The space of Cauchy-Riemann structures on 3-D compact contact manifolds
TTHE SPACE OF CAUCHY-RIEMANN STRUCTURES ON 3-D COMPACTCONTACT MANIFOLDS
J. BLAND AND T. DUCHAMP
Abstract.
We study the action of the group of contact diffeomorphisms on CR deformations ofcompact three-dimensional CR manifolds. Using anisotropic function spaces and an anisotropicstructure on the space of contact diffeomorphisms, we establish the existence of local transverseslices to the action of the contact diffeomorphism group in the neighbourhood of a fixed embeddablestrongly pseudoconvex CR structure. Introduction
Cauchy-Riemann manifolds arise naturally as the boundary of a bounded domain D ⊂ C n +1 . Inthis case, the Cauchy-Riemann structure is simply that residual complex structure which is inheritedfrom the complex structure on C n . Local coordinates for ∂D are said to be CR (for Cauchy-Riemann) if they are the restriction of holomorphic coordinates in C n +1 , and they define a conjugateCR tangent space for ∂D in the same manner that the holomorphic coordinates on C n define aconjugate holomorphic tangent space for C n +1 . Intrinsically, one can define the Cauchy-Riemann structure on ∂D by specifying the space of conjugate CR tangent vectors in the same manner asone defines the complex structure on C n +1 by specifying the conjugate holomorphic tangent space.All questions which arise for abstract complex structures on a smooth manifold are equally validfor Cauchy-Riemann manifolds: for example, the embeddability and local embeddability (or theexistence of holomorphic (CR) coordinates, or how many structures exist up to equivalence.The significance of generalizing from complex structures on manifolds to studying Cauchy-Riemann structures can easily be seen from the following considerations. When D is a boundeddomain in C n +1 , n ≥
1, then holomorphic functions on D which extend smoothly to ∂D restrict to ∂D as CR functions; on the other hand, a slight generalization of Hartog’s phenomenon in severalcomplex variables states that CR functions on ∂D extend uniquely to D as holomorphic functions;that is, ∂D with its Cauchy-Riemann structure completely determines D with its complex struc-ture. On the other hand, if we generalize to Σ a complex analytic space with an isolated singularityat p ∈ Σ, then the boundary of a small neighbourhood of Σ inherits a smooth Cauchy Riemannstructure whereas the space Σ is singular. On the basis of this observation, Kuranishi proposed[Ku] to study the deformation space for isolated singularities by studying the deformation space forCauchy-Riemann structures on the boundary of the neighbourhood, a smooth compact manifold.A case of particular interest is that in which the domain D is strongly convex (more generally,strongly pseudoconvex). In this case, the boundary admits a natural family of positive definite Date : June 30, 2010.2000
Mathematics Subject Classification.
Key words and phrases.
Cauchy-Riemann structure, contact structure, contact diffeomorphism, Folland-Steinspace.The first author was partially supported by an NSERC grant. The second author was partially supported by anNSF grant. a r X i v : . [ m a t h . DG ] J u l J. BLAND AND T. DUCHAMP metrics which are adapted to the CR structure, and play much the same role that K¨ahler metricsplay in complex geometry. One consequence of particular importance is that when M is compact,strongly pseudoconvex and n ≥ M ≥ M is embeddable. This is definitely notthe case when n = 1, and this case has many deep and interesting features which have yet to befully understood.In this paper, we fix a smooth compact underlying manifold, and study the space of CR structureson the manifold up to equivalence. In particular, we study the local deformation theory for thespace of CR structures, and the local action of the contact diffeomorphism group on the space ofsuch structures. Although for much of the paper we set up the machinery to work in arbitrarydimensions, our main interest is in the three dimensional case, and we restrict our attention to thiscase in the latter sections of this paper This was largely a matter of expedience, since in higherdimensions integrability factors play a role, and require the introduction of new operators andsignificantly different treatment than in the three dimensional case.Most of the results in this paper rely heavily on [BD1] in which we developed the machinery todo analysis on contact manifolds using intrinsically defined anisotropic functions spaces.The outline of the paper is as follows. In Section 2, we give a quick review of strongly pseudo-convex Cauchy-Riemann structures and the relevant deformation theory. In Section 3, we definethe weighted or anisotropic function spaces in which we will work, and recall the results from [BD1]on the space of weighted contact diffeomorphisms which we will need throughout the remainderof the paper. The inclusion of these two sections is to fix notation and to help make the paperself-contained. In Section 4, we study the action of contact diffeomorphisms on CR structures,computing both the linear and the fully nonlinear action; it is also in this section that we introducethe notion of complex contact vector fields, and explain their relation to the symmetry group. InSection 5, we collect results on homotopy operators for the ¯ ∂ b -complex on compact CR manifoldsand adapt them to our particular situation; we also indicate how to split complex contact vectorfields into real contact vector fields and a transverse vector field. Section 6 contains the main resultsof the paper. In this section, we obtain normal forms for CR structures under the action of thegroup of contact diffeomorphisms with sharp regularity results. This is accomplished in two steps:first we obtain a weak normal form with a loss of regularity, and then using a priori estimateswe recover the lost regularity. It is believed that this approach to studying the action of infinitedimensional symmetry groups on underlying structures is new, and may have applications in othersituations.Earlier results in this direction were obtained in [CL] and and [B]. The main idea in bothpapers was to study the linearized action, and to construct appropriate function spaces in whichone can solve the linearized equation with good estimates. Since the ¯ ∂ b –operator appears in thelinearized equation, the anisotropic function spaces appear naturally. In [CL], they avoided usingthe anisotropic spaces by working in the Nash Moser category; they obtained a transverse slice forsmooth CR structures. In [B], we restricted our attention to the case of the standard S ⊂ C ,and used explicit information to construct an anisotropic Hilbert space structure on contact diffeo-morphisms near the identity; the description of transverse slices follows easily from the linearizedanalysis. However in [B], the action described for the contact diffeomorphism group was incorrectlyasserted to be C , a necessary condition to apply the inverse function theorem in Banach spaces andobtain the transverse slices; a modified action is used in Section 6 of the current paper to correctthis error. With this modification and the generalization of the weighted function space structurefor contact diffeomorphisms to arbitrary compact contact manifolds (see [BD1]), we are now ableto obtain local transverse slices to the action of the contact diffeomorphism group on the space of R DEFORMATIONS 3
CR structures for an arbitrary compact embeddable strongly pseudoconvex three dimensional CRstructure.1.1.
Notation.
Throughout the paper, M will denote a smooth compact 2 n + 1 dimensional man-ifold equipped with a fixed contact distribution H ⊂ T M and a fixed contact one form η . As usual, T M and T ∗ M denote the tangent and cotangent bundles of M , respectively, Λ p M denotes the p -thexterior power of T ∗ M , Ω p ( M ) the space of smooth p -forms on M , L X β the Lie derivative of theform β with respect to the vector field X , and X β interior evaluation.We give M a fixed Riemannian metric g compatible with η (see Equation (2.1.3) for details),and let | X | denote the norm of the tangent vector X with respect to g , and we let exp : T M → M denote the exponential map of the g .We let π H : T ∗ M → H ∗ denote the projection map. The characteristic (or Reeb ) vector field T is the unique vector field sat-isfying the conditions T η = 1 and
T dη = 0. We can then identify the dual contact distributionwith the annihilator of T , i.e. H ∗ = { β ∈ T ∗ M : T β = 0 } ⊂ T ∗ M ;more generally Λ p H ∗ = { β ∈ Λ p ( M ) : T β = 0 } , and we have the identity(1.1.1) π H ( β ) = T ( η ∧ β ) . We endow R n +1 with the contact structure defined by the one-form η = dx n +1 − n (cid:88) j =1 x n + j dx j , where ( x , . . . , x n , x n +1 , . . . , x n , x n +1 ) are the standard coordinates on R n +1 , and we let dV denote the standard volume form: dV = 1 n ! η ∧ ( dη ) n . We denote the contact distribution of η by H ⊂ T R n +1 and we set T = ∂∂x n +1 , X j = ∂∂x j + x n + j ∂∂x n +1 , and X n + j = ∂∂x n + j , ≤ j ≤ n . Observe that the collection { X j , ≤ j ≤ n } is a global framing for H . Note also that the 1-forms η , dx j , dx n + j , ≤ j ≤ n, are the dual coframe to T , X j , X n + j , 1 ≤ j ≤ n .Let f = ( f , . . . , f m ) be a smooth, R m -valued function defined on the closure of a domain D (cid:98) R n +1 . We define X I f = (cid:40) X i X i . . . X i t f for t > f for t = 0 , where we have introduced the multi-index notation I = ( i , . . . , i t ), 1 ≤ i j ≤ n and X I f =( X I f , . . . , X I f m ). (For t = 0, I denotes the empty index I = ().) The integer t is called the order of I and written | I | . J. BLAND AND T. DUCHAMP
Remark 1.1.2.
We will often have to work in local coordinates adapted to the contact structureon M . An adapted coordinate chart for M is a chart φ : U → R n +1 for which η = φ ∗ η . It followsthat φ ∗ T = T and φ ∗ H = H . An adapted atlas consists of the following data: a fixed finite opencover V (cid:96) (cid:98) U (cid:96) , (cid:96) = 1 , , . . . , m and an atlas { φ (cid:96) : U (cid:96) → R n +1 } , consisting of adapted coordinatecharts. We set D (cid:96) = φ (cid:96) ( V (cid:96) ). By compactness of M and Darbourx’s Theorem for contact structures[Arn, page 362], M has an adapted atlas. We shall fix once and for all an adapted atlas and apartition of unity ρ (cid:96) a partition of unity subordinate to { V (cid:96) } .If F : A → B is a map between Banach spaces, with norms (cid:107) · (cid:107) A and (cid:107) · (cid:107) B , respectively, thenthe expression (cid:107)F ( f ) (cid:107) B ≺ (cid:107) f (cid:107) A means that there is a constant C > (cid:107)F ( f ) (cid:107) B ≤ C (cid:107) f (cid:107) A for all f ∈ A .2. CR structures
Deformation theory of CR structures.
We begin with a quick review of the deformationtheory of CR structures as presented in the paper of Akahori, Garfield, and Lee [AGL]. See also[BD2, Section 16], where the special case of the deformation theory of S n +1 is studied using asimilar framework. Definition 2.1.1.
Let M be a 2 n + 1 dimensional manifold. A (rank n ) Cauchy-Riemann structure(CR-structure) on M is a rank n complex subbundle H (1 , ⊂ T C M of the complexified tangentbundle of M such that(i) H (1 , ∩ H (1 , = { } ,(ii) the integrability condition is satisfied:[Γ ∞ ( H (1 , ) , Γ ∞ ( H (1 , )] ⊂ Γ ∞ ( H (1 , ) . The bundle H (1 , is called the holomorphic tangent bundle of the CR-structure. As usual, welet H (0 , denote the conjugate bundle H (1 , . The transversality condition (i) implies that H C = H (1 , (cid:76) H (0 , ⊂ T C M has complex codimension one. Remark 2.1.2.
We recall that when n = 1, the bundle H (1 , is a complex line bundle, andcondition (ii) is automatic. To see this let Z be section of H (1 , that does not vanish on an openset U ⊂ M . Then since [ Z, Z ] = 0,[ f Z, gZ ] = ( f Z ( g ) − gZ ( f )) Z for any two sections, say X = f Z and Y = gZ of H (1 , .Two CR-structures H (1 , and ˆ H (1 , are said to be equivalent if there is a diffeomorphism F : M → M such that F ∗ H (1 , = ˆ H (1 , . We are only interested in CR-structures up to equivalence.Observe that H C is the complexification of a real codimension one subbundle H ⊂ T M consistingof vectors of the form X + X , X ∈ H (1 , . Let η be a real one-form dual to H . The CR-structure H (1 , is said to be strongly pseudoconvex if − i dη ( X, X ) > X ∈ H (1 , . In thiscase, η ∧ ( dη ) n is a nowhere vanishing (2 n + 1)-form. In other words, ( M, H ) is a contact manifoldand η is a contact one-form.The most common examples of CR-structures are those arising from domains in C n +1 . Let D = { z ∈ C n +1 : ρ ( z ) < } be a smoothly bounded domain in C n +1 with connected boundary, R DEFORMATIONS 5 where ρ is a smooth nonnegative function defined on a neighbourhood of D , and dρ (cid:54) = 0 on ∂D .The boundary ∂D is a CR-manifold for which the holomorphic tangent bundle is the intersectionof the complexified tangent bundle of ∂D with the holomorphic tangent bundle of C n +1 , and if thepullback to ∂D of the one-form i ¯ ∂ρ is a contact form, then it is strongly pseudoconvex .We will assume, henceforth, that M is a contact manifold equipped with a fixed strongly pseudo-convex CR-structure H (1 , ⊂ T C M such that H C is the complexification of the contact distributionof M . We shall refer to this CR-structure as the reference CR-structure on M .The reference CR structure determines an endomorphism J : H → H satisfying the condition J = − Id , which in turn defines a Riemannian metric g by the formula(2.1.3) g ( X, Y ) = η ( X ) η ( Y ) + dη ( X, J Y ) . The metric g is said to be adapted to the CR structure . We let exp : T M → M denote theexponential map of g . Objects associated to any other CR-structure on M will be decoratedwith hats. Two strongly pseudoconvex CR-structures on M are said to be isotopic if they can beconnected by a smooth 1-parameter family of strongly pseudoconvex CR-structures. We consideronly strongly pseudoconvex CR-structures which are isotopic to the reference CR-structure.2.2. Representation by Deformation Tensors.
Every CR-structure that is isotopic to thereference one can be represented by a deformation tensor that takes values in H (1 , . The proof ofthis fact relies on a theorem of John Gray [G] which states that isotopic contact structures on acompact manifold are equivalent. Theorem 2.2.1 (Gray) . Let η t be a differentiable family of contact forms on a compact n + 1 dimensional manifold M . Then there is a differentiable family of diffeomorphisms F t : M → M and a family of non-vanishing functions p t such that F t ∗ ( η t ) = p t η . Corollary 2.2.2.
