A note on the closed range of ∂ ¯ b on q-convex manifolds
aa r X i v : . [ m a t h . C V ] A p r A NOTE ON THE CLOSED RANGE OF ¯ ∂ b ON q -CONVEXMANIFOLDS LUCA BARACCO AND ALEXANDER TUMANOV
To the memory of Nicholas Hanges
Abstract.
We prove that the tangential Cauchy-Riemann operator ¯ ∂ b hasclosed range on Levi-pseudoconvex CR manifolds that are embedded in a q -convex complex manifold X . Our result generalizes the known case when X isa Stein manifold (in particular, when X = C n ). Introduction
The ¯ ∂ operator in complex analysis is very important because it is involved inthe explanation of many phenomena concerning holomorphic functions and maps.The solvability of ¯ ∂ on forms gives a characterization of domains of holomorphyand of Stein manifolds. Moreover, the regularity of solutions of the ¯ ∂ -equation givesa different perspective on the regularity up to the boundary of biholomorphisms.Similarly, the study of ¯ ∂ b , which is the restriction to a hypersurface of ¯ ∂ , is tightlyconnected with geometric aspects of CR manifolds, like embeddability in C n .The natural environment to study these differential operators is the space offorms with L coefficients. In these spaces the operators ¯ ∂ and ¯ ∂ b are naturallydefined in the distribution sense. If Ω is a domain in C n , then for any d < n we have aclosed densily defined unbounded operator ¯ ∂ d : L ,d ) (Ω) −→ L ,d +1) (Ω). It is easyto check that ¯ ∂ j +1 ◦ ¯ ∂ j = 0, and thus ( L ,j (Ω) , ¯ ∂ j ) j =0 ,...,n is a complex. Similarly,for ¯ ∂ b the same hold. Forms in the kernel of ¯ ∂ (resp. ¯ ∂ b ) are called ¯ ∂ -closed (resp.¯ ∂ b -closed), and those in the image ¯ ∂ -exact (resp. ¯ ∂ b -exact). In the ¯ ∂ b problem, one isgiven a (0 , q )-form f that is ¯ ∂ b -closed and wants to find a (0 , q − u such that¯ ∂ b u = f (eventually with regularity requirements on the solution). The question hasbeen studied and solved for ¯ ∂ by Kohn and H¨ormander for pseudoconvex domainsin C n . For ¯ ∂ b , the first result of this kind was proved for strictly pseudoconvexhypersurface-type CR manifolds.The starting point in tackling these problem is proving that the range of ¯ ∂ ( ¯ ∂ b )is closed, and this is done starting from the Kohn-Morrey-H¨ormander identity. Themain difference between the two situations is that in dealing with ¯ ∂ b one need tocontrol a mixed term that appears when integrating by parts. This mixed terminvolves the derivative of the coefficients of the form in the so called “totally real”direction. The presence of this term is what makes the closed range for ¯ ∂ b harder tocheck. For strictly pseudoconvex manifolds, the fact that ¯ ∂ b has closed range wasbeen proved by Kohn (indeed he proved closed range not only in L , but also in Research of the second author is partially supported by Simons Foundation grant.
Sobolev spaces). Shaw [18] and Boas-Shaw [4] proved closed range for ¯ ∂ b on bound-aries of pseudoconvex domains in C n . Their method does not generalize to domainsin manifolds. Kohn [11], using microlocal analysis, proved closed range of ¯ ∂ b onboundaries of pseudoconvex domains in Stein manifolds. Similar, from the complexpoint of view, to the boundaries of domains are the CR manifolds of hypersurfacetype (see Definition 2.3). Nicoara [15] proved closed range on pseudoconvex CR manifolds of hypersurface type in C n whose real dimension is larger than 3. Fi-nally, Baracco [2] proved the 3 dimensional case using the technique of Kohn anda desingularization argument of complexification.The pseudoconvex case in Stein manifolds is well understood. Most of the tech-niques adopted rely on the existence of a plurisubharmonic weight. This is indeeda distinctive tract of Stein manifolds. Yet there are many other important mani-folds that one encounter in complex analysis which do not have a plusubharmonicweight like for instance the compact manifolds. In the attempt to bridge the gapbetween these two extreme cases in [1] the notion of completely q -convex manifoldsis introduced. Roughly speaking a complex manifold X of dimension n is said tobe completely q -convex if X is endowed with an exhaustion function which has acontrolled number of positive Levi eigenvalues (namely greater than q + 1 see Defi-nitions 2.1 and 2.2 ). Since their introduction these manifolds have been intensivelystudied. Most of the main tools in complex analysis can be considered in this newsetting (see [5, 7, 16] and the references therein) and the main difficulty is of coursethe lack of convexity of the exhaustion funcion. In this paper we want to extendfurther the results on the range of ¯ ∂ b on pseudoconvex manifolds of hypersurfacetype when these are contained in a completely q -convex manifold. We will assumethe existence of a q + 1-convex weight defined only around M . Here is the statementof our main result. Theorem 1.1.
