aa r X i v : . [ m a t h . S G ] A ug HOLOMORPHIC OPEN BOOK DECOMPOSITIONS
CASIM ABBAS
Abstract.
Emmanuel Giroux showed that every contact structure on a closedthree dimensional manifold is supported by an open book decomposition. Wewill extend this result by showing that the open book decomposition can bechosen in such a way that the pages are solutions to a homological perturbedholomorphic curve equation.
Contents
1. Introduction 12. Existence and local foliations 72.1. Local model near the binding orbits 72.2. Functional Analytic set-up and the Implicit Function Theorem 163. From local foliations to global ones 193.1. The Beltrami equation 243.2. A uniform L –bound for the harmonic forms and uniform convergence 293.3. A uniform L p –bound for the gradient 323.4. Convergence in W ,p ( B ′ ) 353.5. Improving regularity using both the Beltrami and the Cauchy-Riemannequations 364. Conclusion 37Appendix A. Some local computations near the punctures 38References 401. Introduction
This paper is the starting point of a larger program by the author, HelmutHofer and Samuel Lisi investigating a perturbed holomorphic curve equation in thesymplectisation of a three dimensional contact manifold [3], [4]. One aim of thisprogram is to provide an alternative proof of the Weinstein conjecture in dimensionthree as outlined in [2] complementing Clifford Taubes’ gauge theoretical proof[37], [38]. A special case of this paper’s main result has been used in the proof ofthe Weinstein conjecture for planar contact structures in [2]. Another reason forstudying this equation is to construct foliations by surfaces of section with nontrivialgenus. This is usually impossible to do with the unperturbed holomorphic curveequation since solutions generically do not exist.Consider a closed three dimensional manifold M equipped with a contact form λ .This is a 1-form which satisfies λ ∧ dλ = 0 at every point of M . We denote theassociated contact structure by ξ = ker λ and the Reeb vector field by X λ . Recall that the Reeb vector field is defined by the two equations i X λ dλ = 0 and i X λ λ = 1 . Definition 1.1. (Open Book Decomposition)
Assume that K ⊂ M is a link in M , and that τ : M \ K → S is a fibration sothat the fibers F ϑ = τ − ( ϑ ) are interiors of compact embedded surfaces ¯ F ϑ with ∂ ¯ F ϑ = K , where ϑ is the coordinate along K . We also assume that K has a tubularneighborhood K × D , D ⊂ R being the open unit disk, such that τ restricted to K × ( D \{ } ) is given by τ ( ϑ, r, φ ) = φ , where ( r, φ ) are polar coordinates on D .Then we call τ an open book decomposition of M , the link K is called the binding of the open book decomposition, and the surfaces F ϑ are called the pages of theopen book decomposition.It is a well-known result in three dimensional topology that every closed threedimensional orientable manifold admits an open book decomposition. Indeed, J.Alexander proved the following theorem in 1923, see [10] or [33]: Theorem 1.2.
Every closed, orientable manifold M of dimension is diffeomor-phic to W ( h ) [ Id ( ∂W × D ) where D is the closed unit disk in R , W is an orientable surface with boundaryand h : W → W is an orientation preserving diffeomorphism which restricts tothe identity near ∂W . W ( h ) denotes the manifold obtained from W × [0 , π ] byidentifying ( x, with ( h ( x ) , π ) . (cid:3) The above decomposition is an open book decomposition, the pages are givenby F ϑ := ( W × { ϑ } ) [ Id ( ∂W × I ϑ ) , ≤ ϑ < π, where I ϑ := { re iϑ ∈ D | < r < } , and the binding is given by K = ∂W × { } .Note that we allow ∂W to be disconnected.Emmanuel Giroux introduced the notion of an open book decomposition sup-porting a contact structure: Definition 1.3. (Supporting Open Book Decomposition [17] ) Assume that M is a closed three dimensional manifold endowed with a contact form λ . Let τ be an open book decomposition with binding K . We say that τ supportsthe contact structure ξ if there exists a contact form λ ′ with the same kernel as λ so that dλ ′ induces an area-form on each fiber F ϑ with K consisting of closed orbitsof the Reeb vector field X λ ′ , and λ ′ orients K as the boundary of ( F ϑ , dλ ′ ).We will refer to a contact form λ ′ above as a ’ Giroux contact form ’. Note that λ ′ is not unique and that it is in general different from the original contact form λ . The following theorem by E. Giroux guarantees existence of such open bookdecompositions, and it contains a uniqueness statement as well, see [17]: Theorem 1.4.
Every co-oriented contact structure ξ = ker λ on a closed threedimensional manifold is supported by some open book. Conversely, if two contactstructures are supported by the same open book then they are diffeomorphic. OLOMORPHIC OPEN BOOK DECOMPOSITIONS 3 (cid:3)
In the topological category it is possible to modify an open book decompositionsuch that the pages of the new decomposition have lower genus at the expense ofincreasing the number of connected components of K . It was not known for sometime whether a similar statement can also be made in the context of supportingopen book decompositions. In particular, it was unclear whether every contactstructure is supported by an open book decomposition whose pages are puncturedspheres ( planar pages ). The author and his collaborators could resolve the Wein-stein conjecture for contact forms inducing a ’planar contact structure’ in 2005 (see[2]). So the question whether all contact structures are planar became a priority,which prompted John Etnyre to address it in [15]. He showed that overtwistedcontact structures always admit supporting open book decompositions with planarpages, but many contact structures do not. Since then planar open book decom-positions have become an important tool in contact geometry.In this paper we will prove that every contact structure has a supporting openbook decomposition such that the pages solve a homological perturbed Cauchy-Riemann type equation which we will now describe after introducing some notation.We write π λ = π : T M → ξ for the projection along the Reeb vector field X λ . Fixa complex multiplication J : ξ → ξ so that the map ξ ⊕ ξ → R defined by( h, k ) → dλ ( h, Jk )defines a positive definite metric on the fibers. We will call such complex mul-tiplications compatible (with dλ ). The equation of interest here is the followingnonlinear first order elliptic system. The solutions consist of 5-tuplets ( S, j, Γ , ˜ u, γ )where ( S, j ) is a closed Riemann surface with complex structure j , Γ ⊂ S is a finitesubset, ˜ u = ( a, u ) : ˙ S → R × M is a proper map with ˙ S = S \ Γ, and γ is a one-formon S so that(1.1) π ◦ T u ◦ j = J ◦ π ◦ T u on ˙ S ( u ∗ λ ) ◦ j = da + γ on ˙ Sdγ = d ( γ ◦ j ) = 0 on SE (˜ u ) < ∞ . Here the energy E (˜ u ) is defined by E (˜ u ) = sup ϕ ∈ Σ Z ˙ S ˜ u ∗ d ( ϕλ ) , where Σ consists of all smooth maps ϕ : R → [0 ,
1] with ϕ ′ ( s ) ≥ s ∈ R .Note that the above equation reduces to the usual pseudoholomorphic curveequation in the symplectisation R × M if we set γ = 0. The following proposition,which is a modification of a result by Hofer, [18], shows that solutions to the problem(1.1) approach cylinders over periodic orbits of the Reeb vector field. Proposition 1.5.
Let ( M, λ ) be a closed three-dimensional manifold equipped witha contact form λ . Then the associated Reeb vector field has periodic orbits if andonly if the associated PDE-problem (1.1) has a non-constant solution.Proof. Let (
S, j, Γ , ˜ u, γ ) be a non-constant solution of (1.1). If Γ = ∅ then theresults in [18] and [22] imply that near a puncture the solution is asymptotic to a CASIM ABBAS periodic orbit (see also [6] for a complete proof). Here we use that γ is exact nearthe punctures. The aim is now to show that in the absence of punctures the map a is constant while the image of u lies on a periodic Reeb orbit. Assume that Γ = ∅ .Since u ∗ λ = − da ◦ j − γ ◦ j, we find after applying d that∆ j a = − d ( da ◦ j ) = u ∗ dλ. In view of the equation π ◦ T u ◦ j = J ◦ π ◦ T u we see that u ∗ dλ is a non-negativeintegrand. Applying Stokes’ theorem we obtain R S u ∗ dλ = 0 implying that π ◦ T u ≡ . Hence a is a harmonic function on S and therefore constant. So far, we also knowthat the image of u lies on a Reeb trajectory, and it remains to show that thistrajectory is actually periodic.Let τ : ˜ S → S be the universal covering map. The complex structure j lifts to acomplex structure ˜ j on ˜ S . Pick now smooth functions f, g on ˜ S such that dg = τ ∗ γ =: ˜ γ , − df = τ ∗ ( γ ◦ j ) = ˜ γ ◦ ˜ j. Then the map u ◦ τ : ˜ S → M satisfies( u ◦ τ ) ∗ λ = df. The image of u ◦ τ lies on a trajectory x of the Reeb vector field in view of D ( u ◦ τ )( z ) ζ = Df ( z ) ζ · X λ (( u ◦ τ )( z )) , hence ( u ◦ τ )( z ) = x ( h ( z )) for some smooth function h on ˜ S , and it follows that,after maybe adding a constant to f , we have( u ◦ τ )( z ) = x ( f ( z )) . The function f does not descent to S . If it did it would have to be constant since itis harmonic. On the other hand this would imply that u is constant in contradictionto our assumption that it is not. Therefore, there is a point q ∈ S and two lifts z , z ∈ ˜ S such that f ( z ) > f ( z ). Let ℓ : S → S be a loop which lifts to a path α : [0 , → ˜ S with α (0) = z and α (1) = z . Considering the map v := u ◦ ℓ : S −→ M we see that v ( t ) = ( u ◦ τ ◦ α )( t ) = x ( f ( α ( t ))) and x ( f ( z )) = x ( f ( z )), i.e. thetrajectory x is a periodic orbit. Hence the image of u is a periodic orbit for theReeb vector field. (cid:3) The following is the main result of this paper.
Theorem 1.6.
Let M be a closed three dimensional manifold, and let λ ′ be acontact form on M . Then the following holds for a suitable contact form λ = f λ ′ where f is a positive function on M : There exists a smooth family ( S, j τ , Γ τ , ˜ u τ =( a τ , u τ ) , γ τ ) τ ∈ S of solutions to (1.1) for a suitable compatible complex structure J : ker λ → ker λ such that • all maps u τ have the same asymptotic limit K at the punctures, where K is a finite union of periodic trajectories of the Reeb vector field X λ , • u τ ( ˙ S ) ∩ u τ ′ ( ˙ S ) = ∅ if τ = τ ′ OLOMORPHIC OPEN BOOK DECOMPOSITIONS 5 • M \ K = [ τ ∈ S u τ ( ˙ S ) • the projection P onto S defined by p ∈ u τ ( ˙ S ) τ is a fibration • The open book decomposition given by ( P, K ) supports the contact structure ker λ , and λ is a Giroux form. Here is a very brief outline of the argument. The reader is invited to skip fowardto the section ’Conclusion’ to see in more detail how all the partial results of thispaper are tied together to prove the main result. In section 2 we will find a Girouxcontact form which has a certain normal form near the binding. Following an ar-gument by Chris Wendl [40], [41] we will then almost be able to turn the Girouxleaves into solutions of (1.1) without harmonic form except for the fact that wehave to accept a confoliation form instead of a contact form. Pick one of theseGiroux leaves as a starting point. The next step is to prove a result which permitsus to perturb the Giroux leaf into a genuine solution of (1.1) while simultaneouslyperturbing the confoliation form slightly into a contact form. This is where theharmonic form in (1.1) is required. We actually obtain a local family of nearbysolutions, not just one. In section 3 we prove a compactness result which extendsthe local family of solutions into a global one. The remarkable fact is that thereis a compactness result in the context of this paper although there is none in gen-eral for the perturbed holomorphic curve equation. The special circumstances inthis paper imply a crucial apriori bound which implies that a sequence of solutionshas a pointwise convergent subsequence with a measureable limit. The objectiveis then to show that the regularity of this limit is much better, it is actually smooth.We consider two solutions (
S, j, Γ , ˜ u, γ ) and ( S ′ , j ′ , Γ ′ , ˜ u ′ , γ ′ ) equivalent if thereexists a biholomophic map φ : ( S, j ) → ( S ′ , j ′ ) mapping Γ to Γ ′ (preserving theenumeration) so that ˜ u ′ ◦ φ = ˜ u . We will often identify a solution ( S, j, Γ , ˜ u, γ )of (1.1) with its equivalence class [ S, j, Γ , ˜ u, γ ]. We note that we have a natural R -action on the solution set by associating to c ∈ R and [ S, j, Γ , ˜ u, γ ] the newsolution c + [ S, j, Γ , ˜ u, γ ] = [ S, j, Γ , ( a + c, u ) , γ ] , ˜ u = ( a, u ) . A crucial concept for our discussion will be the notion of a finite energy foliation F . Definition 1.7. (Finite Energy Foliation)
A foliation F of R × M is called a finite energy foliation if every leaf F is the imageof an embedded solution [ S, j, Γ , ˜ u, γ ] of the equations (1.1), i.e. F = ˜ u ( ˙ S ) , so that u ( ˙ S ) ⊂ M is transverse to the Reeb vector field, and for every leaf F ∈ F we also have c + F ∈ F for every c ∈ R , i.e. the foliation is R -invariant.We recall the concept of a global surface of section. Let M be a closed three-manifold and X a nowhere vanishing smooth vector field. Definition 1.8. (Surface of section) a) A local surface of section for (
M, X ) consists of an embedded compact surfaceΘ ⊂ M with boundary, so that ∂ Θ consists of a finite union of periodic orbits
CASIM ABBAS (called the binding orbits). In addition the interior ˙Θ = Θ \ ∂ Θ is transverse to theflow.b) A local surface of section is called a global surface of section if in addition everyorbit other than a binding orbit hits ˙Θ in forward and backward time. In additionthe globally defined return map Ψ : ˙Θ → ˙Θ has a bounded return time, i.e. thereexists a constant c > x ∈ ˙Θ hits ˙Θ again in forward time notexceeding c .Using proposition 2.5 below, the existence part of Giroux’s theorem can berephrased as follows: Theorem 1.9.