Every strongly pseudoconvex CR-structure on M that is isotopic to the referenceone is CR-equivalent to one of the form ˆ H (1 , where (2.2.3) ˆ H (0 , = { X − φ ( X ) : X ∈ H (0 , } and φ : H (0 , → H (1 , is a map of complex vector bundles, called the deformation tensor for ˆ H (1 , .Proof. The fact that the CR-structure is equivalent to one satisfying the inclusion relation ˆ H (0 , ⊂ H C follows immediately from Gray’s theorem. Thus, there is a family ˆ H (0 , ( t ), t ∈ [0 , H (0 , to ˆ H (0 , . For t small, it is clear that there are bundle maps φ ( t ) such that ˆ H (0 , ( t ) is thegraph of − φ ( t ). The integrability conditions for CR-structures imply that φ ( t ) satisfies certainsymmetry properties, and when combined with the transversality condition they imply an a priori bound on the size of φ ( t ), from which the result follows.We explain in brief. Choose a local basis Z α for H (1 , , and let iη [ Z α , Z ¯ β ] = − idη ( Z α , Z ¯ β ) = h α ¯ β define the Levi form. Then integrability implies in particular that iη [ Z ¯ β − φ α ¯ β Z α , Z ¯ δ − φ γ ¯ δ Z γ ] = 0.Since η [ Z α , Z γ ] = η [ Z ¯ β , Z ¯ δ ] = 0, it follows that iη [ − φ α ¯ β Z α , Z ¯ δ ]+ iη [ Z ¯ β , − φ γ ¯ δ Z γ ] = φ γ ¯ δ h γ ¯ β − φ α ¯ β h α ¯ δ = 0; The fact that D is bounded forces the Levi form to be positive at some point on ∂D , hence by the non-degeneracyof dη everywhere on the connected manifold ∂D . Here, and for the remainder of this section, we employ the Einstein summation conventions, with Greek indicesranging from 1 to n , and the conventions for raising and lowering indices by contraction with the hermitian form h α ¯ δ and its inverse, with φ γ ¯ β h γ ¯ δ = φ ¯ β ¯ δ . J. BLAND AND T. DUCHAMP this is the symmetry condition φ γ ¯ δ h γ ¯ β = φ α ¯ β h α ¯ δ . It follows that the operator φ ◦ ¯ φ : H (1 , → H (1 , has non-negative eigenvalues since iη [( φ ◦ ¯ φ ) Z α , Z ¯ µ ] = ( φ ◦ ¯ φ ) δα h δ ¯ µ = ( ¯ φ ) ¯ γα φ δ ¯ γ h δ ¯ µ = h ¯ γβ ( ¯ φ ) βα φ ¯ γ ¯ µ is hermitian positive semi-definite.Next note that the transversality condition for CR structures (Definition 2.1.1(ii)) implies thatnone of the eigenvalues of φ ◦ ¯ φ can be equal to one. Indeed, suppose to the contrary that ( ¯ φ ) ¯ γα φ δ ¯ γ − δ δα is a degenerate matrix. Then there exists v α such that v α ( ¯ φ ) ¯ γα φ δ ¯ γ = v δ , from which one obtains therelation v α ( Z α − ( ¯ φ ) ¯ βα Z ¯ β ) = v α ( ¯ φ ) ¯ βα φ δ ¯ β Z δ − v α ( ¯ φ ) ¯ βα Z ¯ β = − v α ( ¯ φ ) ¯ βα (cid:16) Z ¯ β − φ δ ¯ β Z δ (cid:17) ;that is, the transversality condition is violated for the subspace ˆ H (1 , ( t ) and its conjugate.Since φ ◦ ¯ φ is isotopic to the zero map by assumption, has non-negative eigenvalues, and ( φ ◦ ¯ φ − I )is nondegenerate, it follows that the eigenvalues of the operator ( φ ◦ ¯ φ ) are bounded between 0 and1, which implies the norm condition. (See [BD2, page 83] where a similar argument is given.) (cid:3) Remark 2.2.4.
The choice to refer to the map φ : H (0 , → H (1 , as the deformation tensor(rather than the conjugate map) is consistent with the deformation theory for complex structures,and has the advantage that φ may be thought of as a “vector-valued (0 , − form”, thus fittingnaturally within a ¯ ∂ − complex (or in this case, a ¯ ∂ b − complex).In light of Corollary 2.2.2, we identify the space of CR structures with the subset of the spaceof H (1 , -valued (0 , ω α is a local coframe of H (1 , with dual frame Z α of H (1 , such that dη = i δ α ¯ β ω α ∧ ω ¯ β , then the CR deformation tensor can be written as φ = φ α ¯ β ω ¯ β ⊗ Z α ;it uniquely determines the space of (0 , H C annihilated by the one-forms ˆ ω α := ω α + φ α ¯ β ω ¯ β . The space of all smooth deformation tensors is given by(2.2.5) D ef = Ω (0 , ( H (1 , ) (cid:39) Γ ∞ (cid:0) H (0 , ⊗ H (1 , (cid:1) . The deformation complex.
Each deformation of a CR structure can be expressed as a H (1 , -valued (0,1)-form. In [Aka], Akahori studied CR deformations by developing the Hodgetheory of a certain complex of vector-valued forms. A similar complex was studied in [BD2] andused to show that CR deformations of the standard CR structure on S n +1 can be parameterizedby complex Hamiltonian vector fields .The space of smooth forms of type (0 , q ), written Ω (0 ,q ) ( M ), is the space of sections of thebundle Λ q H (0 , , where H (0 , denotes the dual bundle of the complex vector bundle H (0 , . By theintegrability condition for the CR structure, the exterior differential operator d naturally inducesan operator ¯ ∂ b : Ω (0 ,q ) ( M ) → Ω (0 ,q +1) ( M ) . R DEFORMATIONS 7
Set T (1 , M = T C M/H (0 , , where T C M = T M ⊗ R C is the complexified tangent bundle of M ,and let π (1 , : T C M → T (1 , M denote the quotient map. The space of T (1 , M -valued forms oftype (0 , q ) is the space of homomorphisms of complex vector bundlesΩ (0 ,q ) ( T (1 , M ) = Γ ∞ (cid:0) Hom C (Λ q H (0 , M, T (1 , M ) (cid:1) . By virtue of the integrability condition (Definition 2.1.1(ii)), the operator ¯ ∂ b extends to anoperator on the space of T (1 , M -valued forms [BlEp, BuMi], which by abuse of notation we againdenote by ¯ ∂ b :(2.3.1) ¯ ∂ b : Ω (0 ,q ) ( T (1 , M ) → Ω (0 ,q +1) ( T (1 , M ) . This operator is characterized by the following properties:¯ ∂ b = 0 ;(2.3.2a) ¯ ∂ b ( X )( Z ) = π (1 , [ Z, X ] , (2.3.2b)for X ∈ Ω (0 , ( T (1 , M ) = Γ( T (1 , M ) and Z ∈ Γ( H (0 , M );¯ ∂ b ( α ∧ β ) = ( ¯ ∂ b α ) ∧ β + ( − q α ∧ ¯ ∂ b β , (2.3.2c)for α ∈ Ω (0 ,q ) ( M ) and β ∈ Ω (0 ,q ) ( T (1 , M ). Remark 2.3.3.
The operator defined by equation (2.3.2b) further lifts to an operator¯ ∂ b : Γ( T C M ) → Ω (0 , ( T (1 , M )via the formula ¯ ∂ b X := ¯ ∂ b ( π (1 , X ) . In particular, ¯ ∂ b X is well-defined in the special case where X is a real vector field. By abuse ofnotation we again denote the lifted operator by ¯ ∂ b . Remark 2.3.4.
When (
M, η, H (1 , ) is embedded, it bounds a strongly pseudoconvex complexspace Σ, and there is a natural identification between T (1 , M and the restriction of the holomorphictangent bundle from Σ. In this case, ¯ ∂ b is naturally identified with the restriction of the ¯ ∂ operatorto the boundary.3. The group of Folland-Stein contact diffeomorphisms
In the previous section, we showed that every CR structure isotopic to a reference CR structurecan be represented by a deformation tensor. In Section 4, we study the action the group of contactdiffeomorphisms on to space of CR structures. In this section, we recall the results from [BD1] thatwe need. Details can be found in [BD1].3.1.
Folland-Stein spaces.
We begin by recalling the anisotropic function spaces Γ s ( M ) on M ,introduced by Folland and Stein in [FS], and their generalizations. These spaces are the naturalones in which to work in order to obtain sharp estimates for the various operators which will arise.Consider an open domain D (cid:98) R n +1 . The Folland-Stein space Γ s = Γ s ( D ) is the Hilbert spacecompletion of the set of smooth functions on D with respect to the inner product( f, g ) D,s := (cid:88) ≤| I |≤ s (cid:90) D | X I f | | X I g | dV , J. BLAND AND T. DUCHAMP with associated norm written (cid:107) f (cid:107) D,s = (cid:112) ( f, f ), where X I and dV are as in Section 1.1. LetΓ s ( D, R m ) denote the closure of the smooth R m valued functions on D with inner product( f, g ) D,s = m (cid:88) j =1 ( f j , g j ) s for smooth functions f = ( f , . . . , f m ) and g = ( g , . . . , g m ).Let ( M, η ) be a smooth compact contact manifold, and let { φ (cid:96) : U (cid:96) → R n +1 } be an adapted atlasas in Section 1.1). A function f : M → R is said to be a Γ s function if the functions f (cid:96) = f ◦ φ − (cid:96) lie in Γ s ( D (cid:96) ) for all (cid:96) . For functions f, g ∈ Γ s ( M ), we define the inner product( f, g ) s := (cid:88) (cid:96) ( ρ (cid:96) f (cid:96) , ρ (cid:96) g (cid:96) ) D (cid:96) ,s . Similar definitions hold for Γ s ( M, R m ), f, g ∈ Γ s ( M, R m ). The definition of the function spacesis independent of the choice of adapted atlas and the local framings X I and dV . Although thedefinition of the inner products depend upon the choices involved, different choices lead to equivalentnorms.Let F : M → (cid:102) M be a C -map from M into a smooth (cid:101) m -dimensional manifold (cid:102) M . Choose anadapted atlas { ( φ (cid:96) , U (cid:96) , V (cid:96) ) } for M and a smooth atlas { (cid:101) φ (cid:96) : (cid:101) U (cid:96) → R (cid:101) m } for (cid:102) M such that F ( U (cid:96) ) ⊂ (cid:101) U (cid:96) , and F (cid:96) ( ¯ D (cid:96) ) ⊂ (cid:101) D (cid:96) for all (cid:96) , where F (cid:96) = (cid:101) φ (cid:96) ◦ F ◦ φ − (cid:96) : φ (cid:96) ( U (cid:96) ) → R (cid:101) m and (cid:101) D (cid:96) (cid:98) φ (cid:96) ( (cid:101) U (cid:96) ) is a collection of open domainssuch that { (cid:101) φ − (cid:96) ( (cid:101) D (cid:96) ) } covers (cid:102) M . The map F is said to be a Γ s map if F (cid:96) restricts to an element F (cid:96) ∈ Γ s ( D (cid:96) , R (cid:101) m ) for all (cid:96) . It is not difficult to show that the notion of Γ s map is independent ofthe choice of atlases and that F (cid:96) restricts to an element in Γ s ( D ) for any open set D ⊂⊂ φ (cid:96) ( U (cid:96) ).Let Γ s ( M, (cid:102) M ) for s ≥ n + 4 denote the topological space of Γ s maps between M and (cid:102) M . Therestriction s ≥ n + 4 ensures that the maps are C . More generally, consider a smooth fibre bundle π : P → M , with base a compact contact manifold. The space Γ s ( P ) of Γ s sections of π is definedin the obvious way by choosing an adapted atlas for M such that π − ( U (cid:96) ) → U (cid:96) is trivial for all (cid:96) and requiring the local coordinate representations of sections to be Γ s maps from U (cid:96) into the fiberof π . (See [BD1] for details.)3.2. The smooth manifold of Folland-Stein diffeomorphisms.
Let D s ( M ) ⊂ Γ s ( M, M ) de-note the space of Γ s diffeomorphisms of M . We showed in [BD1] that D s ( M ) is an open subsetof Γ s ( M, M ) for all s ≥ n + 4. Let D scont ( M ) ⊂ D s ( M ) denote the subspace of Γ s contact diffeo-morphisms of M . In [BD1], we obtained a local coordinate chart for contact diffeomorphisms in aneighbourhood of the identity, and we showed that D scont ( M ) is a topological group with respectto composition, provided that s ≥ n + 4.More precisely, let g be a metric adapted to the contact structure such as the one constructedin the Section 1.1. The exponential map induces various maps between Γ s spaces that we need toparameterize contact diffeomorphisms. If X is a vector field, we use the notation F X to denote themap(3.2.1) F X := exp ◦ X : M → M .
Recall that because M is compact, the map F X is a diffeomorphism for X sufficiently small.The following proposition summarizes various smoothness properties of the maps that we need toconstruct our local coordinate charts for contact diffeomorphisms. R DEFORMATIONS 9
Proposition 3.2.2.
Let Γ s ( T M ) denote the space of Γ s sections of T M . For s ≥ (2 n + 4) , themap exp : Γ s ( T M ) → Γ s ( M, M ) : X (cid:55)→ F X = exp ◦ X is smooth. Moreover, there is a neighbourhood U ⊂ Γ n +4 ( T M ) such that F X is in D s ( M ) for all X ∈ U s and all s ≥ n + 4 , where U s := U ∩ Γ s ( T M ) ; and the restriction exp : U s → D s ( M ) is a homeomorphism from U s to a neighbourhood of the identity diffeomorphism. In general, the diffeomorphism F X of Proposition 3.2.2 will not be a contact diffeomorphism.However in [BD1], we showed that the subset of U s for which it is a contact diffeomorphism issmoothly parameterized by the set of contact vector fields in a neighbourhood of the zero section.As shown in [BD1], this implies that the space of Γ s contact diffeomorphisms is a smooth Hilbertmanifold.We now introduce some notation that will be necessary to express the sharp estimates used laterin the paper. Choose an adapted atlas φ (cid:96) : U (cid:96) → R n +1 for M and a collection of open sets V (cid:96) (cid:98) U (cid:96) covering M as in Section 1.1 and let ρ (cid:96) be a partition of unity subordinate to { V (cid:96) } . By compactnessof M , there is a constant c > x, X ) ∈ U (cid:96) for all x ∈ V (cid:96) , all X ∈ T M x , with | X | < c ,and all (cid:96) . Let X be a C vector field with | X | < c .Fix a chart, say φ (cid:96) , and set U = U (cid:96) and V = V (cid:96) . To simplify notation, we adopt the Einsteinsummation conventions, letting Roman indices range from 1 to 2 n + 1. As explained in [BD1], bythe second order Taylor’s formula with integral remainder, there exist smooth functions B kij ( x, X )(locally defined) on T M such that(3.2.3) F kX := exp k ( x, X ) = x k + X k + B kij ( x, X ) X i X j . A standard computation using Equation (3.2.3) then yields the following expansion for the pull-backof a q -form by F X . Lemma 3.2.4 ([BD1]) . Let ψ be a smooth q -form on M and choose a coordinate patch U = U (cid:96) ,with V = V (cid:96) (cid:98) U . Let c > be chosen so that exp( x, X ) ∈ U for all x ∈ V and all X ∈ T x M with | X | < c . Then there are (locally defined) smooth fibre bundle maps Q ij : BM | V → Λ q M | V and Q ij : BM | V → Λ q − M (cid:12)(cid:12) V , where BM = { X ∈ T M : | X | < c } , such that for any C vector field X : M → BM ⊂ T M theequation F ∗ X ψ = ψ + L X ψ + Q ij ( X ) X i X j + Q ij ( X ) ∧ X i dX j is satisfied on all of V . Henceforth, we will use the notation(3.2.5) Q ψ ( X ) := F ∗ X ( ψ ) − ψ − L X ψ to denote the non-linear part of the pull-back F ∗ X ψ . The lemma states that in local coordinates(3.2.6) Q ψ ( X ) = Q ij ( X ) X i X j + Q ij ( X ) ∧ X i dX j where Q ij and Q ij are smooth differential forms on BM | V ⊂ T M , which depend on the smoothform ψ and on the coordinate chart φ (cid:96) . Because the maps Q aij are smooth differential forms forany smooth q -form ψ , and because M is compact, we have the following corollary to Lemma 3.2.4,which we prove in [BD1]: Lemma 3.2.7.
