Fiber Structure and Local Coordinates for the Teichmueller Space of a Bordered Riemann Surface
aa r X i v : . [ m a t h . C V ] J un FIBER STRUCTURE AND LOCAL COORDINATES FOR THETEICHM ¨ULLER SPACE OF A BORDERED RIEMANN SURFACE
DAVID RADNELL AND ERIC SCHIPPERS
Abstract.
We show that the infinite-dimensional Teichm¨uller space of a Riemann surfacewhose boundary consists of n closed curves is a holomorphic fiber space over the Teichm¨ullerspace of n -punctured surfaces. Each fiber is a complex Banach manifold modeled on a two-dimensional extension of the universal Teichm¨uller space. The local model of the fiber,together with the coordinates from internal Schiffer variation, provides new holomorphiclocal coordinates for the infinite-dimensional Teichm¨uller space. Introduction
Statement of results.
Let D = { z ∈ C : | z | < } , D = D \ { } , and D = { z ∈ C :0 < | z | ≤ } . Definition 1.1.
We say that Σ B is a bordered Riemann surface of type ( g, n ) if 1) its bound-ary consists of n ordered closed curves homeomorphic to S and 2) it is biholomorphicallyequivalent to a compact Riemann surface of genus g with n simply-connected non-overlappingregions, biholomorphic to D , removed. We say that Σ P is a punctured Riemann surface oftype ( g, n ) if it is biholomorphically equivalent to a compact Riemann surface with n points p , . . . , p n removed.Note that it is assumed that bordered Riemann surfaces have no punctures, and thatthe boundary components of punctured Riemann surfaces consist only of single points. Wewill denote bordered Riemann surfaces of type ( g, n ) with a superscript B and puncturedRiemann surfaces with a superscript P . Remark . One can also view a punctured Riemann surface Σ P as a compact Riemannsurface with n distinguished points. Remark . Throughout the paper, we consider only Riemann surfaces with no non-trivialautomorphisms that are homotopic to the identity. That is, we do not consider the specialcases where 2 g − n ≤ B of type ( g, n ), we can obtain a punctured Riemannsurface Σ P in the following way. For details see [12, Section 3]. Denote the boundary com-ponents by ∂ i Σ B . Let τ = ( τ , . . . , τ n ) where each τ i : ∂ D → ∂ i Σ B is a fixed quasisymmetricmapping. For the purposes of this paper, we say that such a τ i is quasisymmetric if τ i extendsto a quasiconformal map of { z : 1 < | z | < r } into a doubly-connected neighborhood of ∂ i Σ B [12, Definition 2.12]. We sew on n copies of the punctured unit disc D , denoted D ,i , to Σ B Date : November 23, 2018.2000
Mathematics Subject Classification.
Primary 30F60, 58B12 ; Secondary 81T40.
Key words and phrases.
Teichm¨uller spaces, quasiconformal mappings, sewing, rigged Riemann surfaces,conformal field theory. s follows. Consider the disjoint union of Σ B and D , ⊔ · · · ⊔ D ,n . Identifying boundarypoints using τ , the result is a compact surface Σ P with n punctures p i corresponding to eachpuncture in D . That is, let Σ P = (Σ B ⊔ D , · · · ⊔ D ,n ) / ∼ where p ∼ q if p ∈ ∂ i Σ B and q ∈ ∂ D i , and p = τ i ( q ). By [12, Theorem 3.3] Σ P has a uniquecomplex structure which is compatible with that of both Σ B and D ,i for all i . (Note thatthe punctures of Σ P are also ordered.) If Σ P is obtained from Σ B in this way we will saythat Σ P is obtained by “sewing caps on Σ B via τ ” and we writeΣ P = Σ B τ D n . The parametrizations τ i can be extended to maps ˜ τ i : D → Σ P to the caps of Σ P by(1.1) ˜ τ i ( x ) = ( τ i ( x ) for x ∈ ∂ D x for x ∈ D . These maps ˜ τ i have quasiconformal extensions to a neighborhood of D .The aim of this paper is to show that the infinite-dimensional Teichm¨uller space T (Σ B )of Σ B is a holomorphic fiber space over the finite-dimensional Teichm¨uller space T (Σ P ) ofΣ P . The fibers can be explicitly described as follows. Definition 1.4 (of O qc (Σ P ∗ )) . Let Σ P ∗ be a compact Riemann surface of type ( g, n ), withdistinguished points p , . . . , p n . Define O qc (Σ P ∗ ) to be the set of n -tuples ( φ , . . . , φ n ) where φ i : D → Σ P ∗ are maps with the following properties:(1) φ i (0) = p i (2) φ i is conformal on D (3) φ i has a quasiconformal extension to a neighborhood of D (4) φ i ( D ) ∩ φ j ( D ) = ∅ whenever i = j .It is convenient to single out the following case. Definition 1.5.
Let O qc denote the set of holomorphic univalent functions f : D → C withquasiconformal extensions to C satisfying the normalization f (0) = 0.In [13] it was shown that O qc possesses a natural complex structure related to that ofthe universal Teichm¨uller space, and that O qc (Σ P ∗ ) is a complex Banach manifold which islocally modeled on O n qc = O qc × · · · × O qc .Let Σ P be the punctured Riemann surface obtained from Σ B by sewing on caps via τ ,and let Σ P ∗ be a marked Riemann surface representing an element of the Teichm¨uller space T (Σ P ) of Σ P . We show that the fiber in T (Σ B ) over this element of T (Σ P ), modulo theaction of Dehn twists around curves homotopic to boundary curves, is in a natural one-to-one correspondence with O qc (Σ P ∗ ).We summarize the main results. Precise definitions and careful statements of the theoremscan be found in subsequent sections. Summary of results (1) If Σ B is a bordered Riemann surface of type ( g, n ) (with 2 g − n >
0) then theTeichm¨uller space T (Σ B ) is a complex fiber space over T (Σ P ).
2) Let F B ([Σ P , f, Σ P ∗ ]) be the fiber over [Σ P , f, Σ P ∗ ] ∈ T (Σ P ). Let DB be the subgroup ofthe mapping class group of Σ B corresponding to Dehn twists around curves homotopicto boundary curves. Then F B ([Σ P , f, Σ P ∗ ]) / DB is biholomorphic to O qc (Σ P ∗ ). Inparticular F B ([Σ P , f, Σ P ∗ ]) is locally biholomorphic to the function space O n qc .(3) Schiffer variation coordinates on T (Σ P ) together with open subsets of O qc (Σ P ∗ ) givelocal holomorphic coordinates for T (Σ B ).Some recent results are a key part of the formulation and proofs of these theorems. Firstis the authors’ construction in [12] of a complex structure on the “rigged moduli space”and its explicit relation to the Teichm¨uller space T (Σ B ). This construction uses ideas fromtwo-dimensional conformal field theory in an essential way. A further crucial result from[12] is that the operation of sewing two Riemann surfaces with a quasisymmetric boundaryidentification is holomorphic in Teichm¨uller space. The complex structure on O qc (Σ P ∗ ) wasconstructed in [13].A further tool is Gardiner’s construction of coordinates on the Teichm¨uller space of asurface of finite type using Schiffer variation [2]. We use his method in order to constructa section of the projection of T (Σ B ) onto T (Σ P ). In the first author’s thesis [11], Schiffervariation was used in an analogous way, although for the different purpose of defining acomplex structure on the analytically rigged Teichm¨uller space.The results above are an application of Segal’s [14] formulation of conformal field theoryto Teichm¨uller theory. 2. fiber structure of T (Σ B )2.1. Teichm¨uller spaces.
We now define the relevant Teichm¨uller spaces, to fix notation.Let Σ B be a bordered Riemann surface of of type ( g, n ). Consider the set of triples { (Σ B , f , Σ B ) } where Σ B is a fixed Riemann surface, Σ B is another Riemann surface and f :Σ B → Σ B is a quasiconformal map (the “marking”). We say that (Σ B , f , Σ B ) ∼ (Σ B , f , Σ B )if there exists a biholomorphism σ : Σ B → Σ B such that f − ◦ σ ◦ f is homotopic to theidentity “rel boundary”, i.e. in such a way that the restriction of f − ◦ σ ◦ f to the boundaryis the identity throughout the homotopy. Definition 2.1.