Let M be a closed orientable three-manifold and ˜ λ a contact formon M . Then there exists a smooth function f : M → (0 , ∞ ) so that the contactform λ = f ˜ λ has a Reeb vector field admitting a global surface of section. (cid:3) Existence results for finite energy foliations with a given contact form λ are hardto come by since they usually have striking consequences. In the article [26] forexample, H. Hofer, K. Wysocki and E. Zehnder show that every compact strictlyconvex energy hypersurface S in R carries either two or infinitely many closedcharacteristics. The proof relies on constructing a special finite energy foliation. Inspecial cases they were established by H. Hofer, K. Wysocki and E. Zehnder [25]and by C. Wendl [41], [42]. Proofs usually require a ’starting point’, i.e. a finiteenergy foliation for a slightly different situation as the given one. Then some kindof continuation argument is employed where all kinds of things can and do happento the original foliation. In [25] the authors start with an explicit finite energy folia-tion for the round three dimensional sphere S ⊂ R which is then deformed. ChrisWendl’s papers also use a rather special manifold as a starting point. The mainresult of this paper, theorem 1.6, provides a ’starting finite energy foliation’ forany closed three dimensional contact manifold ( M, ker λ ) since it is obtained fromdeforming the leaves of Giroux’s open book decomposition. The pages are usuallynot punctured spheres, and generically there are no pseudoholomorphic curves onpunctured surfaces with genus which are transverse to the Reeb vector field. Thismakes the introduction of the harmonic form in (1.1) a necessity. The price to bepaid is that compactness issues are more complicated.Chris Wendl [40] published a proof of theorem 1.6 for the special case where ker λ is a planar contact structure, i.e. the surfaces ˙ S are punctured spheres. This resultwas outlined in the article [2]. Regardless of whether the contact structure is planaror not, there are two main steps in the proof: Existence of a solution and Compact-ness of a family of solutions. While the author established the compactness part fortheorem 1.6 long before the article [2] appeared we will use the same argument as inWendl’s article [40] for the existence part since it simplifies the proof considerably.The main theorem of this article was the first step in the proof of the Weinsteinconjecture for the planar case in [2]. Recall that the Weinstein conjecture statesthe following: Conjecture (A. Weinstein, 1978):
Every Reeb vector field X on a closed contact manifold M admits a periodic orbit. OLOMORPHIC OPEN BOOK DECOMPOSITIONS 7
In fact, Weinstein added the additional hypothesis that the first cohomologygroup H ( M, R ) with real coefficients vanishes, but there seems to be no indicationthat this additional hypothesis is needed.Moreover, theorem 1.6 is also the starting point for the construction of globalsurfaces of section in the forthcoming paper [4]. Another application will be an al-ternative proof of the Weinstein conjecture in dimension three [4] as outlined in thepaper [2]. This complements Clifford Taubes’ recent proof of the Weinstein conjec-ture in dimension three using a perturbed version of the Seiberg-Witten equations[37], [38]. The main issue with the homological perturbed holomorphic curve equa-tion (1.1) is that there is no natural compactification of the space of solutions unlessthe harmonic forms are uniformly bounded. In the forthcoming papers [3], [4] thelack of compactness is investigated, and bounds for the harmonic forms are derivedin particular cases.I am very grateful to Richard Siefring and Chris Wendl for explaining some oftheir work to me. Their results are indispensable for the arguments in this article.I would also like to thank Samuel Lisi for having numerous discussions with meabout the subject of this article.2. Existence and local foliations
Local model near the binding orbits.
We will use the same approach as in[40] and [41] to prove existence of a solution to (1.1). Given a closed contact threemanifold (
M, ξ ), Giroux’s theorem implies that there is an open book decompositionas in theorem 1.2 supporting ξ . On the other hand, any other contact structure ξ ′ supported by the same open book is diffeomorphic to ξ . Starting with an openbook decomposition for M , we construct a contact structure supported by it withGiroux contact form λ which has a certain normal form near the binding. Definition 2.1.
Let θ ∈ S = R / π Z , denote polar coordinates on the unit disk D ⊂ R by ( r, φ ), and let γ , γ : [0 , + ∞ ) → R be smooth functions. A 1-form λ = γ ( r ) dθ + γ ( r ) dφ is called a local model near the binding if the following conditions are satisfied:(1) The functions γ , γ and γ ( r ) /r are smooth if considered as functions onthe disk D . In particular, γ ′ (0) = γ ′ (0) = γ (0) = 0.(2) µ ( r ) := γ ( r ) γ ′ ( r ) − γ ′ ( r ) γ ( r ) > r > γ (0) > γ ′ ( r ) < r > r → µ ( r ) r = γ (0) γ ′′ (0) > κ := γ ′′ (0) γ ′′ (0) / ∈ Z and κ ≤ − CASIM ABBAS (6) A ( r ) = 1 µ ( r ) ( γ ′′ ( r ) γ ′ ( r ) − γ ′′ ( r ) γ ′ ( r ))is of order r for small r > λ ∧ dλ = µ ( r ) dθ ∧ dr ∧ dφ = µ ( r ) r dθ ∧ dx ∧ dy the form λ is a contact form on S × D . The Reeb vector field is given by X ( θ, r, φ ) = γ ′ ( r ) µ ( r ) ∂∂θ − γ ′ ( r ) µ ( r ) ∂∂φ =: α ( r ) ∂∂θ + β ( r ) ∂∂φ . The trajectories of X all lie on tori T r = S × ∂D r :(2.1) θ ( t ) = θ + α ( r ) t , φ ( t ) = φ + β ( r ) t. We compute lim r → α ( r ) = lim r → γ ′′ ( r ) µ ′ ( r ) = γ ′′ (0) γ (0) γ ′′ (0) = 1 γ (0)and lim r → β ( r ) = − lim r → γ ′′ ( r ) µ ′ ( r ) = − γ ′′ (0) γ (0) γ ′′ (0)Recalling that ∂∂φ = x ∂∂y − y ∂∂x we obtain for r = 0 X = 1 γ (0) ∂∂θ i.e. the central orbit has minimal period 2 πγ (0). If the ratio α ( r ) /β ( r ) is irrationalthen the torus T r carries no periodic trajectories. Otherwise, T r is foliated withperiodic trajectories of minimal period τ = 2 πmα = 2 πnβ where α/β = m/n or β/α = n/m for suitable integers m, n (choose whatever makessense if either α or β is zero). We calculatelim r → dαdr = lim r → γ ′′ ( r ) µ ( r ) − γ ′ ( r ) µ ′ ( r ) µ ( r )= lim r → γ ′′′ ( r )2 µ ′ ( r ) − lim r → µ ′′ ( r )2 µ ′ ( r ) γ ′ ( r ) r rµ ( r )= γ ′′′ (0)2 µ ′ (0) − γ (0) γ ′′′ (0) γ ′′ (0)2( µ ′ (0)) = 0since µ ′′ (0) = γ (0) γ ′′′ (0) and µ ′ (0) = γ (0) γ ′′ (0) >
0. Converting to Cartesiancoordinates on the disk we get X ( θ, x, y ) = α ( x, y ) ∂∂θ − β ( x, y ) y ∂∂x + β ( x, y ) x ∂∂y , OLOMORPHIC OPEN BOOK DECOMPOSITIONS 9 and linearizing the Reeb vector field along the center orbit yields DX ( θ, ,
0) = − β (0)0 β (0) 0 The linearization of the Reeb flow is given by(2.2) Dφ t ( θ, ,
0) = β (0) t − sin β (0) t β (0) t cos β (0) t with Φ( t ) = e β (0) tJ , J = (cid:18) −
11 0 (cid:19) . the spectrum of Φ( t ) is giving by σ (Φ( t )) = { e ± iβ (0) t } . The binding orbit has period 2 πγ (0). Since γ (0) β (0) = − γ ′′ (0) γ ′′ (0) / ∈ Z it is nondegenerate and elliptic. Example 2.2.
For the contact form
T dθ + k ( x dy − y dx ) = T dθ + r k dφ the centralorbit S × { } is degenerate, but λ = (1 − r )( T dθ + r k dφ ) is a local model near the binding if k, T > , kT / ∈ Z and kT ≥ In this case µ ( r ) = 2 rTk (1 − r ) > and γ ′′ (0) γ ′′ (0) = − kT, and we note that A ( r ) = 1 µ ( r ) ( γ ′′ ( r ) γ ′ ( r ) − γ ′′ ( r ) γ ′ ( r )) = 4 krT (1 − r ) . If α ( r ) β ( r ) = − γ ′ ( r ) γ ′ ( r ) = 1 − r kT = mn for integers n, m then the invariant torus T r is foliated with periodic orbits. Thecase m = 0 is only possible if r = √ . If r is sufficiently small then | m | ≥ .Indeed, we would otherwise be able to find sequences r l ց and { n l } ⊂ Z such that kT / (1 − r l ) = n l which is impossible. The binding orbit has period πT while theperiodic orbits close to the binding orbit have much larger periods equal to τ = 2 πT m (1 − r ) − r . Example 2.3.
Consider the contact form λ = T (1 − r ) dθ + r k dφ on S × D .It is also a local model near the binding if k, T > , kT ≥ and kT is not aninteger. We even have A ( r ) ≡ . In contrast to the previous example, if kT / ∈ Q the invariant tori T r carry no periodic orbits. If kT = nm ∈ Q , but not in Z , allinvariant tori are foliated with periodic orbits of period πmT with | m | ≥ whilethe binding orbit has period πT . The function A ( r ) is identically zero. This is thecontact form on the ’irrational ellipsoid’ in R . The following proposition is essentially proposition 1 from [40]. The constructionin the proof was used by Thurston and Winkelnkemper [39] to show existence ofcontact forms on closed three manifolds.
Proposition 2.4.
Let M be a three dimensional manifold given by an open bookdecomposition M = W ( h ) [ Id ( ∂W × D ) as described in theorem 1.2. We denote the pages by F α := ( W × { α } ) [ Id ( ∂W × I α ) , ≤ α < π, where I α := { re iα ∈ D | < r < } , and the binding ∂W × { } by K . Moreover, let λ be a contact form on ∂W × D which is a local model near the binding on eachconnected component of ∂W × D .Then there is a smooth family of 1-forms ( λ δ ) ≤ δ< on M such that • The form λ is a confoliation 1-form, i.e. λ ∧ dλ ≥ , and ker λ agreeswith the tangent spaces to the pages F ϑ away from the binding • For δ > the forms λ δ are contact forms such that ker λ δ is supported bythe above open book. In particular, the Reeb vector fields X λ δ are transverseto the pages F α , and the binding K consists of periodic orbits of X λ δ . • The forms λ δ agree with the local model λ near the binding. In particular,the binding orbits are nondegenerate and elliptic.Proof. We will first construct contact/confoliation forms λ on W ( h ), depend-ing smoothly on a parameter δ ≥
0, that we control well near the boundary ∂W ( h ) ≈ ∂W × S . Then we will glue these forms together with λ in a smoothway to obtain a contact form on W ( h ) S Id( ∂W × D ) for δ > δ = 0. This procedure was used by Thurston and Winkelnkemper [39]where they showed that every open book is supported by some contact structure.Starting with an open book as above, we can find a collar neighborhood C of ∂W so that h ( t, θ ) = ( t, θ ) for all ( t, θ ) ∈ C . Here we identify ( C, ∂W ) with([0 , ε ] × ( ˙ S n S ) , { } × ( ˙ S n S )) where we take an n–fold disjoint union of circles S ≈ R / π Z according to the number n of components of ∂W .We claim that there is an area form Ω on W that satisfies • R W Ω = 2 πn , • Ω | C = dt ∧ dθ. Indeed, start with any area form Ω ′ so that R W Ω ′ = 2 πn . Then we have Ω ′ | C = f ′ ( t, θ ) dt ∧ dθ with a positive smooth function f ′ (after switching signs if necessary).Pick now a new smooth positive function f which is equal to some constant c if t ≤ ε and agrees with f ′ if t ≥ ε so that the resulting area form Ω still satisfies R W Ω = 2 πn . Do one component of ∂W at a time. Rescaling the t–coordinate we OLOMORPHIC OPEN BOOK DECOMPOSITIONS 11 may assume that c = 1.Let α be any 1-form on W which equals (1 + t ) dθ near ∂W . Then we obtainby Stokes’ theorem: Z W (Ω − dα ) = 2 πn − Z ∂W α = 2 πn + Z ∂W dθ = 0 . The 2–form Ω − dα on W is closed and vanishes near ∂W . Then there exists a1-form β on W with dβ = Ω − dα and β ≡ ∂W . Define now α := α + β . Then α satisfies:(2.3) dα is an area form on W inducing the same orientation as Ω , (2.4) α = (1 + t ) dθ near ∂W . The set of 1-forms on W satisfying (2 .
3) and (2 .
4) is therefore nonempty and alsoconvex. We define the following 1-form on W × [0 , π ], where α is any 1–form on W satisfying (2 .