Let ψ be a smooth q form. Then the following estimates are satisfied for all X ∈ Γ s ( T M ) , s ≥ n + 6 , such that | X | < c : (cid:107) F ∗ X ψ (cid:107) s − ≺ (cid:107) ψ (cid:107) s − + (cid:107)L X ψ (cid:107) s − + (cid:107) X (cid:107) s − (cid:107) X (cid:107) s , (a) (cid:107) ( F ∗ X ψ ) ∧ η (cid:107) s − ≺ (cid:107) ψ ∧ η (cid:107) s − + (cid:107)L X ψ ∧ η (cid:107) s − + (cid:107) X (cid:107) s − (cid:107) X (cid:107) s . (b) Moreover, the estimate (cid:107) ( Q ψ ( X ) − Q ψ ( X )) ∧ η (cid:107) s − ≺ (cid:107) X − X (cid:107) s − ( (cid:107) X (cid:107) s + (cid:107) X (cid:107) s )+ (cid:107) X − X (cid:107) s ( (cid:107) X (cid:107) s − + (cid:107) X (cid:107) s − ) , (c) holds for any two vector fields X i , i = 1 , with | X i | < c . Remark 3.2.9.
As shown in [BD1], for ψ a smooth p -form, the maps X (cid:55)→ F ∗ X ψ and X (cid:55)→ η ∧ F ∗ X ψ define smooth mapsΓ s ( T M ) → Γ s − (Λ p M ) , for s ≥ n + 6 , and Γ s ( T M ) → Γ s − (Λ p +1 M ) , for s ≥ n + 5.Recall that the condition for the diffeomorphism F X to be a contact diffeomorphism is thevanishing of the one-form F ∗ X η mod η . Hence by Equation (3.2.5), F X is a contact diffeomorphismif and only if it satisfies the condition(3.2.10) L X η + Q η ( X ) = 0 mod η . Furthermore, by Equation (3.2.6), the linearization of this condition at the zero vector field is thecondition L X η = 0 mod η , i.e. X is a contact vector field . Remark 3.2.11.
Using the characteristic vector field T for the contact form η , we may expressany vector field X as X = X T + X H , where X H belongs to the contact distribution. Applyingthe Cartan identity L X ( η ) = X dη + d ( X η ) = X H dη + dX , yields the well known facts that (i) the vector field X is a contact vector field if and only if(3.2.12) dX = − X H dη mod η ;and (ii) that X is completely determined by the real-valued function X = X η . For this reason
X η is called the generating function for X and is denoted by g X . In [BD1], we proved that thereis an isomorphism Γ scont ( T M ) → Γ s +1 ( M ) : X (cid:55)→ g X := X η .
The main result of [BD1] is the construction of a smooth parameterization Ψ of the space ofΓ s -contact diffeomorphisms near the identity diffeomorphism by contact vector fields near the zerovector field. The parameterization Ψ in turn induces a smooth structure on the space D scont ( M ) ofall Γ s -contact diffeomorphisms. Theorem 3.2.13 ([BD1]) . For all s ≥ n + 4 , and for U ⊂ Γ n +4 ( T M ) sufficiently small, there isa smooth map Ψ : Γ scont ( T M ) ∩ U → U s ⊂ Γ s ( T M ) such that the following holds: for all Y ∈ U ∩ Γ s ( T M ) , F Y is a contact diffeomorphism if and onlyif Y = Ψ( X ) for some X ∈ Γ scont ( T M ) ∩ U . Moreover, the map Ψ is of the form Ψ( X ) = X + B ( X )( X, X ) , R DEFORMATIONS 11 where B : (Γ scont ( T M ) ∩ U ) × Γ scont ( T M ) × Γ scont ( T M ) → Γ s ( T M ) is smooth and bilinear in the lasttwo factors. This theorem implies the following global result:
Theorem 3.2.14 ([BD1]) . Let ( M, η ) be a compact contact manifold. For s ≥ (2 n + 4) , the spaceof Γ s contact diffeomorphisms is a smooth Hilbert manifold. We close this section with the a priori estimates for the nonlinear term B ( X )( X, X ), which weproved in [BD1] and which we require in Section 6:
Proposition 3.2.15 ([BD1]) . For X ∈ V s = Γ scont ( T M ) ∩ U s , (a) (cid:107) Ψ( X ) − X (cid:107) s ≺ (cid:107) X (cid:107) s (cid:107) X (cid:107) s − . Moreover, for all X , X ∈ V s , (cid:107) (Ψ( X ) − X ) − (Ψ( X ) − X ) (cid:107) s ≺ (cid:107) X − X (cid:107) s − ( (cid:107) X (cid:107) s + (cid:107) X (cid:107) s )(b) + (cid:107) X − X (cid:107) s ( (cid:107) X (cid:107) s − + (cid:107) X (cid:107) s − ) . The action of the contact diffeomorphism group
There is a natural action of contact diffeomorphisms on the space of CR deformations: (cid:40) D ∞ cont ( M ) × Ω (0 , ( H (1 , ) → Ω (0 , ( H (1 , )( F, φ ) (cid:55)→ µ = F ∗ φ . The main result of this section (Proposition 4.1.12) is a formula for F ∗ φ in the special case where F = F Ψ( X ) .Let F be a contact diffeomorphism, and let φ be a deformation tensor. Let ˆ H (0 , ⊂ H C = H (0 , ⊕ H (1 , denote the anti-holomorphic tangent bundle of the strongly pseudoconvex CR structureassociated to φ , and define the pull-back CR structure F ∗ ˆ H (0 , ⊂ H C to be the CR structure withanti-holomorphic subbundle F ∗ ( ˆ H (0 , ) := (cid:110) Z ∈ H C : F ∗ Z ∈ ˆ H (0 , (cid:111) . It is straightforward to check that if F and F are two contact diffeomorphisms then the identity( F ◦ F ) ∗ ( ˆ H (0 , ) = F ∗ ( F ∗ ( ˆ H (0 , ))holds. By Corollary 2.2.2, if F is isotopic to the identity, then F ∗ ˆ H (0 , is represented by a defor-mation tensor, which we call the pull-back CR deformation , denoted by F ∗ φ .4.1. Local formulæ.
We need a local formula for F ∗ Ψ( X ) φ that exhibits the non-linear dependenceon the contact vector field X . It will also prove important to single out terms involving compositionof the components of the tensor φ with F Ψ( X ) ; we accomplish this by introducing an auxiliarycontact vector field Y into some formulæ.Choose an adapted atlas and subordinate partition of unity as in Remark 1.1.2. By smoothnessof the map X (cid:55)→ F Ψ( X ) and compactness of M , for all sufficiently small X , the condition F Ψ( X ) ( V (cid:96) ) (cid:98) U (cid:96) holds for all (cid:96) . Next let η, ω α(cid:96) , ω ¯ α(cid:96) = ω α(cid:96) be a coframing for T C M on U (cid:96) , with H (0 , the annihilatorof η, ω α .For ease of notation, we temporarily suppress the index (cid:96) and set F = F Ψ( X ) . Then(4.1.1) F ∗ ( ˆ H (0 , ) = (cid:110) Z ∈ H C : Z F ∗ ( ω α + φ α ¯ β ω ¯ β ) = 0 , for α = 1 , , . . . , n (cid:111) . One sees immediately that(4.1.2) F ∗ ( ω α + φ α ¯ β ω ¯ β ) = A αβ ω β + B α ¯ β ω ¯ β mod η , where A αβ = Z β F ∗ ( ω α + φ α ¯ γ ω ¯ γ ) = ( Z β F ∗ ω α ) + ( φ α ¯ γ ◦ F ) (cid:0) Z β F ∗ ω ¯ γ (cid:1) (4.1.3a) B α ¯ β = Z ¯ β F ∗ ( ω α + φ α ¯ γ ω ¯ γ ) = (cid:0) Z ¯ β F ∗ ω α (cid:1) + ( φ α ¯ γ ◦ F ) (cid:0) Z ¯ β F ∗ ω ¯ γ (cid:1) . (4.1.3b)By Lemma 3.2.4, one has the formulæ F ∗ ω α = ω α + L X ω α + L (Ψ( X ) − X ) w α + Q ω α (Ψ( X ))(4.1.4a)and F ∗ ω ¯ α = ω ¯ α + L Ψ( X ) ω ¯ α + Q ω ¯ α (Ψ( X )) , (4.1.4b)for one-forms Q ω α and Q ω ¯ α as in Equation (3.2.6) Consequently,(4.1.5) F ∗ (cid:0) ω α + φ α ¯ γ ω ¯ γ (cid:1) = ω α + L X ω α + ( φ α ¯ γ ◦ F Ψ( X ) ) ω ¯ γ + Q α ( X, X, φ ) , where the expression Q α ( X, Y, φ ) is defined by the formula Q α ( X, Y, φ ) := L Ψ( X ) − X ω α + ( φ α ¯ γ ◦ F Ψ( Y ) ) L Ψ( X ) ω ¯ γ + (cid:8) Q ω α (Ψ( X )) + φ α ¯ γ ◦ F Ψ( Y ) ) Q ω ¯ γ (Ψ( X )) (cid:9) (4.1.6)for Y a second, sufficiently small, contact vector field.To single out the terms of the form φ α ¯ γ ◦ F Ψ( X ) , we replace the term φ α ¯ γ ◦ F in Equations (4.1.3a)and (4.1.3b) by φ α ¯ γ ◦ F Ψ( Y ) to get matrix-valued functions(4.1.7) A = A ( X, Y, φ ) and B = B ( X, Y, φ ) . Using the identity Z ¯ β ( L X ω α ) = − ( L X Z ¯ β ) ω α = ( L Z ¯ β X ) ω α = ( ¯ ∂ b X ) α ¯ β , and the expression for ¯ ∂ b X in Remark 2.3.3, yields the following formulæ for the entries of A ( X, Y, φ )and B ( X, Y, φ ) A αβ = δ αβ + Z β L X ω α + Z β Q α ( X, Y, φ )(4.1.8a)and B α ¯ β = ( ¯ ∂ b X ) α ¯ β + ( φ α ¯ β ◦ F Ψ( Y ) ) + Z ¯ β Q α ( X, Y, φ ) . (4.1.8b)Finally, expressing A − B in the form A − B = B + A − ( I − A ) B yields the identity F ∗ Ψ( X ) φ = ¯ ∂ b X + (cid:16) φ α ¯ β ◦ F Ψ( X ) + E α ¯ β ( X, X, φ ) (cid:17) ω ¯ β ⊗ Z α where(4.1.9) E α ¯ β ( X, Y, φ ) = Z ¯ β Q α ( X, Y, φ ) + [ A − ( I − A ) B ] α ¯ β . and where A = A ( X, Y, φ ), B = B ( X, Y, φ ). R DEFORMATIONS 13
Using the partition of unity, we globalize these local formulæ to obtain the vector-valued oneforms(4.1.10) φ ◦ F Ψ( X ) := (cid:88) (cid:96) ρ (cid:96) · ( φ α(cid:96), ¯ β ◦ F Ψ( X ) ) ω ¯ β(cid:96) ⊗ Z (cid:96),α and(4.1.11) E ( X, Y, φ ) := (cid:88) (cid:96) ρ (cid:96) · E α(cid:96), ¯ β ( X, Y, φ ) ω ¯ β(cid:96) ⊗ Z (cid:96),α . Noting that µ = (cid:80) (cid:96) ρ (cid:96) · µ and ¯ ∂ b X = (cid:80) (cid:96) ρ (cid:96) ¯ ∂ b X then immediately gives the next proposition, whichwe need to prove the normal form theorem of Section 6. Proposition 4.1.12.
Let ( F Ψ( X ) , φ ) ∈ D s +1 cont ( M ) × Γ s (cid:0) ( H (0 , ⊗ H (1 , (cid:1) be near ( id M , . Then F ∗ Ψ( X ) φ is given by (a) F ∗ Ψ( X ) φ = ¯ ∂ b X + φ ◦ F Ψ( X ) + E ( X, X, φ ) . The linearized action at the identity map and the zero deformation tensor is (b) ( X, ˙ φ ) (cid:55)→ ¯ ∂ b X + ˙ φ. Remark 4.1.14.
These equations require some care in interpretation. First, notice that the terms F ∗ Ψ( X ) φ and ¯ ∂ b X are in fact globally defined tensors, and make invariant sense. On the other hand, φ ◦ F Ψ( Y ) , and E ( X, Y, φ ) have been defined using local coordinates and are coordinate dependent.
Remark 4.1.15.
Observe that for s ≥ n + 4 the map ( X, Y, φ ) (cid:55)→ E ( X, Y, φ ) extends to the map E : Γ s +1 cont ( T M ) × Γ s +1 cont ( T M ) × Γ s (cid:16) ( H (0 , ⊗ H (1 , (cid:17) → Γ s (cid:16) ( H (0 , ⊗ H (1 , (cid:17) between Folland-Stein spaces. In Section 6.2 we obtain estimates for E that play a critical role inthe proof or our normal form theorem.4.2. Complex contact vector fields.
By equation (b), the action of the group of contact dif-feomorphisms suggests normalizing deformation tensors by the image of ¯ ∂ b X where X is a realcontact vector field. On the other hand, since ¯ ∂ b X = ¯ ∂ b ( π (1 , X ), it is natural to normalize thedeformation tensor by the image of ¯ ∂ b X for X ∈ T (1 , M . We accomplish this by introducing thenotion of complex contact vector fields. Begin by recalling that T (1 , M is defined as the quotient bundle0 → H (0 , (cid:44) → T C M π (1 , −→ T (1 , M → . Noting that T C M = H (1 , ⊕ H (0 , ⊕ C · T , where T is the Reeb vector field, we see that therestriction of π (1 , to H (1 , ⊕ C · T is an isomorphism of complex vector bundles. Thus, we shallidentify T (1 , M with H (1 , ⊕ C · T when it is convenient.Next observe that the composite map T M (cid:44) → T C M π (1 , −→ T (1 , M is injective with image the subbundle { Z ∈ T (1 , M : η ( Z ) ∈ R } . Consequently, there are naturalidentifications(4.2.1) T M = H (1 , ⊕ R · T := { X ∈ T (1 , M : η ( X ) ∈ R } ;and it is easy to check that the inclusion Γ s ( T M ) ⊂ Γ s ( T (1 , M ) is norm preserving. This corresponds to the notion of
Hamiltonian vector fields as used in [BD2].
Because H (0 , is contained in the annihilator of η , the quantity η ( Z ) is well-defined for all Z ∈ T (1 , M . In addition, the quantity π (0 , ( Z dη ) is well-defined, where π (0 , : T C M ∗ → H (0 , denotes the natural projection map. More precisely, let π (0 , ( Z dη ) := π (0 , ( (cid:101) Z dη ) , for any (cid:101) Z ∈ T C M such that π (1 , (cid:101) Z = Z .Finally, recall from Remark 3.2.12 that a real vector field X is a contact vector field if and onlyif it satisfies the identity dX + X dη = 0 mod η , where X = X η . This is equivalent to the two conditions W (cid:0) dX + X dη (cid:1) = 0 and W (cid:0) dX + X dη (cid:1) = 0 for all W ∈ H (1 , .Since X is real, W (cid:0) dX + X dη (cid:1) = W ( dX + X dη ). This leads us to the followingdefinition.
Definition 4.2.2.
We say that a (1 , Z ∈ Γ( T (1 , M ) is a complex contact vector field if it satisfies the condition ¯ ∂ b ( Z η ) + π (0 , ( Z dη ) = 0 . We denote by Γ scont ( T (1 , M ) the Folland-Stein completion of the space of complex contact vectorfields.The following lemma places this definition in context. Lemma 4.2.3.
