Monodromy, liftings of holomorphic maps, and extensions of holomorphic motions
aa r X i v : . [ m a t h . C V ] A p r MONODROMY, LIFTINGS OF HOLOMORPHIC MAPS, ANDEXTENSIONS OF HOLOMORPHIC MOTIONS
YUNPING JIANG AND SUDEB MITRA
Abstract.
We study monodromy of holomorphic motions and show the equiv-alence of triviality of monodromy of holomorphic motions and extensions ofholomorphic motions to continuous motions of the Riemann sphere. We alsostudy liftings of holomorphic maps into certain Teichm¨uller spaces. We usethis “lifting property” to prove that, under the condition of trivial monodromy,any holomorphic motion of a closed set in the Riemann sphere, over a hyper-bolic Riemann surface, can be extended to a holomorphic motion of the sphere,over the same parameter space. We conclude that this extension can be donein a conformally natural way.
Introduction
Throughout this paper, we shall use the following notations: C for the complexplane, b C = C ∪ {∞} for the Riemann sphere, and ∆ for the open unit disk { z ∈ C : | z | < } .The subject of holomorphic motions was introduced in the study of the dynamicsof rational maps; see [13]. Since its inception, an important topic has been thequestion of extending holomorphic motions. The papers [5] and [20] containedpartial results. Subsequently, Slodkowski showed that any holomorphic motion of aset in b C , over ∆ as the parameter space, can be extended to a holomorphic motionof b C over ∆; see [19] and [6]. The paper [9] used a group-equivariant version ofSlodkowski’s theorem to prove results in Teichm¨uller theory.The main purpose of our paper is to study necessary and sufficient topologicalconditions for extending holomorphic motions. We study monodromy of a holo-morphic motion φ of a finite set E in b C , defined over a connected complex Banachmanifold V . We show the equivalence of the triviality of this monodromy and theextendability of φ to a continuous motion of b C over V . We then show that, if V isa hyperbolic Riemann surface, and E is any set in b C , then, under the condition oftrivial monodromy, any holomorphic motion of E extends to a holomorphic motionof b C over V . The main technique is to study the liftings of holomorphic maps froma hyperbolic Riemann surface into Teichm¨uller spaces of b C with punctures.Our paper is organized as follows. In Section 1, we give all precise definitions anddiscuss the useful facts that will be necessary in the proofs of the main theorems Mathematics Subject Classification.
Primary 32G15, Secondary 30C62, 30F60, 30F99.
Key words and phrases.
Teichm¨uller spaces, Holomorphic maps, Universal holomorphicmotions.This material is based upon work supported by the National Science Foundation. The firstauthor is also partially supported by a collaboration grant from the Simons Foundation (grantnumber 523341) and PSC-CUNY awards and a grant from NSFC (grant number 11571122). of this paper. In Section 2, we give precise statements of the main theorems. InSections 3, 4, 5, and 6, we prove the main theorems.1.
Definitions and some facts
Definition 1.1.
Let V be a connected complex manifold with a basepoint x andlet E be any subset of b C . A holomorphic motion of E over V is a map φ : V × E → b C that has the following three properties:(1) φ ( x , z ) = z for all z in E ,(2) the map φ ( x, · ) : E → b C is injective for each x in V , and(3) the map φ ( · , z ) : V → b C is holomorphic for each z in E .We say that V is a parameter space of the holomorphic motion φ . We will alwaysassume that φ is a normalized holomorphic motion; i.e. 0, 1, and ∞ belong to E and are fixed points of the map φ ( x, · ) for every x in V . It is sometimes useful towrite φ ( x, z ) as φ x ( z ).If E is a proper subset of b E and φ : V × E → b C , b φ : V × b E → b C are twoholomorphic motions, we say that b φ extends φ if b φ ( x, z ) = φ ( x, z ) for all ( x, z ) in V × E . Remark . Let V and W be connected complex manifolds with basepoints, and f be a basepoint preserving holomorphic map of W into V . If φ is a holomorphicmotion of E over V its pullback by f is the holomorphic motion(1.1) f ∗ ( φ )( x, z ) = φ ( f ( x ) , z ) , ∀ ( x, z ) ∈ W × E, of E over W . Definition 1.3.