The Teichm¨uller space of Σ B is T (Σ B ) = { (Σ B , f , Σ B ) } / ∼ . We denote the equivalence classes by [Σ B , f , Σ B ]. The case T ( D ∗ ), where D ∗ = { z : | z | > } ∪ {∞} , is the universal Teichm¨uller space.It is well known that T (Σ B ) is a complex Banach manifold with complex structure com-patible with the space L ∞− , (Σ B ) of Beltrami differentials µ on Σ B in the following sense.Let L ∞− , (Σ B ) denote the unit ball in the space of Beltrami differentials. LetΦ : L ∞− , (Σ B ) → T (Σ B )be the fundamental projection, given by taking a Beltrami differential µ to [Σ B , f µ , Σ µ ] where f µ is a quasiconformal map of dilatation µ . Φ is well-defined and Theorem 2.2 ([7],[10]) . Φ is holomorphic and possesses local holomorphic sections. Similarly, we can define the Teichm¨uller space of a punctured Riemann surface. Let Σ P bea punctured Riemann surface of type ( g, n ). Consider the set of triples (Σ P , f , Σ P ) where Σ P s a fixed punctured Riemann surface, and f : Σ P → Σ P is a quasiconformal map onto theRiemann surface Σ P . We say that (Σ P , f , Σ P ) ∼ (Σ P , f , Σ P ) if there is a biholomorphism σ : Σ P → Σ P such that f − ◦ σ ◦ f is homotopic to the identity. Definition 2.3.
The Teichm¨uller space of Σ P is defined by T (Σ P ) = { (Σ P , f , Σ P ) } / ∼ . We denote the equivalence classes by [Σ P , f , Σ P ]. The Teichm¨uller space T (Σ P ) is acomplex manifold of dimension 3 g − n . Remark . Observing that quasiconformal and holomorphic maps must extend to thepunctures, one obtains an alternate description of punctured Riemann surfaces and theirTeichm¨uller spaces. One regards a punctured Riemann surface as a compact surface withdistinguished points, and the definition of T (Σ P ) is altered to require that the quasiconformalmappings take punctures to punctures, and the homotopy is “rel punctures”, that is itpreserves the punctures throughout. Remark . Of course, the Teichm¨uller spaces of punctured and bordered Riemann surfacesare special cases of the general definition of the Teichm¨uller space of any Riemann surfacecovered by the disc.As in [12, section 2.1] we introduce a certain subgroup of the mapping class group. Thepure mapping class group of Σ B is the group of homotopy classes of quasiconformal self-mappings of Σ B which preserve the ordering of the boundary components. Let PModI(Σ B )be the subgroup of the mapping class group consisting of equivalence classes of mappingsthat are the identity on the boundary ∂ Σ B . The group PModI(Σ B ) is finitely generated byDehn twists. Definition 2.6.
Let DB(Σ B ) be the subgroup of PModI(Σ B ) generated by the equivalenceclasses of mappings which are Dehn twists around curves that are homotopic to boundarycurves. We write DB when the surface is clear from context.If Σ B is not an annulus or a disk then DB(Σ B ) is isomorphic to Z n and is in the center ofPModI(Σ B ).The mapping class group acts on T (Σ B ) by [ ρ ] · [Σ B , f, Σ B ] = [Σ B , f ◦ ρ, Σ B ]. Proposition 2.7 ([12, Lemmas 5.1 and 5.2]) . The group
PModI(Σ B ) , and hence its subgroup DB(Σ B ) , acts properly discontinuously and fixed-point freely by biholomorphisms on T (Σ B ) . Description of the fibration.
In this section we describe the fibers. This requires theuse of a “rigged Teichm¨uller space”, a concept motivated by conformal field theory, whichwe will define in the next paragraph. Details and proofs of the statements in this sectionwere given in [12].
Definition 2.8.
Let Σ P be a type ( g, n ) punctured Riemann surface. Consider the set ofquadruples (Σ P , f , Σ P , φ ) where Σ P is a Riemann surface, f : Σ P → Σ P is a quasiconformalmarking map as in the definition of T (Σ P ), and φ ∈ O qc (Σ P ). Define the equivalence relation(Σ P , f , Σ P , φ ) ∼ (Σ P , f , Σ P , φ ) if and only if there exists a biholomorphism σ : Σ P → Σ P which preserves the punctures and their ordering such that σ ◦ φ = φ on ∂ D (and henceon D ) and f − ◦ σ ◦ f is homotopic to the identity. We then define e T P (Σ P ) = { (Σ P , f , Σ P , φ ) } / ∼ . he rigged Teichm¨uller space e T P (Σ P ) possesses a complex structure obtained in the fol-lowing way. First, we have a bijection e T P (Σ P ) ∼ = T (Σ B ) / DBwhere DB and its action are defined in Section 2.1. By Proposition 2.7, e T P (Σ P ) inherits acomplex structure from T (Σ B ) and the bijection is in fact a biholomorphism [12, Theorem5.7 parts (3) and (4)].Next define the map P : T (Σ B ) → e T P (Σ P ) from [12, Section 5.4] in the following way(note that there, the map is denoted P DB ). Fix a base welding τ : ∂ D n → ∂ Σ B and a baseRiemann surface Σ P = Σ B τ D n . Given [Σ B , h, Σ B ] ∈ T (Σ B ), let Σ P = Σ B h ◦ τ D n , anddefine ˜ h : Σ P → Σ P by(2.1) ˜ h = ( h on Σ B id on D n , Define ˜ τ : D n → Σ P by (1.1), and set(2.2) P ([Σ B , h, Σ B ]) = [Σ P , ˜ h, Σ P , ˜ h ◦ ˜ τ ] . This map satisfies P ( p ) = P ( q ) if and only if p is equivalent to q under the action of DB.Furthermore P is a holomorphic map which is locally a biholomorphism in the sense that forany point in w ∈ T (Σ B ) there’s an open set U containing w on which P is a biholomorphismonto an open subset of e T P (Σ P ). Remark . The biholomorphism between T B (Σ B ) /DB and e T P (Σ P ) is given explicitly by G : T B (Σ B ) /DB → e T P (Σ P )[Σ B , h, Σ B ]
7→ P ([Σ B , h, Σ B ])That G is well-defined and a biholomorphism was established in [12]. It is not possible towrite the inverse explicitly, although we can say the following: G − ([Σ P , f, Σ P ∗ , φ ]) = [Σ B , f φ , Σ P ∗ \ φ ( D n )]where it is understood that the right hand side is a representative of a point in T B (Σ B ) /DB ,and f φ is a quasiconformal map satisfying (1) f φ is homotopic to f , and (2) h ◦ τ = φ on ∂ D . Details are found in [12, Theorems 5.5 and 5.6]. Remark . Similarly, although P has a local holomorphic inverse, one cannot be com-pletely explicit about it. Many of the difficulties in this paper can be traced to this fact.However, we are able to write the restriction of the inverse to specified holomorphic curves.The local invertibility of P was established with the help of an existence theorem forquasiconformal maps, which is ultimately based on the λ -lemma. See [12, Sections 4 and5.4].Next we define two fiber projections. Firstly, let F : e T P (Σ P ) → T (Σ P )(2.3) [Σ P , f, Σ P , φ ] [Σ P , f, Σ P ] . ote that in [12] we call this map F T . Secondly, we have the sewing map C : T (Σ B ) −→ T (Σ P )(2.4) [Σ B , h, Σ B ] [Σ B τ D n , ˜ h, Σ B h ◦ τ D n ] . It was proved in [12, Section 6] that
F ◦ P = C and C is a holomorphic map. Since P is alocal biholomorphism it follows immediately that F is a holomorphic map.We define the fibers by F B ([Σ P , f, Σ P ]) = C − ([Σ P , f, Σ P ]) . and F P ([Σ P , f, Σ P ]) = F − ([Σ P , f, Σ P ]) . We will often denote a particular fiber by F B or F P , if there is no fear of confusion. Theaction of DB preserves each fiber F B since F ◦ P = C and thus we can conclude that F P = G ( F B / DB).