Let X ∈ Γ( T C M ) be a real vector field. Then the following are equivalent: (a) X is a contact vector field. (b) X satisfies the identity ¯ ∂ b ( X η ) + π (0 , ( X dη ) = 0 where π (0 , : T ∗ C M → H ∗ (0 , M is the restriction operator. (c) The vector-valued one-form ¯ ∂ b X takes values in H (1 , .Proof. By the observations above, a real vector field X is contact if and only if W ( d ( X η ) +
X dη ) = 0 ∀ W ∈ H (0 , . This is simply condition 4.2.3(b), thus establishing the equivalence of 4.2.3(a) and 4.2.3(b). Theequivalence of 4.2.3(b) and 4.2.3(c) is a special case of Lemma 4.2.4 below. (cid:3)
The next lemma gives a useful characterization of complex contact vector fields. Before stat-ing the lemma, we remark that the quotient bundle T (1 , M has a naturally defined subbundledetermined by the vanishing of η , that is H C /H (0 , = { Z ∈ T (1 , M : η ( Z ) = 0 } ⊂ T (1 , M .
A simple computation shows that the map π (1 , defined above restricts to an isomorphism H (1 , (cid:39) H C /H (0 , . Hence, we may identify H C /H (0 , -valued forms with H (1 , -valued forms. Independence of the choice of (cid:101) Z is an immediate consequence of the identity dη ( W , W ) = 0 for all W , W ∈ H (0 , . R DEFORMATIONS 15
Lemma 4.2.4.
The vector field Z ∈ Γ( T (1 , M ) is a complex contact vector field if and only if ¯ ∂ b Z is an H (1 , -valued (0 , -form.Proof. Suppose that ¯ ∂ b Z takes its values in H (1 , ; that is, that(4.2.5) η ( ¯ ∂ b Z ( W )) = 0 ∀ W ∈ H (0 , . By equation (2.3.2b), η (cid:0) ¯ ∂ b Z ( W ) (cid:1) = − η (cid:0) π (1 , [ Z, W ] (cid:1) = − η (cid:0) [ Z, W ] (cid:1) ;but η (cid:0) [ Z, W ] (cid:1) = − dη ( Z, W ) + Zη ( W ) − W η ( Z )= − dη ( Z, W ) − W ( Z η )= − W (cid:0) Z dη + ¯ ∂ b ( Z η ) (cid:1) . Thus, ¯ ∂ b Z takes its values in H (1 , if and only if W (cid:0) Z dη + ¯ ∂ b ( Z η ) (cid:1) = 0 for all W ∈ H (0 , ,which is equivalent to Z being complex contact. (cid:3) Homotopy operators for CR manifolds
In this section, we will collect various results concerning the existence and regularity of homotopyoperators on embedded strongly pseudoconvex CR manifolds. We restrict our statements to thespecial case of embedded, three dimensional CR manifolds. More details of these constructions andtheir generalizations can be found in e.g. [BuMi], [Miy1].5.1.
Miyajima’s homotopy operators.
First, we have the following result for the ¯ ∂ b complex.It follows immediately from the vector bundle valued version contained in [Miy2], where the vectorbundle is the trivial line bundle, and P = N ¯ ∂ ∗ b , but the result is essentially contained in [BeGr].Roughly speaking, it says that there exists a partial inverse and a Szeg¨o projector with goodestimates. Theorem 5.1.1.
There exist linear operators H : C ∞ ( M ) → C ∞ ( M ) , P : Ω (0 , ( M ) → C ∞ ( M ) , and S : Ω (0 , ( M ) → Ω (0 , ( M ) , such that the following identities and estimates are satisfied: ¯ ∂ b ◦ H = 0 , P ◦ S = 0 , S ◦ ¯ ∂ b = 0(a) u = P ¯ ∂ b u + Hu and α = ¯ ∂ b P α + Sα , (b) (cid:107) H ( u ) (cid:107) s ≺ (cid:107) u (cid:107) s (cid:107) P ( α ) (cid:107) s ≺ (cid:107) α (cid:107) s +1 , (cid:107) S ( α ) (cid:107) s ≺ (cid:107) α (cid:107) s , (c) for all u ∈ C ∞ ( M ) , α ∈ Ω (0 , ( M ) , and s ≥ . H extends to a self-adjoint, projection operator on L ( M, C ) . (d)Similarly, homotopy operators for T (1 , M -valued (0 ,
1) forms also exist, with similar estimates[Miy2]. These estimates work in general for vector valued forms, where the vector bundle is therestriction of a complex vector bundle which extends to the complex manifold bounded by M asa holomorphic bundle. (If the complex space X bounded by M is singular, we first resolve thesingularities of X and then apply the above definition.) Theorem 5.1.2 (Miyajima [Miy1], [Miy2]) . There exist linear operators ρ : Γ ∞ ( T (1 , M ) → Γ ∞ ( T (1 , M ) ,P : Ω (0 , ( T (1 , M ) → Γ ∞ ( T (1 , M ) ,Q : Ω (0 , ( T (1 , M ) → Ω (0 , ( T (1 , M ) satisfying the following identities and estimates: ¯ ∂ b ◦ ρ = 0 , P ◦ Q = 0 Q ◦ ¯ ∂ b = 0(a) Z = P ¯ ∂ b Z + ρZ and φ = ¯ ∂ b P φ + Qφ (b) (cid:107) P φ (cid:107) s +1 ≺ (cid:107) φ (cid:107) s , (cid:107) Qφ (cid:107) s ≺ (cid:107) φ (cid:107) s , (cid:107) ρ ( Z ) (cid:107) s ≺ (cid:107) Z (cid:107) s , (c) for all Z ∈ Γ ∞ ( T (1 , M ) , φ ∈ Ω (0 , ( T (1 , M ) , and s ≥ .Finally, there exist linear operators L : Ω (0 , ( T (1 , M ) → Γ ∞ ( T (1 , M ) and N : Γ ∞ ( T (1 , M ) → Γ ∞ ( T (1 , M ) , with L a smooth horizontal linear first order differential operator such that P = N ◦ L , (d) and N satisfies the estimate (cid:107) N ( Z ) (cid:107) s +2 ≺ (cid:107) Z (cid:107) s for all Z ∈ Γ ∞ ( T (1 , M ) , s ≥ . (e)5.2. Homotopy operators for complex contact vector fields.
These homotopy formulæ donot single out contact vector fields in any significant manner. We now show how to modify thehomotopy operators in order to do so. We begin by introducing the raising and lowering operatorsinduced by the nondegenerate two form dη : Definition 5.2.1.
The lowering operator is the vector bundle map( ) (cid:91) : T M → H ∗ : X (cid:55)→ X (cid:91) = X dη whose restriction to H ⊂ T M is an isomorphism between the contact distribution and its dualspace. The raising operator is the inverse( ) (cid:93) : H ∗ → H : φ (cid:55)→ φ (cid:93) . Remark 5.2.2.
The maps ( ) (cid:91) and ( ) (cid:93) of Definition 5.2.1 induce (complex) linear maps( ) (cid:91) : T (1 , M → H (0 , : Z (1 , (cid:55)→ Z (1 , dη and ( ) (cid:93) : H (0 , → H (1 , , where the map ( ) (cid:93) is an isomorphism of complex vector bundles. Observe that by construction,(5.2.3) Z = η ( Z ) T + ( Z (cid:91) ) (cid:93) for all Z ∈ Γ ∞ ( T (1 , M ). Notice also, that by Definition 4.2.2, Z is an element of the spaceΓ ∞ cont ( T (1 , M ) of complex contact vector fields if and only if it satisfies the identity(5.2.4) ¯ ∂ b ( η ( Z )) + Z (cid:91) = 0 . R DEFORMATIONS 17
Thus, every complex contact vector field is of the form(5.2.5) Z f = f T − ( ¯ ∂ b f ) (cid:93) for f a smooth complex valued function. Moreover, the inclusion T M (cid:44) → T (1 , M induces theinclusion Γ ∞ cont ( T M ) (cid:44) → Γ ∞ cont ( T (1 , M ) : X (cid:55)→ Z g X , where g X = η ( X ). (See Remark 3.2.11.) Proposition 5.2.6.
There exist smooth linear operators ˆ P , ˆ S : Γ ∞ ( T (1 , M ) → Γ ∞ ( T (1 , M ) satisfying the following: Z = ˆ P Z + ˆ S Z for all Z ∈ Γ ∞ ( T (1 , M ) . (a) range( ˆ P ) = ker( ˆ S ) = Γ ∞ cont ( T (1 , M ) . (b) ˆ P ◦ ˆ S = 0 , ˆ S ◦ ˆ P = 0 , ˆ P ◦ ˆ P = ˆ P , and ˆ S ◦ ˆ S = ˆ S . (c) (cid:107) ˆ P Z (cid:107) s ≺ (cid:107) Z (cid:107) s and (cid:107) ˆ S Z (cid:107) s ≺ (cid:107) Z (cid:107) s for all Z ∈ Γ ∞ ( T (1 , M ) . (d) Proof.
Choose a vector field Z ∈ Γ( T (1 , M ), and compute as follows using the homotopy operatorsfrom Theorems 5.1.1 and 5.1.2: Z = η ( Z ) T + ( Z (cid:91) ) (cid:93) = ( H ( η ( Z )) + P ( ¯ ∂ b ( η ( Z )))) T + (cid:110) ¯ ∂ b P ( Z (cid:91) ) + S ( Z (cid:91) ) (cid:111) (cid:93) . Add and subtract the term P ( Z (cid:91) ) T and rearrange to get Z = (cid:110) ( H ( η ( Z )) − P ( Z (cid:91) )) T + ( ¯ ∂ b P ( Z (cid:91) )) (cid:93) (cid:111) + (cid:26)(cid:16) P ( ¯ ∂ b ( η ( Z ))) + P ( Z (cid:91) ) (cid:17) T + (cid:16) S ( Z (cid:91) ) (cid:17) (cid:93) (cid:27) . (5.2.7)Define ˆ P , ˆ S : Γ ∞ ( T (1 , M ) → Γ ∞ ( T (1 , M ) to be the linear operators given by the formulæˆ P ( Z ) = (cid:16) H ( η ( Z )) − P ( Z (cid:91) ) (cid:17) T + (cid:16) ¯ ∂ b P ( Z (cid:91) ) (cid:17) (cid:93) ˆ S ( Z ) = (cid:16) P ( ¯ ∂ b ( η ( Z ))) + P ( Z (cid:91) ) (cid:17) T + (cid:16) S ( Z (cid:91) ) (cid:17) (cid:93) . By construction, Z = ˆ P Z + ˆ S Z .We claim that ˆ P Z is a smooth complex contact vector field. This follows from Equation (5.2.4)and the computation¯ ∂ b (cid:16) η ( ˆ P ( Z )) (cid:17) + ˆ P ( Z ) (cid:91) = ¯ ∂ b H ( η ( Z )) − ¯ ∂ b P ( Z (cid:91) ) + ¯ ∂ b P ( Z (cid:91) ) = ¯ ∂ b H ( η ( Z )) = 0 . Observe also that by (5.2.4)ˆ S ( Z ) = (cid:16) P ( ¯ ∂ b ( η ( Z ))) + P ( Z (cid:91) ) (cid:17) T + (cid:16) S ( Z (cid:91) ) (cid:17) (cid:93) = (cid:16) P ( − Z (cid:91) ) + P ( Z (cid:91) ) (cid:17) T + (cid:0) S ( − ¯ ∂ b ( η ( Z ))) (cid:1) (cid:93) = ( − (cid:93) = 0 . for all Z ∈ Γ ∞ cont ( T (1 , M ).We have shown that ˆ P takes values in Γ ∞ cont ( T (1 , M ) and that ˆ S vanishes on Γ ∞ cont ( T (1 , M ).These facts, combined with Equation (a) imply that ˆ P and ˆ S satisfy the identities :ˆ P ◦ ˆ S = 0 , ˆ S ◦ ˆ P = 0 , ˆ P ◦ ˆ P = ˆ P , and ˆ S ◦ ˆ S = ˆ S , as well as the equalities Γ ∞ cont ( T (1 , M ) = range( ˆ P ) = ker( ˆ S ) . The estimates follow from the estimates in Theorems 5.1.1 and 5.1.2. (cid:3)
Remark 5.2.8.
Because the projection operators ˆ P , ˆ S in Proposition 5.2.6 preserve the Folland-Stein regularity, they extend to projection operators on the Folland-Stein space Γ s ( T (1 , M ) andthey induce a direct sum decompositionΓ s ( T (1 , M ) = Γ scont ( T (1 , M ) ⊕ ker( ˆ P )with Γ scont ( T (1 , M ) = ker( ˆ S ) = range( ˆ P ) ⊂ Γ s ( T (1 , M ).The following variant of Theorem 5.1.2, highlights the role of contact vector fields. Theorem 5.2.9.
There exist linear operators P : Ω (0 , ( T (1 , M ) → Γ ∞ cont ( T (1 , M ) and H : Ω (0 , ( T (1 , M ) → Ω (0 , ( T (1 , M ) such that: Z = P ¯ ∂ b Z + ρZ for all Z ∈ Γ ∞ cont ( T (1 , M )(a) φ = ¯ ∂ b P φ + H φ for all φ ∈ Ω (0 , ( T (1 , M )(b) ¯ ∂ b P ◦ H = 0 , H ◦ ¯ ∂ b P = 0(c) H ◦ ¯ ∂ b Z = 0 for all Z ∈ Γ ∞ cont ( T (1 , M )(d) (cid:107)P φ (cid:107) s +1 ≺ (cid:107) φ (cid:107) s , (cid:107)H φ (cid:107) s ≺ (cid:107) φ (cid:107) s for all φ ∈ Ω (0 , ( T (1 , M ) . (e) Moreover, (cid:107) ρ ( Z ) (cid:107) s ≺ (cid:107) Z (cid:107) s for all Z ∈ Γ ∞ cont ( T (1 , M ) , s ≥ . (f) Finally, there exists a smooth linear operators L : Ω (0 , ( T (1 , M ) → Γ ∞ ( T (1 , M ) and N : Γ ∞ ( T (1 , M ) → Γ ∞ ( T (1 , M ) with L a horizontal, first order differential operator, such that P = N ◦ L (g) (cid:107)N ( Z ) (cid:107) s +2 ≺ (cid:107) Z (cid:107) s for all Z ∈ Γ ∞ ( T (1 , M ) , s ≥ . (h) Proof.
The key step in the proof is to express the homotopy operator P of Theorem 5.1.2 as thesum of two operators P and S , defined by the formulas P = ˆ P ◦ P and S = ˆ S ◦
P .