Let V be a connected complex manifold with a basepoint. Let G be a group of M¨obius transformations, let E ⊂ b C be G -invariant, which means, g ( E ) = E for each g in G . A holomorphic motion φ : V × E → b C is G -equivariant if for any x ∈ V, g ∈ G there is a M¨obius transformation, denoted by θ x ( g ), suchthat φ ( x, g ( z )) = ( θ x ( g ))( φ ( x, z ))for all z in E .It is well-known that if φ : V × E → b C is a holomorphic motion, where V is aconnected complex manifold with a basepoint x , then φ extends to a holomorphicmotion of the closure E , over V ; see [13] and [3]. Hence, throughout this paper, wewill assume that E is a closed set in b C (that contains the points 0, 1, and ∞ ).Recall that a homeomorphism of b C is called normalized if it fixes the points 0, 1,and ∞ . The blanket assumption that E is a closed set in b C containing the points0, 1, and ∞ holds. Definition 1.4.
Two normalized quasiconformal self-mappings f and g of b C aresaid to be E -equivalent if and only if f − ◦ g is isotopic to the identity rel E .The Teichm¨uller space T ( E ) is the set of all E -equivalence classes of normalizedquasiconformal self-mappings of b C . ONODROMY, LIFTING, AND HOLOMORPHIC MOTIONS 3
Let M ( C ) be the open unit ball of the complex Banach space L ∞ ( C ). Each µ in M ( C ) is the Beltrami coefficient of a unique normalized quasiconformal homeo-morphism w µ of b C onto itself. The basepoint of M ( C ) is the zero function.We define the quotient map P E : M ( C ) → T ( E )by setting P E ( µ ) equal to the E -equivalence class of w µ , written as [ w µ ] E . Clearly, P E maps the basepoint of M ( C ) to the basepoint of T ( E ).In his doctoral dissertation [12], G. Lieb proved that T ( E ) is a complex Banachmanifold such that the projection map P E : M ( C ) → T ( E ) is a holomorphic splitsubmersion; see [9] for the details. Remark . Let E be a finite set. Its complement Ω = b C \ E is the Riemann spherewith punctures at the points of E . Then, T ( E ) is biholomorphic to the classicalTeichm¨uller space T eich (Ω); see Example 3.1 in [14] for the proof. This canonicalidentification will be very important in our paper.
Proposition 1.6.
There is a continuous basepoint preserving map s from T ( E ) to M ( C ) such that P E ◦ s is the identity map on T ( E ) . See [9], [10] for all details.
Definition 1.7.
The map s from T ( E ) to M ( C ) is called the Douady-Earle section of P E for the Teichm¨uller space T ( E ). Remark . When E is finite, T ( E ) is canonically identified with the classicalTeichm¨uller space T eich ( b C \ E ), and hence s is the continuous section studied inLemma 5 in [7]. Definition 1.9.
The universal holomorphic motion Ψ E : T ( E ) × E → b C is definedas follows: Ψ E ( P E ( µ ) , z ) = w µ ( z ) for µ ∈ M ( C ) and z ∈ E. It is clear from the definition of P E that the map Ψ E is well-defined. It is aholomorphic motion because P E is a holomorphic split submersion and µ w µ ( z )is a holomorphic map from M ( C ) to b C for every fixed z in b C , by Theorem 11 in [2].This holomorphic motion is “universal” in the following sense: Theorem 1.10.
Let φ : V × E → b C be a holomorphic motion. If V is a simplyconnected complex Banach manifold with a basepoint, there is a unique basepointpreserving holomorphic map f : V → T ( E ) such that f ∗ (Ψ E ) = φ . For a proof see Section 14 in [14].
Remark . Let φ : V × E → b C be a holomorphic motion where V is a connectedcomplex Banach manifold with a basepoint x . Suppose there exists a basepointpreserving holomorphic map f : V → T ( E ) such that f ∗ (Ψ E ) = φ . Let e f : V → M ( C ) where e f = s ◦ f . By Proposition 1.6, e f is a basepoint preserving continuousmap. Then, for all ( x, z ) ∈ V × E , we have φ ( x, z ) = Ψ E ( f ( x ) , z ) = Ψ E ( P E ( s ( f ( x ))) , z ) = w s ( f ( x )) ( z ) = w e f ( x ) ( z ) . YUNPING JIANG AND SUDEB MITRA
Definition 1.12.