Let X be a complex manifold and M ⊂⊂ X a smooth compact,pseduconvex-oriented CR submanifold of hypersurface type of dimension p − .Assume that there exists a ( q + 1) -convex function φ defined in a neighborhood of M in X . Then ¯ ∂ b : Dom ( ¯ ∂ b ) r,s − −→ L r,s ( M ) has closed range if n − q ≤ s ≤ p + q − n and p > n − q ) . Moreover, if X iscompletely q -convex, then the same conclusion holds for p = 2( n − q ) . The paper is organized as follows. In Section 2 recall some basic definitions andwe show how to realize a CR manifold of hypersurface type as the boundary of acomplex manifold Y . In Section 3 we present the proof of Theorem 1.1.This paper was written for a special volume of CASJ dedicated to the memory ofNicholas Hanges. He did pioniering work on propagation of holomorphic extendibil-ity of CR functions, and we use related results in this paper. We will remember himas a prominent mathematician, a wonderful person, and a great colleague.2. Definitions and construction of a partial complexification
Let X be a complex manifold of dimension n endowed with a Hermitian product,which we denote by ( · , · ) p : T (1 , p X × T (1 , p X → C . We first extend this scalarproduct to forms in the following way. Let L , ..., L n be a local orthonormal basisof (1 , ω , ..., ω n be the dual basis of (1 , ,
0) forms by declaring that ω , ..., ω n is an orthonormal NOTE ON THE CLOSED RANGE OF ¯ ∂ b ON q -CONVEX MANIFOLDS 3 basis and we will denote this product again by ( · , · ) p . Such product do not dependon the choice of the basis L . For forms of higher bi-degree, say ( r, s ) we proceed inthe same way by declaring that ω I ∧ ω J is an orthonormal basis where I and J aremulti-indexes of lenght r and s respectively.The volume form of X is thus given by dV = i n ω ∧ ¯ ω ∧· · ·∧ ω n ∧ ¯ ω n . If Ω ⊂ X isa relatively compact open subset with smooth boundary we define the space L (Ω)of square integrable functions on Ω as the set of all complex-valued measurablefunctions f such that Z Ω | f | dV < ∞ .L (Ω) is a Hilbert space with scalar product given by h f, g i := Z Ω f ¯ g dV. We extend this definition to forms and define for two integers 0 ≤ r, s ≤ n , thespace L r,s ) (Ω) which is the space of forms f such that f can be written lo-cally as P | I | = r | J | = s f IJ ω I ∧ ¯ ω J where f IJ are measurable functions and such that R Ω ( f, f ) p dV ( p ) < ∞ . The scalar product is defined similarly by h f, g i = Z Ω ( f, g ) p dV ( p ) . Let D (Ω) be the space of functions on Ω which are smooth up to the boundary andlet D r,s (Ω) be the corresponding space of ( r, s )-forms with coefficients in D (Ω). Onthese spaces, the operator ¯ ∂ ( r,s ) is defined in the usual way as¯ ∂ r,s : D r,s (Ω) −→ D r,s +1 (Ω)In local coordinates we have¯ ∂f ( z ) = X IJj ∂∂ ¯ z j f IJ d ¯ z j ∧ dz I ∧ d ¯ z J . If instead of a local coordinate system we use a local system of orthonormal vectorfields then we have¯ ∂f ( z ) = X I,J,j L j f IJ ¯ ω j ∧ ω I ∧ ¯ ω J + · · · = A ( f ) + . . . (2.1)where dots stand for terms which do not involve derivatives of f and A denote theoperator formed with all the terms containing the derivatives of f . Since D r,s (Ω)is dense in L r,s ) (Ω), we consider the maximal closed extension of ¯ ∂ (still called ¯ ∂ ),and its L -adjoint ¯ ∂ ∗ . Particularly useful is the formal adjoint operator ϑ whichon smooth compactly supported forms is characterized by the property h ϑf, g i = h f, ¯ ∂g i . In a local frame we have ϑf = X IK X j L j f I,jK ω I ∧ ω K + . . . where dots stand for terms that do not contain derivatives of f . If φ : X → R isa continuous functions we define the weighted Hermitian product with weight φ inthe following way h f, g i φ := Z Ω e − φ ( f, g ) dV. LUCA BARACCO AND ALEXANDER TUMANOV