Remark . C and F , and hence the fibrations themselves, depend on the choice of τ .For the convenience of the reader we summarize this section with a theorem. Theorem 2.12. (1) e T P (Σ P ) possesses a complex structure, (2) P is holomorphic, and for every point w ∈ T (Σ B ) , there is an open neighborhood U of w such that P is a biholomorphism onto its open image. (3) F ◦ P = C . (4) F and C are holomorphic and onto. Fibers are complex submanifolds of T (Σ B ) . In this section we demonstrate that thefibers are complex submanifolds of T (Σ B ). This requires the construction of local sections.The construction relies on a method of Gardiner [2] (see also Nag [10]), who used Schiffervariation to construct local coordinates on Teichm¨uller spaces of punctured Riemann surfacesof finite type.We outline some necessary facts about complex submanifolds of Banach spaces. Thesecan be found in for example [10, Section 1.6.2], and also [5] in the differentiable setting.Let E and E be Banach spaces, and Y a complex Banach manifold. Let U and U beopen subsets of E and E respectively. We say that g : U × U → Y is a projection, if thereis a map h : U → Y such that g = h ◦ pr where pr : U × U → U is the projection ontothe first component. Definition 2.13.
Let X and Y be complex Banach manifolds. f : X → Y is a holomorphicsubmersion if f is holomorphic, for all x ∈ X there is a chart ( U, φ ) on X with x ∈ U andopen sets U ⊂ E and U ⊂ E of Banach spaces E and E , such that (1) φ : U → U × U is a biholomorphism and (2) there is a chart ( V, ψ ) with f ( x ) ∈ V , such that ψ ◦ f ◦ φ − isa projection.A holomorphic fiber space is defined as follows. Definition 2.14.
A holomorphic fiber space is a pair of complex Banach manifolds (
X, Y )together with a holomorphic, submersive and surjective map π : X → Y . t follows immediately from the definition of a submersion that the fibers of a submersionare complex submanifolds. Lemma 2.15.
A holomorphic submersion F : X → Y is an open mapping and the fibers F − ( y ) are complex submanifolds of X . We will use the following characterization of submersions.
Lemma 2.16.
A holomorphic mapping F : X → Y between Banach spaces is submersive ifand only if it possesses local holomorphic sections passing through every point x ∈ X . We now briefly describe Schiffer variation of Riemann surfaces. For details see [2] or [10,Section 4.3]. Let Σ be either a punctured Riemann surface of type ( g, n ) or a borderedRiemann surface of type ( g, n ). Let (
V, ζ ) be a holomorphic chart on Σ such that D ⊂ ζ ( V ),and let U = ζ − ( D ) be a parametric disk on Σ.For ǫ in a sufficiently small disk centered at 0 ∈ C , the map v ǫ : ∂ D → C given by v ǫ ( z ) = z + ǫ/z is a biholomorphism on some neighborhood of ∂ D . Let D ǫ denote the regionbounded by v ǫ ( ∂ D ). We obtain a new Riemann surface Σ ǫ as follows:Σ ǫ = (Σ \ U ) ⊔ D ǫ / ∼ where x ∈ ∂U and x ′ ∈ ∂D ǫ are equivalent, x ∼ x ′ , if x ′ = v ǫ ◦ ζ ( x ). The complex structureon Σ ǫ is compatible with D ǫ and Σ \ U .The quasiconformal map w ǫ : D → D ǫ given by w ǫ ( z ) = z + ǫ ¯ z has complex dilatation ǫ .Let ν ǫ : Σ → Σ ǫ be defined by ν ǫ ( x ) = ( x if x ∈ Σ \ Uw ǫ ◦ ζ ( x ) if x ∈ U. This map is quasiconformal with dilatation 0 on Σ \ U and dilatation ǫd ¯ ζ/dζ on U .Let Ω be some polydisc centered at 0 ∈ C n and ǫ = ( ǫ , . . . , ǫ n ) ∈ Ω. The above variationprocedure can be applied to n non-overlapping parametric disks U , . . . , U n to obtain aRiemann surface Σ ǫ and a quasiconformal map ν ǫ : Σ → Σ ǫ . We thus have a map S : Ω → T (Σ) ǫ [Σ , ν ǫ , Σ ǫ ] . The key result of [2] and [10, Theorem 4.3.2] is the following:
Theorem 2.17. (1)
The map S is holomorphic for any type of Riemann surface Σ . (2) Let Σ P be a punctured surface of type ( g, n ) , and let d = 3 g − n . For an essentiallyarbitrary choice of parametric disks U , . . . , U n , the parameters ( ǫ , . . . , ǫ d ) providelocal holomorphic coordinates for T (Σ P ) in a neighborhood of [Σ , id , Σ] . That is, S isa biholomorphism onto its image. The first result follows directly from the fact that dilatation of ν ǫ is holomorphic in ǫ . Forthe second result, it is non-trivial to prove that the variations give independent directions in T (Σ P ).Getting coordinates at an arbitrary point [Σ P , f, Σ P ] in Teichm¨uller space follows by ap-plying a change of base surface biholomorphism from T (Σ P ) to T (Σ P ) in the following way. irst we apply Schiffer variation to Σ P . This gives a neighborhood of the base point in T (Σ P ). Let f ∗ : T (Σ P ) → T (Σ P ) be the change of base surface biholomorphism correspond-ing to f . That is, f ∗ ([Σ P , ν ǫ , Σ P ]) = [Σ P , ν ǫ ◦ f, Σ P ]. Thus, the image under f ∗ of the Schifferneighborhood in T (Σ P ) is the Schiffer neighborhood of [Σ P , f, Σ P ] ∈ T (Σ P ).We can now show that Theorem 2.18.
The map C (2.4) possess local holomorphic sections through every point.Proof. Let [Σ B , f, Σ B ] be an arbitrary point in T (Σ B ) and let [Σ P , ˜ f , Σ P ] = C [Σ B , f, Σ B ]where C is the sewing map defined in (2.4). Recall that Σ P = Σ B f ◦ τ D n . Let d = 3 g + 3 − n .By Theorem 2.17, we can choose d disjoint disks on Σ B such that performing Schiffer variationon Σ P using these disks results in a biholomorphic map S : Ω −→ S (Ω) ⊂ T (Σ P ) ǫ [Σ P , ( ˜ f ) ǫ , Σ P,ǫ ] , where ( ˜ f ) ǫ = ν ǫ ◦ ˜ f , ǫ = ( ǫ , . . . , ǫ d ), and Ω is an open neighborhood of 0 ∈ C d .Performing the Schiffer variation on Σ B using the same disks produces a holomorphic map S B : Ω −→ T (Σ B ) ǫ [Σ B , f ǫ , Σ B,ǫ ] , where f ǫ = ν ǫ ◦ f .By Theorem 2.17, η = S B ◦ S − : S (Ω) → T (Σ B ) is holomorphic. To show that it is asection of C through [Σ B , f, Σ B ], it remains to show that C ◦ η is the identity.Note that f = f ǫ on ∂ Σ, and by definition of ˜ f , ( ˜ f ) ǫ = e f ǫ . Because the disks on which toperform Schiffer variation were chosen to be away from the caps of Σ P ,Σ B,ǫ f ǫ ◦ τ D n = Σ P,ǫ . So we have(
C ◦ η ) (cid:16) [Σ P , ( ˜ f ) ǫ , Σ P,ǫ ] (cid:17) = C (cid:16) [Σ B , f ǫ , Σ B,ǫ ] (cid:17) = [Σ P , e f ǫ , Σ B,ǫ f ǫ ◦ τ D n ] = [Σ P , ( ˜ f ) ǫ , Σ P,ǫ ]and thus C ◦ η is the identity. (cid:3) We now have the following key results.