Let W be a path-connected Hausdorff space with a basepoint x .A (normalized) continuous motion of b C over W is a continuous map φ : W × b C → b C such that:(a) φ ( x , z ) = z for all z ∈ b C , and(b) for each x in W , the map φ ( x, · ) := φ x ( · ) is a homeomorphism of b C ontoitself that fixes the points 0, 1, and ∞ .In [15] it was shown that: Theorem 1.13.
Let φ : V × E → b C be a holomorphic motion where V is aconnected complex Banach manifold with a basepoint x . Then the following areequivalent: (i) There is a continuous motion e φ : V × b C → b C that extends φ . (ii) There exists a basepoint preserving holomorphic map F : V → T ( E ) suchthat F ∗ (Ψ E ) = φ . The following corollary will be useful in this paper; see [15].
Corollary 1.14.
If the holomorphic motion φ can be extended to a continuousmotion e φ , then e φ can be chosen so that: (i) the map e φ x : b C → b C is quasiconformal for each x in V , and (ii) its Beltrami coefficient µ x is a continuous function of x . Let w be a normalized quasiconformal self-mapping of b C , and let e E = w ( E ).By definition, the allowable map g from T ( e E ) to T ( E ) maps the e E -equivalenceclass of f to the E -equivalence class of f ◦ w for every normalized quasiconformalself-mapping f of b C . Lemma 1.15.
The allowable map g : T ( e E ) → T ( E ) is biholomorphic. If µ is theBeltrami coefficient of w , then g maps the basepoint of T ( e E ) to the point P E ( µ ) in T ( E ) . See § § Lemma 1.16.
Let B be a path-connected topological space. Let f and g be twocontinuous maps from B to T ( E ) , satisfying: (i) Ψ E ( f ( t ) , z ) = Ψ E ( g ( t ) , z ) for all z in E , and (ii) f ( t ) = g ( t ) for some t .Then, f ( t ) = g ( t ) for all t in B . See §
12 in [14] for the proof.If f ( t ) = [ w µ ] E and g ( t ) = [ w ν ] E , Condition (i) of the lemma means that w µ ( z ) = w ν ( z ) for all z in E .If E is a subset of the closed set b E and µ is in M ( C ), then the b E -equivalenceclass of w µ is contained in the E -equivalence class of w µ . Therefore, there is awell-defined ‘forgetful map’(1.2) p b E,E : T ( b E ) T ( E )such that P E = p b E,E ◦ P b E . It is easy to see that this is a basepoint preservingholomorphic split submersion. ONODROMY, LIFTING, AND HOLOMORPHIC MOTIONS 5
The following is a consequence of Lemma 1.16. Here, Ψ E is the universal holo-morphic motion of E and Ψ b E is the universal motion of b E . Proposition 1.17.
Let V be a connected complex Banach manifold with basepoint,and let f and g be basepoint preserving holomorphic maps from T ( E ) and T ( b E ) respectively. Then p b E,E ◦ g = f if and only if g ∗ (Ψ b E ) extends f ∗ (Ψ E ) . See §
13 in [14] for the proof. We say that the holomorphic map g lifts theholomorphic map f .We now discuss the concept of monodromy of a holomorphic motion. We closelyfollow the discussion in § φ : V × E → b C be a holomorphic motion, where V is a connected complex Banach manifold with a basepoint x . Let π : e V → V bea holomorphic universal covering, with the group of deck transformations Γ. Wechoose a point e x in e V such that π ( e x ) = x . Let π ( V, x ) denote the fundamentalgroup of V with basepoint x .Let Φ = π ∗ ( φ ). Then, Φ : e V × E → b C is a holomorphic motion of E over e V with e x as the basepoint. By Remark 1.11, there exists a basepoint preservingcontinuous map e f : e V → M ( C ) such thatΦ( x, z ) = w e f ( x ) ( z )for each x ∈ e V and each z ∈ E .For each z ∈ E and for each γ ∈ Γ, we have w e f ◦ γ ( e x ) ( z ) = Φ( γ ( e x ) , z ) = φ ( π ◦ γ ( e x ) , z ) = φ ( x , z ) = z. Therefore, w e f ◦ γ ( e x ) keeps every point of E fixed. Lemma 1.18.