3) and (2 . α ( x, τ ) := τ α ( x ) + (2 π − τ )( h ∗ α )( x ) . This 1-form descends to the quotient W ( h ) and the restriction to each fiber of thefiber bundle W ( h ) π → S satisfies condition (2 . h ≡ Id near ∂W we have ˜ α ( x, τ ) = 2 π (1 + t ) dθ for all ( x, τ ) = (( t, θ ) , τ ) near ∂W ( h ) = ∂W × S .Let dτ be a volume form on S . We claim that λ := − δ ˜ α + π ∗ dτ are contact forms on W ( h ) whenever δ > x, τ ) ∈ W ( h )and let { u, v, w } be a basis of T ( x,τ ) W ( h ) with π ∗ u = π ∗ v = 0. Then( λ ∧ dλ )( x, τ )( u, v, w )= δ (˜ α ∧ d ˜ α )( x, τ )( u, v, w ) − δ [ dτ ( π ∗ w ) d ˜ α ( x, τ )( u, v )] = 0for sufficiently small δ >
0, and dλ is a volume form on W . Now we have tocontinue the contact forms λ beyond ∂W ( h ) ≈ ∂W × S onto ∂W × D . Atthis point it is convenient to change coordinates. We identify C × S with ∂W × ( D ε \ D ), where D ρ is the 2-disk of radius ρ . Using polar coordinates ( r, φ ) on D ε with 0 ≤ φ ≤ π and 0 < r ≤ ε , our old coordinates are related to thenew ones by ∂W × ( D ε \ D ) ∋ ( θ, r, φ ) ≈ ( θ, t, τ ) ∈ C × S and λ is given by λ = − δ π r dθ + dφ on ∂W × ( D ε \ D ), with ε sufficiently small so that (2.4) holds. We will from nowon drop the factor 1 / π , absorbing it into the constant δ . We have to extend thisnow smoothly to a contact form on ∂W × D ε which agrees with λ near { r = 0 } .We set λ = γ ( r ) dθ + γ ( r ) dφ where γ , γ satisfy the conditions in definition 2.1 for small r , say r ≤ ε , and γ ( r ) = − δr , γ ( r ) = 1 for r ≥ − ε If we write γ ( r ) = γ ( r ) + iγ ( r ) = ρ ( r ) e iα ( r ) then µ ( r ) := γ ( r ) γ ′ ( r ) − γ ′ ( r ) γ ( r ) = ℜ ( iγ ( r ) γ ′ ( r )) = ρ ( r ) α ′ ( r )which has to be positive. Also recall from definition 2.1 that γ (0) > γ ′ ( r ) < r > γ = γ δ have to turn counterclockwise in the first quadrant startingat the point ( γ (0) ,
0) and later connecting with ( − δ (1 − ε ) , δ > X δ ( θ, r, φ ) = γ ′ ( r ) µ ( r ) ∂∂θ − γ ′ ( r ) µ ( r ) ∂∂φ , and in particular(2.5) X δ ( θ, r, φ ) = ∂∂φ for r ≥ − ε which implies that the Reeb vector fields X δ converge as δ ց
0. In addition to λ δ being contact forms for δ > λ δ , hence X δ needs to be transverse to the pages of the openbook decomposition which is equivalent to γ ′ ( r ) = 0. A curve γ ( r ) fulfilling theseconditions can clearly be constructed. (cid:3) The following result shows that we can always assume that a Giroux contactform is equal to any of the forms provided by 2.4.
Proposition 2.5.
Let M be a closed three dimensional manifold with contact struc-ture ξ . Then for every δ > there is a diffeomorphism ϕ δ : M → M such that ker λ δ = ϕ ∗ ξ where λ δ is given by proposition 2.4.Proof. Existence of an open book decomposition supporting ξ follows from the ex-istence part of Giroux’s theorem. On the other hand, proposition 2.4 yields contactforms λ δ such that ker λ δ is also supported by the same open book decompositionas ξ for any δ >
0. By the uniqueness part of Giroux’s theorem, ξ and ker λ δ arediffeomorphic. (cid:3) It follows from our previous construction of the forms λ δ that λ satisfies λ ∧ dλ > ∂W × D − ε , and λ = dφ otherwise. For δ → X δ will converge to some vector field X which is the Reeb vector field of λ if r < − ε and which equals ∂∂φ everywhere else. Proposition 2.6.
Let M be a closed three dimensional manifold with an open bookdecomposition and a family of 1-forms λ δ , δ ≥ as in proposition 2.4. Then thereare • a smooth family ( ˜ J δ ) δ ≥ of almost complex structures on T ( R × M ) whichare R -independent, which satisfy ˜ J δ ( X δ ) = − ∂/∂τ where τ denotes thecoordinate on R , so that J δ := ˜ J δ | ker λ δ are dλ δ -compatible whenever λ δ isa contact form • A parametrization of the Giroux leaves u α : ˙ S → M , α ∈ [0 , π ] , where ˙ S = S \{ p , . . . , p n } , and S is a closed surface. OLOMORPHIC OPEN BOOK DECOMPOSITIONS 13 • A smooth family of smooth functions a α : ˙ S → R such that ˜ u α = ( a α , u α ) : ˙ S → R × M is a family of embedded ˜ J -holomorphiccurves for a suitable smooth family of complex structures j α on S which restrict tothe standard complex structure on the cylinder [0 , + ∞ ) × S after introducing polarcoordinates near the punctures. Moreover, all the punctures are positive and thefamily (˜ u α ) ≤ α ≤ π is a finite energy foliation, and the curves ˜ u α are ˜ J δ -holomorphicnear the punctures.Proof. We parameterize the leaves of the open book decomposition u α : ˙ S → M ,0 ≤ α < π , and we assume that they look as follows near the binding(2.6) u α : [0 , + ∞ ) × S −→ S × D u α ( s, t ) = (cid:0) t, r ( s ) e iα (cid:1) where r are smooth functions with lim s →∞ r ( s ) = 0 to be determined shortly. Weuse the notation ( r, φ ) for polar coordinates on the disk D = D . We identify someneighborhood U of the punctures of ˙ S with a finite disjoint union of half-cylinders[0 , + ∞ ) × S . Recall that the binding orbit is given by x ( t ) = (cid:18) tγ (0) , , (cid:19) , ≤ t ≤ πγ (0)and it has minimal period T = 2 πγ (0). We define smooth functions a α : ˙ S → R by a α ( z ) := (cid:26) R s γ ( r ( s ′ )) ds ′ if z = ( s, t ) ∈ [0 , + ∞ ) × S ⊂ U z / ∈ U so that u ∗ α λ ◦ j = da α , where j is a complex structure on ˙ S which equals the standard structure i on[0 , + ∞ ) × S , i.e. near the punctures. We want to turn the maps ˜ u α = ( a α , u α ) :˙ S → R × M into ˜ J -holomorphic curves for a suitable almost complex structure ˜ J on R × M . Recall that the contact structure is given byker λ δ = Span { η , η } = Span (cid:26) ∂∂r , − γ ( r ) ∂∂θ + γ ( r ) ∂∂φ (cid:27) . We define complex structures J δ : ker λ δ → ker λ δ by(2.7) J δ ( θ, r, φ ) (cid:18) − γ ( r ) ∂∂θ + γ ( r ) ∂∂φ (cid:19) := − h ( r ) ∂∂r and J δ ( θ, r, φ ) ∂∂r := h ( r ) (cid:18) − γ ( r ) ∂∂θ + γ ( r ) ∂∂φ (cid:19) where h : (0 , → R \{ } are suitable smooth functions. Also recall that γ , γ depend on δ away from the binding orbit. We want J δ to be compatible with d λ δ ,i.e. dλ δ ( η , Jη ) = h ( r ) µ ( r ) > dλ δ ( η , Jη ) = µ ( r ) h ( r ) > A puncture p j is called positive for the curve ( a α , u α ) if lim z → p j a α ( z ) = + ∞ . so that h ( r ) >
0. We also demand that J δ extends smoothly over the binding { r = 0 } . Expressing the vectors η and η in Cartesian coordinates, we have η = 1 r (cid:18) x ∂∂x + y ∂∂y (cid:19) and η = − γ ( r ) ∂∂θ + γ ( r ) x ∂∂y − γ ( r ) y ∂∂x . We introduce the following generators of the contact structure: ε := γ ( r ) ∂∂y − xγ ( r ) r ∂∂θ = yγ ( r ) r η + xr η and ε := γ ( r ) ∂∂x + yγ ( r ) r ∂∂θ = xγ ( r ) r η − yr η . We compute from this η = 1 rγ ( r ) ( y ε + x ε ) , η = x ε − y ε . Now J δ ε = yγ ( r ) h ( r ) r η − xr h ( r ) η = (cid:18) r xyγ ( r ) h ( r ) − xyr h ( r ) γ ( r ) (cid:19) ε −− (cid:18) r y γ ( r ) h ( r ) + x r γ ( r ) h ( r ) (cid:19) ε and J δ ε = xγ ( r ) h ( r ) r η + yr h ( r ) η = (cid:18) − r xyγ ( r ) h ( r ) + xyr h ( r ) γ ( r ) (cid:19) ε + (cid:18) r x γ ( r ) h ( r ) + y r γ ( r ) h ( r ) (cid:19) ε . Inserting x = r cos φ , y = r sin φ and demanding for a φ -independent limit as r → r h ( r ) γ ( r ) ≡ ± r . Recalling that weneed h > h ( r ) = 1 r γ ( r ) for small r. As usual, we continue J δ to an almost complex structure ˜ J δ on R × M by setting˜ J δ ( θ, r, φ ) ∂∂τ := X δ ( θ, r, φ ) OLOMORPHIC OPEN BOOK DECOMPOSITIONS 15 where τ denotes the coordinate in the R -direction. We emphasize that ˜ J δ alsomakes sense for δ = 0. We will now arrange r ( s ) in (2.6) such that the Girouxleaves ˜ u α = ( a α , u α ) become ˜ J -holomorphic curves . We compute for r ≤ − ε ∂ s ˜ u α + ˜ J ( u α ) ∂ t ˜ u α = γ ( r ) ∂∂τ + r ′ ∂∂r + ˜ J ( u α ) (cid:18) ∂∂θ (cid:19) = γ ( r ) ∂∂τ + r ′ ∂∂r + ˜ J ( u α )( γ ( r ) X δ ( u α )) ++ ˜ J ( u α ) (cid:18) ∂∂θ − γ ( r ) X δ ( u α ) (cid:19) = r ′ ∂∂r + ˜ J ( u α ) (cid:18) γ ′ ( r ) µ ( r ) (cid:18) γ ( r ) ∂∂φ − γ ( r ) ∂∂θ (cid:19)(cid:19) = (cid:18) r ′ − γ ′ ( r ) µ ( r ) h ( r ) (cid:19) ∂∂r hence the Giroux leaves satisfy the equation if we choose r to be a solution of theordinary differential equation r ′ ( s ) = γ ′ ( r ( s )) µ ( r ( s )) h ( r ( s )) . Note that r ′ ( s ) <
0. We choose also h ( r ) ≡ r ≥ − ε . We continuethe almost complex structures J δ : ker λ δ → ker λ δ (which were only defined nearthe binding) smoothly to all of M . Away from the binding we have X δ = ∂/∂φ ,and we extend J δ as before to T ( R × M ). Away from the binding, if δ = 0, wehave that ker λ coincides with the tangent spaces of the pages of the open bookdecomposition. Because a α is constant away from the binding, the solutions ˜ u α which we constructed near the binding fit together smoothly with the pages of theopen book decomposition and solve the holomorphic curve equation for the almostcomplex structure J . (cid:3) Remark . Near the binding orbit the function r ( s ) satisfies a differential equationof the form r ′ ( s ) = Λ( r ( s )) r ( s ) := γ ′ ( r ( s )) γ ( r ( s )) µ ( r ( s )) r ( s )and lim r → Λ( r ) = γ ′′ (0) γ ′′ (0) =: κ. Writing r ( s ) = c ( s ) e κs the function c ( s ) satisfies c ′ ( s ) = (Λ( r ( s )) − κ ) c ( s ), hence itis a decreasing function which converges to a constant as s → + ∞ .We will return to examples 2.2 and 2.3 and compute r ( s ) for large s . Thedifferential equation in the case of example 2.3 for large s is r ′ ( s ) = γ ′ ( r ( s )) γ ( r ( s )) µ ( r ( s )) r ( s ) = − kT (1 − r ( s )) r ( s ) , so that r ( s ) = 1 √ c e kT s , The calculation shows that we can make them J δ -holomorphic for all δ ≥ where c is a constant. In example 2.2 the differential equation reads r ′ ( s ) = γ ′ ( r ( s )) γ ( r ( s )) µ ( r ( s )) r ( s ) = − kT − r ( s ) r ( s ) , and solutions satisfy r ( s ) = c e − kT s e r ( s ) . Functional Analytic set-up and the Implicit Function Theorem.
Inthe following theorem we will prove the existence of a smooth family of solutionsnear a given solution. In proposition 2.6 we constructed a finite energy foliationfor the data ( λ , J ) with vanishing harmonic form. The form λ however is onlya confoliation form. We will produce solutions for the perturbed data ( λ δ , J δ ), andharmonic forms will appear if the surface S is not a sphere. The key result is anapplication of the implicit function theorem in a suitable setting. Theorem 2.8.