Corollary 2.19.
The Teichm¨uller space T (Σ B ) is a holomorphic fiber space over T (Σ P ) with fiber structure given by the sewing map C : T (Σ B ) T (Σ P ) . Proof.
Theorem 2.12 states that C is holomorphic and onto. Lemma 2.16 and Theorem 2.18show that C is a submersion. (cid:3) From Lemma 2.15 we obtain:
Corollary 2.20.
The fibers F B are complex submanifolds of T (Σ B ) . Corollary 2.21.
The fibers F P are complex submanifolds of e T P (Σ P ) .Proof. From Theorem 2.12 F is holomorphic. It possesses local holomorphic sections, sinceif ρ : T (Σ P ) → T (Σ B ) is a local holomorphic section of C , then P ◦ ρ is a local holomorphicsection of F . (cid:3) . Local model of fibers
Complex structure on O qc (Σ P ) . The authors defined a complex structure on O qc (Σ P )in [13], with the help of a model of the universal Teichm¨uller curve due to Teo [15]. We outlinethe definition of the complex structure in this section.The complex structure O qc (Σ P ) is locally biholomorphic to O n qc = O qc × · · · × O qc . So wemust first describe the complex structure on O qc . To do this, we define an injection of O qc onto an open subset of a Banach space. Consider the Banach space A ∞ ( D ) = { v ( z ) : D → C | v holomorphic , || v || , ∞ = sup z ∈ D (1 − | z | ) | v ( z ) | < ∞} . For f ∈ O qc define A ( f ) = f ′′ ( z ) f ′ ( z ) . We then have a natural one-to-one map χ : O qc → A ∞ ( D ) ⊕ C (3.1) f ( A ( f ) , f ′ (0)) , where A ∞ ( D ) ⊕ C is a Banach space with the direct sum norm || ( φ, c ) || = || φ || , ∞ + | c | . It was shown in [13] that the image of O qc is open and thus O qc inherits a complex structurefrom A ∞ ( D ) ⊕ C .We now describe the complex structure on O qc (Σ P ) in terms of local charts into O n qc . Fixa point ( φ , . . . , φ n ) ∈ O qc (Σ P ). For i = 1 , . . . , n , let D i = φ i ( D ) be open sets in Σ P . Choosedomains B i ⊂ Σ P with the following properties: 1) φ i ( D ) ⊂ B i , 2) B i ∩ B j = ∅ for i = j and3) B i are open and simply connected. Let ζ i : B i → C be a local biholomorphic parametersuch that ζ i ( p i ) = 0. We then have that ζ i ◦ φ i ∈ O qc .Let K i be a compact set, which is the closure of an open set containing ζ i ◦ φ i ( D ), suchthat ζ − i ( K i ) ⊂ B i . By [13, Corollary 3.5], for each i there is an open neighborhood U i of ζ i ◦ φ i in O qc such that ψ i ( D ) ⊂ K i for all ψ i ∈ U i , and so ( ζ − ◦ ψ , . . . , ζ − n ◦ ψ n ) is anelement of O qc (Σ P ).Thus, U × · · · × U n is an open subset of O qc × · · · × O qc with the product topology. Let V i = { ζ − i ◦ ψ i | ψ i ∈ U i } ;then sets of the form V = V × · · · × V n form a base of the topology of O qc (Σ P ). Let T i : V i → O qc g ζ i ◦ g. and note that T i ( V i ) = U i . The local coordinates T = ( T , . . . , T n ) : V → U define a complex Banach manifold structure on O qc (Σ P ). Remark . In fact, fixing B i and ζ i , i = 1 , . . . , n , T is a valid chart on the set of all elementsof O qc (Σ P ) which map all n copies of the disk into the corresponding set B i . et U i = { ψ i ∈ O qc | ψ i ( D ) ⊂ ζ i ( B i ) } and observe that U = U × · · · × U n is open by[13, Corollary 3.5]. Furthermore, the set V = { φ ∈ O qc (Σ P ) | φ i ( D ) ⊂ B i } is open sinceevery point is contained in an open set of the form described above. The map T is clearly abiholomorphism on all of V . Hence, ( T, V ) is a local coordinate.3.2.
The biholomorphism L : O qc (Σ P ∗ ) → F P ([Σ P , f, Σ P ∗ ]) . Fix a point [Σ P , f, Σ P ∗ ] ∈ T (Σ P ) and recall from Section 2.2 that F P ([Σ P , f, Σ P ∗ ]) ⊂ e T P (Σ P ) is the fiber over thispoint. Define the map L : O qc (Σ P ∗ ) → F P ([Σ P , f, Σ P ∗ ]) φ [Σ P , f, Σ P ∗ , φ ] . Our goal is to prove
Theorem 3.2.
The map L : O qc (Σ P ∗ ) → F P ([Σ P , f, Σ P ∗ ]) is a biholomorphism.Proof. We will prove this in three stages: first, we show that L is a bijection onto F P (Lemma3.9 ahead), second that L is holomorphic (Lemma 3.11 ahead). Lastly, we show that theinverse is holomorphic.To show that the inverse is holomorphic, in Section 3.4 we will construct near any pointin T (Σ B ) a local inverse Λ to the lift P − ◦ L where P − is a local inverse of P . By Theorem2.12 it is enough to show that Λ is holomorphic, which we will establish in Theorem 3.23ahead. (cid:3) Theorem 3.2 has the following consequences.
Corollary 3.3.
Fix any [Σ P , f, Σ P ∗ ] ∈ T (Σ P ) . The map O qc (Σ P ∗ ) → F B ([Σ P , f, Σ P ∗ ]) / DB φ [Σ B , f φ , Σ P ∗ \ φ ( D n )] is a biholomorphism, where f φ is the restriction of a map f ′ φ : Σ P → Σ P which is homotopicto f and satisfies f ′ φ ◦ ˜ τ = φ .Proof. This follows from Theorem 3.2 and Remark 2.9. (cid:3)
Corollary 3.4.
The fibers F B ([Σ P , f, Σ P ]) and F P ([Σ P , f, Σ P ]) are locally biholomorphic to O n qc for any [Σ P , f, Σ P ] via the maps T ◦ L − ◦ P and T ◦ L − respectively. In the next few sections we require some facts regarding infinite-dimensional holomorphy(see for example [1], [7, V.5.1] or [10, Section 1.6]). Let E and F be Banach spaces and let U be an open subset of E . Definition 3.5.
A map f : U → F is holomorphic if for each x ∈ U there is a continuouscomplex linear map Df ( x ) : E → F such thatlim h → || f ( x + h ) − f ( x ) − Df ( x )( h ) || F || h || E = 0 . Definition 3.6.
A map f : U → F is called Gˆateaux holomorphic if f is holomorphic oncomplex lines. That is, if for all a ∈ U and all x ∈ E , the map z f ( a + zx ) is holomorphicon { z ∈ C | a + zx ∈ U } . heorem 3.7 ([1, p 198]) . Let f : U → F . The following are equivalent. (1) f is holomorphic. (2) f is Gˆateaux-holomorphic and continuous. (3) f is Gˆateaux-holomorphic and locally bounded on U . A subset H of the (continuous) dual space F ′ is called separating if for all non-zero x ∈ F there exists α ∈ H such that α ( x ) = 0. The following theorem gives another characterizationof holomorphicity. See [3] for a statement in a more general setting. Theorem 3.8.
Let f : Ω → F be a function on a domain Ω in C . If (1) α ◦ f is holomorphic for each continuous linear functional α from a separating subsetof the dual space F ′ , and (2) f is locally bounded,then f is holomorphic. We now turn to first step in the proof of Theorem 3.2.