The homotopy class of w e f ◦ γ ( e x ) relative to E does not depend onthe choice of the continuous map e f . See Lemma 2.12 in [4].We now assume that E is a finite set containing n points where n ≥
4; as usual,0, 1, and ∞ are in E . Let φ : V × E → b C be a holomorphic motion. The map w e f ◦ γ ( e x ) is quasiconformal selfmap of the hyperbolic Riemann surface X E := b C \ E .Therefore, it represents a mapping class of X E , and by Lemma 1.18, we have ahomomorphism ρ φ : π ( V, x ) → Mod(0 , n ) given by ρ φ ( c ) = [ w e f ◦ γ ( e x ) ]where Mod(0 , n ) is the mapping class group of the n -times punctured sphere, γ ∈ Γis the element corresponding to c ∈ π ( V, x ), and [ w ] denotes the mapping classgroup of X E for w . Definition 1.19.
We call the homomorphism ρ φ the monodromy of φ the holo-morphic motion φ of the finite set E. The monodromy is called trivial if it mapsevery element of π ( V, x ) to the identity of Mod(0 , n ).2. Statements of the main theorems
In this Section, we give the precise statements of the main theorems of our paper.
Theorem A.
Let φ : V × E → b C be a holomorphic motion of a finite set E ,containing the points , , and ∞ , where V is a connected complex Banach manifoldwith basepoint x . The following are equivalent: YUNPING JIANG AND SUDEB MITRA (i)
There exists a continuous motion e φ : V × b C → b C , such that e φ extends φ . (ii) The monodromy ρ φ is trivial. In the next theorem, let b E = E ∪{ ζ } , where, E is a finite set containing 0, 1, and ∞ , and ζ ∈ C \ E . Let V be a connected complex Banach manifold with basepoint x . Theorem B.
Suppose every holomorphic map from V into T ( E ) lifts to a holo-morphic map from V into T ( b E ) . Then, if φ : V × E → b C is a holomorphic motionthat has trivial monodromy, there exists a holomorphic motion b φ : V × b E → b C suchthat b φ extends φ and also has trivial monodromy. In the next three theorems, X is a hyperbolic Riemann surface with a basepoint x , and E is a closed set in b C containing the points 0 ,
1, and ∞ .Let E n = { , , ∞ , ξ , · · · , ξ n } where n ≥
1, and E n +1 = E n ∪ { ξ n +1 } where ξ n +1 ∈ b C \ E n . Let p : T ( E n +1 ) → T ( E n ) denote the forgetful map in (1.2). Let φ n : X × E n → b C be a holomorphic motion, that has trivial monodromy. Wewill see in the proof of Theorem A that there exists a unique basepoint preservingholomorphic map f n : X → T ( E n ) such that f ∗ n (Ψ E n ) = φ n . The following theoremis a key result in our paper. Theorem C.
Let φ n : X × E n → b C be a holomorphic motion. If the monodromyof φ n is trivial, there exists a basepoint preserving holomorphic map f n +1 : X → T ( E n +1 ) such that p ◦ f n +1 = f n . The following corollary is an immediate consequence. Here Ψ E n +1 : T ( E n +1 ) × E n +1 → b C is the universal holomorphicmotion of E n +1 . Corollary 2.1.
Let φ n +1 := f ∗ n +1 (Ψ E n +1 ) . Then φ n extends φ n +1 and has thetrivial monodromy. Theorem D.
Let φ : X × E → b C be a holomorphic motion such that φ restrictedto X × E ′ has trivial monodromy, or extends to a continuous motion of b C (over X ), where E ′ is any finite subset of E , containing the points , , and ∞ . Then,there exists a holomorphic motion b φ : X × b C → b C such that b φ extends φ .Remark . H. Shiga [18] has recently announced a completely different approachto a part of this theorem. Our methods are totally independent and more direct.The crucial point in our approach is the lifting property as given in Theorem C.3.