Assume one of the following: (1)
Let ( a , u ) : ˙ S → R × M be one of the ˜ J -holomorphic curves described inproposition 2.6 with complex structure j on S (we refer to such u as a’Giroux leaf ’) and confoliation form λ or (2) let ( ˙ S, j , a , u , γ ) be a solution of the differential equation (1.1) for some dλ -compatible complex structure J : ker λ → ker λ which, near the bind-ing orbit, agrees with (2.7), and where λ is a contact form which is a localmodel near the binding. Assume that u is an embedding and that it isof the form u = φ g ( v ) where g : S → R is a smooth function, φ is theflow of the Reeb vector field and where v : ˙ S → M is a Giroux leaf as inproposition 2.6Let J δ be a smooth family of dλ δ -compatible complex structures also agreeing with(2.7) near the binding orbit, where ( λ δ ) − ε<δ< + ε , ε > is a smooth family of 1-forms which are contact forms for δ = 0 and local models near the binding. Thenthere is a smooth family ( S, j δ,τ , a δ,τ , u δ,τ , γ δ,τ , J δ ) − ε<δ,τ< + ε of solutions of equation (1.1) so that u δ,τ ( ˙ S ) ∩ u δ,τ ′ ( ˙ S ) = ∅ whenever τ = τ ′ andeach u δ,τ is an embedding.Proof. In both cases we wish to find solutions of (1.1) for the data ( λ δ , J δ ) of theform u δ ( z ) = φ f δ ( z ) ( u ( z )) , a δ ( z ) = b δ ( z ) + a ( z )where t φ t = φ δt is the flow of the Reeb vector field X δ of λ δ , and where b δ + if δ : S → C is a smooth function defined on the unpunctured surface. We willderive an equation for the unknown function b δ + if δ . From now we will suppressthe superscript δ in the notation unless for δ = 0. Because of the first equation in(1.1) the complex structure on S is then determined by f , denote it by j = j f , andit is given by(2.8) j f ( z ) = ( π λ T u ( z )) − ◦ J ( u ( z )) ◦ π λ T u ( z )Note that this is well defined because u is transverse to the Reeb vector field sothat π λ T u ( z ) : T z S → ker λ ( u ( z )) is an isomorphism. By the second equation of OLOMORPHIC OPEN BOOK DECOMPOSITIONS 17 (1.1), we then have to solve the equation df ◦ j f + u ∗ λ ◦ j f = da + γ for a, f, γ on˙ S which is equivalent to the equation(2.9) ¯ ∂ j f ( a + if ) = u ∗ λ ◦ j f − i ( u ∗ λ ) − γ − i ( γ ◦ j f ) . Recalling that we are looking for a of the form a = a + b where b is a suitable realvalued function defined on the whole surface S . We obtain the differential equation(2.10) ¯ ∂ j f ( b + if ) = u ∗ λ ◦ j f − i ( u ∗ λ ) − ¯ ∂ j f a − γ − i ( γ ◦ j f ) . and it follows from a straight forward calculation (see appendix A) that all expres-sions on the right hand side of equation (2.10) are bounded near the punctures, inparticular they are contained in the spaces L p ( T ∗ S ⊗ C ) for any p . This is what theassumption κ ≤ − from definition 2.1 is needed for. We will work in the functionspace b + if ∈ W ,p ( S, C ) where p >
2. For any complex structure j on S the space L p ( T ∗ S ⊗ C ) of complex valued 1–forms of class L p decomposes into complex linearand complex antilinear forms (with respect to j ). We use the notation L p ( T ∗ S ⊗ C ) = L p ( T ∗ S ⊗ C ) , j ⊕ L p ( T ∗ S ⊗ C ) , j . The operator b + if ¯ ∂ j f ( b + if ) is then a section in the vector bundle L p ( T ∗ S ⊗ C ) , := [ b + if ∈ W ,p ( S, C ) { b + if } × L p ( T ∗ S ⊗ C ) , j f → W ,p ( S, C ) . This vector bundle is of course trivial, but here are some explicit local trivializationsfor f , g ∈ W ,p ( S, R ) sufficiently close to each other:(2.11) Ψ fg : L p ( T ∗ S ⊗ C ) , j f g −→ L p ( T ∗ S ⊗ C ) , j g τ τ + i ( τ ◦ j g ) . If we write τ + i ( τ ◦ j g ) = τ ◦ (Id T S − j f ◦ j g ) , we see that Ψ fg is invertible withΨ − fg τ = τ ◦ (Id T S − j f ◦ j g ) − . It follows from the Hodge decomposition theorem that every cohomology class[ σ ] ∈ H ( S, R ) has a unique harmonic representative ψ j ( σ ) ∈ H j ( S ) where H j ( S )is defined as(2.12) H j ( S ) := { γ ∈ E ( S ) | dγ = 0 , d ( γ ◦ j ) = 0 } and where E ( S ) denotes the space of all (smooth) real valued 1–forms on S , andwe write E , ( S ) = E , j ( S ) for the space of complex antilinear 1–forms on S withrespect to j , i.e. complex valued 1–forms σ such that i σ + σ j = 0. Note that ourdefinition coincides with the set of closed and co-closed 1-forms on S . Moreover, byelliptic regularity, we may also consider Sobolev forms. We will identify H ( S, R )with R g , and we consider the following parameter dependent section in the bundle L p ( T ∗ S ⊗ C ) , → W ,p ( S, C )(2.13) F : W ,p ( S, C ) × R g −→ L p ( T ∗ S ⊗ C ) , F ( b + if, σ ) := ¯ ∂ j f ( b + if ) − u ∗ λ ◦ j f + i ( u ∗ λ ) ++ ¯ ∂ j f a + ψ j f ( σ ) + i ( ψ j f ( σ ) ◦ j f ) with j f as in (2.8). Recalling that z j f ( z ) may not be differentiable, we interpretthe equation d ( γ ◦ j f ) = 0 in the sense of weak derivatives. The solution set of (2.10)is then the zero set of F . We consider the real parameter δ which we dropped fromthe notation, fixed at the moment. For g ≡ b + if small in the W ,p –normwe consider the composition ˆ F ( b + if, σ ) = Ψ fg ( F ( b + if, σ )). Its linearizationin the point ( b + if, σ ) = (0 , σ ) where σ is defined by ψ j ( σ ) = γ , and where F (0 , σ ) = 0, is D ˆ F (0 , σ ) : W ,p ( S, C ) × R g −→ L p ( T ∗ S ⊗ C ) , j D ˆ F (0 , σ )( ζ, σ ) = ¯ ∂ j ζ + ψ j ( σ ) + i ( ψ j ( σ ) ◦ j ) + L ζ, where L : W ,p ( S, C ) → W ,p ( T ∗ S ⊗ C ) , j ֒ → L p ( T ∗ S ⊗ C ) , j is a compact linear map because we are working on a compact domain S . Thelinear term L therefore does not contribute to the Fredholm–index of D ˆ F (0 , σ ).We claim that the operator W ,p ( S, C ) × R g −→ L p ( T ∗ S ⊗ C ) , j ( ζ, σ ) ¯ ∂ j ζ + ψ j ( σ ) + i ( ψ j ( σ ) ◦ j )is a surjective Fredholm operator of index two. Then we would have ind( D ˆ F (0 , σ )) =2 as well. Here is the argument: The Riemann Roch theorem asserts that the kerneland the cokernel of the Cauchy-Riemann operator ¯ ∂ j (acting on smooth complexvalued functions on S ) are both finite dimensional and thatdim R ker ¯ ∂ j − dim R ( E , ( S ) / Im ¯ ∂ j ) = 2 − g, where g is the genus of the surface S . The only holomorphic functions on S are theconstant functions, hence E , ( S ) / Im ¯ ∂ j has dimension 2 g .On the other hand, the vector space H j ( S ) of all (real–valued) harmonic 1–formson S also has dimension 2 g (see [16]). We consider now the linear mapΨ : H j ( S ) −→ E , ( S ) / Im ¯ ∂ j Ψ( γ ) := [ γ + i ( γ ◦ j )] , where [ . ] denotes the equivalence classes of (0 , γ ) = [0],i.e. there is a complex–valued smooth function f = u + iv on S such that ¯ ∂ j f = γ + i ( γ ◦ j ). Since γ is a harmonic 1–form, we conclude that d ( dv ◦ j ) = d ( du ◦ j ) = 0,i.e. both u and v are harmonic. Since there are only constant harmonic functionson S we obtain γ = 0, i.e. Ψ is injective and also bijective. Hence (0 , γ + i ( γ ◦ j ) with γ ∈ H j ( S ) make up the cokernel of ¯ ∂ j : C ∞ ( S, C ) → E , ( S ). The linear map ζ Lζ only depends on the imaginary part of ζ . It is actually given by Lζ = − u ∗ λ ◦ ( Aζ + j Aζj ) + i u ∗ λ ◦ ( j Aζ − Aζj ) + Bζ + i Bζ j where Bζ = ddτ (cid:12)(cid:12)(cid:12)(cid:12) τ =0 ψ j τ k ( σ ) , ζ = h + ik and Aζ = ddτ (cid:12)(cid:12)(cid:12)(cid:12) τ =0 j τ k = h ( π λ T u ) − h J ( u ) DX λ ( u ) − DX λ ( u ) J ( u ) + DJ ( u ) X λ ( u ) i ( π λ T u ) . OLOMORPHIC OPEN BOOK DECOMPOSITIONS 19
This proves the claim that the operator D ˆ F (0 , σ ) is Fredholm of index two. Wewill now show that the operator D ˆ F (0 , σ ) is surjective. Using the decomposition L p ( T ∗ S ⊗ C ) , j = R ( ¯ ∂ j ) ⊕ H j ( S )and denoting the corresponding projections by π , π we see that it suffices to provesurjectivity of the operator T : W ,p ( S, C ) → R ( ¯ ∂ j ) T ζ := ¯ ∂ j ζ + π ( Lζ )which is a Fredholm operator of index 2. Assume ζ ∈ ker T . Unless ζ ≡ { z ∈ S | ζ ( z ) = 0 } consists of finitely many points by the Similarity Principle [19]and the local degree of each zero is positive. On the other hand, the sum of all thelocal degrees has to be zero, hence elements in the kernel of T are nowhere zero.Actually, if h + ik ∈ ker T then even k is nowhere zero because h + c + ik ∈ ker T forany real constant c since the zero order term L only depends on the imaginary partof ζ . Therefore dim ker T ≤
2, and since the Fredholm index of T equals 2, weactually have dim ker T = 2. This proves surjectivity of T and also of D ˆ F (0 , σ )so that the set M of all pairs ( b + if, γ ) solving the differential equation (2.10)is a two dimensional manifold with T (0 ,γ ) M = ker D ˆ F (0 , γ ). If we add a realconstant to b + if then we obtain again a solution of (2.10). If we divide M by this R –action then we obtain a one–dimensional family of solutions (˜ u τ ) − ε<τ<ε with˜ u τ = ( a τ , u τ ) for which u τ = φ f τ ( u ), and the functions f τ do not vanish at anypoint. Therefore, we have u ( ˙ S ) ∩ u τ ( ˙ S ) = ∅ and also u τ ′ ( ˙ S ) ∩ u τ ( ˙ S ) = ∅ if τ = τ .Moreover, the maps u τ are transverse to the Reeb vector field by construction. (cid:3) From local foliations to global ones
The aim of this section is to show that a family of solutions produced by theimplicit function theorem (theorem 2.8) can be enlarged further. For this purposea compactness result is needed for which we are setting the stage now.First, we will summarize a result by Richard Siefring (theorem 2.2 in [35]) whichwill be used later on:
Theorem 3.1.
Let ˜ u ∈ M ( P, J ) and ˜ v ∈ M ( P, J ) , let maps U, V : [ R, ∞ ) × S → C ∞ ( P ∗ ξ ) be asymptotic representatives of ˜ u and ˜ v , respectively, and assume that U − V does not vanish identically. Then there exists a negative eigenvalue λ of theasymptotic operator A P,J and an eigenvector e with eigenvalue λ so that U ( s, t ) − V ( s, t ) = e λs ( e ( t ) + r ( s, t )) where the map r satisfies for every ( i, j ) ∈ N , a decay estimate of the form |∇ is ∇ jt r ( s, t ) | ≤ M ij e − ds with M ij and d positive constants. Indeed, otherwise we would be able to find three linearly independent elements in the kernel ζ , ζ , ζ . Because C has real dimension two we can find real numbers α , α , α , not all simul-taneously zero, and a point z ∈ S such that P j =1 α j ζ j ( z ) = 0. Then ζ = P j =1 α j ζ j is in thekernel of T and ζ ( z ) = 0, a contradiction. (cid:3) Our situation is less general than in [35], so we will explain the notation in thecontext of this paper. The setup is a manifold M with contact form λ and contactstructure ξ = ker λ . Consider a periodic orbit ¯ P of the Reeb vector field X λ withperiod T , and we may assume here that T is its minimal period. We introduce P ( t ) := ¯ P ( T t/ π ) such that P (0) = P (2 π ). If J : ξ → ξ is a dλ -compatiblecomplex structure the set of all ˜ J -holomorphic half-cylinders˜ u = ( a, u ) : [ R, ∞ ) × S → R × M , S = R / π Z for which | a ( s, t ) − T s/ π | and | u ( s, t ) − P ( t ) | decay at some exponential rate (inlocal coordinates near the orbit P ( S )) is denoted by M ( P, J ). Note that it isassumed here that the domain [ R, ∞ ) × S is endowed with the standard complexstructure. A smooth map U : [ R, ∞ ) × S → P ∗ ξ for which U ( s, t ) ∈ ξ P ( t ) is calledan asymptotic representative of ˜ u if there is a proper embedding ψ : [ R, ∞ ) × S → R × S asymptotic to the identity so that˜ u ( ψ ( s, t )) = ( T s/ π, exp P ( t ) U ( s, t )) ∀ ( s, t ) ∈ [ R, ∞ ) × S (exp is the exponential map corresponding to some metric on M , for example theone induced by λ and J ). Every ˜ u ∈ M ( P, J ) has an asymptotic representative(see [35]). The asymptotic operator A P,J is defined as follows:( A P,J h )( t ) := − T π J ( P ( t )) (cid:18) dds (cid:12)(cid:12)(cid:12)(cid:12) s =0 Dφ − s ( φ s ( P ( t ))) h ( φ s ( P ( t ))) (cid:19) where φ s is the flow of the Reeb vector field, and where h is a section in P ∗ ξ → S . Because the Reeb flow preserves the splitting T M = R X λ ⊕ ξ we have also( A P,J h )( t ) ∈ ξ P ( t ) .We compute the asymptotic operator A P,J for the binding orbit¯ P ( t ) = (cid:18) tγ (0) , , (cid:19) ∈ S × R . Recall that the above periodic orbit has minimal period T = 2 πγ (0). Using φ s ( P ( t )) = φ s ( t, ,