Lemma 3.9. L is a bijection between O qc (Σ P ∗ ) and F P ([Σ P , f, Σ P ∗ ]) .Proof. By the definitions of e T P (Σ P ) and T (Σ P ), every element in F P has a representative ofthe form [Σ P , ˜ f , Σ P ∗ , φ ]. L is thus clearly a surjection.Assume that L ( φ ) = L ( ψ ). If F ([Σ P , f, Σ P ∗ , φ ]) = F ([Σ P , f, Σ P ∗ , ψ ]), then there exists abiholomorphism σ : Σ P ∗ → Σ P ∗ that is homotopic to the identity. Since 2 g − n > σ is the identityand so φ = ψ . (cid:3) Remark . The bijection is not canonical since O qc (Σ P ∗ ) depends on the choice of repre-sentative in the Teichm¨uller equivalence class. However, if [Σ P , f , Σ P ] = [Σ P , f , Σ P ] and σ : Σ P → Σ P is the biholomorphism realizing the equivalence, then we have the bijection σ ∗ : O qc (Σ P ) → O qc (Σ P ) defined by σ ∗ ( φ ) = σ ◦ φ , and L = L ◦ σ ∗ . It will follow fromTheorem 3.2 (once the proof is complete) that O qc (Σ P ) and O qc (Σ P ) are biholomorphicunder σ ∗ .3.3. L is holomorphic.Lemma 3.11. L is holomorphic.Proof. Fix φ ∈ O qc (Σ P ∗ ). We will show that L is holomorphic in a neighborhood of φ byfirst proving that L is Gˆateaux holomorphic and then that the lift of L to L ∞− , (Σ B ) islocally bounded.To simplify notation we assume that Σ P has only one puncture, p . The proof of thegeneral case is identical with the exception of the notation. Let ( T, V ) be a chart on O qc (Σ P ∗ )containing φ and let B = B , D = D and ζ = ζ be as in Section 3.1. Recall that the map χ (3.1) defines the complex structure on O qc .To show L is Gˆateaux holomorphic we must show that L ◦ T − ◦ χ − : A ∞ ( D ) ⊕ C → e T P (Σ P )is Gˆateaux holomorphic. Since the holomorphic structure on e T P (Σ P ) is obtained from T (Σ B )we proceed by producing a lift of L ◦ T − ◦ χ − to T (Σ B ). et ψ = T ( φ ) = ζ ◦ φ ∈ O qc , u = A ( ψ ) ∈ A ∞ , and choose an element ( v, c ) ∈ A ∞ ( D ) ⊕ C . Let N be an open neighborhood of 0 ∈ C , and consider the map N → χ ( T ( V )) ⊂ A ∞ ⊕ C given by t ( u + tv, q t ) where q t = ψ ′ (0) + tc . That is, we have an complex line through χ ( ψ ) = ( u , ψ ′ (0)).Let ψ t = χ − ( u + tv, q t ) . From the definition (3.1) of χ we have a differential equation for ψ t , whose solution is(3.2) ψ t ( z ) = q t ψ ′ (0) Z z ψ ′ ( ξ ) exp (cid:18) t Z ξ g ( w ) dw (cid:19) dξ. It is clear from this expression that ψ t ( z ) is holomorphic in t for fixed z ∈ D .Now, ψ t ◦ ψ − is a holomorphic motion of ψ ( D ). By the extended lambda-lemma [17] itextends to a holomorphic motion of C , and in particular the continuous extension of ψ t ◦ ψ − to ψ ( D ) is a holomorphic motion. Thus ψ t ◦ ψ − restricted to ψ ( ∂ D ) is a holomorphicmotion.Let φ t = T − ( ψ t ) = ζ − ◦ ψ t , and let A t = B \ φ t ( D ) be the annular region on Σ P ∗ bounded by ∂B and φ t ( ∂ D ). Applying [12, Lemma 7.1] we obtain a holomorphic motion H t : ζ ( A ) → ζ ( A t ) such that H t | ζ ( ∂B ) is the identity and H t | ψ ( ∂ D ) = ψ t ◦ ψ − .Let Σ B ∗ ,t = Σ P ∗ \ φ t ( D ). By [12, Proposition 7.1] the map F t : Σ B ∗ , → Σ B ∗ ,t defined by(3.3) F t = ( id on Σ B ∗ \ A ζ − ◦ H t ◦ ζ on A . is quasiconformal and holomorphic in t for fixed z . On φ ( ∂ D ), F t = ζ − ◦ ψ t ◦ ψ − ◦ ζ = φ t ◦ φ − .From [12, Proposition 7.2], t µ ( F t ) is a holomorphic function N → L ∞− , (Σ B ∗ , ) . Therefore,by the holomorphicity of the fundamental projection, t [Σ B ∗ , , F t , Σ B ∗ ,t ] is holomorphic.Let [Σ B , h , Σ B ∗ , ] = P − ([Σ P , f, Σ P ∗ , φ ]) where P − is a local inverse of P and ˜ h ◦ ˜ τ = φ from the definition of P in (2.2). For g : Σ B → Σ B ∗ , , let g ∗ : T (Σ B ∗ , ) → T (Σ B )[Σ B ∗ , , h , Σ B ] [Σ B , h ◦ g, Σ B ] . be the change of base point biholomorphism (see [10, Sections 2.3.1 and 3.2.5]). In particular, g ∗ ([Σ B ∗ , , F t , Σ B ∗ ,t ]) = [Σ B , F t ◦ g, Σ B ∗ ,t ] is a biholomorphism, and therefore the map t [Σ B , F t ◦ h , Σ B ∗ ,t ] from N into T (Σ B ) is holomorphic. Moreover, t µ ( F t ◦ h ) is also holomorphic.From the boundary values of F t and h we see that F t ◦ h ◦ τ = φ t on ∂ D . Furthermore,extending F t , h and τ by the identity to the caps as in equation (2.1), we have that ˜ F t ◦ ˜ h ◦ ˜ τ is homotopic to f (after identifying Σ B ∗ ,t D with Σ P ∗ ). Thus P ([Σ B , F t ◦ h , Σ B ∗ ,t ]) = [Σ P , ˜ F t ◦ ˜ h , Σ B ∗ ,t φ t D , φ t ] = [Σ P , f, Σ P ∗ , φ t ]and so N → e T (Σ P ) t ( L ◦ T − ◦ χ − )( h + tg, q t ) = [Σ P , f, Σ P ∗ , φ t ]is holomorphic. That is, L ◦ T − ◦ χ − is Gˆateaux holomorphic. e have that 1) the fundamental projection Φ : L ∞− , (Σ B ) → T (Σ B ) is holomor-phic and possesses local holomorphic sections and 2) P : T (Σ B ) → e T P (Σ P ) is holomor-phic and possesses local holomorphic sections. Thus there is an open neighborhood M of[Σ P , f, Σ P ∗ , φ ] ∈ e T P (Σ P ) and a local holomorphic section σ : M → L ∞− , (Σ B ) of P ◦
Φ. Con-tinuity in t guarantees that for | t | sufficiently small, [Σ P , f, Σ P ∗ , φ t ] ⊂ M . The function σ ◦ L is Gˆateaux holomorphic and locally bounded since σ maps into the open unit ball. Thus, byTheorem 3.7, σ ◦ L is holomorphic, and therefore L = P ◦ Φ ◦ σ ◦ L is holomorphic. (cid:3) Remark . For any fixed z ∈ D , the point evaluation map O qc → C given by f f ( z ) isholomorphic. The proof is an immediate consequence of Theorem 3.7 noting that Gˆateauxholomorphy follows from equation (3.2), and continuity is proved in [13, Corollary 3.4.]. Thisresult is also included in the proof of [13, Lemma 3.10] but it is not mentioned explicitly.3.4. The local inverse of L . In order to show that L − is biholomorphic, we show that P − ◦ L : O qc (Σ P ∗ ) → T (Σ B ) has a local holomorphic inverse for any local inverse P − of P .The description of the inverse to P − ◦ L is somewhat lengthy and deserves its own section.In fact we are only able to explicitly describe the inverse on specified holomorphic curves.The source of the trouble can be partly traced to the fact that local inverses of P cannot beexplicitly defined (see Remark 2.10).It is necessary to make the following change of base point. Recall that Σ P = Σ B τ D n asin Section 2.2, which we now think of as a punctured surface. Let U be the upper half-plane,and choose a Fuchsian group G such that Σ G = U /G is an n -punctured surface biholomorphicto Σ P . Let α : Σ P → Σ G be a fixed biholomorphism.Let A be an open set of T (Σ B ) such that P| A is a biholomorphism. Let(3.4) A B = F B ∩ A, where F B is the fiber in T (Σ B ) above the fixed point [Σ P , f, Σ P ∗ ] (see Section 2.2).Given [Σ B , h, Σ B ] ∈ A B we have that P ([Σ B , h, Σ B ]) = ([Σ P , ˜ h, Σ B h ◦ τ D n ]). The changeof base point biholomorphism ( α ∗ ) − : T (Σ P ) → T (Σ G ) is defined by( α ∗ ) − ([Σ P , ˜ h, Σ B h ◦ τ D n ]) = [Σ G , ˜ h ◦ α − , Σ B h ◦ τ D n ] . Now define the Beltrami differential µ = µ (˜ h ◦ α − ) = ( µ ( h ◦ α − ) on α (Σ B ) ⊂ Σ G α ( D ) . in L ∞− , (Σ G ) . Let L ∞ ( U , G ) be the space of Beltrami differentials compatible with G , andfollowing [10, p. 51] we identify µ with its unique lift to L ∞ ( U , G ) . Let w µ be the uniquesolution to the Beltrami equation on C fixing 0, 1 and ∞ and having dilatation 0 on thelower half-plane. Let G µ = w µ ◦ G ◦ ( w µ ) − andΣ µ = w µ ( U ) /G µ . Let T ( G ) = L ∞ ( U , G ) / ∼ , where µ ∼ ν if and only if w µ = w ν on R , be the Teichm¨ullerspace of G . Let π µ : w µ ( U ) → w µ ( U ) /G µ be the canonical projection. It is a standard fact(see [10, Sections 2.2.2 and 3.3.1]) that: emma 3.13. If [ µ ] = [ ν ] in T ( G ) then w µ ( U ) = w ν ( U ) , G µ = G ν , and π µ = π ν . Proposition 3.14.