The corresponding norm will be denoted by k · k φ and the adjoint of ¯ ∂ in thisproduct will be denoted by ¯ ∂ ∗ φ . The formal adjoint operator ϑ φ in a local frame is ϑ φ f := ( − r − X IK X j δ φj f I,jK ω I ∧ ¯ ω K + · · · = B ( f ) + . . . (2.2)where δ φj u = L j ( u ) − L j ( φ ) u and dots stand for terms without derivatives of f and B is the operator defined by the summation in the term in the middle of2.2. In the sequel we shall use weighted scalar product with weights of the form tλ , where λ is a convenient function and t a real parameter. When this choice ismade we shall indicate the corrisponding scalar product with h· , ·i t and similarlyfor k · k t , ¯ ∂ ∗ t , ϑ t , δ tj . Let φ : X → R be a smooth function. We denote by ∂ ¯ ∂φ the(1 , z ∈ X by ∂ ¯ ∂φ ( z ) = n X i,j =1 ∂ z i ∂ ¯ z j φ ( z ) dz i ∧ d ¯ z j , where z , . . . , z n are local coordinates for X at z . If an orthonormal basis ω i of(1 , ∂ ¯ ∂φ = X ij φ ij ω i ∧ ¯ ω j . The form ∂ ¯ ∂φ defines a Hermitian form, called the Levi form , on the holomorphictangent bundle T , X of X . The Levi form will also be denoted by ∂ ¯ ∂φ , and isdefined in the following way. For z ∈ X and vectors X = P a i ∂ z i and Y = P i b i ∂ z j in T , z X , we have ∂ ¯ ∂φ ( z )( X, Y ) = n X i,j =1 a i ¯ b j ∂ z i ∂ ¯ z j φ ( z ) . Definition 2.1.
We say that φ is q -convex if the Levi form ∂ ¯ ∂φ has at least q positive eigenvalues. Definition 2.2.
We say that X is completely q -convex if there exists a smoothexhaustion function φ : X → R which is ( q + 1)-convex.Let J : T X → T X be the standard complex structure induced by the multipli-cation by i and let M be a real submanifold of X . Definition 2.3.
The complex tangent space to M at a point z ∈ M is the subspace T C z M := T z M ∩ JT z M. We will say that M is a CR manifold if T C z M has constant dimension. The bundleso formed is called the complex tangent bundle of M and is denoted by T C M . Wesay that M is of hypersurface type if T MT C M has rank 1.Let M be a smooth, compact CR submanifold of X equipped with the inducedCR structure T , M = C T M ∩ T , X . The De Rham exterior derivative induces acomplex on skew-symmetric antiholomorphic forms on M . We denote such complexby ¯ ∂ b . Assume that M is of hypersurface type. Hence the complexified tangentbundle C T M is spanned by T , M , its conjugate T , M and a single additionalvector field T . We can assume T to be purely imaginary, that is, satisfying T = − T .Let η be a purely imaginary 1-form which annihilates T , M ⊕ T , M and nor-malized so that h η, T i = −
1. The manifold M is orientable if there exists a global NOTE ON THE CLOSED RANGE OF ¯ ∂ b ON q -CONVEX MANIFOLDS 5 η (or vector field T ) and is pseudoconvex if the hermitian form de-fined on T , M by dη ( X, Y ) = h dη, X ∧ Y i is positive semidefinite. We say that M is pseudoconvex-oriented if both properties are satisfied at the same time.A CR curve γ on M is a real curve such that T γ ⊂ T C M . A CR orbit is theunion of all piecewise smooth CR curves issued from a point of M . We denote by O ( z ) the CR orbit of a point z ∈ M , and we say that a set S is CR invariant if O ( z ) ⊂ S for all z ∈ S . By Sussmann’s Theorem [14], the orbit O ( z ) has thestructure of an immersed variety of X . Following [2] and [8], we prove that themanifold M in question consists of a single orbit. The difference with [2, 8] is thatinstead of holomorphic coordinate functions that are not available here, we use thegiven (q+1)-convex function. Proposition 2.4.