0) = (cid:18) t + sγ (0) , , (cid:19) , formula (2.2) for the linearization of the Reeb flow with h ( t ) = (0 , ζ ( t ) , η ( t )) andthe fact that J ( t, , ∂∂x = ∂∂y and J ( t, , ∂∂y = − ∂∂x we compute( A P,J h )( t ) = − γ (0) J ( t, , (cid:18) h ′ ( t ) γ (0) + (cid:18) β (0) − β (0) 0 (cid:19) (cid:18) ζ ( t ) η ( t ) (cid:19)(cid:19) = − J h ′ ( t ) − γ (0) β (0) h ( t )= − J h ′ ( t ) + κ h ( t )where J = (cid:18) −
11 0 (cid:19) and κ = γ ′′ (0) /γ ′′ (0) ∈ ( − , λ ∈ σ ( A P,J )precisely if h ′ ( t ) = ( λ − κ ) J h ( t ) and h (2 π ) = h (0) , i.e. σ ( A P,J ) = { κ + l | l ∈ Z } , and the largest negative eigenvalue is given by κ .The corresponding eigenspace consists of all constant vectors h ( t ) ≡ const ∈ R .The eigenspace for the eigenvalues κ + l consists of all h ( t ) = e J l t h with h ∈ R . OLOMORPHIC OPEN BOOK DECOMPOSITIONS 21
Theorem 3.2. (Compactness)
Let λ be a contact form on M which is a local model near the binding (of the Girouxleaf v ), and let J : ker λ → ker λ be a dλ -compatible complex structure. Considera smooth family of solutions ( S, j τ , a τ , u τ , γ τ , J ) ≤ τ<τ to equation (1.1) satisfyingthe following conditions: • u τ = φ f τ ( v ) where v : ˙ S → M is a Giroux leaf as in proposition 2.6 and f τ : S → R are suitable smooth functions. • For any ≤ τ < τ there is δ > such that u τ ( ˙ S ) ∩ u τ ′ ( ˙ S ) = ∅ whenever < | τ − τ ′ | < δ. • Assume that u and u τ never have identical images whenever < τ < τ .Then the functions f τ converge uniformly with all derivatives to a smooth function f τ : S → R as τ ր τ . The harmonic 1–forms γ τ also converge in C ∞ ( S ) to a1–form γ τ which is harmonic with respect to the complex structure j τ on ˙ S givenby j τ ( z ) := ( π λ T u τ ( z )) − ◦ J ( u τ ( z )) ◦ π λ T u τ ( z ) , where u τ := φ f τ ( v ) . Moreover, we can find a smooth function a τ on ˙ S so that ( S, j τ , a τ , u τ , γ τ , J ) solves the differential equation (1.1).Remark . We may assume without loss of generality that v ≡ u and f ≡ z ∈ ˙ S we denote by T ( z ) > u ( z ), i.e. T ( z ) := inf { T > | φ T ( u ( z )) ∈ u ( ˙ S ) } < + ∞ . We claim that the return time z T ( z ) extends continously over the punctures ofthe surface, and therefore there is an upper bound T := sup z ∈ ˙ S T ( z ) < ∞ Using (2.1) and (2.6), we note that, asymptotically near the punctures, φ T ( u ( s, t )) ∈ S × R has the following structure: φ T ( u ( s, t )) = (cid:0) t + α ( r ( s )) T , r ( s ) exp (cid:2) i ( α + β ( r ( s )) T ) (cid:3)(cid:1) where r ( s ) is a strictly decreasing function, α is some constant, and α ( r ) , β ( r ) aresuitable functions for which the limits lim r → β ( r ) and lim r → α ( r ) exist and arenot zero. Hence, if T = T ( u ( s, t )) is the positive return time at the point u ( s, t ),then T ( u ( s, t )) = 2 π | β ( r ( s )) | , and therefore the limit for s → + ∞ exists.The remainder of this section is devoted to the proof of theorem 3.2. We recallthat the functions a τ and f τ satisfy the Cauchy-Riemann type equation (2.9) whichis ¯ ∂ j τ ( a τ + if τ ) = u ∗ λ ◦ j τ − i ( u ∗ λ ) − γ τ − i ( γ τ ◦ j τ ) , where the complex structure j τ is given by (2.8) or j τ ( z ) = ( π λ T u ( z )) − ( T φ f τ ( z ) ( u ( z ))) − ·· J ( φ f τ ( z ) ( u ( z ))) T φ f τ ( z ) ( u ( z )) π λ T u ( z )and γ τ is a closed 1–form on S with d ( γ τ ◦ j τ ) = 0. The following L ∞ -bound is the crucial ingredient for the compactness result: Weclaim that(3.1) sup ≤ τ<τ k f τ k L ∞ ( ˙ S ) ≤ T. Restricting any of the solutions to a simply connected subset U ⊂ ˙ S we can write γ τ = dh τ for a suitable function h τ : U → R , and the maps˜ u τ : U → R × M , ˜ u τ = ( a τ + h τ , u τ )are ˜ J -holomorphic curves. If two such curves ˜ u τ and ˜ u τ ′ have an isolated intersec-tion then the corresponding intersection number is positive (see [31] or [7], [30] forpositivity of (self)intersections for holomorphic curves). We claim that u ( ˙ S ) ∩ u τ ( ˙ S ) = ∅ ∀ < τ < τ and not just for small τ as assumed. If we can show this then (3.1) follows. Indeed,for any z ∈ ˙ S the function τ f τ ( z ) is strictly increasing from f ( z ) = 0, andequality f τ ( z ) = T ( z ) would imply that u τ ( z ) ∈ u ( ˙ S ). Arguing indirectly, weassume that the set O := { τ ∈ (0 , τ ) | u τ ( ˙ S ) ∩ u ( ˙ S ) = ∅} is not empty. We denote its infimum by ˜ τ which must be a positive number since u τ ( ˙ S ) ∩ u ( ˙ S ) = ∅ for all sufficiently small τ > u ˜ τ and u cannot intersect. If u τ ( p ) = u ( q ) for suitable points p, q ∈ ˙ S then we consider locallynear these points the corresponding holomorphic curves ˜ u τ and ˜ u . Adding someconstant to the R -component of one of them we may assume that ˜ u τ ( p ) = ˜ u ( q ).If this intersection point is not isolated then p and q have open neighborhoods U and V respectively on which the holomorphic curves ˜ u τ and ˜ u agree. This impliesthat the set of all points p ∈ ˙ S such that ˜ u τ ( p ) is a non-isolated intersection pointbetween ˜ u τ and ˜ u , is open and closed, i.e. it is either empty or all of ˙ S . Sincewe assumed that each set u τ ( ˙ S ), τ > u ( ˙ S ) we conclude that if u τ and u intersect then the intersection point of the corresponding holomorphiccurves ˜ u τ and ˜ u must be isolated. But this implies on the other hand that u τ ′ and u would also intersect for all τ ′ sufficiently close to τ by positivity of theintersection number showing that the set O is open.We conclude from the above that there are a sequence τ k ց ˜ τ and points p k , q k ∈ ˙ S such that u τ k ( p k ) = u ( q k ). Passing to a suitable subsequence we may assumeconvergence of the sequences ( p k ) k ∈ N and ( q k ) k ∈ N to points p, q ∈ S . Because of u ˜ τ ( ˙ S ) ∩ u ( ˙ S ) = ∅ the points p, q must be punctures, and they have to be equal z = p = q ∈ S \ ˙ S . The reason for this is the following: The maps u τ k , u areasymptotic near the punctures to a disjoint union of finitely many periodic Reeborbits which are not iterates of other periodic orbits. Also, different puncturesalways correspond to different periodic orbits. This follows from E. Giroux’s resultand our constructions in section 2 of this paper.We now derive a contradiction using Richard Siefring’s result. The harmonic forms γ τ k in equation (1.1) are defined on all of S , hence they are exact on some openneighborhood U of the puncture z and γ τ k = dh τ k for suitable functions h τ k on U and similarly γ ˜ τ = dh ˜ τ . We may also assume that j ˜ τ | U = j τ k | U = j after changinglocal coordinates near z . Then, on the set U , the maps ˜ u τ k = ( a τ k + h τ k , u τ k ) and OLOMORPHIC OPEN BOOK DECOMPOSITIONS 23 ˜ u = ( a + h , u ) are holomorphic curves with ˜ u τ k ( p k ) = ˜ u ( q k ) while the imagesof ˜ u ˜ τ and ˜ u have empty intersection. Let now U ˜ τ , U τ k , U : [ R, ∞ ) × S → R be asymptotic representatives of the holomorphic curves ˜ u ˜ τ , ˜ u τ k , ˜ u respectively.Invoking theorem 3.1 and our subsequent computation of the asymptotic operatorand its spectrum we obtain the following asymptotic formulas(3.2) U τ ( s, t ) − U ( s, t ) = e λ τ s ( e τ ( t ) + r τ ( s, t )) , τ = ˜ τ , τ k , s ≥ R τ where R τ > λ τ < A P,J . It is of the form λ τ = κ + l τ where l τ is an integer, κ = γ ′′ (0) /γ ′′ (0)is not an integer, and where e τ ( t ) = e J l τ t h τ , h τ ∈ R \{ } is an eigenvectorcorresponding to the eigenvalue λ τ = κ + l τ . Note that the above formula appliessince U τ − U can not vanish identically. We will actually show that l τ ≡
0. Theasymptotic representative U is given by u ( s, t ) = ( t, r ( s ) e iα ) = ( t, U ( s, t )) , using equation (2.6), and we recall that r ( s ) = c ( s ) e κs where c ( s ) → c ∞ > s → + ∞ . An asymptotic representative of ˜ u τ is however given by an expressionsuch as u τ ( ψ ( s, t )) = ( t, U τ ( s, t ))where ψ : [ R, ∞ ) × S → R × S is a proper embedding converging to the identitymap as s → + ∞ . Writing ( s ′ , t ′ ) = ψ ( s, t ) we get using formula equations (2.1) forthe Reeb flow U τ ( s, t ) = c ( s ′ ) e κs ′ e i ( α + β ( r ( s ′ )) f τ ( s ′ ,t ′ )) = e κs ( e τ + r τ ( s, t )) . The asymptotic formula for U τ apriori allows for other decay rates but κ is the onlypossible one. Dividing by e κs and passing to the limit s → + ∞ we obtain e τ = c ∞ e iα e iβ (0) f τ ( ∞ ) where f τ ( ∞ ) = lim s → + ∞ f τ ( s, t ) which is independent of t since f τ extends contin-uously over the punctures. Hence the difference U τ − U has decay rate λ τ ≡ κ asclaimed unless the two eigenvectors e τ and e agree which is equivalent to f τ ( ∞ ) ∈ πβ (0) Z or τ = ˜ τ in our case. The maps U τ − U satisfy a Cauchy-Riemann type equationto which the Similarity Principle applies so that for every zero ( s, t ) of U τ − U themap σ ( U τ − U )( s + ǫ cos σ, t + ǫ sin σ ) has positive degree for small ǫ >
0. TheCauchy-Riemann type equation mentioned above is derived in [35] in section 5.3as well as in section 3 of [1] in a slightly different context and also in [23]. If R issufficiently large then the map S → S , t W τ ( R, t ) := U τ − U | U τ − U | ( R, t )is well-defined, and it has degree l τ because the remainder term r τ ( s, t ) decaysexponentially in s . Zeros of U τ − U contribute in the following way: If R ′ < R such that ( U τ − U )( R ′ , t ) = 0 then(3.3) deg W τ ( R, · ) = deg W τ ( R ′ , · ) + X { z | U τ ( z ) − U ( z )=0 } o ( z ) . We know already that l τ = 0 whenever τ = ˜ τ . Arguing indirectly, we assume that l ˜ τ is not zero. It would have to be negative then. Choose then R ′ > W ˜ τ ( R ′ , · ) = l ˜ τ <
0. For τ sufficiently close to ˜ τ we also have deg W τ ( R ′ , · ) = l ˜ τ .On the other hand we have deg W τ ( R, · ) = 0 for R > R ′ sufficiently large. Equation(3.3) implies that the map U τ − U must have zeros in [ R ′ , R ] × S to accountfor the difference in degrees, but we know that there are none for τ < ˜ τ . Thiscontradiction shows that l ˜ τ = 0 is impossible. Choose again R ′ > W ˜ τ ( R ′ , · ) = 0. The degree does not change if we slightly alter τ . In particular,we have deg W τ ( R ′ , · ) = 0 for τ > ˜ τ close to ˜ τ as well. For R >> R ′ we havedeg W ˜ τ ( R, · ) = 0, and we recall that( U τ k − U )( s k , t k ) = 0 , τ k ց ˜ τ for a suitable sequence ( s k , t k ) with s k → + ∞ , and the set of zeros of U τ k − U is discrete. This however contradicts equation (3.3) since the zeros have positiveorders.Summarizing, we have shown that the assumption O 6 = ∅ leads to a contradictionwhich implies the apriori bound (3.1).The monotonicity of the functions f τ in τ and the bound (3.1) imply that thefunctions f τ converge pointwise to a measureable function f τ as τ ր τ . We alsoknow that k f τ k L ∞ ( ˙ S ) ≤ T . We then obtain a complex structure j τ on ˙ S by j τ ( z ) = ( π λ T u ( z )) − ( T φ f τ ( z ) ( u ( z ))) − ·· J ( φ f τ ( z ) ( u ( z ))) T φ f τ ( z ) ( u ( z )) π λ T u ( z ) . By definition the complex structure j τ is also of class L ∞ and j τ ( z ) → j ( z )pointwise. Our task is to improve the regularity of the limit f τ and the characterof the convergence f τ → f τ . We also have to establish convergence of the functions a τ for τ ր τ . The complex structures j τ are of course all smooth, but the limit j τ might only be measureable.3.1. The Beltrami equation.