By Proposition 5.2.6, P = P + S and the image of P is contained in the space Γ ∞ cont ( T (1 , M ) ofsmooth complex contact vector fields. Next let H = ¯ ∂ b ◦ S + Q , where Q is as in Theorem 5.1.2.To prove (a), let Z be a complex contact vector field and note that by 5.1.2(b) Z = P ¯ ∂ b Z + ρZ = P ¯ ∂ b Z + S ¯ ∂ b Z + ρZ . We need only show that S ¯ ∂ b Z = 0, for Z complex contact. First observe that whenever Z is a com-plex contact vector field, then P ¯ ∂ b Z is also complex contact. This follows easily from Lemma 4.2.4,the formula P ¯ ∂ b Z = Z − ρ ( Z ), and ¯ ∂ b ρ ( Z ) = 0. Consequently, S ( ¯ ∂ b Z ) = ˆ S ( P ¯ ∂ b Z ) = 0, for all Z ∈ Γ ∞ cont ( T (1 , M ). R DEFORMATIONS 19
To prove the homotopy formula (b), notice that Proposition 5.2.6(a) implies the decomposition P = P + S ;then use the homotopy formula 5.1.2(b) to compute as follows: φ = ¯ ∂ b P φ + Qφ = ¯ ∂ b P + ¯ ∂ b S φ + Qφ = ¯ ∂ b P φ + H φ . We now prove parts (c) and (d). First observe that
S ◦ ¯ ∂ b P = 0. Since P φ is complex contact, P ¯ ∂ b P φ is complex contact. Therefore, S ¯ ∂ b P φ = ˆ S ( P ¯ ∂ b P φ ) = 0. Next observe that ¯ ∂ b ( P ◦ ¯ ∂ b S ) = 0as follows: For φ ∈ Ω (0 , ( T (1 , M ), compute as follows:¯ ∂ b P φ = ¯ ∂ b P (cid:0) ¯ ∂ b P φ + ¯ ∂ b S φ + Qφ (cid:1) = ¯ ∂ b P ¯ ∂ b P φ + ¯ ∂ b P ¯ ∂ b S φ ;on the other hand ¯ ∂ b P φ = ¯ ∂ b P ( ¯ ∂ b P φ ) + ¯ ∂ b S ( ¯ ∂ b P φ ) + Q ( ¯ ∂ b P φ ) = ¯ ∂ b P ¯ ∂ b P φ . Thus, ¯ ∂ b ( P ◦ ¯ ∂ b S ) = 0. Finally, the identities Q ◦ ¯ ∂ b = P ◦ Q = S ◦ Q = 0 follow immediately fromProposition 5.2.6 and Theorem 5.1.2. Then the identities ¯ ∂ b P ◦ H = 0 and
H ◦ ¯ ∂ b P = 0 follow fromthe identities ¯ ∂ b ( P ◦ ¯ ∂ b S ) = 0 and S ◦ ¯ ∂ b P = 0.To prove part (g), set L = L and N = ˆ P ◦ N . Since P = ˆ P ◦ P and by (5.1.2c) P = N ◦ L , itfollows that P = N ◦ L .The estimates (e), (f), and (h) follow immediately from the estimates in Theorems 5.1.1 and 5.1.2. (cid:3)
Notice that in the last theorem, since Ω (0 , ( H (1 , ) ⊂ Ω (0 , ( T (1 , M ), it follows that for φ ∈ Ω (0 , ( H (1 , ), we have φ = ¯ ∂ b P φ + H φ . Moreover, since the range of P is the space of complexcontact vector fields, then ¯ ∂ b P φ ∈ Ω (0 , ( H (1 , ) (see Lemma 4.2.4). It follows that H restricts toan operator H : Ω (0 , ( H (1 , ) → Ω (0 , ( H (1 , ). Therefore, we can restrict the homotopy formulato the horizontal vector valued forms. We state this next, using the same symbols to denote therestricted operators without risk of confusion. Corollary 5.2.10.
There exist homotopy operators P : Ω (0 , ( H (1 , ) → Γ ∞ cont ( T (1 , M ) , H : Ω (0 , ( H (1 , ) → Ω (0 , ( H (1 , ) such that: φ = ¯ ∂ b P φ + H φ for all φ ∈ Ω (0 , ( H (1 , )(a) ¯ ∂ b P ◦ H = 0
H ◦ ¯ ∂ b P = 0(b) H ◦ ¯ ∂ b Z = 0 for all Z ∈ Γ ∞ cont ( T (1 , M )(c) (cid:107)P φ (cid:107) s +1 ≺ (cid:107) φ (cid:107) s (cid:107)H φ (cid:107) s ≺ (cid:107) φ (cid:107) s for all φ ∈ Ω (0 , ( H (1 , ) . (d) Moreover, noting that the harmonic projection ρ restricts to a map ρ : Γ ∞ cont ( T (1 , M ) → Γ ∞ cont ( T (1 , M ) : Z = P ¯ ∂ b Z + ρZ for all Z ∈ Γ ∞ cont ( T (1 , M )(e) (cid:107) ρ ( Z ) (cid:107) s ≺ (cid:107) Z (cid:107) s for all Z ∈ Γ ∞ cont ( T (1 , M ) , s ≥ . (f) Harmonic decomposition of complex contact vector fields.
In this section, we obtaina decomposition of complex contact vector fields into real contact vector fields and a complemen-tary subspace. Recall from Equation (5.2.5) that the space of complex contact vector fields isparameterized by complex valued functions as follows: f (cid:55)→ Z f = f T − ( ¯ ∂ b f ) (cid:93) . The observation that this parametrization agrees with the parametrization of real contact vec-tor fields as introduced in Remark 3.2.11 suggests constucting the decomposition using the na¨ıveprojection operator π Re : Z f (cid:55)→ Z Re( f ) . Unfortunately, this projection map is not continuous in theFolland-Stein norm. We see this as follows. By virtue of the identification T (1 , M = H (1 , ⊕ C · T ,the Folland-Stein structure on the space of complex contact vector fields is( Z f , Z g ) s = (cid:16) ( f T − ( ¯ ∂ b f ) ) , ( gT − ( ¯ ∂ b g ) ) (cid:17) s = ( f T, gT ) s + (cid:16) ( ¯ ∂ b f ) , ( ¯ ∂ b g ) (cid:17) s = ( f, g ) s + (cid:0) ¯ ∂ b f, ¯ ∂ b g (cid:1) s . (cid:107) Z f (cid:107) s = (cid:107) f (cid:107) s + (cid:107) ¯ ∂ b f (cid:107) s . On the other hand, since Z Re( f ) = 1 / Z f + Z f ), (cid:107) Z Re( f ) (cid:107) s = 1 / (cid:107) Z f + Z f (cid:107) s = (cid:107) Re( f ) (cid:107) s + 1 / (cid:107) ¯ ∂ b f + ∂ b f (cid:107) s . Let f k be a sequence of CR functions with (cid:107) ∂f k (cid:107) s → ∞ and (cid:107) f k (cid:107) s bounded. Then (cid:107) Z f k (cid:107) s isbounded, but (cid:107) Z Re( f k ) (cid:107) s = (cid:107) Re( f k ) (cid:107) s + 12 (cid:107) ¯ ∂ b f k (cid:107) s = (cid:107) Re( f k ) (cid:107) s + 12 (cid:107) ∂ b f k (cid:107) s → ∞ . Therefore, to obtain a bounded projection, we have to proceed differently. We need the followingregularly lemma.
Lemma 5.3.1.
The estimate (cid:107) u (cid:107) s +2 ≺ (cid:107) Re( I + (cid:3) b ) u (cid:107) s holds for any smooth, real-valued function u . In particular, if Re( I + (cid:3) b ) u is smooth, then so is u .Proof. One easily verifies that for u real, the identity Re( u + (cid:3) b u ) = u + n +2 ∆ R u holds, where∆ R is the Laplace operator in the Rumin complex. The estimate follows from the correspondingestimate for ∆ R , proved in [R, BD3]. (cid:3) Next let f be a smooth, complex valued function f . Then Re( f + (cid:3) b f ) is smooth, and Lemma 5.3.1implies that there is a unique, smooth, real-valued function u , satisfying the equation( I + 12 n + 2 ∆ R ) u = Re( f + (cid:3) b f ) . Proposition 5.3.2.
For all s ≥ n + 4 , the map f (cid:55)→ u := ( I + 12 n + 2 ∆ R ) − Re( f + (cid:3) b f ) induces a bounded projection operator π Re : Γ scont ( T (1 , M ) → Γ scont ( T (1 , M ) : Z f (cid:55)→ Z u with image Γ scont ( T M ) . R DEFORMATIONS 21
Proof.
By construction π Re ( Z u ) = Z u for u real. Consequently, π Re is a projection operator, asclaimed. To prove that π Re is bounded, note that regularity for ∆ R justifies estimating as follows: (cid:107) Z u (cid:107) s ≺ (cid:107) u (cid:107) s + (cid:107) ¯ ∂ b u (cid:107) s ≺ (cid:107) u (cid:107) s +1 ≺ (cid:107) ( I + 12 n + 2 ∆ R ) u (cid:107) s − ≺ (cid:107) Re( f + (cid:3) b f ) (cid:107) s − ≺ (cid:107) f (cid:107) s − + (cid:107) (cid:3) b f (cid:107) s − . But (cid:107) (cid:3) b f (cid:107) s − = (cid:107) ¯ ∂ ∗ b ¯ ∂ b f (cid:107) s − ≺ (cid:107) ¯ ∂ b f (cid:107) s implies the estimate (cid:107) Z u (cid:107) s ≺ (cid:107) f (cid:107) s − + (cid:107) ¯ ∂ b f (cid:107) s ≺ (cid:107) Z f (cid:107) s . (cid:3) The projection map π Re induces the decompositionΓ ∞ cont ( T (1 , M ) = Γ ∞ cont ( T M ) ⊕ iV , where(5.3.3) V := { Y ∈ Γ ∞ cont ( T (1 , M ) : π Re ( iY ) = 0 } . Let V s denote the closure of V in the Γ s norm. It will prove convenient to adopt the notationalconvention(5.3.4) Z f = X f − i Y f , where X f := π Re ( Z f ) ∈ Γ ∞ cont ( T M ) and Y f := π Im ( Z f ) is the projection π Im := i (Id − π Re ) : Z f (cid:55)→ Y f . Moreover, the estimate (cid:107) X f (cid:107) s + (cid:107) Y f (cid:107) s ≺ (cid:107) Z (cid:107) s holds for all f ∈ Γ s ( M, C ), with s ≥ n + 4. Remark 5.3.5.
We caution the reader that although X f is real, it is not the real part of Z f . Remark 5.3.6.
We could at this point let iV be a rather arbitrary complement to Γ ∞ cont ( T M ).The only properties for V that are important in what follows are:(a) Γ ∞ cont ( T (1 , M ) ∼ = Γ ∞ cont ( T M ) ⊕ iV ,(b) Γ scont ( T (1 , M ) ∼ = Γ scont ( T M ) ⊕ iV s ,(c) (cid:107) X (cid:107) s + (cid:107) Y (cid:107) s ≺ (cid:107) X − iY (cid:107) s ,for all s ≥ n + 4. 6. Normal form for CR deformations
In this section, we study the action of the contact diffeomorphism group on the space of defor-mations of a fixed embeddable CR structure (
M, H (1 , ) on a compact three dimensional manifold M .There are significant differences in the analysis between the three dimensional case and higherdimensions. These arise since first, there are no integrability conditions in dimension three, andsecond, the relevant operators are not subelliptic in three dimensions. While the analysis general-izes to higher dimensions, the details are numerous and everything requires a separate statement,including the introduction of new operators to take into account the integrability conditions. Sincein dimensions at least five, it is well known that all compact, strongly pseudoconvex CR manifoldsare embeddable, our main interest is in the three dimensional case where the situation is moresubtle and less well understood. Henceforth, we will restrict our attention to this case.Before beginning the statement and proof of the main results, we make some comments tomotivate the definitions and statements. The contact diffeomorphism group acts on the space ofdeformation tensors, and the linearization of the action at the identity map and the zero deformation tensor is ( X, ˙ φ ) (cid:55)→ ( ¯ ∂ b X + ˙ φ ), where X is a contact vector field. On the other hand, the Hodgedecomposition of Corollary 5.2.10 shows that a deformation tensor can be split as φ = ¯ ∂ b P φ + H φ , where P φ is a complex contact vector field, and H φ serves as the “harmonic part” of thedeformation. If we split the complex contact vector fields as P φ = X − iY , where X is a realcontact vector field, and Y lies in a transverse subspace (see Section 5.3), then Y can be heuristicallythought of as infinitesimally arising from one of Kuranishi’s “wiggles” of the embedded CR manifoldwithin its ambient surface. The normal form should then be i ¯ ∂ b Y + φ H , that is, a harmonic formplus a wiggle.This overview suggests that we should consider a map Γ ∞ cont ( T M ) ⊕ iV ⊕ ker P → D ef and showthat for all φ ∈ D ef , there exist ( X, Y, ψ ) such that F ∗ Ψ( X ) φ = i ¯ ∂ b Y + ψ or ( F − X ) ) ∗ ( i ¯ ∂ b Y + ψ ) = φ ;here, F Ψ( X ) is the contact diffeomorphism defined by Ψ( X ) as in Theorem 3.2.13. Unfortunately,the linearization of this map loses regularity, since it involves differentiation with respect to X ,which has a component in the direction of X . To circumvent this difficulty, we carry along a copyof φ and consider the modified map ( φ, X, Y, ψ ) (cid:55)→ ( φ, F ∗ Ψ( X ) φ − ( i ¯ ∂ b Y + ψ )). This map is nowinvertible (modulo a kernel – the CR vector fields – which is easily incorporated) giving a weaknormal form: for every φ , there is a triple ( X, Y, ψ ) such that F ∗ Ψ( X ) φ = i ¯ ∂ b Y + ψ . However, in the proof, the normal form i ¯ ∂ b Y + ψ has less regularity than φ . This can be viewed asa weak Hodge decomposition for the nonlinear theory. However, taking our lead from the proof ofregularity for the standard linear Hodge theory, we then obtain a priori estimates in Section 6.2to improve the regularity and establish a strong normal form: if F ∗ Ψ( X ) φ = i ¯ ∂ b Y + ψ with φ ∈ Γ s , then X, Y ∈ Γ s +1 , ψ ∈ Γ s . Remark.
We expect that this approach of first using linear analysis to obtain a weak normal formand then a priori estimates to obtain the strong normal form will find a wide range of use in otherapplications.6.1.
Statement of the Normal Form Theorem.
Throughout the remainder of the paper,(
M, H (1 , ) is a fixed embeddable compact three dimensional CR manifold.We first establish notation. Let H = ker P ⊂ Ω (0 , ( H (1 , ) represent the “harmonic deformationtensors”, where P is as in Corollary 5.2.10 and denote the the CR vector fields by Γ s +1 CR ( T M ) =ker ¯ ∂ b ∩ Γ s +1 ( T (1 , M ). Let Γ s (H ) denote the Folland-Stein completion of H in Γ s (Ω (0 , ( H (1 , )).Notice that Γ s (H ) is closed in the space of deformation tensors Γ s ( D ef ) = Γ s (Ω (0 , ( H (1 , )) andthat by Corollary 5.2.10 Γ s (Ω (0 , ( H (1 , )) = range( ¯ ∂ b ) ⊕ Γ s (H ) . We define the map :(6.1.1a) Φ : Γ s +2 ( D ef ) ⊕ Γ s +1 cont ( T M ) ⊕ V s +1 ⊕ Γ s (H ) −→ Γ s +2 ( D ef ) ⊕ Γ s ( D ef ) ⊕ Γ s +1 CR ( T M )by the formula(6.1.1b) ( φ, X, Y, ψ ) (cid:55)→ (cid:16) φ, F ∗ Ψ( X ) φ − i ¯ ∂ b Y − ψ, ρ ( X − iY ) (cid:17) . This corrects a mistake in [B] when we mistakenly asserted the map Φ to be C if we take the first factor on eachside to be in Γ s ( D ef ). In the Section 6.2, we obtain a priori estimates to establish a local nonlinear Hodge theoryand recover the lost regularity. R DEFORMATIONS 23
Proposition 6.1.2.