Proof of Theorem A
Let π : e V → V be a holomorphic universal covering with the group Γ of decktransformations, so that, V = e V /
Γ, and π ( e x ) = x .Suppose φ can be extended to a continuous motion e φ of b C over V . Then, byCorollary 1.14, there exists a continuous map f : V → M ( C ) such that e φ ( x, z ) = w f ( x ) ( z ) for all ( x, z ) ∈ V × b C . Let e f = f ◦ π . Then, for any c ∈ π ( V, x ) withcorresponding γ ∈ Γ, we have ρ φ ( c ) = [ w e f ◦ γ ( e x ) ] = [ w f ◦ π ◦ γ ( e x ) ] = [ w f ( x ) ] = [ Id ] . This shows that the monodromy ρ φ is trivial. ONODROMY, LIFTING, AND HOLOMORPHIC MOTIONS 7
Let φ : V × E → b C be a holomorphic motion with trivial monodromy. Let φ e V := π ∗ ( φ ) be the holomorphic motion of E over e V . By Theorem 1.10, thereexists a unique basepoint preserving holomorphic map f e V : e V → T ( E ), such that φ e V = f ∗ e V (Ψ E ). For any element γ ∈ Γ, we also have f e V ◦ γ : e V → T ( E ). Note that( f e V ◦ γ ) ∗ (Ψ E )( x, z ) = Ψ E (( f e V ◦ γ ) x, z ) = φ e V ( γ ( x ) , z ) = φ ( π ( γ ( x )) , z )= φ ( π ( x ) , z ) = φ e V ( x, z ) = ( f e V ) ∗ (Ψ E )( x, z ) . By the triviality of the monodromy, we have f e V ◦ γ ( x ) = f e V ( x ) for all γ ∈ Γ.Lemma 1.16 implies that f e V ◦ γ = f e V for all γ ∈ Γ. Thus, f e V defines a uniquebasepoint preserving holomorphic map f : V → T ( E ) such that φ = f ∗ (Ψ E ). Itthen follows from Theorem 1.13 that there exists a continuous motion of b C over V that extends φ . This completes the proof. (cid:3) Proof of Theorem B
Let φ : V × E → b C be a holomorphic motion such that it has trivial monodromy.By the proof of Theorem A, there exists a basepoint preserving holomorphic map f : V → T ( E ) such that φ = f ∗ (Ψ E ). Let p : T ( b E ) → T ( E ) denote the forgetfulmap defined in (1.2). By hypothesis, there exists a basepoint preserving holomor-phic map b f : V → T ( b E ) such that p ◦ b f = f . Let b φ := b f ∗ (Ψ b E ). By Proposition 1.17, b φ extends φ . Note that, for x ∈ V , and z ∈ b E , we have b φ ( x, z ) = w α ( x ) ( z ) where α = s b E ◦ b f and s b E is the continuous section of the projection P b E : M ( C ) → T ( b E );see Remark 1.8.Let π : e V → V be the holomorphic universal cover, and π ( e x ) = x . Then, π ∗ ( b φ ) : e V × b E → b C is a holomorphic motion. Since e V is simply connected, thereexists a basepoint preserving continuous map β : e V → M ( C ) such that β ∗ (Ψ b E ) = π ∗ ( b φ ) (see Remark 1.11). That implies π ∗ ( b φ )( x, z ) = w β ( x ) ( z ). Recall that themonodromy ρ : π ( V ) → Mod(0 , n + 1) is defined by ρ ( c ) = [ w β ◦ γ ( x ) ]for any c ∈ π ( V, x ) with the corresponding γ ∈ Γ. Furthermore, it is independentof the choice of β . In particular, if we choose β = α ◦ π , we see that ρ ( c ) = [ w α ◦ π ◦ γ ( x ) ] = [ w α ( x )] = [ Id ] . Note that π ◦ γ ( x ) = x . This implies that ρ is trivial. (cid:3) Proofs of Theorem C and Corollary 2.1
We recall the following result, due to S. Nag, that will be fundamental in ourpaper; see [16].
Theorem 5.1.