The equivalence class [ µ ] and the Riemann surface Σ µ are independentof [Σ B , h, Σ B ] ∈ A B .Proof. By definition of the fiber F B , [Σ P , f, Σ P ∗ ] = [Σ P , ˜ h, Σ B h ◦ τ D ], and applying ( α ∗ ) − leads to [Σ G , f ◦ α − , Σ P ∗ ] = [Σ G , ˜ h ◦ α − , Σ B h ◦ τ D ]Therefore µ = µ (˜ h ◦ α − ) is equivalent to the fixed element µ ( f ◦ α − ). The result now followsfrom Lemma 3.13. (cid:3) We may therefore let σ : Σ µ → Σ P ∗ be a fixed biholomorphism, and defineΛ : A B → O qc (Σ P ∗ )(3.5) [Σ B , h, Σ B ] σ ◦ f µ ◦ α ◦ ˜ τ where µ = µ (˜ h ◦ α − ) and f µ is the unique quotient map f µ : Σ G → Σ µ . corresponding to w µ . Remark . The argument above also shows that for any [Σ B , h , Σ B ] and [Σ B , h , Σ B ] in A B , the corresponding w µ and w µ are homotopic rel ∂ U , and similarly f µ and f µ arehomotopic in Σ G [10, Sections 2.2.2 and 3.3.1]. Remark . The map L ∞− , (Σ B ) → L ∞− , (Σ G ) µ ( h ) µ (˜ h ◦ α − )is holomorphic. This follows from the fact that µ ( h ) µ (˜ h ) is holomorphic (see [12, Lemma6.2]), and the map µ (˜ h ) µ (˜ h ◦ α − ) is holomorphic by the change of base point holomor-phicity (see [10, Sections 2.3.1 and 3.2.5]). Proposition 3.17. If (Σ B , h , Σ B ) is equivalent to (Σ B , h , Σ B ) in A B ⊂ T (Σ B ) , then thecorresponding maps f µ and f µ are equal on α ( ∂ Σ B ) . In particular, Λ is well-defined.Proof. Let [Σ B , h , Σ B ] = [Σ B , h , Σ B ], so that there exists a biholomorphism γ : Σ B → Σ B such that h − ◦ γ ◦ h is homotopic to the identity rel ∂ Σ B . Since the dilatation of γ ◦ h and h are equal, composition by γ does not change the resulting f µ in the definition of Λ.Thus we may absorb γ into h and assume that Σ B = Σ B and h − ◦ h is homotopic to theidentity rel ∂ Σ B . In particular, h = h on ∂ Σ B and moreover ˜ h is homotopic to ˜ h on Σ P .Now let σ : Σ µ → Σ P ∗ be the fixed biholomorphism in the definition of Λ. Let f µ i : Σ G → Σ µ be the pair of maps with dilatations µ i = µ (˜ h i ◦ α − ) respectively.Since ˜ h i ◦ α − and f µ i have the same dilatations for i = 1 ,
2, there is a pair of biholomorphicmaps δ i : Σ Bi h i ◦ τ D n → Σ µ such that(3.6) δ i ◦ ˜ h i ◦ α − = f µ i n Σ G . Because h = h on ∂ Σ B and Σ B = Σ B , Σ B h ◦ τ D n = Σ B h ◦ τ D n .By Remark 3.15, we know that ( f µ ) − ◦ f µ is homotopic to the identity on Σ G . Thus δ − ◦ δ is homotopic to the identity. Since Σ P is of type ( g, n ) with 2 g − n > δ − ◦ δ must be the identity. We can conclude that δ = δ . Therefore,since h = h on ∂ Σ B , by (3.6) f µ = f µ on α ( ∂ Σ B ), soΛ([Σ B , h , Σ B ]) = Λ([Σ B , h , Σ B ]) . (cid:3) Let P − denote a locally defined inverse of P (see (2.2)) in a neighborhood of [Σ P , f, Σ P ∗ , φ ].We need to show that Λ ◦ P − ◦ L = idon some neighborhood of φ . For notational simplicity we work with the case of a singlepuncture on Σ P . Let [Σ B , h , Σ B ∗ , ] = P − ([Σ P , f, Σ P ∗ , φ ]) and recall that Σ B ∗ , = Σ P ∗ \ φ ( D )and h ◦ τ = φ on ∂ D .Choose a compact set E ⊂ Σ P ∗ such that E ⊂ φ ( D ). Let V ⊂ O qc (Σ P ∗ ) be an openneighborhood of φ such that for all φ ∈ V , E ⊂ φ ( D ). This is possible by [13, Corollary3.4].Let ι : Σ B ∗ , h ◦ τ D → Σ P ∗ be the biholomorphism defined by(3.7) ι ( z ) = ( z z ∈ Σ B ∗ , φ ( z ) z ∈ D . Since h ◦ τ = φ on ∂ D , we have(3.8) ι ◦ ˜ h ◦ ˜ τ = φ . Let σ = ι ◦ δ − : Σ µ → Σ P ∗ , where δ : Σ B ∗ , h ◦ τ D → Σ µ is the unique biholomorphism satisfying (3.6). Use the map σ for the biholomorphism in the definition of Λ. Now let ψ = Λ([Σ B , h , Σ B ∗ , ]). It followsdirectly that ψ = φ : ψ = σ ◦ f µ ◦ α ◦ ˜ τ = ι ◦ δ − ◦ ( δ ◦ ˜ h ◦ α − ) ◦ α ◦ ˜ τ = ι ◦ ˜ h ◦ ˜ τ . Thus by equation (3.8) ψ = φ .For φ ∈ V , let [Σ B , h φ , Σ B ∗ ,φ ] = ( P − ◦ L )( φ ) where Σ B ∗ ,φ = Σ P ∗ \ φ ( D ). We eventually wantto show that for any φ ∈ V , Λ([Σ B , h φ , Σ B ∗ ,φ ]) = φ .Choose a holomorphic curve φ t ∈ V joining φ and φ . That is, φ = φ and φ is as above.By Remark 3.12, φ t ( z ) is holomorphic in t and so the construction in the proof of Lemma3.11 can be repeated. Let Σ B ∗ ,t = Σ B ∗ ,φ t and define F t ◦ h : Σ B → Σ B ∗ ,t where F t is as in(3.3). As before, t µ ( F t ◦ h ) is holomorphic. Let h t = F t ◦ h , µ t = µ (˜ h t ◦ α − ), and let δ t : Σ B ∗ ,t h t ◦ τ D → Σ µ be the biholomorphism defined by δ t = f µ t ◦ α ◦ ˜ h − t as in (3.6). Define ι t as in (3.7) by replacing φ with φ t . emma 3.18. The map δ t : Σ B ∗ ,t h t ◦ τ D → Σ µ is holomorphic in t for z ∈ D .Proof. Since t µ ( h t ) is holomorphic in t , Remark 3.16 shows that µ t = µ (˜ h t ◦ α − ) isholomorphic in t . Thus for fixed z , w µ t ( z ) is holomorphic in t . Because π µ is independentof µ by Lemma 3.13 and Proposition 3.14, f µ t ( z ) is also holomorphic in t for fixed z .For z ∈ D , δ t ( z ) = ( f µ t ◦ α ◦ ˜ h − t )( z ) = ( f µ t ◦ α )( z )from the definition of δ t , and the conclusion follows immediately. (cid:3) Lemma 3.19. σ = ι ◦ δ − = ι t ◦ δ − t Proof.