The map Φ is a local diffeomorphism in a neighbourhood of the origin.Proof. By the inverse function theorem for Banach spaces, it is sufficient to establish that:(1) Φ is locally C ;(2) d Φ | (0 , , , is invertible.To establish (1), notice that all terms in the map Φ are linear, and smooth (see Theorem 5.1.2), ex-cept F ∗ Ψ( X ) φ , so it suffices to check the regularity of this term. By Remark 3.2.9, X (cid:55)→ ( Z β F ∗ Ψ( X ) ω )and X (cid:55)→ ( Z β F ∗ Ψ( X ) ¯ ω ) are smooth maps from Γ s +1 contact vector fields to Γ s functions. Weproved in [BD1] that the map Γ s +2 ( M ) ⊕ D s +1 cont ( M ) → Γ s ( M ) : ( u, F ) (cid:55)→ u ◦ F is C . From the local expressions in formulæ (4.1.3a) and (4.1.3b) and the fact that the matrix A in these formulæ is invertible, it follows that the term ( φ, X ) (cid:55)→ F ∗ Ψ( X ) φ is C , completing theproof that the map Φ is C .We next check that d Φ is invertible at the origin. Let ( ˙ φ, ˙ X, ˙ Y , ˙ ψ ) be a tangent vector at theorigin. Then d Φ( ˙ φ, ˙ X, ˙ Y , ˙ ψ ) = (cid:16) ˙ φ, ¯ ∂ b ˙ X − i ¯ ∂ b ˙ Y + ˙ φ − ˙ ψ, ρ ( ˙ X − i ˙ Y ) (cid:17) . It is clear that this map has trivial kernel and that it is surjective. In fact, using the homotopyoperators P , H , we can verify that the inverse map ( d Φ) − is given by( d Φ) − : (cid:40) Γ s +2 ( D ef ) ⊕ Γ s ( D ef ) ⊕ Γ s +1 CR ( T M ) −→ Γ s +2 ( D ef ) ⊕ Γ s +1 cont ( T M ) ⊕ V s +1 ⊕ Γ s (H )( ˙ φ, χ, ξ ) (cid:55)→ ( ˙ φ, π Re ( P ( χ − ˙ φ ) + ξ ) , π Im ( P ( χ − ˙ φ ) + ξ ) , −H ( χ − ˙ φ )) . To verify that this is the inverse of d Φ (0 , , , compute as follows:( d Φ) − ( ˙ φ, ¯ ∂ b ˙ X − i ¯ ∂ b ˙ Y + ˙ φ − ˙ ψ, ρ ( ˙ X − i ˙ Y )) = ( ˙ φ, π Re ( P ( ¯ ∂ b ˙ X − i ¯ ∂ b ˙ Y − ˙ ψ ) + ρ ( ˙ X − i ˙ Y )) ,π Im ( P ( ¯ ∂ b ˙ X − i ¯ ∂ b ˙ Y − ˙ ψ ) + ρ ( ˙ X − i ˙ Y )) , − H ( ¯ ∂ b ˙ X − i ¯ ∂ b ˙ Y − ˙ ψ ))= ( ˙ φ, ˙ X, ˙ Y , ˙ ψ ) . (cid:3) By the implicit function theorem, inverting Φ gives rise to the C map(6.1.3a) Γ s +2 ( D ef ) → D s +1 cont ( M ) ⊕ V s +1 ⊕ Γ s (H ) : φ (cid:55)→ ( F φ , Y φ , ψ φ )defined by the constraint(6.1.3b) ( φ, X φ , Y φ , ψ φ ) = Φ − ( φ, , , with F φ = F Ψ X φ and φ in a sufficiently small neighbourhood of the origin. Corollary 6.1.4.
There exist neighbourhoods ∈ U ⊂ Γ s +2 ( D ef ) and id M ∈ ˜ U ⊂ D s +1 cont ( M ) suchthat for any φ ∈ U , there is a contact diffeomorphism F φ ∈ (cid:101) U such that F ∗ φ φ is contained in thesubspace ¯ ∂ b (cid:0) iV s +1 (cid:1) ⊕ Γ s (H ) ⊂ Γ s ( D ef ) . The equation F ∗ φ φ = i ¯ ∂ b Y φ + ψ φ ∈ Γ s ( D ef ) Notice that the differential of this map is the Lie derivative of u , which explains the loss in regularity on u . determines F φ , Y φ and ψ φ up to the CR -vector field ρ ( X φ − iY φ ) , which is in turn determined bythe additional constraint ρ ( X φ − iY φ ) = 0 . We call the deformation tensor F φ ∗ φ = i ¯ ∂ b Y φ + ψ φ ∈ Γ s ( D ef )the normal form of φ . The following theorem, which is proved using a priori estimates, givesincreased regularity for the normal form. It is an immediate corollary to Theorem 6.2.7 below. Theorem 6.1.5.
The map φ (cid:55)→ ( F φ , Y φ , ψ φ ) defines a C map of the form Γ s +2 ( D ef ) → D s +3 cont ( M ) ⊕ V s +3 ⊕ Γ s +2 (H ) , for sufficiently small φ ∈ Γ s +2 ( D ef ) . In particular, the normal form F ∗ φ φ = (cid:0) i ¯ ∂ b Y φ + ψ φ (cid:1) is contained in ¯ ∂ b (cid:0) iV s +3 (cid:1) ⊕ Γ s +2 (H ) ⊂ Γ s +2 ( D ef ) . Remark 6.1.6.
As noted in Remark 5.3.6, we have some freedom in the choice of V s +3 , thecomplementary subpace to Γ s +3 cont ( T M ) in Γ s +3 cont ( T (1 , M ). If the original CR manifold admits a free S action as a symmetry, we can choose all homotopy operators to be S equivariant. Complexcontact vector fields then have Fourier expansions, and we can choose our complement V to consistof complex vector fields of the form Z f , where f has only positive (respectively negative) Fouriercoefficients. In [B], we made these choices to obtain the interior (respectively exterior) normal form.In general, since M is embeddable it follows that M (cid:44) → Σ for some compact complex surfaceΣ as a separating hypersurface (see [Le].) The elements of V correspond on the infinitesimal levelto Kuranishi’s “wiggles”, that is, CR structures which are induced on M through infinitesimalisotopies of M within Σ. In this regard, one expects the factor ψ to correspond to deformationsof the singularities of the “fill-in” of M (that is, the pseudoconvex side of Σ bounded by M ) or tonon-embeddable structures on M .6.2. A priori estimates for the action on CR structures.
We now proceed to establish the a priori regularity estimates for the action of the contact diffeomorphism group on the space ofdeformation tensors that we need to establish Theorem 6.1.5.Let X be a contact vector field and let φ be a CR deformation, expressed relative to a localframe Z α and dual coframe ω ¯ β as φ = φ α ¯ β ω ¯ β ⊗ Z α . For X and φ sufficiently small, we will obtainestimates for the deformation tensor for the pull-back CR-structure µ = F ∗ φ . Remark 6.2.1.
Since we are restricting ourselves to a small neighbourhood of the embeddablestructure, we may choose the neighbourhood small enough to have the following uniform estimates: (cid:107) φ (cid:107) s +2 < C , (cid:107) µ (cid:107) s +2 < C , and (cid:107) X (cid:107) s +1 < C where C is a fixed (sufficiently small) constant. Since when (cid:107) X (cid:107) s +1 < C one has (cid:107) X (cid:107) s +1 ∼(cid:107) F Ψ( X ) (cid:107) s +1 , and one can choose C such that in addition (cid:107) F Ψ( X ) (cid:107) s +1 < C ;here and in what follows we use the norm on contact diffeomorphisms (cid:107) F Ψ( X ) (cid:107) s +1 := (cid:107) Ψ( X ) (cid:107) s +1 ,where F Ψ( X ) = exp ◦ Ψ( X ). Although we have restricted to the three dimensional case n = 1, we continue to use index notation to helpdistinguish between functions and coefficients of tensors. R DEFORMATIONS 25
Remark 6.2.2.
We will repeatedly use the estimates (cid:107) f g (cid:107) s ≺ (cid:107) f (cid:107) s (cid:107) g (cid:107) s − + (cid:107) f (cid:107) s − (cid:107) g (cid:107) s and (cid:107) g ◦ F (cid:107) s ≺ ( (cid:107) g (cid:107) s + (cid:107) g (cid:107) s (cid:107) F (cid:107) s − + (cid:107) g (cid:107) s − (cid:107) F (cid:107) s ) · (1 + (cid:107) F (cid:107) s − ) s − ≺ (cid:107) g (cid:107) s + (cid:107) g (cid:107) s (cid:107) F (cid:107) s − + (cid:107) g (cid:107) s − (cid:107) F (cid:107) s for all s > n + 4 , f, g ∈ Γ s ( M ), and F ∈ D scont ( M ) , (cid:107) F (cid:107) ≺
1, without comment. The first estimatewas proved in [BD1]. The second estimate follows easily by writing g ◦ F in local coordinatesand computing (cid:107) g ◦ F (cid:107) U,s in a coordinate neighborhood U ⊂ M using the chain rule. In the lastestimate, we used the fixed bound on X to conclude that (1 + (cid:107) F (cid:107) s − ) s − ≺ µ = F ∗ Ψ( X ) φ . Lemma 6.2.3.
Let F = F Ψ( X ) and s > n + 4 . Then (cid:107) φ ◦ F (cid:107) s ≺ (cid:107) φ (cid:107) s + (cid:107) φ (cid:107) s · (cid:107) X (cid:107) s − + (cid:107) φ (cid:107) s − · (cid:107) X (cid:107) s . Proof.
Observe that on each coordinate patch U (cid:96) (cid:107) ρ (cid:96) ( φ ◦ F ) (cid:96) (cid:107) U (cid:96) ,s ≺ (cid:107) φ (cid:107) U (cid:96) ,s + (cid:107) φ (cid:107) U (cid:96) ,s (cid:107) F (cid:96) (cid:107) U (cid:96) ,s − + (cid:107) φ (cid:107) U (cid:96) ,s − (cid:107) F (cid:96) (cid:107) U (cid:96) ,s ≺ (cid:107) φ (cid:107) s + (cid:107) φ (cid:107) s (cid:107) F (cid:107) s − + (cid:107) φ (cid:107) s − (cid:107) F (cid:107) s ≺ (cid:107) φ (cid:107) s + (cid:107) φ (cid:107) s (cid:107) Ψ( X ) (cid:107) s − + (cid:107) φ (cid:107) s − (cid:107) Ψ( X ) (cid:107) s ≺ (cid:107) φ (cid:107) s + (cid:107) φ (cid:107) s (cid:107) X (cid:107) s − + (cid:107) φ (cid:107) s − (cid:107) X (cid:107) s The result follows from finiteness of the cover U (cid:96) . (cid:3) Next let E ( X, Y, φ ) be the vector-valued one-form defined by Equation (4.1.11). Then we havethe following estimates:
Lemma 6.2.4.
