a degree d d 1 branched covering of the Q 0 sphere which is locally quasiconformal awayC from its branch points. Moreover, A h 1. / is a “topological Herman ring” of rotation number for F , D \ 0 Q and F is holomorphic in a neighborhood of F 1.A/. CO r To conjugate F to a rational map, define a Beltrami form on as follows. Q CO First define on A by ( 0 on A ext. / \ D h 0 on A int. / \ Clearly, F A A preserves . Extend to the union S F n.A/ by Q W ! n 1 Q pulling back via the appropriate iterate of F . Note that only the first pull-back Q to F 1.A/ A can possibly increase the dilatation of ; all further pull-backs Q r are taken by iterates of F which are holomorphic and so do not change the Q dilatation. On the complement of this union, set 0. The Beltrami form defined this way is clearly F -invariant and has boundedD dilatation. It follows that Q F 'Œ F .'Œ/ 1 is a rational map with a Herman ring 'Œ.A/ of rotation Q numberD .ı ı

2.5. The no wandering domain theorem We present a simpliﬁed version of Sullivan’s proof of Fatou’s no wandering domain conjecture, following N. Baker and C. McMullen. 2.5 The no wandering domain theorem 23

THEOREM 2.6 (Sullivan). Let f Ratd with d 2. Then every Fatou component U of f is eventually periodic,2 i.e., there exist n > m > 0 such that f n.U / f m.U /. ı D ı The idea of the proof is as follows: Assuming there exists a wandering Fatou component U (or simply a wandering domain), we change the conformal structure of the sphere along the grand orbit of U to find an infinite-dimensional family of rational maps of degree d, all quasiconformally conjugate to f . This is a contradiction since the space Ratd of rational maps of degree d, as a Zariski 2d 1 open subset of CP C , is finite-dimensional. The eventual periodicity statement for entire maps is false. For example, the transcendental map z z sin.2z/ has wandering domains. 7! C

LEMMA 2.7 (Baker). If U is a wandering domain for a rational map f , then n f ı .U / is simply connected for all large n.

n PROOF. Let Un f ı .U /. Replacing U by Uk for some large k if necessary, D n we may assume that no Un contains a critical point of f , so that f U Un ı W ! is a covering map for all n. We can also arrange that U . Then the Un 1 2 are disjoint subsets of C r U for n 1, so area.Un/ 0 as n . But n ! ! 1 f ı U is a normal family, so every convergent subsequence of this sequence f j g n must be a constant function. In particular, diam.f ı .K// 0 for every compact n ! set K U . Take any loop U and set n f . / Un. By the above D ı argument diam. n/ 0. If Bn is the union of the bounded components of Cr n, ! it follows that diam.Bn/ 0 also. Since f .Bn/ is open, @f.Bn/ n 1, and ! C diam f .Bn/ 0, we must have f .Bn/ Bn 1 for large n. In particular, the ! C iterated images of Bn are subsets of C r U for large n. Montel’s theorem then implies Bn F .f /, which gives Bn Un. Thus n is null-homotopic in Un for n large n. Since f U Un is a covering map, we can lift this homotopy to U , ı W ! which proves U is simply connected. Let a rational map f have a wandering domain U . In view of the above lemma, n we can assume that Un f ı .U / is simply connected and f Un Un 1 is a conformal isomorphismD for all n 0. Given an L Beltrami formW ! definedC in 1 U , we can construct an f -invariant L Beltrami form on as follows. Use the 1 CO forward and backward iterates of f to spread along the grand orbit n GO.U / z C f ı .z/ Um for some n; m 0 : D f 2 O W 2 g On the complement C r GO.U /, set 0. The resulting Beltrami form is O D defined a.e. on bC, it satisfies f by the way it is defined, and < D k k1 1 24 2 Some Applications of Quasiconformal Maps

since spreading U along GO.U / by the iterates of the holomorphic map f does not change thej dilatation. Now consider the deformation t for complex t with t < ", where " > 0 is small enough to guarantee t < 1. Since f j j k k1 Œt is holomorphic, f acts as a linear rotation, so f .t/ t. Let 't ' D D W C C be the unique solution of the Beltrami equation wz .t/ wz which O ! O 1 N D fixes 0; 1; . Then ft 't f ' is a rational map of degree d and t ft 1 D ı ı t 7! is holomorphic, with f0 f . The infinitesimal variation D d ˇ w.z/ ˇ ft .z/ D dt ˇt 0 D defines a holomorphic vector field whose value at z lies in the tangent space Tf .z/bC. In other words, w is a holomorphic section of the pull-back bundle f .T bC/ which in turn can be identified with a tangent vector in Tf Ratd . This is the so-called infinitesimal deformation of f induced by . We say that induces a trivial deformation if w 0 everywhere. D Another way of describing w is as follows: First consider the unique quasiconformal vector field solution to the equation

@v with v.0/ v.1/ v. / 0: D D D 1 D

This is precisely the inﬁnitesimal variation .d=dt/ t 0 't .z/ of the normalized j D solution of the Beltrami equation. It is not hard to check that w ıf v, where D

ıf v.z/ v.f .z// f 0.z/v.z/ D

measures the deviation of v from being f -invariant. Note in particular that ıf v is holomorphic even though v is only quasiconformal, and that w ıf v depends linearly on , a fact that is not immediately clear from the ﬁrst descriptionD of w. It follows that induces a trivial deformation if and only if v is f -invariant.