Given n > , choose a point ( ζ , · · · , ζ n ) in the domain Y n = { ( z , · · · , z n ) } ∈ C n : z i = z j for i = j and z i = 0 , for all i = 1 , · · · , n } and let E n = { , , ∞ , ζ , · · · , ζ n } . Then, the map p n : T ( E n ) → Y n defined by p n ([ w µ ] E n ) = ( w µ ( ζ ) , · · · , w µ ( ζ n )) for all µ ∈ M ( C ) is a holomorphic universal covering. YUNPING JIANG AND SUDEB MITRA
Let E n = { , , ∞ , ζ , · · · , ζ n } and E n +1 = E ∪ { ζ n +1 } where ζ n +1 ∈ b C \ E n . Wehave also a holomorphic universal covering p n +1 : T ( E n +1 ) → Y n +1 where Y n +1 = { ( z , ··· , z n +1 ) ∈ C n +1 : z i = z j for i = j and z i = 0 , i = 1 , ··· , n +1 } Let p : T ( E n +1 ) → T ( E n ) denote the forgetful map defined in (1.2).We need some preliminaries. The reader is referred to Lemmas 3.1–3.5 in [11]for all details. Let C ( C ) denote the complex Banach space of bounded, continuousfunctions φ on C with the norm k φ k = sup z ∈ C | φ ( z ) | . In § K : C ( C ) → C ( C ) . By Lemma 3.2 in [11], there exists a constant C > kKk ≤ C for all f ∈ C ( C ) . For ζ n +1 , let B = { f ∈ C ( C ) : k f k ≤ | ζ n +1 | + C } . It is a bounded convex subset in C ( C ). The continuous compact operator ζ n +1 + K maps B into itself. By Schauder fixed point theorem (see Theorem 2A on page 56of [21]; also page 557 of [11]), ζ n +1 + K has a fixed point in B . This says that wecan find a g n +1 ∈ B such that g n +1 ( z ) = ζ n +1 + K g n +1 ( z ) for all z ∈ C . The reader is referred to § g n +1 ( z ) is the unique fixed point of the operator ζ n +1 + K .Suppose φ n : X × E n → b C is a holomorphic motion, that has trivial monodromy.By Theorem A, it can be extended to a continuous motion e φ : X × b C → b C . ByTheorem 1.13, that there exists a basepoint preserving holomorphic map f n : X → T ( E n ) such that f ∗ n (Ψ E ) = φ n . Proof of Theorem C.
Let p n : T ( E n ) → Y n and p n +1 : T ( E n +1 ) → Y n +1 be the two holomorphic coverings in Theorem 5.1.Since X is a hyperbolic Riemann surface, its universal cover space is the openunit disk ∆. Let π : ∆ → X be the universal cover. Let Γ be the group of decktransformations such that X = ∆ / Γ.Define the holomorphic map f ∆ ,n = f n ◦ π : ∆ → T ( E n )and for any given γ ∈ Γ, consider the holomorphic map f ∆ ,n ◦ γ : ∆ → T ( E n ) . This give us two holomorphic maps F n = p n ◦ f ∆ ,n : ∆ → Y n and F n ◦ γ = p n ◦ ( f ∆ ,n ◦ γ ) : ∆ → Y n . Let us write F n = ( h , · · · , h n ) and F n ◦ γ = ( h ◦ γ, · · · , h n ◦ γ ) . ONODROMY, LIFTING, AND HOLOMORPHIC MOTIONS 9
Each h i (as well as h i ◦ γ ) is holomorphic in ∆. In § g i (as well as a map g i ◦ γ ) by using h i (as well as h i ◦ γ ) which is holomorphicoutside ∆ and continuous on C . By using g i for all 1 ≤ i ≤ n , we constructed acontinuous compact operator K = K ( F n ) : C ( C ) → C ( C )in § g i ◦ γ for all 1 ≤ i ≤ n , we have acontinuous compact operator K γ = K ( F n ◦ γ ) : C ( C ) → C ( C ) . The main point in § g n +1 for ξ n +1 + K and the unique fixed point g n +1 ,γ for ξ n +1 + K γ . That is,(5.1) g n +1 ( z ) = ξ n +1 + K g n +1 ( z )and(5.2) g n +1 ,γ ( z ) = ξ n +1 + K γ g n +1 ,γ ( z ) . We also have(5.3) g n +1 ◦ γ ( z ) = ξ n +1 + K γ ( g n +1 ◦ γ ( z )) . From g n +1 (as well as g n +1 ,γ ), which is holomorphic outside ∆ and continuousin C , we get a holomorphic map h i (as well as h n +1 ,γ ) in ∆, for i = 1 , · · · , n . Thenwe form two holomorphic maps(5.4) F n +1 = ( h , · · · , h n , h n +1 ) : ∆ → Y n +1 and(5.5) F n +1 ,γ = ( h ◦ γ, · · · , h n ◦ γ, h n +1 ,γ ) : ∆ → Y n +1 . Since ∆ is simply connected and since p n +1 : T ( E n +1 ) → Y n +1 is the universalcover, we can lift F n +1 and F n +1 ,γ to two two holomorphic maps f ∆ ,n +1 : ∆ → T ( E n +1 ) and f ∆ ,n +1 ,γ : ∆ → T ( E n +1 )such that p n +1 ◦ f ∆ ,n +1 = F n +1 and and p n +1 ◦ f ∆ ,n +1 ,γ = F n +1 ,γ .Under the assumption of the trivial monodromy, we know that f ∆ ,n = f ∆ ,n ◦ γ (see the proof of Theorem A). That is, h i = h i ◦ γ and g i = g i ◦ γ for all 1 ≤ i ≤ n .Thus K γ = K . Since the fixed point is unique, we get g n +1 ◦ γ = g n +1 ,γ = g n +1 . This implies that h n +1 ◦ γ = h n +1 and F n +1 ◦ γ = F n +1 and f ∆ ,n +1 ◦ γ = f ∆ ,n +1 .Since this holds for all γ ∈ Γ, the map f ∆ ,n +1 defines a holomorphic map f n +1 : X → T ( E n +1 ) such that p ◦ f n +1 = f n . Therefore, f n +1 is a lift of f n . Thiscompletes the proof. (cid:3) Proof of Corollary 2.1.
This follows at once from Theorem C, Proposition 1.17, andTheorem B. (cid:3) Proof of Theorem D
Let E be an arbitrary closed set in b C such that 0, 1, and ∞ are in E , and let X be a hyperbolic Riemann surface with a basepoint x .We will need the following result; see Theorem C in [4]. Theorem 6.1.
Let φ : X × E → b C be a holomorphic motion. Suppose therestriction of φ to X × E ′ extends to a holomorphic motion e φ : X × b C → b C ,whenever { , , ∞} ⊂ E ′ ⊂ E and E ′ is finite. Then φ can be extended to aholomorphic motion of b C over X .Proof of Theorem D. Let φ : X × E → b C be a holomorphic motion with the prop-erty that φ restricted to X × E ′ has trivial monodromy, where E ′ is any finite subsetof E , containing 0, 1, and ∞ .Fix some E ′ ⊂ E such that E ′ contains the points 0 , , ∞ and φ restricted to X × E ′ has trivial mondromy. Let E = E ′ . Consider E = E ∪ { ζ } for ζ E .Inductively, consider E n +1 = E n ∪ { ζ n +1 } for ζ n +1 E n for all n ≥
0. Then weeventually get a countable set E ∞ = ∪ ∞ n =0 E n in b C . We can assume that E ∞ isdense in b C .Define φ := φ restricted to X × E . Using Theorem C and Corollary 2.1inductively, we know that the holomorphic motion φ n : X × E n → b C can beextended to a holomorphic motion φ n +1 : X × E n +1 → b C with trivial monodromyfor all n ≥
0. Thus we can extend φ to a holomorphic φ ∞ : X × E ∞ → b C . Since E ∞ is dense in b C , it can be further extended to a holomorphic motion e φ : X × b C → b C .The conclusion now follows from Theorem 6.1. (cid:3) Remark . Let G be a group of M¨obius transformations, such that the closedset E is G -invariant. Let φ : X × E → b C be a G -equivariant holomorphic motion;see Definition 1.3. If φ has the property that φ restricted to X × E ′ has trivialmonodromy, where E ′ is any subset of E, containing the points 0 ,
1, and ∞ , then φ can be extended to a holomorphic motion e φ : X × b C → b C which is also G -equivariant.The proof is very similar to the proof of Theorem 1 in [8] or of Theorem B in [3]. References [1] L. V. Ahlfors,
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Department of Mathematics, Queens College of the City University of NewYork, USA, and Department of Mathematics, The Graduate Center, CUNY, USA
E-mail address : [email protected] (Mitra) Department of Mathematics, Queens College of the City University of NewYork, USA, and Department of Mathematics, The Graduate Center, CUNY, USA
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