For s > n + 4 , let φ ∈ Γ s ( D ef ) be a deformation tensor with (cid:107) φ (cid:107) s < C and let X, Y ∈ Γ s +1 cont ( T M ) be vector fields with (cid:107) X (cid:107) s +1 < C , (cid:107) Y (cid:107) s +1 < C , for C chosen as in Remark (6.2.1) . Then (cid:107)E ( X, Y, φ ) (cid:107) s ≺ ( (cid:107) X (cid:107) s + (cid:107) φ ◦ F Ψ( Y ) (cid:107) s ) · (cid:107) X (cid:107) s +1 . Let φ j ∈ Γ s ( D ef ) , j = 1 , be two deformation tensors with (cid:107) φ j (cid:107) s < C , and let X j , Y j ∈ Γ s +1 cont ( T M ) , j = 1 , be contact vector fields with (cid:107) X j (cid:107) s +1 < C , (cid:107) Y j (cid:107) s +1 < C . Then (cid:107)E ( X , Y , φ ) − E ( X , Y , φ ) (cid:107) s ≺ ( (cid:107) X (cid:107) s +1 + (cid:107) X (cid:107) s +1 ) · (cid:107) X − X (cid:107) s + ( (cid:107) X (cid:107) s + (cid:107) X (cid:107) s ) · (cid:107) X − X (cid:107) s +1 + ( (cid:107) X (cid:107) s +1 + (cid:107) X (cid:107) s +1 ) · (cid:107) φ ◦ F − φ ◦ F (cid:107) s + ( (cid:107) φ ◦ F (cid:107) s + (cid:107) φ ◦ F (cid:107) s ) · (cid:107) X − X (cid:107) s +1 . where F j = F Ψ( Y j ) , j = 1 , .Proof. By Equation (4.1.11), our proof amounts to obtaining sufficiently good estimates on theentries of the matrices A ( X, Y, φ ) and B ( X, Y, φ ) defined in Equations (4.1.7) Recall the localformulæ for A and B : A αβ = δ αβ + Z β L X ω α + Z β Q α ( X, Y, φ ) B α ¯ β = ( ¯ ∂ b X ) α ¯ β + ( φ α ¯ β ◦ F Ψ( Y ) ) + Z ¯ β Q α ( X, Y, φ ) . The estimate (cid:107)E α ¯ β (cid:107) s ≺ (cid:107) Z ¯ β Q α ( X, Y, φ ) (cid:107) s + (cid:107) A − (cid:107) s (cid:107) [( I − A ) B ] α ¯ β (cid:107) s follows immediately from the formula for E α ¯ β .We estimate each term on the right-hand side. First, using Proposition 3.2.15 to estimateΨ( X ) − X and observing that the estimate (cid:107) Z β Q ω α (Ψ( X )) (cid:107) s ≺ (cid:107) Ψ( X ) (cid:107) s (cid:107) Ψ( X ) (cid:107) s +1 followsimmediately from the local formula (3.2.6), we obtain (cid:107) Z β Q α ( X, Y, φ ) (cid:107) s ≺ (cid:107) Z β L Ψ( X ) − X ω α (cid:107) s + (cid:107) ( φ α ¯ γ ◦ F Ψ( Y ) ) Z β L Ψ( X ) ω ¯ γ (cid:107) s + (cid:107) Z β Q ω α (Ψ( X )) (cid:107) s + (cid:107) ( φ α ¯ γ ◦ F Ψ( Y ) ) Z β Q ω ¯ γ (Ψ( X )) (cid:107) s ≺ (cid:107) Ψ( X ) − X (cid:107) s +1 + (cid:107) φ ◦ F Ψ( Y ) (cid:107) s (cid:107) Z β L Ψ( X ) ω ¯ γ (cid:107) s − + (cid:107) φ ◦ F Ψ( Y ) (cid:107) s − (cid:107) Z β L Ψ( X ) ω ¯ γ (cid:107) s + (cid:107) Z β Q ω α (Ψ( X )) (cid:107) s + (cid:107) φ ◦ F Ψ( Y ) (cid:107) s − (cid:107) Z β Q ω ¯ γ (Ψ( X )) (cid:107) s + (cid:107) φ ◦ F Ψ( Y ) (cid:107) s (cid:107) Z β Q ω ¯ γ (Ψ( X )) (cid:107) s − ≺ (cid:107) X (cid:107) s (cid:107) X (cid:107) s +1 + (cid:107) φ ◦ F Ψ( Y ) (cid:107) s (cid:107) Ψ( X ) (cid:107) s + (cid:107) φ ◦ F Ψ( Y ) (cid:107) s − (cid:107) Ψ( X ) (cid:107) s +1 + (cid:107) Ψ( X ) (cid:107) s (cid:107) Ψ( X ) (cid:107) s +1 + (cid:107) φ ◦ F Ψ( Y ) (cid:107) s − (cid:107) Ψ( X ) (cid:107) s (cid:107) Ψ( X ) (cid:107) s +1 + (cid:107) φ ◦ F Ψ( Y ) (cid:107) s (cid:107) Ψ( X ) (cid:107) s − (cid:107) Ψ( X ) (cid:107) s ≺ (cid:107) X (cid:107) s (cid:107) X (cid:107) s +1 + (cid:107) φ ◦ F Ψ( Y ) (cid:107) s (cid:107) X (cid:107) s + (cid:107) φ ◦ F Ψ( Y ) (cid:107) s − (cid:107) X (cid:107) s +1 + (cid:107) X (cid:107) s (cid:107) X (cid:107) s +1 + (cid:107) φ ◦ F Ψ( Y ) (cid:107) s − (cid:107) X (cid:107) s (cid:107) X (cid:107) s +1 + (cid:107) φ ◦ F Ψ( Y ) (cid:107) s (cid:107) X (cid:107) s − (cid:107) X (cid:107) s ≺ ( (cid:107) X (cid:107) s + (cid:107) φ ◦ F Ψ( Y ) (cid:107) s ) · (cid:107) X (cid:107) s +1 with a similar estimate for (cid:107) Z ¯ β Q α ( X, Y, φ ) (cid:107) s . Next (cid:107) [( I − A )] αγ (cid:107) s = (cid:107) ( Z γ L X ω α + Z γ Q α ( X, Y, φ )) (cid:107) s ≺ (cid:107) X (cid:107) s +1 + ( (cid:107) X (cid:107) s + (cid:107) φ ◦ F Ψ( Y ) (cid:107) s ) · (cid:107) X (cid:107) s +1 , which implies in particular that A = I − ( I − A ) is invertible. More precisely, because the matrix A = (cid:104) A αβ (cid:105) is the of the form I + (small matrix), a series expansion for A − yields the estimate (cid:107) A − (cid:107) s ≺ (cid:107) X (cid:107) s +1 + (cid:107) φ ◦ F Ψ( Y ) (cid:107) s · (cid:107) X (cid:107) s +1 which is uniformly bounded by a constant dependingonly on the constant C in Remark 6.2.1. Also (cid:107) B γ ¯ β (cid:107) s = (cid:107) (( ¯ ∂ b X ) γ ¯ β + ( φ γ ¯ β ◦ F Ψ( Y ) ) + Z ¯ β Q γ ( X, Y, φ )) (cid:107) s ≺ (cid:107) X (cid:107) s +1 + (cid:107) φ ◦ F Ψ( Y ) (cid:107) s + ( (cid:107) X (cid:107) s + (cid:107) φ ◦ F Ψ( Y ) (cid:107) s ) · (cid:107) X (cid:107) s +1 , so (cid:107) [( I − A ) B ] α ¯ β (cid:107) s ≺ (cid:107) [( I − A )] αγ (cid:107) s (cid:107) [ B ] γ ¯ β (cid:107) s − + (cid:107) [( I − A )] αγ (cid:107) s − (cid:107) [ B ] γ ¯ β (cid:107) s ≺ (cid:0) (cid:107) X (cid:107) s +1 + ( (cid:107) X (cid:107) s + (cid:107) φ ◦ F Ψ( Y ) (cid:107) s ) · (cid:107) X (cid:107) s +1 (cid:1) · (cid:0) (cid:107) X (cid:107) s + (cid:107) φ ◦ F Ψ( Y ) (cid:107) s − + ( (cid:107) X (cid:107) s − + (cid:107) φ ◦ F Ψ( Y ) (cid:107) s − ) · (cid:107) X (cid:107) s (cid:1) , + (cid:0) (cid:107) X (cid:107) s + ( (cid:107) X (cid:107) s − + (cid:107) φ ◦ F Ψ( Y ) (cid:107) s − ) · (cid:107) X (cid:107) s (cid:1) · (cid:0) (cid:107) X (cid:107) s +1 + (cid:107) φ ◦ F Ψ( Y ) (cid:107) s + ( (cid:107) X (cid:107) s + (cid:107) φ ◦ F Ψ( Y ) (cid:107) s ) · (cid:107) X (cid:107) s +1 (cid:1) , ≺ (cid:107) X (cid:107) s +1 · ( (cid:107) X (cid:107) s + (cid:107) ( φ ◦ F Ψ( Y ) ) (cid:107) s − )+ (cid:107) X (cid:107) s · ( (cid:107) X (cid:107) s +1 + (cid:107) ( φ ◦ F Ψ( Y ) ) (cid:107) s ) ≺ ( (cid:107) X (cid:107) s + (cid:107) φ ◦ F Ψ( Y ) (cid:107) s ) · (cid:107) X (cid:107) s +1 . R DEFORMATIONS 27
This completes the proof of the first estimate.To prove the second estimate, let A j = A ( X j , Y j , φ j ), B j = B ( X j , Y j , φ j ), E j = E ( X j , Y j , φ j ), j = 1 ,
2. Then A − B − A − B = A − ( B − B ) − ( A − − A − ) B = [( B − B ) + A − ( I − A )( B − B )] − [ A − ( A − A ) A − B ] . Using this in Equation (4.1.11), we obtain the equality[ E − E ] α ¯ β = Z ¯ β ( Q α ( X , Y , φ ) − Q α ( X , Y , φ ))+ [ A − ( I − A )( B − B )] α ¯ β − [ A − ( A − A ) A − B ] α ¯ β . Choose the constant C in Remark 6.2.1 sufficiently small to ensure that (cid:107) A − j (cid:107) s < C (cid:48) for somefixed constant C (cid:48) . The triangle inequality, then gives(6.2.5) (cid:107) [ E − E ] α ¯ β (cid:107) s ≺ (cid:107) Z ¯ β Q α ( X , Y , φ ) − Z ¯ β Q α ( X , Y , φ ) (cid:107) s + (cid:107) ( I − A )( B − B ) (cid:107) s + (cid:107) ( A − A ) (cid:107) s (cid:107) B (cid:107) s − + (cid:107) ( A − A ) (cid:107) s − (cid:107) B (cid:107) s . We estimate all four terms on the right-hand side of (6.2.5) in a similar manner. We present theestimate of the first term in detail and leave the verification of the estimates of the remaining twoterms to the reader. Rearranging terms and simplifying gives Z ¯ β Q α ( X , Y , φ ) − Z ¯ β Q α ( X , Y , φ )= (cid:8) Z ¯ β L Ψ( X ) − X ω α + ( φ α , ¯ γ ◦ F ) Z ¯ β L Ψ( X ) ω ¯ γ + Z ¯ β Q ω α (Ψ( X )) + ( φ α , ¯ γ ◦ F ) Z ¯ β Q ω ¯ γ (Ψ( X )) (cid:9) − (cid:8) Z ¯ β L Ψ( X ) − X ω α + ( φ α , ¯ γ ◦ F ) Z ¯ β L Ψ( X ) ω ¯ γ + Z ¯ β Q ω α (Ψ( X )) + ( φ α , ¯ γ ◦ F ) Z ¯ β Q ω ¯ γ (Ψ( X )) (cid:9) = Z ¯ β L (Ψ( X ) − X ) − (Ψ( X ) − X ) ( ω α ) + ( Z ¯ β Q ω α (Ψ( X )) − Z ¯ β Q ω α (Ψ( X ))+ (cid:8) ( φ α , ¯ γ ◦ F ) Z ¯ β L Ψ( X ) ω ¯ γ − ( φ α , ¯ γ ◦ F ) Z ¯ β L Ψ( X ) ω ¯ γ (cid:9) + (cid:8) ( φ α , ¯ γ ◦ F ) Z ¯ β Q ω ¯ γ (Ψ( X )) − ( φ α , ¯ γ ◦ F ) Z ¯ β Q ω ¯ γ (Ψ( X )) (cid:9) By our previous estimates, we may estimate as follows: (cid:107) Z ¯ β Q α ( X , Y , φ ) − Z ¯ β Q α ( X , Y , φ ) (cid:107) s ≺ (cid:107) Z ¯ β L (Ψ( X ) − X ) − (Ψ( X ) − X ) ω α (cid:107) s + (cid:107) Z ¯ β Q ω α (Ψ( X )) − Z ¯ β Q ω α (Ψ( X )) (cid:107) s + (cid:107) ( φ α , ¯ γ ◦ F ) Z ¯ β L Ψ( X ) ω ¯ γ − ( φ α , ¯ γ ◦ F ) Z ¯ β L Ψ( X ) ω ¯ γ (cid:107) s + (cid:107) ( φ α , ¯ γ ◦ F ) Z ¯ β Q ω ¯ γ (Ψ( X )) − ( φ α , ¯ γ ◦ F ) Z ¯ β Q ω ¯ γ (Ψ( X )) (cid:107) s ≺ (cid:107) (Ψ( X ) − X ) − (Ψ( X ) − X ) (cid:107) s +1 + (cid:107) X − X (cid:107) s · ( (cid:107) X (cid:107) s +1 + (cid:107) X (cid:107) s +1 ) + (cid:107) X − X (cid:107) s +1 · ( (cid:107) X (cid:107) s + (cid:107) X (cid:107) s )+ (cid:107) ( φ α , ¯ γ ◦ F ) Z ¯ β L Ψ( X ) ω ¯ γ − ( φ α , ¯ γ ◦ F ) Z ¯ β L Ψ( X ) ω ¯ γ (cid:107) s + (cid:107) ( φ α , ¯ γ ◦ F ) Z ¯ β Q ω ¯ γ (Ψ( X )) − ( φ α , ¯ γ ◦ F ) Z ¯ β Q ω ¯ γ (Ψ( X )) (cid:107) s where we have used Lemma 3.2.7(c) ≺ (cid:107) X − X (cid:107) s ( (cid:107) X (cid:107) s +1 + (cid:107) X (cid:107) s +1 ) + (cid:107) X − X (cid:107) s +1 · ( (cid:107) X (cid:107) s + (cid:107) X (cid:107) s ) , + (cid:107) ( φ α , ¯ γ ◦ F ) Z ¯ β L Ψ( X ) ω ¯ γ − ( φ α , ¯ γ ◦ F ) Z ¯ β L Ψ( X ) ω ¯ γ (cid:107) s + (cid:107) ( φ α , ¯ γ ◦ F ) Z ¯ β Q ω ¯ γ (Ψ( X )) − ( φ α , ¯ γ ◦ F ) Z ¯ β Q ω ¯ γ (Ψ( X )) (cid:107) s , where we have used Proposition 3.2.15(b). Observe that (cid:107) ( φ α , ¯ γ ◦ F ) Z ¯ β L Ψ( X ) ω ¯ γ − ( φ α , ¯ γ ◦ F ) Z ¯ β L Ψ( X ) ω ¯ γ (cid:107) s ≺ ( (cid:107) φ ◦ F (cid:107) s − + (cid:107) φ ◦ F (cid:107) s − ) · (cid:107) X − X (cid:107) s +1 + ( (cid:107) φ ◦ F (cid:107) s + (cid:107) φ ◦ F (cid:107) s ) · (cid:107) X − X (cid:107) s + ( (cid:107) X (cid:107) s + (cid:107) X (cid:107) s ) · (cid:107) φ ◦ F − φ ◦ F (cid:107) s + ( (cid:107) X (cid:107) s +1 + (cid:107) X (cid:107) s +1 ) · (cid:107) φ ◦ F − φ ◦ F (cid:107) s − ≺ ( (cid:107) φ ◦ F (cid:107) s + (cid:107) φ ◦ F (cid:107) s ) · (cid:107) X − X (cid:107) s +1 + ( (cid:107) X (cid:107) s +1 + (cid:107) X (cid:107) s +1 ) · (cid:107) φ ◦ F − φ ◦ F (cid:107) s , where we have used the identity f g − f g = f ( g − g ) + ( f − f ) g and the correspondingestimate (cid:107) f g − f g (cid:107) s ≺ ( (cid:107) f (cid:107) s − + (cid:107) f (cid:107) s − ) · (cid:107) g − g (cid:107) s + ( (cid:107) f (cid:107) s + (cid:107) f (cid:107) s ) · (cid:107) g − g (cid:107) s − + ( (cid:107) g (cid:107) s − + (cid:107) g (cid:107) s − ) · (cid:107) f − f (cid:107) s + ( (cid:107) g (cid:107) s + (cid:107) g (cid:107) s ) · (cid:107) f − f (cid:107) s − . A similar argument yields the estimate (cid:107) ( φ ◦ F ) Z ¯ β Q ω (Ψ( X )) − ( φ ◦ F ) Z ¯ β Q ω (Ψ( X )) (cid:107) s ≺ ( (cid:107) φ ◦ F (cid:107) s + (cid:107) φ ◦ F (cid:107) s ) · (cid:107) X − X (cid:107) s +1 + ( (cid:107) X (cid:107) s +1 + (cid:107) X (cid:107) s +1 ) · (cid:107) φ ◦ F − φ ◦ F (cid:107) s . Thus (cid:107) Z ¯ β Q α ( X , Y , φ ) − Z ¯ β Q α ( X , Y , φ ) (cid:107) s ≺ ( (cid:107) X (cid:107) s +1 + (cid:107) X (cid:107) s +1 ) · (cid:107) X − X (cid:107) s + ( (cid:107) X (cid:107) s + (cid:107) X (cid:107) s ) · (cid:107) X − X (cid:107) s +1 + ( (cid:107) X (cid:107) s +1 + (cid:107) X (cid:107) s +1 ) · (cid:107) φ ◦ F − φ ◦ F (cid:107) s + ( (cid:107) φ ◦ F (cid:107) s + (cid:107) φ ◦ F (cid:107) s ) · (cid:107) X − X (cid:107) s +1 . (cid:3) Our proof of the a priori estimates from which Theorem 6.1.5 follows requires one more technicallemma. For k > n + 4 and (cid:15) > φ ∈ Γ k ( D ef ) and X ∈ Γ kcont ( T M ) with (cid:107) φ (cid:107) k < (cid:15) and (cid:107) X (cid:107) < (cid:15) . Then the map T k +1 φ,X : Γ k +1 cont ( T (1 , M ) → Γ k +1 cont ( T (1 , M ) : Z (cid:55)→ Z + P ( E ( π Re ( Z ) , X , φ ) , is defined for all Z in a sufficiently small ball about the origin. Lemma 6.2.6.
There exists a sufficiently small (cid:15) > such that the following holds. For all φ ∈ Γ k ( D ef ) and X ∈ Γ kcont ( T M ) such that (cid:107) X (cid:107) k < (cid:15) and (cid:107) φ (cid:107) k < (cid:15) , the equation T k +1 φ,X ( Z ) = W has a unique solution Z ∈ Γ k +1 cont ( T (1 , M ) for all W ∈ Γ k +1 cont ( T (1 , M ) with (cid:107) W (cid:107) k +1 < (cid:15) . Moreover,the solution satisfies the estimate (cid:107) Z (cid:107) k +1 ≤ (cid:107) W (cid:107) k +1 . R DEFORMATIONS 29
Proof.