Let β t = σ ◦ δ t ◦ ι − t : Σ P ∗ → Σ P ∗ . We will show that β t is the identity. First we claim that β t is holomorphic in t for z ∈ E .We have that φ t ( z ) is holomorphic as a function of t and z and thus so is φ − t ( z ) by a directapplication of the implicit function theorem. For z ∈ E , and using ι t ◦ ˜ h t ◦ ˜ τ = φ t from (3.8), β t ( z ) = ( σ ◦ f µ t ◦ α ◦ ˜ h − t ◦ ι − t )( z )= ( σ ◦ f µ t ◦ α ◦ ˜ τ ◦ φ − t )( z )which is holomorphic in t .The fact that β t is the identity follows from the following four observations. (1) ByHurwitz’s theorem, the automorphism group of Σ P ∗ is finite. (2) β t is the identity if and onlyif β t | E is the identity. (3) For z ∈ E , β t ( z ) is continuous in t . (4) From the definition of σ , β is the identity. (cid:3) Theorem 3.20.
For all φ ∈ V , and any local inverse P − of P , (Λ ◦ P − ◦ L )( φ ) = φ .Proof. We join φ to φ by a holomorphic curve φ t such that φ = φ . From Lemma 3.19 andequation (3.6) we have:(Λ ◦ P − ◦ L )( φ t ) = Λ([Σ B , h t , Σ B ∗ , ])= σ ◦ f µ t ◦ α ◦ ˜ τ = ( ι t ◦ δ − t ) ◦ ( δ t ◦ ˜ h t ◦ α − ) ◦ α ◦ ˜ τ = ι t ◦ ˜ h t ◦ ˜ τ = φ t where the last equality follows as both ˜ τ and ˜ h t are the identity on D . Setting t = 1 completesthe proof. (cid:3) L − is holomorphic.Lemma 3.21. Let [Σ B , h , Σ B ∗ , ] = P − ([Σ P , f, Σ P ∗ , φ ]) for some local inverse P − of P . Let B be an open set in Σ P ∗ such that ( σ ◦ f µ ◦ α ◦ ˜ τ )( D ) ⊂ B . There exists a neighborhood A ⊂ F B of [Σ B , h , Σ B ∗ , ] such that Λ (cid:0) [Σ B , h, Σ B ] (cid:1) ( D ) ⊂ B for all [Σ B , h, Σ B ] ∈ A . roof. Let γ be a local holomorphic section of the fundamental projection Φ in a neighbor-hood of [Σ B , h , Σ B ∗ , ], and choose U to be an open set in the domain of γ containing thispoint. Given u ∈ U we choose a representative [Σ B , h u , Σ B ] = u such that µ ( h u ) = γ ( u ).By Remark 3.16, µ ( h u ) µ (˜ h u ◦ α − ) is holomorphic. Therefore, since γ is holomorphic,the map U → L ∞− , (Σ G ) u µ (˜ h u ◦ α − )is holomorphic and in particular continuous.We now show in general that w µ ( z ) and hence f µ ( z ) are jointly continuous in µ and z .Let D r = { z : | z | < r } for some r >
1. Fix µ and z . By [4, Theorem 4.7.4], for any ǫ > δ so that if k µ − µ k < δ then | w µ ( z ) − w µ ( z ) | < ǫ/ z ∈ D r . On theother hand, since w µ is continuous in z there is a δ such that | w µ ( z ) − w µ ( z ) | < ǫ/ | z − z | < δ . We can assume that δ is small enough that the disk of radius δ centeredon z is contained in D r . Thus for | z − z | < δ and k µ − µ k < δ | w µ ( z ) − w µ ( z ) | ≤ | w µ ( z ) − w µ ( z ) | + | w µ ( z ) − w µ ( z ) | < ǫ. So w µ ( z ) is jointly continuous in µ and z for all z ∈ D and hence so is f µ ( z ).By applying this fact with µ = µ (˜ h ◦ α − ), we have that( u, z ) f µ ( z )is continuous.In the following let D ( ζ , R ) denote the disc of radius R centered on ζ . Let B be as in Section3.1 (recall that we are assuming that there is a single puncture). Since each ( σ ◦ f µ ◦ α )( ∂ Σ B )is compact, there is an r so that D ( ξ, r ) ⊂ B for all ξ ∈ ( σ ◦ f µ ◦ α )( ∂ Σ B ). By the continuityof f µ ( ζ ) in both µ and ζ for each ζ ∈ α ( ∂ Σ B ), and by letting ξ = ( σ ◦ f µ )( z ), one can choosean open neighborhood U ξ = D ( z, δ ξ ) × W ξ of ( z, [Σ B , h , Σ B ∗ , ]) so that f µ ( ζ ) ⊂ B ( ξ, r ) for all ζ ∈ D ( z, δ ξ ) and [Σ B , h u , Σ B ] ∈ W ξ . Since α ( ∂ Σ B ) is compact, its open cover by the union of D ( z, δ ξ ) has a finite subcovering D ( z , δ ξ ) , . . . , D ( z m , δ ξ m ). The open set A = W ξ ∩· · ·∩ W ξ m has the desired properties. (cid:3) Fix a point z ∈ α ( D ). Choose Q ⊂ D to be an open neighborhood of z such that thereexists a local holomorphic section s of π G : U → U/G = Σ G defined on ( α ◦ τ )( Q ). Let π = π µ , which is independent of µ by Lemma 3.13 and Proposition 3.14. For z ∈ Q we cannow write Λ in terms of w µ and fixed maps as follows:(3.9) Λ([Σ B , h, Σ B ]) = σ ◦ ( π ◦ w µ ◦ s ) ◦ α ◦ ˜ τ . With the aid of Theorem 3.8, we can now proceed with the proof that Λ is holomorphic.
Lemma 3.22.