Let X be a complex manifold and M a smooth, compact, con-nected CR submanifold of hypersurface type. Let φ be a ( q + 1) -convex functiondefined on a neighborhood of M in X . Assume that the dimension of M is p − ,with p > n − q . Then M consists of a single CR orbit.Proof. Let S ⊂ M be a closed, non empty CR invariant subset of M . Since M is compact we can assume that S is the smallest of such sets i.e. that it doesn’tcontain any smaller closed non-empty CR invariant subset. We will now prove that S = M . Assume by contradiction that S = M . For a point x ∈ S , we have onlytwo possibilities: either x is minimal in the sense of Tumanov (that is, the local CR orbit of x contains a neighborhood of x in M ) or there exists a complex manifoldof dimension p − M that passes through x . No point x ∈ S can beminimal in the sense of Tumanov, otherwise the set S \O ( x ) would be proper, closed, CR invariant, and strictly smaller than S . Hence S is foliated by complex manifoldsof dimension p −
1. Since S is compact, there exists a point ¯ x of S where φ achievesits maximum value. In particular, φ has a maximum on the complex leaf passingthrough ¯ x . This is impossible, because the Levi form of φ has at least one positivedirection in the complex tangent space of M . We have reached a contradiction, thusproving that S = M . Let now x ∈ M be the point where φ reaches its maximum.By the same reasoning as above, we can rule out the case in which there exists acomplex manifold of dimension p − M that passes through x . Hence x must be a minimal point in the sense of Tumanov. It follows that O ( x ) is open in M . Since M is the smallest closed CR invariant subset element, we conclude that O ( x ) = M . (cid:3) Proposition 2.5.
Let M ⊂⊂ X be a smooth, compact, connected, pseudoconvex-oriented CR manifold of hypersurface type of dimension p − . Let φ be a ( q + 1) -convex function defined on a neighborhood of M in X , with q > n − p . Then M is endowed with a partial one-sided complexification in X . That is, there exists acomplex manifold Y ⊂⊂ X which has M as the smooth connected component of itsboundary on the pseudoconvex side.Proof. The proof follows closely [2], since some geometric details will be neededin the next section we repeat the proof here for the reader’s convenience. The setof points of M for which there exists a neighborhood where M has a one-sided,positive, partial complexification is obviously open. We show that this set is alsonon-empty and closed. Let z ∈ M be a point where there is no ( p −
1) dimen-sional complex submanifold S ⊂ M (see the proof of Proposition 2.4). Considera local coordinate patch U ⊂ C n of X at z in which the projection π z : C n → LUCA BARACCO AND ALEXANDER TUMANOV T z M + iT z M ∼ = C p induces a diffeomorphism between M and π z ( M ). Since π z ( M ) is part of a mininimal and pseudoconvex hypersurface, then ( π z | M ) − ex-tends holomorphically to the pseudoconvex side π z ( M ) + by [20] and [21]. Moreover,the map ( π z | M ) − parametrizes a one-sided complex manifold which has a neigh-borhood of z in M as its boundary. By global pseudoconvexity and by uniquenessof holomorphic functions having the same trace on a real hypersurface, one-sidedcomplex neighborhoods glue together into a complex neighborhood of a maximalopen subset M ⊂ M . This is indeed also closed. In fact, let z ∈ M . Since M consists of a single CR orbit by Proposition 2.4, then z is connected to any otherpoint of M by a piecewise smooth CR curve γ . The statement now follows fromthe lemma below whose proof can be found in [2] and [22]. (cid:3) Lemma 2.6.
Let M ⊂⊂ X be a smooth, pseudoconvex-oriented CR manifold ofhypersurface type. Let γ be a piecewise smooth CR curve connecting two points z and z of M . If M has complex extension in direction + JiT ( z ) at z , then M alsohas complex extension in direction + JiT ( z ) at z .Remark . In the proof of Proposition 2.4 we have used [20] and [21] to build aone sided complexification of M near minimal points. The results in [20] and [21],however, do not specify on which side of M this complexification lies. Our hypothe-ses on the pseudoconvexity of M assures that the side of the complexification atminimal points is the pseudoconvex side of M , namely the side pointed by JiT .3.
Proof of the main result
We follow the same proof as in [2], and first prove a closed range theorem for ¯ ∂ on an annulus-like domain: Proposition 3.1.