For the convenience of the reader we briefly sum-marize a few classical facts from the theory of quasiconformal mappings (see thepaper by L. Ahlfors and L. Bers [8], [9]).The punctured surface ˙ S carries metrics g τ , also of class L ∞ for τ = τ and smoothotherwise, so that g τ ( z )( j τ ( z ) v, j τ ( z ) w ) = g τ ( z )( v, w ) for all v, w ∈ T z ˙ S. In fact, g τ is given by g τ ( z )( v, w ) = dλ ( u τ ( z )) (cid:16) π λ T u τ ( z ) v, J ( u τ ( z )) π λ T u τ ( z ) w (cid:17) . In the case τ = τ we replace π λ T u τ ( z ) by T φ f τ ( z ) ( u ( z )) π λ T u ( z ). We havesup τ k g τ k L ∞ ( ˙ S ) < ∞ and g τ → g τ pointwise as τ ր τ . Our considerations aboutthe regularity of the limit are of local nature, so we may replace ˙ S with a ball OLOMORPHIC OPEN BOOK DECOMPOSITIONS 25 B ⊂ C centered at the origin. Denoting the metric tensor of g τ by ( g τkl ) ≤ k,l ≤ wedefine the following complex–valued smooth functions: µ τ ( z ) := ( g τ ( z ) − g τ ( z )) + i g τ ( z ) ( g τ ( z ) + g τ ( z )) + p g τ ( z ) g τ ( z ) − ( g τ ( z )) , and we note that sup τ k µ τ k L ∞ ( ˙ S ) < µ τ → µ τ pointwise. We view the functions µ τ as functions on thewhole complex plane by trivially extending them beyond B . Then they are also τ –uniformly bounded in L p ( C ) for all 1 ≤ p ≤ ∞ and µ τ → µ τ in L p ( C ) for1 ≤ p < ∞ by Lebesgue’s theorem. If we solve now the Beltrami equation ∂α τ = µ τ ∂α τ for τ < τ so that ∂α τ (0) = 0 then α τ is a diffeomorphism of the plane onto itselfso that g τ ( α τ ( z ))( T α τ ( z ) v, T α τ ( z ) w ) = λ τ h v, w i if z ∈ B, where h . , . i denotes the standard Euclidean scalar product on R and λ τ is apositive function. We then get T α τ ( z ) ◦ i = j τ ( α τ ( z )) ◦ T α τ ( z ) , ≤ τ < z ∈ B. For H¨older continuous µ τ the map α τ exists, and it is a C –diffeomorphism. Thisis a classical result by A. Korn and L. Lichtenstein [29]. More modern proofsmay be found for example in [11] and [14]. In our case we have smooth solutions α τ belonging to smooth µ τ , but we only know that the µ τ converge pointwise as τ ր
1. On the other hand, we would like to derive a decent notion of convergencefor the transformations α τ . An interesting case for us is the one where µ is onlya measureable function. Results in this direction were obtained by C. B. Morrey[32], L. Ahlfors and L. Bers [9] and also by L. Bers and L. Nirenberg [12]. We alsorefer to the book [8] by Lars Ahlfors. We summarize now a few results from thepaper [9] about the Beltrami equation for measureable µ which we will need lateron. The first result concerns the inhomogeneous Beltrami equation ∂u = µ ∂ u + σ, where u : C → C , µ is a complex–valued measureable function on C with k µ k L ∞ ( C ) < σ ∈ L p ( C ) for a suitable p > p areadmissible). We consider the following operators acting on smooth functions withcompact support in the plane:( Ag )( z ) := 12 πi Z C g ( ξ ) (cid:18) ξ − z − ξ (cid:19) dξ dξ, (Γ g )( z ) := 12 πi lim ε ց Z C \ B ε (0) g ( ξ ) − g ( z )( ξ − z ) dξ dξ. Both operators can be extended to continuous operators L p ( C ) → L p ( C ) for all2 < p < ∞ . They have the following properties:(1) ∂ ( Ag ) = A ( ∂g ) = g, (2) ∂ ( Ag ) = A ( ∂g ) = Γ g, (3) | Ag ( z ) − Ag ( z ) | ≤ C p k g k L p ( C ) | z − z | − p , (4) k Γ g k L p ( C ) ≤ c p k g k L p ( C ) with c p → p → ∂ = ( ∂ s + i∂ t ) and ∂ = ( ∂ s − i∂ t ). Properties (1)and (2) above should be understood in the sense of distributions. The proof of (4)involves the Calder´on–Zygmund inequality and the Riesz–Thorin convexity theorem(see [28] and [36]). Following [9] we define B p with p >
2, to be the space of alllocally integrable functions on the plane which have weak derivatives in L p ( C ),vanish in the origin and which satisfy a global H¨older condition with exponent1 − p . For u ∈ B p we then define a norm by k u k B p := sup z = z | u ( z ) − u ( z ) || z − z | − p + k ∂u k L p ( C ) + k ∂u k L p ( C ) , so that B p becomes a Banach space. We will usually choose p > c p sup τ k µ τ k L ∞ ( C ) < c p is the constant from item (4) above. Theorem 3.4. (see [9] , theorem 1)
Assume that p > such that c p sup τ k µ τ k L ∞ ( C ) < . If σ ∈ L p ( C ) then the equation ∂u = µ ∂u + σ has a unique solution u = u µ,σ ∈ B p . For the existence part of the theorem, one first solves the following fixed pointproblem in L p ( C ) q = Γ( µ q ) + Γ σ. This is possible because the map L p ( C ) −→ L p ( C ) q Γ( µq + σ )is a contraction in view of c p k µ k L ∞ ( C ) <
1. Then u := A ( µ q + σ )is the desired solution. The following estimate is also derived in the paper [9](3.4) k q k L p ( C ) ≤ c ′ p k σ k L p ( C ) , with c ′ p = c p / (1 − c p k µ k L ∞ ( C ) ) which follows from k q k L p ( C ) ≤ k Γ( µq ) k L p ( C ) + k Γ σ k L p ( C ) ≤ c p k µ k L ∞ ( C ) k q k L p ( C ) + c p k σ k L p ( C ) . Recalling our original situation we have the following result which shows thatthere is some sort of conformal mapping for j on the ball B . Theorem 3.5. (see [9] , theorem 4)
Let µ : C → C be an essentially bounded measureable function with µ | C \ B ≡ and p > such that c p k µ k L ∞ ( C ) < . Then there is a unique map α : C → C with α (0) = 0 such that ∂α = µ∂α in the sense of distributions with ∂α − ∈ L p ( C ) . OLOMORPHIC OPEN BOOK DECOMPOSITIONS 27
The desired map α is given by α ( z ) = z + u ( z ) . where u ∈ B p solves the equation ∂u = µ∂u + µ . In particular, α ∈ W ,p ( B ).Lemma 8 in [9] states that α : C → C is a homeomorphism. We can apply thetheorem to all the µ τ , 0 < τ ≤ µ τ –conformal mappings α τ together with the associated maps u τ . Lemma 3.6.
Let µ n : C → C be a sequence of measureable functions so that µ n | C \ B ≡ and sup n k µ n k L ∞ ( C ) < . Assume that µ n → µ pointwise almosteverywhere. Then the corresponding quasiconformal mappings α n , α as in theorem3.5 satisfy k α n − α k W ,p ( B ) −→ as n → ∞ , for any p > such that c p sup n k µ n k L ∞ ( C ) < and any compact set B ⊂ C . Proof:
We first estimate with g ∈ L p ( C ) and z = 0 | Ag ( z ) | = 12 π (cid:12)(cid:12)(cid:12)(cid:12)Z C g ( ξ ) zξ ( ξ − z ) (cid:12)(cid:12)(cid:12)(cid:12) dξ d ¯ ξ ≤ | z | π k g k L p ( C ) (cid:13)(cid:13)(cid:13)(cid:13) ξ ( ξ − z ) (cid:13)(cid:13)(cid:13)(cid:13) L pp − ( C ) (3.5) ≤ C p k g k L p ( C ) | z | − p , where the last estimate holds in view of Z C | ξ ( ξ − z ) | − pp − dξ d ¯ ξ ζ = z − ξ = Z C | z ζ − z ζ | − pp − | z | dζ d ¯ ζ = | z | − pp − Z C | ζ ( ζ − | − pp − dζ d ¯ ζ | {z } πC p . If q solves q = Γ( µq + µ ) then¯ ∂ ( α n − α ) = µ n ∂ ( α n − α ) − µ ∂α + µ n ∂α = µ n ∂ ( α n − α ) + µ n − µ + ( µ n − µ )Γ( µq + µ ) , i.e. the difference α n − α again satisfies an inhomogeneous Beltrami equation. Bytheorem 3.4 we have α n − α = A ( µ n q n + λ n ) , where λ n = µ n − µ + ( µ n − µ )Γ( µq + µ ) and where q n ∈ L p ( C ) solves q n =Γ( µ n q n + λ n ). Combining this with the estimates (3.5) and (3.4) we obtain | α n ( z ) − α ( z ) | ≤ C p k µ n q n + λ n k L p ( C ) | z | − p ≤ (cid:16) C p sup n k µ n k L ∞ ( C ) · c ′ p k λ n k L p ( C ) + C p k λ n k L p ( C ) (cid:17) | z | − p . (3.6)Since k µ n − µ k L p ( C ) → k ( µ n − µ )Γ( µq + µ ) k L p ( C ) → k λ n k L p ( C ) → α n → α uniformly on compact sets.Since ¯ ∂ ( α n − α ) = µ n ∂ ( α n − α ) + λ n and α n − α = A ( µ n q n + λ n ) we verify that ∂ ( α n − α ) = Γ( µ n q n + λ n ) = q n and ¯ ∂ ( α n − α ) = µ n q n + λ n . Invoking (3.4) once again we see that both k ∂ ( α n − α ) k L p ( C ) and k ¯ ∂ ( α n − α ) k L p ( C ) can be bounded from above by a constant times k λ n k L p ( C ) which converges to zero. (cid:3) We will also need some facts concerning the classical case where µ ∈ C k,α ( B R (0)), B R (0) = { z ∈ C | | z | < R } , which are not spelled out explicitly in [11] or in [14],but which easily follow from the constructions carried out there. Theorem 3.7.
Let µ, γ, δ ∈ C k,α ( B R ′ (0)) with < α < and sup B R ′ (0) | µ | < .Then for sufficiently small < R ≤ R ′ there is a unique solution w ∈ C k +1 ,α ( B R (0)) to the equation ∂w ( z ) = µ ( z ) ∂w ( z ) + γ ( z ) w ( z ) + δ ( z ) with w (0) = 0 and ∂w (0) = 1 . If w , w solve the above equation with coefficientfunctions µ l , γ l , δ l , l = 1 , then there is a constant c = c ( α, R, k w k C k,α ( B R (0)) , k ) > such that for all w ∈ C k +1 ,α ( B R (0)) k w − w k C k +1 ,α ( B R (0)) ≤ c ( k δ − δ k C k,α ( B R (0)) ++ k µ − µ k C k,α ( B R (0)) + k γ − γ k C k,α ( B R (0)) ) . Sketch of the proof:
The existence proof is a slight generalization of the Korn–Lichtenstein result (seealso [11] or [14]). What we are looking for is the estimate. We define the followingoperator (
T w )( z ) := A ( µ∂w + γw )( z ) − z Γ( µ∂w + γw )(0)and the function g ( z ) := ( Aδ )( z ) − z (Γ δ )(0) + z. A solution to the problem w ( z ) = ( T w )( z ) + g ( z )also solves the equation ∂w ( z ) = µ ( z ) ∂w ( z ) + γ ( z ) w ( z ) + δ ( z ) with w (0) = 0 and ∂w (0) = 1. In lecture 4 of [11] it is shown that T defines a bounded linear operator T : C ,α ( B R (0)) −→ C ,α ( B R (0))with k T k ≤ const · R α = θ , and θ < R > g + P ∞ k =1 T k g converges and the limit w satisfies w = T w + g .Another useful fact is the following: Assume, T , T are operators as above withcoefficient functions µ , γ and µ , γ respectively. Then k T − T k ≤ c ( k µ − µ k C ,α ( B R (0)) + k γ − γ k C ,α ( B R (0)) )for a suitable constant c > α and R . This is only implicitly provedin [11], so we sketch the proof of this inequality. We have( T − T ) h ( z ) = A (( µ − µ ) ∂h + ( γ − γ ) h )( z ) −− z Γ(( µ − µ ) ∂h + ( γ − γ ) h )(0) ,∂ (( T − T ) h )( z ) = Γ(( µ − µ ) ∂h + ( γ − γ ) h )( z ) −− Γ(( µ − µ ) ∂h + ( γ − γ ) h )(0) OLOMORPHIC OPEN BOOK DECOMPOSITIONS 29 and ¯ ∂ (( T − T ) h )( z ) = ( µ − µ )( z ) ∂h ( z ) + ( γ − γ )( z ) h ( z ) . We will need inequalities (21)–(24) from [14]. Adapted to our notation they lookas follows with z, z , z ∈ B R (0) | ( Ah )( z ) | ≤ R k h k C ( B R (0)) | (Γ h )( z ) | ≤ α +1 α R α k h k C ,α ( B R (0)) | ( Ah )( z ) − ( Ah )( z ) || z − z | α ≤ k h k C ( B R (0)) + 2 α +2 α R α k h k C ,α ( B R (0)) | (Γ h )( z ) − (Γ h )( z ) || z − z | α ≤ C α k h k C ,α ( B R (0)) . Recalling that k h k C ,α ( B R (0)) := k h k C ( B R (0)) + k ∂h k C ,α ( B R (0)) + k ¯ ∂h k C ,α ( B R (0)) and k k k C ,α ( B R (0)) := k k k C ( B R (0)) + sup z = z | k ( z ) − k ( z ) || z − z | α and that the H¨older norm satisfies k hk k C ,α ( B R (0)) ≤ C k h k C ,α ( B R (0)) k k k C ,α ( B R (0)) for a suitable constant C depending only on α and R the asserted inequality forthe operator norm of T − T follows. In the same way we obtain k g − g k C ,α ( B R (0)) ≤ c k δ − δ k C ,α ( B R (0)) . Since k w − w k C ,α ( B R (0)) ≤ k ( T − T ) w k C ,α ( B R (0)) ++ θ k w − w k C ,α ( B R (0)) + k g − g k C ,α ( B R (0)) and θ < k = 1. Because derivatives of w satisfy again an equation of the form ∂w ( z ) = µ ( z ) ∂w ( z ) + γ ( z ) w ( z ) + δ ( z ), wecan proceed by iteration. This is carried out in lecture 5 of [11]. (cid:3) A uniform L –bound for the harmonic forms and uniform conver-gence.Proposition 3.8. Let ( S, j , Γ , ˜ u , γ ) be a solution of the PDE (1.1) defined on ˙ S which is everywhere transverse to the Reeb vector field. Assume that ( S, j f , Γ , ˜ u =( a, u ) , γ ) is another smooth solution where u is given by u ( z ) = φ f ( z ) ( u ( z )) for a suitable smooth bounded function f : S → R . Then we have (3.7) k γ k L ,j f ≤ k u ∗ λ k L ,j f where k σ k L ,j f := (cid:18)Z ˙ S σ ◦ j f ∧ σ (cid:19) (with σ a 1–form on ˙ S ). Proof:
Using the differential equation u ∗ λ ◦ j f = da + γ and u ∗ λ = u ∗ λ + df , we compute Z ˙ S u ∗ λ ∧ γ = Z ˙ S u ∗ λ ∧ γ + Z ˙ S d ( f γ )= Z ˙ S u ∗ λ ∧ γ and Z ˙ S u ∗ λ ∧ γ = Z ˙ S da ∧ γ ◦ j f − k γ k L ,j f = −k γ k L ,j f . The integral R S d ( f γ ) vanishes by Stokes’ theorem since f γ is a smooth 1-form onthe closed surface S . The form da ∧ γ ◦ j f is not smooth on S , but the integralvanishes anyway for the following reason. As we have proved in appendix A theform γ ◦ j f is bounded near the punctures, hence in local coordinates near a punctureit is of the form σ = F ( w , w ) dw + F ( w , w ) dw , w + iw ∈ C where F , F are smooth except possibly at the origin but bounded. Passing topolar coordinates via φ : [0 , ∞ ) × S −→ C \{ } φ ( s, t ) = e − ( s + it ) = w + iw we see that φ ∗ σ has to decay at the rate e − s for large s . The form da has γ ( r ( s )) ds as its leading term. Computing the integral R Γ a ( γ ◦ j f ) over small loops Γ aroundthe punctures and using Stokes’ theorem we conclude that the contribution fromneighborhoods of the punctures can be made arbitrarily small. Therefore, theintegral R ˙ S da ∧ γ ◦ j f must vanish.If Ω is a volume form on S then we may write u ∗ λ ∧ γ = g · Ω for a suitable smoothfunction g . Defining Z ˙ S | u ∗ λ ∧ γ | := Z ˙ S | g | Ωwe have k γ k L ,j f = (cid:12)(cid:12)(cid:12)(cid:12)Z ˙ S u ∗ λ ∧ γ (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z ˙ S | u ∗ λ ∧ γ |≤ k u ∗ λ k L ,j f k γ k L ,j f which implies the assertion. (cid:3) We resume the proof of the compactness result, theorem 3.2. All the consider-ations which follow are local. The task is to improve the regularity of the limit f τ and the nature of the convergence f τ → f τ . Because the proof is somewhatlengthy we organize it in several steps. For τ < τ let now α τ : B −→ U τ ⊂ C be the conformal transformations as in the previous section, i.e. T α τ ( z ) ◦ i = j τ ( α τ ( z )) ◦ T α τ ( z ) , z ∈ B. OLOMORPHIC OPEN BOOK DECOMPOSITIONS 31
The L ∞ -bound (3.1) on the family of functions ( f τ ) and the above L –bound implyconvergence of the harmonic forms α ∗ τ γ τ after maybe passing to a subsequence: Proposition 3.9.