The triviality condition ıf v 0 forces v to vanish on the Julia set J.f /. In D fact, let z0 z1 zn z0 be a repelling cycle of f with multiplier 7! 7! 7! D . Then the condition ıf v 0 implies v.zj 1/ f 0.zj /v.zj / for all j 0; : : : ; n 1, so D C D D n 1 n 1 Y Y v.zj / v.zj /: D j 0 j 0 D D The assumption > 1 then implies v.zj / 0 for some j , hence for all j . Since J.f / is thej closurej of repelling cycles andD v is continuous, we conclude that v.z/ 0 for all z J.f /. D 2 2.5 The no wandering domain theorem 25

The above construction gives well-deﬁned linear maps

i D B.U / , B.bC; f / Tf Ratd ! !

Here B.U / is the space of L1 Beltrami forms in U , B.bC; f / is the space of f - invariant L1 Beltrami forms on bC, and D is the linear operator D w ıf v constructed above. D D

LEMMA 2.8. B.U / contains an infinite-dimensional subspace N.U / of compactly supported Beltrami forms with the following property: If N.U / 2 satisfies @v for some quasiconformal vector field v with v @U 0, then 0. D j D D Assuming this lemma for a moment, let us see how we can prove Theorem 2.6. Consider the subspace N.U / for the simply connected wandering domain U and restrict the above diagram to this subspace. If D./ 0 for some N.U /, or in other words if induces a trivial deformation, thatD means the normalized2 solution v to @v is f -invariant. Hence v 0 on J.f / and in particular on the boundaryD of U . By the property of N.UD /, 0. This means that the D infinite-dimensional subspace N.U / injects into Tf Ratd whose dimension is 2d 1. The contradiction shows that no wandering domain can exist. C It remains to prove Lemma 2.8. Let us first consider the corresponding problem for the unit disk D. Let N.D/ B.D/ be the linear span of the Beltrami k dz Q forms k.z/ z N for k 0. The vector field DN dz 8 1 zk 1 @ z < 1 < k 1 C @z C N j j Vk.z/ D : 1 .k 1/ @ k 1 z C @z z 1 C j j solves the equation @Vk k in D. Let @v N.D/ and v @ 0, and D D 2 Q j D D take the appropriate linear combination V of the Vk which solves @V . Then D V v is holomorphic in D and agrees with V on the boundary @D. This is impossible if V @ has any negative power of z in it. Hence 0. To get the j D D compact support condition, let N.D/ B.D/ be obtained by restricting elements of N.D/ to the disk z < 1=2 and extending them to be zero on 1=2 z < 1. If Q j j Ä j j @v N.D/ and v @D 0, then v has to vanish in the annulus 1=2 < z < 1 sinceD it is2 holomorphicj there.D In particular, it is zero on z 1=2. Now thej j same argument applied to the disk z < 1=2 shows 0. j j D j j D 26 2 Some Applications of Quasiconformal Maps

For the general case, consider a conformal isomorphism D Š U with 1 W ! the inverse and define N.U / .N.D//. Let v be a quasiconformal D D vector field such that @v N.U / and v @U 0. Then .v/ .v /= 0 D 2 j D D ı is a quasiconformal vector field in D which is holomorphic near the boundary @D and v. .z// 0 as z 1. By the reflection principle, v is identically ! j j ! ı zero near @D. Since @ .v/ N.D/, we must have 0, which implies 0. D 2 D D REMARK 2.9. Sullivan’s original argument had to deal with two essential difficulties: (i) the possibility of U being non simply connected, perhaps of infinite topological type; (ii) the possible complications near the boundary of U , for example when @U is not locally-connected. He addressed the former by using a direct limit argument, and the latter by using Caratheodory’s´ theory of “prime ends.” Both of these difficulties are miraculously bypassed in the present proof.

2.6. Holomorphic motions

DEFINITION 2.10. Let A C be a set with at least 4 points and T be a O connected complex manifold with base point t0.A holomorphic motion of A over .T; t0/ is a map ' T A C such that W ! O (i) z '.t; z/ is injective for each t T . 7! 2 (ii) t '.t; z/ is holomorphic for each z A. 7! 2 (iii) '.t0; z/ z for every z A. D 2

In other words, 't . / '.t; / t T is a holomorphic family of injections of A f D g 2 into C, with 't0 idA. O D A few remarks on this deﬁnition are in order: There is no assumption on the joint continuity of ' in .t; z/, or even continuity in z for ﬁxed t. They follow automatically from the -Lemma to be discussed below.