We first show that we can choose δ > Z (cid:55)→ P ( E ( π Re ( Z ) , X , φ ) is a contractionmapping in Γ k +1 cont ( T (1 , M ) for (cid:107) Z (cid:107) k +1 < δ . To see this, first note that by Lemma 6.2.4, for φ ∈ Γ k ( D ef ) with (cid:107) φ (cid:107) k < C , for C sufficiently small, the estimate (cid:107)E ( X , X , φ ) − E ( X , X , φ ) (cid:107) k ≺ ( (cid:107) X (cid:107) k + (cid:107) X (cid:107) k +1 + (cid:107) φ ◦ F (cid:107) k ) · (cid:107) X − X (cid:107) k +1 holds for all X , X ∈ Γ k +1 cont ( T M ), with (cid:107) X j (cid:107) k +1 < C , j = 1 ,
2. Thus, (cid:107)P ( E ( X , X , φ ) − P ( E ( X , X , φ )) (cid:107) k +1 ≺ ( (cid:107) X (cid:107) k +1 + (cid:107) X (cid:107) k +1 + (cid:107) φ ◦ F (cid:107) k ) · (cid:107) X − X (cid:107) k +1 . Consequently, for δ (cid:48) > (cid:107)P ( E ( X , X , φ ) − P ( E ( X , X , φ )) (cid:107) k +1 < (cid:107) X − X (cid:107) k +1 , provided (cid:107) φ (cid:107) k < δ , (cid:107) X j (cid:107) k +1 < δ (cid:48) , j = 1 , δ < δ (cid:48) so that (cid:107) π Re ( Z ) (cid:107) k +1 < δ (cid:48) for (cid:107) Z (cid:107) k +1 < δ . Then for X j = π Re ( Z j ), j = 1 , (cid:107)P ( E ( π Re Z , X , φ ) − P ( E ( π Re Z , X , φ )) (cid:107) k +1 < (cid:107) Z − Z (cid:107) k +1 , provided (cid:107) Z j (cid:107) k +1 < δ .Finally, set (cid:15) = δ/
2. Choose any W ∈ Γ k +1 cont ( T (1 , M ) and define the sequence Z n , n = 0 , , , . . . inductively by Z = 0, Z n +1 = W − P ( E ( π Re ( Z n ) , X , φ ). Since E (0 , X , φ ) = 0, Z = W . Conse-quently { Z n } is Cauchy with (cid:107) Z n +1 − Z n (cid:107) k +1 < (cid:107) Z n − Z n − (cid:107) k +1 . Therefore, (cid:107) Z n (cid:107) k +1 < (cid:107) W (cid:107) k +1 .Thus, the sequence converges to a solution Z of the equation T φ,X ( Z ) = W satisfying the estimate (cid:107) Z (cid:107) k +1 ≤ (cid:107) W (cid:107) k +1 . Uniqueness of the solution follows from the contraction mapping property. (cid:3) We are now able to obtain the a priori estimates that we promised and from which Theorem 6.1.5follows.
Theorem 6.2.7.
Fix a smooth background CR structure on M as above, and let P : Ω (0 , ( H (1 , ) → Γ ∞ cont ( T (1 , M ) and H : Ω (0 , ( H (1 , ) → Ω (0 , ( H (1 , ) be the linear operators of Corollary 5.2.10. Then for s > n + 4 , there exists (cid:15) > such that thefollowing holds:Suppose that φ ∈ Γ s +2 ( D ef ) , X ∈ Γ s +1 cont ( T M ) (so F Ψ( X ) ∈ D s +1 cont ( M ) ), Y ∈ V s +1 , and ψ ∈ Γ s ( D ef ) satisfy the conditions ρ ( X − iY ) = 0 , (cid:107) F Ψ( X ) (cid:107) s +1 < (cid:15) , (cid:107) φ (cid:107) s +2 < (cid:15) , and ψ ∈ ker P . If the deformation tensor µ = F ∗ Ψ( X ) φ − i ¯ ∂ b Y − ψ is contained in Γ s +2 ( D ef ) and (cid:107) µ (cid:107) s +2 < (cid:15) then F Ψ( X ) ∈ D s +3 cont ( M ) , Y ∈ V s +3 and ψ ∈ Γ s +2 ( D ef ) . Moreover, the following estimates are satisfied: (cid:107) F Ψ( X ) (cid:107) s +3 ≺ (cid:107) φ (cid:107) s +2 + (cid:107) µ (cid:107) s +2 , (cid:107) Y (cid:107) s +3 ≺ (cid:107) φ (cid:107) s +2 + (cid:107) µ (cid:107) s +2 , (cid:107) ψ (cid:107) s +2 ≺ (cid:107) φ (cid:107) s +2 + (cid:107) µ (cid:107) s +2 . Proof.
Substitution of the expression for F ∗ Ψ( X ) φ given in Proposition 4.1.12 in the formula for µ gives µ = ¯ ∂ b X + φ ◦ F Ψ( X ) − i ¯ ∂ b Y − ψ + E ( X, X, φ )where φ ◦ F and E are defined as in (4.1.10) and (4.1.11). We first prove that (cid:107) X (cid:107) s +3 , (cid:107) Y (cid:107) s +3 , and (cid:107) ψ (cid:107) s +2 are finite. Applying the operator P and usingthe hypothesis P ( ψ ) = 0, gives(6.2.8) P ( µ ) = P ( ¯ ∂ b X − i ¯ ∂ b Y ) + P ( φ ◦ F ) + P ( E ( X, X, φ )) . Since ρ ( X − iY ) = 0, it follows that X − iY = P ( ¯ ∂ b ( X − iY )), and solving for X − iY in the lastequation, we have:(6.2.9) X − iY + P ( E ( X, X, φ )) = P ( µ − φ ◦ F Ψ( X ) ) . Next we “freeze coefficients” in (6.2.9). Let X − iY = X − iY and set W = P ( µ − φ ◦ F Ψ( X ) ) ∈ Γ s +3 cont ( T (1 , M ). Then X − iY is the unique solution in Γ s +1 cont ( T (1 , M ) of the equation(6.2.10) T kφ,X ( Z ) = W for k = s + 1.We now perform the first of two bootstrapping steps. Notice that φ and µ are small in Γ s +2 and,hence, small in Γ s +1 , and that X is also small in Γ s +1 . Consequently the map T kφ,X is defined for k = s + 2. Lemma 6.2.6 then shows that T kφ,X is defined for k = s + 2 and that Equation (6.2.10)with k = s + 2 has a unique solution in Γ s +2 cont ( T (1 , M ). It follows that X − iY is in Γ s +2 cont ( T (1 , M ).Lemma 6.2.3 then gives the a priori estimate (cid:107) X − iY (cid:107) s +2 ≺ (cid:107) W (cid:107) s +2 ≺ (cid:107) µ − φ ◦ F Ψ( X ) (cid:107) s +1 ≺ (cid:107) µ (cid:107) s +1 + (cid:107) φ (cid:107) s +1 . The second bootstrap proceeds as follows. We now know that X and φ are both in Γ s +2 andthat W = P ( µ − φ ◦ F Ψ( X ) ) is in Γ s +3 . By shrinking (cid:15) if necessary, we can solve Equation (6.2.10)with k = s + 3 to conclude that X − iY is in Γ s +3 cont ( T (1 , M ). Finally, we have X − iY = X − iY with the a priori estimate (cid:107) X − iY (cid:107) s +3 ≺ (cid:107) µ (cid:107) s +2 + (cid:107) φ (cid:107) s +2 . Finally, since ψ = F ∗ φ − i ¯ ∂ b Y − µ , it follows that ψ is in Γ s +2 and satisfies the a priori estimate (cid:107) ψ (cid:107) s +2 ≺ (cid:107) µ (cid:107) s +2 + (cid:107) φ (cid:107) s +2 + (cid:107) X (cid:107) s +3 + (cid:107) Y (cid:107) s +3 ≺ (cid:107) µ (cid:107) s +2 + (cid:107) φ (cid:107) s +2 . This establishes the a priori bounds, and hence the a priori estimates for F Ψ( X ) , Y and ψ . (cid:3) Proof of Theorem 6.1.5.
That F φ , Y φ and ψ φ are in the appropriate spaces is an immediate corollaryof Theorem 6.2.7. It remains only to show that the map is continuous.Choose smooth φ j ∈ Γ s +2 ( D ef ) and µ j ∈ Γ s +2 ( D ef ) such that φ j −→ (cid:107)·(cid:107) s +2 φ and µ j −→ (cid:107)·(cid:107) s +2 µ . Bythe analysis above, there exist X j ∈ Γ s +3 cont ( T M ), Y j ∈ V s +3 , and ψ j ∈ Γ s +2 (H ) such that thecontact diffeomorphisms F j = F Ψ( X j ) ∈ D s +3 cont ( M ) satisfy the conditions µ j = F ∗ j φ j − i ¯ ∂ b Y j − ψ j , ρ ( X j − iY j ) = 0 , with F j −→ (cid:107)·(cid:107) s +1 F , Y j −→ (cid:107)·(cid:107) s +1 Y , and ψ j −→ (cid:107)·(cid:107) s ψ . By the a priori estimates (cid:107) F j (cid:107) s +2 ≺ (cid:107) φ j (cid:107) s +1 + (cid:107) µ j (cid:107) s +1 , (cid:107) Y j (cid:107) s +2 ≺ (cid:107) φ j (cid:107) s +1 + (cid:107) µ j (cid:107) s +1 , and (cid:107) ψ j (cid:107) s +1 ≺ (cid:107) φ j (cid:107) s +1 + (cid:107) µ j (cid:107) s +1 established above, we note, inparticular, that F j , Y j , ψ j are bounded sequences in Γ s +2 , Γ s +2 , and Γ s +1 , respectively. Also notethat, by continuity of composition, F j −→ (cid:107)·(cid:107) s +1 F and φ j −→ (cid:107)·(cid:107) s +1 φ together imply φ j ◦ F j −→ (cid:107)·(cid:107) s +1 φ ◦ F .We now show that the sequences X j and Y j are Cauchy in Γ s +2 cont ( T M ). We estimate (cid:107) X j − X i (cid:107) s +2 as follows. Writing µ j = ¯ ∂ b ( X j − i Y j ) + φ j ◦ F j − ψ j + E j (see (4.1.10) and (4.1.11)) R DEFORMATIONS 31 with F j = F Ψ( X j ) , E j = E ( X j , Xj, φ j ) yields the formula µ j − µ i = ¯ ∂ b ( X j − X i ) − i ¯ ∂ b ( Y j − Y i ) − ( ψ j − ψ i ) + ( φ j ◦ F j − φ i ◦ F i ) + ( E j − E i ) . Applying the operator P and using the facts P ( ψ j ) = 0, P ¯ ∂ b ( X j − iY j ) = X j − iY j as above, gives: P ( µ j − µ i ) = P (cid:0) ¯ ∂ b ( X j − X i ) − i ¯ ∂ b ( Y j − Y i ) (cid:1) + P ( φ j ◦ F j − φ i ◦ F i ) + P ( E j − E i )= ( X j − X i ) − i ( Y j − Y i ) + P ( φ j ◦ F j − φ i ◦ F i ) + P ( E j − E i ) . Solving for ( X j − X i ) − i ( Y j − Y i ), we have:( X j − X i ) − i ( Y j − Y i ) = P ( µ j − µ i ) − P ( φ j ◦ F j − φ i ◦ F i ) − P ( E j − E i ) . We can estimate the s + 2 norm for ( X j − X i ) as follows, using our a priori bound (cid:107) X j (cid:107) s +2 ≤ K on the sequence: (cid:107) X j − X i (cid:107) s +2 + (cid:107) Y j − Y i (cid:107) s +2 ≺ (cid:107) ( X j − X i ) − i ( Y j − Y i ) (cid:107) s +2 ≺ (cid:107)P ( µ j − µ i ) (cid:107) s +2 + (cid:107)P ( φ j ◦ F j − φ i ◦ F i ) (cid:107) s +2 + (cid:107)P ( E j − E i ) (cid:107) s +2 ≺ (cid:107) µ j − µ i (cid:107) s +1 + (cid:107) φ j ◦ F j − φ i ◦ F i (cid:107) s +1 + (cid:107)E j − E i (cid:107) s +1 ≺ (cid:107) µ j − µ i (cid:107) s +1 + (cid:107) ( φ j ◦ F j − φ i ◦ F i ) (cid:107) s +1 + ( (cid:107) X i (cid:107) s +2 + (cid:107) X j (cid:107) s +2 ) · (cid:107) ( φ j ◦ F j − φ i ◦ F i ) (cid:107) s +1 + ( (cid:107) φ j ◦ F j (cid:107) s +1 + (cid:107) φ i ◦ F i (cid:107) s +1 ) · (cid:107) X j − X i (cid:107) s +2 + ( (cid:107) X j (cid:107) s +2 + (cid:107) X i (cid:107) s +2 ) · (cid:107) X j − X i (cid:107) s +1 + ( (cid:107) X j (cid:107) s +1 + (cid:107) X i (cid:107) s +1 ) · (cid:107) X j − X i (cid:107) s +2 ≺ (cid:107) µ j − µ i (cid:107) s +1 + (cid:107) ( φ j ◦ F j − φ i ◦ F i ) (cid:107) s +1 + K (cid:107) ( φ j ◦ F j − φ i ◦ F i ) (cid:107) s +1 + ( (cid:107) φ j (cid:107) s +1 + (cid:107) φ j (cid:107) s +1 (cid:107) F j (cid:107) s + (cid:107) φ j (cid:107) s (cid:107) F j (cid:107) s +1 ) · (cid:107) X j − X i (cid:107) s +2 + ( (cid:107) φ i (cid:107) s +1 + (cid:107) φ i (cid:107) s +1 (cid:107) F i (cid:107) s + (cid:107) φ i (cid:107) s (cid:107) F i (cid:107) s +1 ) · (cid:107) X j − X i (cid:107) s +2 + K (cid:107) X j − X i (cid:107) s +1 + C (cid:107) X j − X i (cid:107) s +2 ≺ (cid:107) µ j − µ i (cid:107) s +1 + (cid:107) φ j ◦ F j − φ i ◦ F i (cid:107) s +1 + K ( (cid:107) ( φ j ◦ F j − φ i ◦ F i ) (cid:107) s +1 + (cid:107) X j − X i (cid:107) s +1 ) + C (cid:107) X j − X i (cid:107) s +2 For (cid:107) φ (cid:107) s +1 , (cid:107) µ (cid:107) s +1 sufficiently small (that is for C sufficiently small), we can absorb the last termon the right hand side to obtain an a priori estimate on the sequence: (cid:107) X j − X i (cid:107) s +2 ≺ (cid:107) µ j − µ i (cid:107) s +1 + (cid:107) ( φ j ◦ F j − φ i ◦ F i ) (cid:107) s +1 + (cid:107) X j − X i (cid:107) s +1 ; (cid:107) Ψ( X j ) − Ψ( X i ) (cid:107) s +2 ≺ (cid:107) X j − X i (cid:107) s +2 ≺ (cid:107) µ j − µ i (cid:107) s +1 + (cid:107) ( φ j ◦ F j − φ i ◦ F i ) (cid:107) s +1 + (cid:107) X j − X i (cid:107) s +1 ; (cid:107) Y j − Y i (cid:107) s +2 ≺ (cid:107) µ j − µ i (cid:107) s +1 + (cid:107) ( φ j ◦ F j − φ i ◦ F i ) (cid:107) s +1 + (cid:107) X j − X i (cid:107) s +1 . 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