Let τ ′ k be a sequence converging to τ and B ′ = B ε ′ (0) with B ′ ⊂ B . Then there is a subsequence ( τ k ) ⊂ ( τ ′ k ) such that the harmonic 1–forms α ∗ τ k γ τ k converge in C ∞ ( B ′ ) . Proof:
First, the harmonic 1–forms α ∗ τ γ τ satisfy the same L –bound as in proposition 3.8: k α ∗ τ γ τ k L ( B ) = Z B α ∗ τ γ τ ◦ i ∧ α ∗ τ γ τ = Z B α ∗ τ ( γ τ ◦ j τ ) ∧ α ∗ τ γ τ = Z U τ γ τ ◦ j τ ∧ γ τ ≤ k u ∗ λ k L ,j τ ≤ C, where C is a constant only depending on λ and u sincesup τ k j τ k L ∞ ( ˙ S ) < ∞ . We write α ∗ τ γ τ = h τ ds + h τ dt where h kτ , k = 1 , L ( B ) independent of τ . If y ∈ B and B R ( y ) ⊂ B R ( y ) ⊂ B then the classical mean value theorem h kτ ( y ) = 1 πR Z B R ( y ) h kτ ( x ) dx implies that for any ball B δ = B δ ( y ) with B δ ⊂ B δ ⊂ B we have the rather generousestimate k h kτ k C ( B δ ( y )) ≤ √ π δ k h kτ k L ( B ) ≤ √ C √ π δ . With y ∈ B and ν = ( ν , ν ) being the unit outer normal to ∂B δ ( y ) we get ∂ s h kτ ( y ) = 1 πδ Z B δ ( y ) ∂ s h kτ ( x ) dx = 1 πδ Z B δ ( y ) div( h kτ , dx = 1 πδ Z ∂B δ ( y ) h kτ ν ds and |∇ h kτ ( y ) | = 1 πδ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ∂B δ ( y ) h kτ ν ds (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ δ k h kτ k C ( B δ ( y ))2 CASIM ABBAS so that for B ′ = B ε ′ , r being the radius of B and δ = r − ε ′ k∇ h kτ k C ( B ′ ) ≤ √ C √ π δ . By iterating this procedure on nested balls we obtain τ –uniform C ( B ′ ) boundson all derivatives. Convergence as stated in the proposition then follows from theAscoli–Arzela theorem. (cid:3) A uniform L p –bound for the gradient. The first step of the regularitystory is showing that the gradients of a τ + if τ are uniformly bounded in L p ( B ′ )for some p > B ′ with B ′ ⊂ B . It will become apparent laterwhy this gradient bound is necessary. Since we do not have a lot to start with, theproof will be indirect. Recall the differential equation (2.9):¯ ∂ j τ ( a τ + if τ ) = u ∗ λ ◦ j τ − i ( u ∗ λ ) − γ τ − i ( γ τ ◦ j τ ) , where j τ ( z ) = ( π λ T u ( z )) − ( T φ f τ ( z ) ( u ( z ))) − ·· J ( φ f τ ( z ) ( u ( z ))) T φ f τ ( z ) ( u ( z )) π λ T u ( z ) . We set φ τ ( z ) := a τ ( z ) + i f τ ( z ) , z ∈ U τ . so that for z ∈ B : ∂ ( φ τ ◦ α τ )( z ) = ∂ j τ φ τ ( α τ ( z )) ◦ ∂ s α τ ( z )= ( u ∗ λ ◦ j τ − i ( u ∗ λ )) α τ ( z ) ◦ ∂ s α τ ( z ) −− (cid:18) ( α ∗ τ γ τ )( z ) · ∂∂s + i ( α ∗ τ γ τ )( z ) · ∂∂t (cid:19) (3.8) =: ˆ F τ ( z ) + ˆ G τ ( z )=: ˆ H τ ( z ) , and(3.9) sup τ k ˆ F τ k L p ( B ) < ∞ for some p > α τ → α τ in W ,p ( B ) and sup τ k j τ k L ∞ < ∞ . We also have(3.10) sup τ k ˆ G τ k C k ( B ′ ) < ∞ for any ball B ′ ⊂ B ′ ⊂ B and any integer k ≥ B ′ ⊂ B ′ ⊂ B there is a constant C B ′ > k∇ ( φ τ ◦ α τ ) k L p ( B ′ ) ≤ C B ′ ∀ τ ∈ [0 , τ ) . Arguing indirectly, we may assume that there is a sequence τ k ր τ such that(3.12) k∇ ( φ τ k ◦ α τ k ) k L p ( B ′ ) → ∞ for some ball B ′ ⊂ B ′ ⊂ B. Define now ε k := inf { ε > | ∃ x ∈ B ′ : k∇ ( φ τ k ◦ α τ k ) k L p ( B ε ( x )) ≥ ε p − } OLOMORPHIC OPEN BOOK DECOMPOSITIONS 33 which are positive numbers since ε p − → + ∞ . Because we assumed (3.12) we musthave inf k ε k = 0, hence we will assume that ε k →
0. Otherwise, if ε = inf k ε k > B ′ with finitely many balls of radius ε , and we would get a k –uniform L p –bound on each of them contradicting (3.12). We claim that(3.13) k∇ ( φ τ k ◦ α τ k ) k L p ( B εk ( x )) ≤ ε p − k ∀ x ∈ B ′ . Otherwise we could find y ∈ B ′ so that k∇ ( φ τ k ◦ α τ k ) k L p ( B εk ( y )) > ε p − k , and we would still have the same inequality for a slightly smaller ε ′ k < ε k contra-dicting the definition of ε k . We now claim that there is a point x k ∈ B ′ with(3.14) k∇ ( φ τ k ◦ α τ k ) k L p ( B εk ( x k )) = ε p − k . Indeed, pick sequences δ l ց ε k and y l ∈ B ′ so that k∇ ( φ τ k ◦ α τ k ) k L p ( B δl ( y l )) ≥ δ p − l . We may assume that the sequence ( y l ) converges. Denoting its limit by x k , weobtain k∇ ( φ τ k ◦ α τ k ) k L p ( B εk ( x k )) ≥ ε p − k and (3.14) follows from (3.13). Hence there is a sequence ( x k ) ⊂ B ′ for which (3.14)holds. We may assume that the sequence ( x k ) ⊂ B ′ converges and without loss ofgenerality also that lim k →∞ x k = 0. Let R >
0, and we define for z ∈ B R (0) thefunctions ξ k ( z ) := ( φ τ k ◦ α τ k )( x k + ε k ( z − x k ))which makes sense if k is sufficiently large. The transformationΦ : x x k + ε k ( x − x k )satisfies Φ( B ( x k )) = B ε k ( x k ) and Φ( B ( y )) ⊂ B ε k ( x k + ε k ( y − x k )) so that Z B εk ( x k ) |∇ ( φ τ k ◦ α τ k )( x ) | p dx = ε k Z B ( x k ) |∇ ( φ τ k ◦ α τ k )( x k + ε k ( z − x k )) | p dz = ε k Z B ( x k ) ε − pk |∇ ξ k ( z ) | p dz and k∇ ξ k k L p ( B ( x k )) = ε − p k k∇ ( φ τ k ◦ α τ k ) k L p ( B εk ( x k )) (3.15) = 1 by equation (3 . y for which ξ k | B ( y ) is defined and large enough k (3.16) k∇ ξ k k L p ( B ( y )) ≤ ε − p k k∇ ( φ τ k ◦ α τ k ) k L p ( B εk ( x k + ε k ( y − x k ))) ≤ ξ k satisfy the equation(3.17) ¯ ∂ξ k ( z ) = ε k ˆ F τ k ( x k + ε k ( z − x k )) + ε k ˆ G τ k ( x k + ε k ( z − x k )) =: H τ k ( z )and for every R > k k∇ ξ k k L p ( B R (0)) < ∞ , k∇ ξ k k L p ( B (0)) ≥ because of (3.16) and (3.15) since B ( x k ) ⊂ B (0) for large k . The upper boundon k∇ ξ k k L p ( B R (0)) depends on how many balls B ( y ) are needed to cover B R (0).We compute for ρ > k H τ k k L p ( B ρ ( x k )) = ε − p k k ˆ H τ k k L p ( B ρεk ( x k )) , with ˆ H τ k as in (3.8). We conclude from, p >
2, (3.9) and (3.10) that k H τ k k L p ( B R (0)) −→ R > k → ∞ . Defining X l,p := { ψ ∈ W l,p ( B, C ) | ψ (0) = 0 , ψ ( ∂B ) ⊂ R } , l ≥ , B ⊂ C a ballthe Cauchy-Riemann operator¯ ∂ : X l,p −→ W l − ,p ( B, C )is a bounded bijective linear map. By the open mapping principle we have thefollowing estimate:(3.19) k ψ k l,p,B ≤ C k ∂ψ k l − ,p,B ∀ ψ ∈ X l,p . Let R ′ ∈ (0 , R ). Pick now a smooth function β : R → [0 ,
1] with β | B R ′ (0) ≡ β ) ⊂ B R (0). Define ζ k ( z ) := Re( ξ k ( z ) − ξ k (0)) + iβ ( z ) Im( ξ k ( z ) − ξ k (0)) . We note that sup k k Im( ξ k ) k L p ( B R (0)) ≤ C R with a suitable constant C R > k,R k Im( ξ k ) k L ∞ ( B R (0)) < ∞ . Using (3.19) we then obtain k ξ k − ξ k (0) k ,p,B R ′ (0) ≤ k ζ k k ,p,B R (0) ≤ C k ∂ζ k k L p ( B R (0)) ≤ C R ′ (cid:16) k H τ k k L p ( B R (0)) + k∇ ξ k k L p ( B R (0)) +(3.20) + k Im( ξ k ) − Im( ξ k )(0) k L p ( B R (0)) (cid:17) because of ∂ζ k = H τ k + i ( β − ∂ ( Im( ξ k )) + i ∂β (cid:16) Im( ξ k ) − Im( ξ k )(0) (cid:17) . Hence the sequence ( ξ k − ξ k (0)) is uniformly bounded in W ,p ( B R ′ (0)), in particularit has a subsequence which converges in C α ( B R ′ (0)) for 0 < α < − p and also in L p ( B R ′ (0)). For R ′′ ∈ (0 , R ′ ) we use now the regularity estimate k ξ l − ξ k − ( ξ l − ξ k )(0) k ,p,B R ′′ (0) ≤ c k H τ l − H τ k k L p ( B R ′ (0)) ++ c k ξ l − ξ k − ( ξ l − ξ k )(0) k L p ( B R ′ (0)) , (3.21)where c = c ( p, R ′ , R ′′ ) >
0. This follows from (3.19) applied to ψ = β ( ξ l − ξ k − ( ξ l − ξ k )(0)) where β is a smooth cut-off function with support in B R ′ and β ≡ B R ′′ . We may then assume that the right hand side of (3.21) converges tozero as k, l → ∞ . This argument can be carried out for any triple 0 < R ′′ 1. This contradiction finallydisproves our assertion (3.12). We summarize: Proposition 3.10. For every ball B ′ with B ′ ⊂ B we have sup τ k∇ ( φ τ ◦ α τ ) k L p ( B ′ ) < ∞ . (cid:3) Remark . After establishing the estimates (3.18) for ∇ ξ k we could have de-rived a k –uniform W ,p –bound for ξ k minus its average ξ k over the ball B R (0) viaPoincar´e’s inequality. We could have derived W ,p ( B R (0)) convergence of ξ k − ξ k ,but not convergence in W ,ploc ( C ) since the sequence ( ξ k − ξ k ) depends on the choiceof the ball B R (0). Our sequence ξ k − ξ k (0) has a convergent subsequence on anyball.3.4. Convergence in W ,p ( B ′ ) . Pick a sequence τ k ր τ . We claim that thesequence ( ˆ F τ k ) converges in L p ( B ) maybe after passing to a suitable subsequence(recall that so far we only have the uniform bound (3.9)). The functions ˆ F τ k con-verge pointwise almost everywhere after passing to some subsequence: Indeed, thesequence { ( u ∗ λ ◦ j τ k − i ( u ∗ λ )) α τk ( z ) } converges already pointwise since j τ k and α τ k do (recall that the sequence ( α τ k ) converges in W ,p ( B ) and therefore uniformly).The sequence ( ∂ s α τ k ) converges in L p ( B ) and therefore pointwise almost every-where after passing to a suitable subsequence. Then, by Egorov’s theorem, for any δ > E δ ⊂ B with | B \ E δ | ≤ δ so that the sequence ˆ F τ k convergesuniformly on E δ . Let α be the L p -limit of the sequence ( ∂ s α τ k ), and let ε > 0. Weintroduce C := 2 sup ≤ τ<τ k ( u ∗ λ ◦ j τ − i ( u ∗ λ )) α τ ( z ) k L ∞ ( B ) . Pick now δ > k α k L p ( B \ E δ ) ≤ ε C . Choose now k ≥ k ≥ k k ∂ s α τ k − α k L p ( B ) ≤ ε C and k ˆ F τ k − ˆ F τ l k L ∞ ( E δ ) ≤ ε | B | . Then, if k, l ≥ k , k ˆ F τ k − ˆ F τ l k L p ( B ) ≤ k ˆ F τ k − ˆ F τ l k L p ( E δ ) + k ˆ F τ k − ˆ F τ l k L p ( B \ E δ ) ≤ | E δ | k ˆ F τ k − ˆ F τ l k L ∞ ( E δ ) + 2 sup k ≥ k k ˆ F τ k k L p ( B \ E δ ) ≤ | B | k ˆ F τ k − ˆ F τ l k pL ∞ ( E δ ) + C · sup k ≥ k k ∂ s α τ k k L p ( B \ E δ ) ≤ ε proving the claim. Recalling that φ τ = a τ + if τ and that the family f τ satisfies a uniform L ∞ –boundwe have sup τ k Im( φ τ ◦ α τ ) k