For our purposes, we usually take .T; t0/ .D; 0/ and call ' a holomorphic D motion over D. We can always assume that the motion is normalized in the sense that 0; 1; belong to A and they remain ﬁxed under the motion. To see this, take distinct1 points z1; z2; z3 in A and let ˛; ˇt Aut.C/ be determined by 2 O

˛.0/ z1 ˛.1/ z2 ˛. / z3 D D 1 D 2.6 Holomorphic motions 27

and ˇt .'t .z1// 0 ˇt .'t .z2// 1 ˇt .'t .z3// : D D D 1 1 Then t ˇt 't ˛ is a normalized holomorphic motion of ˛ .A/. D ı ı

EXAMPLE 2.11. Let A 0; 1; ; a and p D C r 0; 1; be the holomorphic D f 1 g W ! O f 1g universal covering map which satisﬁes p.0/ a. Then 't t D deﬁned by D f g 2 't .0/ 0 't .1/ 1 't . / 't .a/ p.t/ D D 1 D 1 D is a holomorphic motion of A over D.

EXAMPLE 2.12. Let A be the lattice Z iZ of Gaussian integers and deﬁne 't t H by ˚ f g 2 't .m in/ m tn C D C is a holomorphic motion of A over .H; i/.

EXAMPLE 2.13. Let f C C be a quasiconformal homeomorphism and f . For t < Œt W O ! O D j j 1, let 't ' be the normalized solution of the Beltrami equation given by Theorem 1.14. Then D ' is a holomorphic motion of over . This shows that every quasiconformal homeomorphism t CO D of the sphere canonically produces a holomorphic motion of the sphere.

EXAMPLE 2.14. Let U C be a Jordan domain. Suppose there are conformal isomorphisms i i 0 1 ft U Ut .i 0; 1/ depending holomorphically on a parameter t D such that Ut and Ut W ! D 2 are disjoint subsets of U . For every finite word i1 in of 0’s and 1’s, let i1 in in i1 U f f .U / t D t ı ı t and define the Cantor sets \ [ i1 in Kt U : D t n 1 i1 in The Kt determine a holomorphic motion of the base Cantor set K0 over D. To see this, take a z K0 and suppose that it is represented by the infinite word i1i2i3 ::: so that 2 z U i1 U i1i2 U i1i2i3 D 0 \ 0 \ 0 \ Define i1 i1i2 i1i2i3 '.t; z/ U U U Kt : D t \ t \ t \ 2 Note that '.t; z/ is the local uniform limit of the sequence of holomorphic functions 'n.t/ i D f in f 1 , so it depends holomorphically on t. It is now easy to check that .t; z/ '.t; z/ t ı ı t 7! is a holomorphic motion of K0 over D.

Let E C be a set with at least 4 points. Let us say that a homeomorphism O f E f .E/ is quasiconformal if there exists an M > 0 such that CO W ! Á dist Œf .z /; f .z /; f .z /; f .z /; Œz ; z ; z ; z M Cr 0;1; 1 2 3 4 1 2 3 4 O f 1g Ä 28 2 Some Applications of Quasiconformal Maps

for all quadruples .z ; z ; z ; z / in E. Here dist is the hyperbolic 1 2 3 4 Cr 0;1; O f 1g distance in the thrice punctured sphere and Œz1; z2; z3; z4 is the cross ratio deﬁned by z3 z1 z4 z2 Œz1; z2; z3; z4 : D z2 z1 z4 z3 It can be shown that when E , this deﬁnition of quasiconformality agrees CO with the standard one given in D1.2.

THEOREM 2.15 (The -Lemma of Mane-Sad-Sullivan˜ and Lyubich). A holomorphic motion ' D A C extends uniquely to a holomorphic motion W ! O ˆ D A C. Moreover, ˆ is continuous on D A and ˆt A ˆt .A/ is a W ! O W ! quasiconformal homeomorphism for each t D. 2 The analogue of the -Lemma for continuous motions is certainly false. As an example, let A 1=k k 1 and deﬁne the continuous motion ' R A C D f g W ! O by '.t; 1=k/ 1=k ikt. Evidently, ' has no continuous extension to R A. D C PROOF. Without losing generality, assume that the motion is normalized. By Montel’s theorem, F t '.t; z/ z A D f 7! W 2 g is a normal family of holomorphic functions D C, so it has compact closure ! O F in Hol.D; C/. Moreover, if f; g F are distinct, then f .t/ g.t/ for all O 2 ¤ t D. To see this, take fn; gn F such that fn gn, fn f and gn g, 2 2 ¤ ! ! and note that t fn.t/ gn.t/ is nowhere vanishing by the injectivity property of holomorphic7! motions. It follows from Hurwitz’s theorem that t f .t/ g.t/ is nowhere vanishing as well. 7! For each t D consider the continuous map 2 pt F C pt .f / f .t/: W ! O D By the above observation, pt is injective. Since F is compact, it follows that pt is a homeomorphism onto its image, which is easily seen to be the closure of 't .A/. Now 1 ˆ.t; z/ pt p0 .z/ .t; z/ D A; D ı 2 extends ' to a motion of A. The deﬁnition of the compact-open topology on F shows that for each r < 1, the family pt t r is equicontinuous, so the same must be true of the family f gj jÄ ˆt t r . It follows that ˆ is continuous on the product D A. f gj jÄ 2.6 Holomorphic motions 29