Let M and φ be as in Proposition 2.5 and assume further that p > n − q ) . Then there exists a complex sub-manifold Y of X of dimension p withsmooth boundary ∂Y such that ∂Y = M ∪ M , where M is CR of hypersurfacetype. Moreover, if ¯ ∂ is the Cauchy Riemann operator on Y , for a suitable weightfunction λ we have, for all f ∈ Dom ( ¯ ∂ ∗ t ) r,s ∩ C ∞ ( r,s ) ( Y ) , that t k f k t ≤ C ( k ¯ ∂f k t + k ¯ ∂ ∗ t f k t )+ C t k f k − ∀ s ≥ n − q, s ≤ p + q − n − , ∀ t > . (3.1) Proof.
First we equip the manifold X with an Hermitian product in such a waythat if φ ( z ) ≤ · · · ≤ φ n ( z ) are the ordered eigenvalues of ∂ ¯ ∂φ ( z ), then φ ( z ) + · · · + φ n − q ( z ) > c > c and for every z in a neighborhood of M . This is possiblebecause φ n − q > q + 1-convexity of φ (see [5, Lemma IX.3.1]). Let Y bethe complex manifold constructed in Proposition 2.5. Note that by the construcionmade there we have that for any point z ∈ M there exists a local coordinate patch U ⊂ C n of X at z in which the projection π z : C n → T z M + iT z M ∼ = C p induces a diffeomorphism between M ∩ U and π z ( M ) and is a biholomorphismbetween Y ∩ U and π z ( M ) + ∩ V where V ia a convenient neighborhood of π ( z )in C p . We shall use π z as a local coordinate chart of Y . Let ρ : Y → R be asmooth, non-negative function such that ρ = 0 on M , and dρ = 0 on M . Since M is pseudoconvex, then the negative eigenvalues of the Levi form of − log( ρ ) arebounded from below. In fact, in the local chart as above around z ∈ M we havethat ρ = hd M where h is a positive and non vanishing smooth function and d M is NOTE ON THE CLOSED RANGE OF ¯ ∂ b ON q -CONVEX MANIFOLDS 7 the Euclidean distance in C p of π z ( z ) from π z ( M ). Then by Oka’s lemma we havethat − log( d M ) is plurisubharmonic then − log( ρ ) = − log( h ) − log( d M ). The Leviform ∂ ¯ ∂φ restricted to T , Y has at least p + q + 1 − n positive eigenvalues. Let ϕ := − log( ρ ) + Cφ . If the constant C is large enough, then ∂ ¯ ∂ϕ restricted to T , Y has at least the same number of positive eigenvalues as ∂ ¯ ∂φ . Since dρ = 0 , then dϕ = 0 in a neighborhood of M in Y . In particular, the subset of Y defined by theequation ρe − Cφ = e − K is a regular hypersurface if K is large enough. We call it M .The Levi form of M has at least p + q − n positive eigenvalues. We consider nowthe annulus-like domain, which we call again Y , defined by ϕ > K . The boundaryof Y consists of the two connected components M and M . We exploit [5, LemmaIX.3.1] once again to choose the Hermitian metric on X in a neighborhood of M so that the following is true: if ϕ ( z ) ≤ · · · ≤ ϕ p − ( z ) are the ordered eigenvaluesof the tangential Levi form ∂ ¯ ∂ϕ (i.e. restricted to T (1 , M ) at a point z ∈ M ,then the sum of any n − q of such eigenvalues is strictly positive. We now follow[19, page 260]. First we choose the weight function. Let λ ∈ C ( Y ) be a functionsuch that λ = φ in a neighborhood of M and λ = − ϕ in a neighborhood of M . Itis enough to prove 3.1 locally in a neighborhood of the boundary ∂Y . Let z ∈ M and U z a small neighborhood of z in X . Choose a local system of orthonormalholomorphic vector fields L , . . . , L p tangent to Y such that L j ( ρ ) = − δ jp , andlet ω , ..., ω p be the corresponding dual frame. Following [19] and [9] we have thefollowing Kohn-H¨ormander-Morrey type formula: k ¯ ∂f k t + k ¯ ∂ ∗ t f k t = X I,J X j k ¯ L j f IJ k t + t X I,K ′ X j,k ( λ jk f I,jK , f
I,kK ) t − X IK ′ X j,k