L ∞ ( B ) < ∞ . Pick now three balls B ′′′ ⊂ B ′′ ⊂ B ′ ⊂ B such that the closure of one is containedin the next. Our aim is to establish W ,p ( B ′′′ )–convergence of a subsequence of thesequence ( φ τ k ◦ α τ k ). By proposition 3.10 we have a uniform L p ( B ′ )–bound on thegradient. If β : R → [0 , 1] is a smooth function with supp ( β ) ⊂ B ′ and β | B ′′ ≡ ζ τ = Re( φ τ ◦ α τ − φ τ (0)) + iβ Im( φ τ ◦ α τ − φ τ (0))then we proceed in the same way as in (3.20), and we obtain k ϕ k k ,p,B ′′ ≤ C (cid:16) k ˆ H τ k k L p ( B ′ ) + k∇ ( φ τ k ◦ α τ k ) k L p ( B ′ ) + k Im( ϕ k ) k L p ( B ′ ) (cid:17) where we wrote ϕ k := φ τ k ◦ α τ k − ( φ τ k ◦ α τ k )(0) , and where C > p, B ′ and B ′′ . The sequence ( ϕ k )is then uniformly bounded in W ,p ( B ′′ ) by proposition 3.10, and it converges in L p ( B ′′ ) after passing to a suitable subsequence. We now use the regularity estimate k ϕ l − ϕ k k ,p,B ′′′ ≤ C (cid:16) k ˆ F τ l − ˆ F τ k k L p ( B ′′ ) +(3.22) + k ˆ G τ l − ˆ G τ k k L p ( B ′′ ) + k ϕ l − ϕ k k L p ( B ′′ ) (cid:17) . Since the right hand side converges to zero as k, l → ∞ we obtain Proposition 3.12. For every ball B ′ ⊂ B ′ ⊂ B the sequence ( φ τ k ◦ α τ k − φ τ k (0)) has a subsequence which converges in W ,p ( B ′ ) . (cid:3) Improving regularity using both the Beltrami and the Cauchy-Riemannequations. In order to improve the convergence of the conformal transformations α τ k we need to improve the convergence of the maps µ τ k → µ τ and the regularityof its limit. It is known that the inverses α − τ k of the conformal transformations α τ k also satisfy a Beltrami equation [9] ∂α − τ = ν τ ∂α − τ where ν τ ( z ) = − ∂α τ ( α − τ ( z )) ∂α τ ( α − τ ( z )) µ τ ( α − τ ( z ))(follows from 0 = ∂ ( α − τ ◦ α τ ) = ∂α − τ ( α τ ) ∂α τ + ∂α − τ ( α τ ) ∂α τ ). After passingto a suitable subsequence we may assume that ∂α τ k and ∂α τ k converge pointwisealmost everywhere since they converge in L p ( B ). Hence we may assume that thesequence ( ν τ k ) also converges pointwise almost everywhere. We also have | ν τ ( z ) | ≤ | µ τ ( α − ( z )) | , hence ν τ satisfies the same L ∞ –bound as µ τ . By lemma 3.6 we conclude that α − τ k −→ α − in W ,p ( B ) OLOMORPHIC OPEN BOOK DECOMPOSITIONS 37 with the same p > µ τ . After passing tosome subsequence, the sequence( ϕ k ) := ( φ τ k − a τ k (0)) ◦ α τ k converges in W ,p ( B ′ ) for any ball B ′ ⊂ B by proposition 3.12. Indeed, the expres-sion φ τ k ◦ α τ k − φ τ k (0) and ϕ k differ by a constant term if τ k (0), but the sequence( if τ k (0)) has a convergent subsequence.We would like to derive a decent notion of convergence for the sequence ( ϕ τ k ◦ α − τ k ),but the space W ,p is not well–behaved under compositions. The composition oftwo functions of class W ,p is only in W ,p/ . Since we can not choose p > ϕ k ) and ( α − τ k )converge in C ,α ( B ′ ) for any ball B ′ ⊂ B ′ ⊂ B and 0 < α ≤ − p . We concludefrom the inequality k f ◦ g k C ,γδ ( B ′ ) ≤ k f k C ,γ ( B ′ ) k g k C ,δ ( B ′ ) ∀ f ∈ C ,γ ( B ′ ) , g ∈ C ,δ ( B ′ )where 0 < γ, δ ≤ φ τ k − a τ k (0)) converges in C ,α ( B ′ ). Inparticular, any sequence ( f τ k ), τ k ր τ , now converges in the C ,α –norm to f τ .H¨older spaces are well–behaved with respect to multiplication, i.e. k f g k C ,γ ( B ′ ) ≤ k f k C ,γ ( B ′ ) k g k C ,γ ( B ′ ) , and composition with a fixed smooth function maps C ,γ ( B ′ ) into itself. It thenfollows from the definition of the complex structure j τ and from the definition of µ τ that µ τ k → µ τ in the C ,α –norm as well. We conclude from theorem 3.7, theclassical regularity result for the Beltrami equation, that α τ k → α τ in the C ,α –norm. The regularity estimate for the Cauchy-Riemann operator (3.22) is also validin H¨older spaces, i.e. k ϕ l − ϕ k k C k +1 ,γ ( B ′′′ ) ≤ C (cid:16) k ˆ F τ l − ˆ F τ k k C k,γ ( B ′′ ) ++ k ˆ G τ l − ˆ G τ k k C k,γ ( B ′′ ) + k ϕ l − ϕ k k C k,γ ( B ′′ ) (cid:17) . The sequence ( ˆ F τ k ) now converges in the C ,α –norm, and the sequence ( ˆ G τ k )converges in any H¨older norm. We obtain with the above regularity estimate C ,α –convergence of the sequence ( ϕ k ), and composing with α − τ k yields C ,α –convergence of ( f τ k ) and ( µ τ l ). Invoking theorem 3.7 again then improves theconvergence of the transformations α τ k , α − τ k to C ,α . We now iterate the pro-cedure using the regularity estimate for the Cauchy-Riemann operator in H¨olderspace and the estimate for the Beltrami equation in theorem 3.7.Theorem 3.2 follows if we apply the Implicit Function theorem to the limitsolution ( S, j τ , ˜ u τ = ( a τ , u τ ) , γ τ ), hence we obtain the same limit for everysequence { τ k } , and we obtain convergence in C ∞ .4. Conclusion The following remarks tie together the loose ends and prove the main result,theorem 1.6. We start with a closed three dimensional manifold with contact form λ ′ . Giroux’s theorem, theorem 1.4, then permits us to change the contact form λ ′ to another contact form λ such that ker λ = ker λ ′ and such that there is a supporting open book decomposition with binding K consisting of periodic orbits ofthe Reeb vector field of λ . Invoking proposition 2.4 we construct a family of 1-forms( λ δ ) ≤ δ< which are contact forms except λ , and the above open book supportsker λ δ as well if δ = 0. By the uniqueness part of Giroux’s theorem, ( M, ker λ )and ( M, ker λ δ ) are diffeomorphic for δ = 0, hence we may assume without lossof generality that λ = λ δ . Proposition 2.6 then permits us to turn the Girouxleaves into holomorphic curves for data associated with the confoliation form λ .Picking one Giroux leaf, the implicit function theorem, theorem 2.8, then allowsus to deform it into a solutions to our PDE (1.1) for small δ = 0. Leaving such aparameter δ fixed from now, and denoting the corresponding solution by (˜ u , γ , j ),theorem 2.8 then delivers more solutions (˜ u τ , γ τ , j τ ) ≤ τ<τ . The leaves u τ ( ˙ S ) are allglobal surfaces of section recalling that they are of the form u τ = φ f τ ( u ). Theorem2.8 also implies that f τ < f τ ′ if τ < τ ′ . The compactness result, theorem 3.2, thenimplies that there is a ’last’ solution for τ = τ as well, and that either u τ ( ˙ S ) isdisjoint from u ( ˙ S ) or agrees with it. In the latter case, the proof of theorem 1.6 iscomplete. In the first case we apply theorem 2.8 again to (˜ u τ , γ τ , j τ ) producinga larger family of solutions. Because τ f τ ( z ) is strictly monotone for each z ∈ S and because the return time for each point on u ( ˙ S ) is finite, the images of u τ and u must agree for some sufficiently large τ concluding the proof. Appendix A. Some local computations near the punctures In this appendix we will present some local computations needed for the proofof theorem 2.8. The issue is to show that the 1-forms u ∗ λ ◦ j f − da and u ∗ λ + da ◦ j f are bounded on ˙ S . We obtain in the second case of the theorem u ∗ λ ◦ j f − da = u ∗ λ ◦ ( j f − j g ) + γ = dg ◦ ( j f − j g ) + γ + v ∗ λ ◦ ( j f − j g ) . The first case can be treated as a special case: Here the objective is to show that the1-form v ∗ λ ◦ ( j f − i ) = v ∗ λ ◦ ( j f − j ) is bounded near the punctures. We again dropthe subscript δ in the notation since we are only concerned with a local analysisnear the binding, and all the forms λ δ are identical there. We use coordinates( θ, r, φ ) near the binding. The contact structure is then generated by η = ∂∂r = (0 , , , η = − γ ∂∂θ + γ ∂∂φ = ( − γ , , γ ) . The projection onto the contact planes along the Reeb vector field is then given by π λ ( v , v , v ) = 1 µ ( v γ ′ + v γ ′ ) η + v η with µ = γ γ ′ − γ ′ γ , and the flow of the Reeb vector field is given by φ t ( θ, r, φ ) = ( θ + α ( r ) t, r, φ + β ( r ) t )where α ( r ) = γ ′ ( r ) µ ( r ) and β ( r ) = − γ ′ ( r ) µ ( r ) . OLOMORPHIC OPEN BOOK DECOMPOSITIONS 39 The linearization of the flow T φ τ ( θ, r, φ ) preserves the contact structure. In thebasis { η , η } it is given by T φ τ ( θ, r, φ ) = (cid:18) τ A ( r ) 1 (cid:19) with A ( r ) = 1 µ ( r ) ( γ ′′ ( r ) γ ′ ( r ) − γ ′′ ( r ) γ ′ ( r )) . The complex structure(s) we chose earlier in (2.7) had the following form near thebinding with respect to the basis { η , η } : J ( θ, r, φ ) = (cid:18) − rγ ( r ) rγ ( r ) (cid:19) . The induced complex structure j τ on the surface is then given by j τ ( z ) = [ π λ T v ( z )] − [ T φ τ ( v ( z ))] − J ( φ τ ( v ( z ))) T φ τ ( v ( z )) π λ T v ( z ) . With v ( s, t ) = ( t, r ( s ) , α ) we find that π λ T v ( s, t ) = r ′ ( s ) 00 γ ′ ( s ) µ ( r ( s )) ! so that j τ = − τ A ( r ) rγ ( r ) − rγ ( r ) γ ′ ( r ) r ′ µ ( r ) r ′ µ ( r ) rγ ( r ) γ ′ ( r ) (1 + τ A ( r ) r γ ( r )) τ A ( r ) rγ ( r ) ! = (cid:18) − τ A ( r ) rγ ( r ) − 11 + τ A ( r ) r γ ( r ) τ A ( r ) rγ ( r ) (cid:19) = j + τ A ( r ) γ ( r ) (cid:18) − τ A ( r ) γ ( r ) 1 (cid:19) . and j τ − j σ = A ( r ) rγ ( r )( τ − σ ) (cid:18) − τ + σ ) A ( r ) rγ ( r ) 1 (cid:19) using the fact that r ( s ) satisfies the differential equation r ′ ( s ) = γ ′ ( r ( s )) γ ( r ( s )) r ( s ) µ ( r ( s )) . With v ∗ λ = γ ( r ) dt we obtain v ∗ λ ◦ ( j τ − j σ ) | ( s,t ) = ( τ − σ ) A ( r ( s )) r ( s ) γ ( r ( s )) ·· (cid:2) ( τ + σ ) A ( r ( s )) r ( s ) γ ( r ( s )) ds + dt (cid:3) . 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