Let t [Σ B , h t , Σ Bt ] be a holomorphic curve in A B . For any z ∈ D , φ t ( z ) =Λ([Σ B , h t , Σ Bt ])( z ) and all its derivatives in z are holomorphic in t .Proof. We first show that the claim holds for a neighborhood of any fixed z in D . By theexistence of holomorphic sections of the fundamental projection the curve t [Σ B , h t , Σ Bt ]is the image of a holomorphic curve in L ∞− , (Σ B ) . We can thus assume without loss ofgenerality that our representatives (Σ B , h t , Σ Bt ) are such that t µ ( h t ) is holomorphic in . Thus the maps t µ t = µ (˜ h t ) and t w µ t ( z ) are holomorphic in t as in the proof ofLemma 3.18.Let C = α ( D ) be the cap on Σ G . Since µ t is zero on C , w µ t ( z ) is holomorphic in z for z ∈ s ( C ). Therefore, it is a holomorphic function of t and z and hence all its derivatives arealso holomorphic functions of both variables. The statement for φ t then follows from (3.9)for all z in some neighborhood Q of z . This proves the claim for z = 0.To prove the claim for z = 0, observe that for all t , φ t (0) = p . So in fact we know that φ t ( z ) is holomorphic in z for fixed t for all z ∈ D . Furthermore, clearly φ t (0) is holomorphicin t . Thus φ is holomorphic in t and z separately and so by Hartog’s theorem φ is jointlyholomorphic in both t and z . Thus all the derivatives of φ with respect to z are holomorphicin t for any fixed z ∈ D . (cid:3) Fix the point [Σ B , h , Σ B ∗ , ] ∈ F B , and let B i i = 1 , . . . , n and A be as in Lemma 3.21;it is possible to choose the B i to be non-overlapping. Let ζ i : B i → D , i = 1 , . . . , n bebiholomorphisms. From Remark 3.1 there is a corresponding chart T : O qc (Σ P ∗ ) → O n qc on the open subset V = { φ ∈ O qc (Σ P ∗ ) | φ i ( D ) ⊂ B i } containing Λ([Σ B , h , Σ B ]), given by T (( φ , . . . , φ n )) = ( ζ ◦ φ , . . . , ζ n ◦ φ n ). Theorem 3.23.
There exists an open neighborhood A of [Σ B , h , Σ B ∗ , ] ∈ A B such that Λ : A → O qc (Σ P ∗ ) is holomorphic.Proof. Choose A as in Lemma 3.21 and the preceding paragraph.Using Theorems 3.7 and 3.8 it is enough to show weak Gˆateaux-holomorphy and localboundedness. As before we temporarily drop the subscript i for ease of notation.Let φ t be as in Lemma 3.22 where it was proved that t φ t ( z ) is holomorphic in t . Thecomplex structure on O qc (Σ P ∗ ) is defined by O qc (Σ P ∗ ) T −→ O qc χ −→ A ∞ ⊕ C . Recall from Section 3.1 that χ ( f ) = ( A ( f ) , f ′ (0)) where A ( f ) = f ′′ /f ′ . Let ψ t = T ( φ t ) = ζ ◦ φ t . We now need to prove that t χ ( T ( φ t )) satisfies condition (1) of Theorem 3.8.This is immediate for the second component of χ because φ t (0) is holomorphic in t . Since φ t ( z ) and its derivatives are holomorphic in t for fixed z , we see that t ( A ( ψ t ))( z )is also holomorphic in t . Define E z : A ∞ → C by E z ( f ) = f ( z ). These point evaluation mapsare continuous linear functionals for z ∈ D . If Ω is an open subset of D then { E z | z ∈ Ω } is a separating set of functionals because the holomorphic functions are determined by theirvalues on an open set.By Theorem 3.8 it remains to show that Λ is locally bounded. From the definition of thecomplex structure on O qc (Σ P ∗ ) we must prove that χ ◦ T ◦ Λ is locally bounded.It suffices to show that each component of χ ◦ T ◦ Λ in A ∞ ⊕ C is bounded uniformly on A . For any element of A the corresponding map ψ = ζ ◦ σ ◦ f µ ◦ α ◦ ˜ τ ∈ O qc is a holomorphic map from D to D satisfying ψ (0) = 0. By the Schwarz lemma | ( ψ ′ (0) | ≤ . urthermore, by an elementary estimate for univalent maps of the disk (using again theabove bound on the first derivative) | (1 − | z | ) A ( ψ )( z ) − z | ≤ ||A ( ψ ) || , ∞ ≤ . (cid:3) By Theorem 3.20 and the preceding theorem we now conclude that Λ ◦ P − is a localholomorphic inverse of L .4. Coordinates on the Teichm¨uller space T (Σ B )Schiffer variation together with coordinates on the fibers F B provide local holomorphiccoordinate charts for the infinite-dimensional Teichm¨uller space T (Σ B ).Let S : Ω → T (Σ P ) be the coordinates in a neighborhood of [Σ P , f, Σ P ∗ ] ⊂ T (Σ P ) obtainedby Schiffer variation as described in Theorem 2.17. Let Σ Pǫ = (Σ P ∗ ) ǫ and recall that f ǫ = ν ǫ ◦ f .Note in particular that for any ǫ = 0 in Ω, S ( ǫ ) = [Σ P , f, Σ P ∗ ]. This implies that Schiffervariation in Σ B is transverse to the fibers F B .Let ( T, V ) be a chart on O qc (Σ P ∗ ) as in Section 3.1. Recall that this chart can be chosenso that for all φ ∈ V , φ ( D ) is contained in some fixed open B ⊂ Σ P ∗ . Now choose theneighborhoods U i , i = 1 , . . . , d , on which the Schiffer variation is to be performed, to bedisjoint from B .Let f φ be as in Corollary 3.3. Theorem 4.1.
The map (Ω × V ) → T (Σ B )( ǫ, φ ) [Σ B , ( f φ ) ǫ , (Σ P ∗ \ φ ( D n )) ǫ ] is biholomorphic onto its image.Proof. Because P is a local biholomorphism, it is sufficient that G : (Ω × V ) → e T P (Σ P )( ǫ, φ ) [Σ P , ν ǫ ◦ f, Σ Pǫ , ν ǫ ◦ φ ]is a biholomorphism onto its image. That G is injective follows directly from the definitionof e T P (Σ P ) and the facts: (1) S : Ω → T (Σ P ) is injective, and (2) Σ P has no non-identityautomorphisms that are homotopic to the identity since 2 g − n > φ , by Theorem 2.18 the map ǫ [Σ B , f ǫφ , (Σ P ∗ \ φ ( D n )) ǫ ] is holomorphic.Thus since P is holomorphic, G is holomorphic in ǫ . Now fix ǫ , and consider the mapcorresponding to the second component of G : H : V → e T P (Σ P ) φ [Σ P , ν ǫ ◦ f, Σ Pǫ , ν ǫ ◦ φ ] . ow H can be written as H ◦ H where H and H are given by H : V → O qc (Σ Pǫ ) φ ν ǫ ◦ φ and H : O qc (Σ Pǫ ) → e T P (Σ P ) ξ [Σ P , ν ǫ ◦ f, Σ Pǫ , ξ ] .H is holomorphic by Theorem 3.2, so it remains to show that H is holomorphic.Let ζ be the collection of local biholomorphisms of neighborhoods of the punctures on Σ P ∗ corresponding to the chart ( T, V ) on O qc (Σ P ∗ ). Let ( T ǫ , V ǫ ) be the chart on O qc (Σ Pǫ ) and ζ ǫ be the corresponding local biholomorphism of Σ Pǫ . We need to show that T ǫ ◦ H ◦ T − isholomorphic, i.e. that the map ζ ◦ φ ζ ǫ ◦ ν ǫ ◦ φ = ( ζ ǫ ◦ ν ǫ ◦ ζ − ) ◦ ( ζ ◦ φ )is holomorphic on O qc . Composition on the left by a biholomorphism is a local biholomor-phism of O qc by [13, Lemma 3.10], which establishes the claim.Finally we need to show that G − is holomorphic. By Theorem 3.2 and the fact thatSchiffer variation provides a section of the fiber projection, it is clear that the derivative of G is an injective linear map at each point for which G is defined. Since G is holomorphicand in particular C , we can apply the inverse function theorem [5] to show that G has a C inverse. The derivative of G − is also complex linear so G − is holomorphic. (cid:3) References
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Department of Mathematics and Statistics, American University of Sharjah, PO BOX26666, Sharjah, UAE
E-mail address , D. Radnell: [email protected]
Department of Mathematics, University of Manitoba, Winnipeg, MB, R3T 2N2, Canada
E-mail address , E. Schippers: eric [email protected] [email protected]