Finally, choose a quadruple .z1; z2; z3; z4/ in A and define a holomorphic map g D C r 0; 1; by W ! O f 1g g.t/ Œˆt .z1/; ˆt .z2/; ˆt .z3/; ˆt .z4/: D By the Schwarz lemma, dist .g.t/; g.0// dist .t; 0/; Cr 0;1; D O f 1g Ä which implies that the distance in C r 0; 1; between the cross ratios O f 1g Œˆt .z1/; ˆt .z2/; ˆt .z3/; ˆt .z4/ and Œz1; z2; z3; z4/ is at most Â1 t Ã log C j j : 1 t j j This proves that each ˆt A ˆt .A/ is quasiconformal. W ! THEOREM 2.16 (The Improved -Lemma of Slodkowski). A holomorphic motion ' D A C extends to a holomorphic motion ˆ D C C. The W ! O W O ! O extended motion ˆ is continuous on D C and ˆt C C is Kt -quasiconformal 1 t O W O ! O for each t D, where Kt 1Cjt j . 2 D j j It was proved by Sullivan and Thurston that there exists a universal constant 0 < a < 1 such that every holomorphic motion of A over D extends to a holomorphic motion of the sphere over the smaller disk D.0; a/. Bers and Royden proved that one can take a 1=3. Moreover, their extended motion over D D.0; 1=3/ has the advantage of being canonical in the sense that the Beltrami form is harmonic in each component of A. (A Beltrami form on a ˆt CO r hyperbolic Riemann surface X is called harmonic if =2, where is a holomorphic quadratic differential on X and is the densityD of the hyperbolic metric.) With this additional property, they proved that the extended motion is unique. As Sullivan and Thurston observed, to obtain the improved -Lemma, it suffices to prove the following holomorphic axiom of choice: Given a finite set A and a point a A, every holomorphic motion of A over D extends to a … holomorphic motion of A a over D. [ f g In the original version of the -Lemma, D can be replaced with an arbitrary connected complex manifold, as essentially the same proof works. In the Bers- Royden version, D can be replaced with the unit ball in any complex normed linear space. In Slodkowski’s improved -Lemma, D cannot be replaced for free, as the next example shows. 30 2 Some Applications of Quasiconformal Maps

EXAMPLE 2.17. (Douady) Let T C r 0; 1; , with the base point t0 2. Let A D O f 1g D D 0; 1; 2; and define a holomorphic motion ' T A C by f 1g W ! O 't .0/ 0 't .1/ 1 't . / 't .2/ t: D D 1 D 1 D This motion is maximal in the sense that it cannot be extended to a holomorphic motion of any larger set over T . To see this, it suffices to show that every holomorphic map f T T has a fixed point. Assume by way of contradiction that such a fixed point free map exists.W By! Picard’s great theorem, none of 0; 1; can be an essential singularity for f , so f extends to a rational 1 map f of degree d 1. As f 1 0; 1; 0; 1; , f acts bijectively on 0; 1; . CO CO By the assumptionW ! all the d 1 fixed pointsf of 1gf belong f to 1g0; 1; . If d 1, f isf a Mobius¨ 1g map which fixes one of 0; 1; C with multiplicity 2 and swapsf the other1g two. ButD any Mobius¨ map with a double fixed point is conjugate1 to z z 1 so it cannot swap a pair of distinct points. If d 2, each fixed point in 0; 1; is a critical7! C point of order d 1 and in particular is a simple (i.e., multiplicity 1) fixed point.f 1g This forces d 1 3, or d 2. But then each of 0; 1; is a simple fixed point of f as well as a critical point.C Ä This is impossibleD since a degree 2 rational1 map has only 2 critical points.

Problems.

(1) Let f Rat have an attracting ﬁxed point p0 with multiplier 0. Show that 2 d for every D there exists f Ratd , quasiconformally conjugate to f , 2 2 with an attracting ﬁxed point p of multiplier .