I,kK i t dS + R ( f ) + E ( f ) . (3.2)where R ( f ) + E ( f ) are terms as in [19, page 263] that arise when manipulating k ¯ ∂f k t + k ¯ ∂ ∗ t f k t . In fact when we replace the terms inside the norms with theterms defined in equations 2.1 and 2.2 we consider first the terms that containsquares of derivatives of f and we group all the other terms in the term indicatedby R ( f ). So R ( f ) contains only terms that can be estimated, uniformly in t , by( k A ( f ) k t + k B ( f ) k t + k f k t ) k f k t . The next step in proving 3.2 is to turn the L derivatives of f in k B ( f ) k t into L derivatives by integration by parts. In doingso some new terms arise. These terms that can be estimated uniformly in t by( k L ( f ) k t k f k t where k L ( u ) k t = P j k L j ( u ) k t + k u k t and we indicate these termswith E ( f ). By the pseudoconvexity of M , the last boundary integral in (3.2) ispositive, and can therefore be dropped. Near M we have that λ = φ , and thus λ jk = φ jk . Moreover, by the choice of the Hermitian product, the sum of any n − q eigenvalues of the Levi form of φ is strictly positive. Hence the second term onthe left side of 3.2 is greater than tc k f k t if the antiholomorphic degree s of f is s ≥ n − q . The terms R ( f ) and E ( f ) can be estimated using the first two term onthe right hand side of 3.2. Let now z be a point of M . We start with the sameformula 3.2 where M is replaced by M and ρ is replaced by a defining equationof M which is of the form ρ = ( λ + K ) h where h is a positive function such that | dρ | = 1 on M . After integrating by parts the terms of type k ¯ L j f IJ k t for j < p LUCA BARACCO AND ALEXANDER TUMANOV we obtain k ¯ ∂f k t + k ¯ ∂ ∗ t f k t = X I,J ′ k ¯ L p ( f IJ ) k t + X I,J ′ X j
I,kK ) t − t X I,J ′ X j
I,J ) t + X I,K ′ X j,k
X J ′ c | f I,J | , (3.5)where c is strictly positive as soon as | J | < p + q − n . In a similar way we canhandle the tangential terms (i.e. those for j, k < p and p / ∈ K, J ) in the second lineof (3.3). The terms with either p ∈ J, K or j = p or k = p can be handled as in[19]. (cid:3) Choosing t large enough, we can pass from (3.3) to a priori estimates of higherSobolev order. As a consequence using the elliptic regularization technique as wasdone in [12] and [10] we obtain that the space of harmonic forms on Y which is thespace H r,st ( Y ) := ker( ¯ ∂ ) ∩ ker( ¯ ∂ ∗ t ), is finite dimensional and moreover we have aHodge decomposition on the space of forms orthogonal to H r,st ( Y ), existence andglobal Sobolev regularity of the ¯ ∂ -Neumann operator N t .We are now in position to prove Theorem 1.1. Proof of Theorem 1.1.
By Proposition 3.1 there exists a complex submanifold Y whose boundary contains M . We follow [11] Paragraph 5 and we have that ¯ ∂ b has closed range in degree s if 3.1 holds in degree s and p − s −
1. Therefore byProposition 3.1 we have closed range of ¯ ∂ b for n − q ≤ s ≤ p + q − n provided p + q − n − ≥ n − q , that is, p > n − q ).If X is q -complete, then by [13] it is possible to extend M to an analytic set E whose boundary, in the sense of currents, is M . By the Hironaka desingularizationtheorem [23] there exists a manifold ˜ E and a proper bimeromorphic map π : ˜ E → E such that π is an isomorphism over the non singular part of E . Near M we havethat E coincides with Y at the regular points of E . Since π is onto and since theregular part of E is dense and connected, it follows that ˜ E contains an isomorphiccopy of Y and π is an actual diffeomorphism near the boundary. Pulling back φ on˜ E , we can repeat the proof of Theorem 3.1, where the boundary of the manifold Y NOTE ON THE CLOSED RANGE OF ¯ ∂ b ON q -CONVEX MANIFOLDS 9 is now just M . Following the same proof we conclude that the range of ¯ ∂ b is closedfor p = 2( n − q ). (cid:3) Acknowledgements
The authors would like to thank Martino Fassina for useful comments on themanuscript and the anonymous referee whose advice improved greatly the exposi-tory quality of the paper.
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Amer-ican Mathematical Society, Providence, RI,
Dipartimento di Matematica Tullio Levi-Civita, Universit`a di Padova, via Trieste 63,35121 Padova, Italy
E-mail address : [email protected] Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 WestGreen Street, Urbana, IL 61801, USA
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