Holomorphic extension from weakly pseudoconcave CR manifolds

Holomorphic Extension from Weakly Pseudoconcave CR Manifolds

ANDREAALTOMANI(*) - C. DENSONHILL(**) - MAURONACINOVICH(***) EGMONTPORTEN(***)

ABSTRACT- LetM be a smooth locally embeddable CR manifold, having some CR dimensionmand some CR codimensiond. We find an improved local geometric condition onM which guarantees, at a pointp on M, that germs of CR dis- tributions are smooth functions, and have extensions to germs of holomorphic functions on a full ambient neighborhood of p. Our condition is a form of weak pseudoconcavity, closely related to essential pseudoconcavity as introduced in [HN1]. Applications are made to CR meromorphic functions and mappings.

Explicit examples are given which satisfy our new condition, but which are not pseudoconcave in the strong sense. These results demonstrate that for codi- mensiond>1there are additional phenomena, which are invisible whendˆ1.

1. Introduction.

The goal of the present article is to give improved geometric conditions on a generic CR manifold MCⁿ which guarantee that all local CR functions extend holomorphically to a full neighborhood of a given point.

This is well known to be true for strictly pseudoconcave CR manifolds,

(*) Indirizzo dell'A.: Research Unity in Mathematics, University of Luxem- bourg, 6, rue Coudenhove-Kalergi, L-1359 Luxembourg.

E-mail address: andrea.altomani@uni.lu

(**) Indirizzo dell'A.: Department of Mathematics, Stony Brook University, Stony Brook NY 11794 (USA).

E-mail address: dhilll@math.sunysb.edu

(***) Indirizzo dell'A.: Dipartimento di Matematica, II UniversitaÁ di Roma

``Tor Vergata'', Via della Ricerca Scientifica, 00133 Roma (Italy).

E-mail address: nacinovi@mat.uniroma2.it

(***) Indirizzo dell'A.: Department of Mathematics, Mid Sweden University, 85170 Sundsvall (Sweden)

E-mail address: Egmont.Porten@miun.se

i.e. in the case where the Levi form has one negative eigenvalue in each characteristic conormal direction. For hypersurfaces this is a classical resultof H. Kneser and H. Lewy; for M of higher codimension itwas proved independently by a number of authors (see [BP], [NV] for CR distributions, and [HN4] for two different proofs, one very short). Despite numerous efforts, the general problem to characterize those weakly pseudoconcave manifolds for which one has extension to full neighbor- hoods is still far from being completely understood, even for real analytic hypersurfaces.

Subtle sufficient conditions (sector and ray property) are known for weakly pseudoconcavehypersurfaces of finite type (see [BT2], [FR] for results and references). Inhigher codimensionthere are several options to approach the weakly pseudoconcave case. Manifolds which are Levi flat at the reference point to a certain order, and have all relevant concavity in the generalized Levi form determined by the next-order terms are stu- died in [Bo]. Here we aim at the opposite case where effects of different orders (counted with respect to bracket length) are combined. Actually our main motivation stems from homogeneous CR manifolds which bi- holomorphically look the same near every point. These higher codimen- sional homogeneous CR manifolds are abundant, occurring naturally in mathematics, and they have a strong tendency to be weakly pseudo- concave (see [MN1], [MN2], [MN3], [MN4], [AMN]). The main resultof the present article reveals that there are additional phenomena which are invisible in codimension one, and indicates that finite type together with a suitable notion of weak pseudoconcavity should imply extension to a full neighborhood. To avoid confusion, we stress that the problem under consideration is different in nature from the problem of holomorphic wedge extension, for which a definitive answer is known ([T1], [Tu1], [J], [M], see also [MP2]). In fact, this definitive answer was obtained without having explicit control on the directions of extension, which is crucial for the problem at hand.

Let MCⁿ be a smooth CR manifold. We denote byJ the complex structure tensor onTCⁿ, byHMˆTM\JTM the holomorphic tangent bundle ofM, which is the real subbundle ofTMinvariantunderJ, and by H⁰MTMthe characteristic bundle, defined fiberwise as the annihilator ofHM. We define thevector valuedLevi form

L_M;pˆ L_p:H_pMH_pM!C(T_pM=H_pM) by

L_M;p(X;Y)ˆ [X;~ JY](p)~ ‡i[X;~ Y](p) mod~ CHpM;

whereX;~ Y~ 2G(M;HM) are smooth extensions ofX;Y, respectively. The usual Levi formLM;jˆ Lj is parameterized by the characteristic codir- ectionsj2H⁰_pM. Itis defined byL_M;j(X;Y)ˆj(Lp(X;Y)), forj2H⁰_pM, X;Y 2H_pM(where we readjas a form onC(T_pM=H⁰_pM) in the cano- nical way). Most often we shall work with the corresponding real hermitian forms Lp(X)ˆ Lp(X;X)2TpM=HpM and Lj(X)ˆ Lj(X;X). The reader should take note of the subtle difference in notation between Lp(X) and L_j(X); the former is vector valued, and the latter is scalar valued.

A CR manifoldMisstrictly (weakly) pseudoconcaveata pointp2Mif for every j2H⁰_pM, j6ˆ0, L_j has a negative (nonpositive) eigenvalue.

Replacingjby j, we see thatL_jhas actually eigenvalues of both signs (in the strictly pseudoconcave case). Following [HN1], we callM trace pseu- doconcaveatp2Mif for everyj2H⁰_pM,L_j is either zero or has eigen- values of both signs. Trace pseudoconcavity isolates one of the properties of essential pseudoconcavity introduced in [HN1]. We refer to that article for background information.

Let G1 be the sheaf of germs of smooth (real) CR vector fields on M (i.e. sections ofHM). For every positive integerkwe define inductively G_k‡1 as the sheaf generated by G_k and [G₁;G_k]. Let G_k;pT_pM be the vector space generated by pointwise evaluations of germs inG_kata point p2M. We say thatMis ofkind katpifG_k;pˆT_pMbutG_j;p4T_pMforj5k.

We say thatMsatisfies theconstant rank conditionif the spacesGk;phave dimension independentofp, i.e. if they form vector bundlesG_k ˆ S

p2MG_k;p. Now we can formulate our main result.

THEOREM 1.1. Let M be a smooth generic CR manifold in Cⁿ and p02M. Assume that in a neighborhood of p0, M is trace pseudoconcave, satisfies the constant rank condition and is of kind less or equal to3. Then for every open neighborhood U of p0in M, there is an open neighborhood V of p₀inCⁿsuch that every CR distribution on U is smooth on M\V and has a unique holomorphic extension to V.

We emphasize that Theorem 1.1 reveals a phenomenon which re- mains invisible in codimension 1. In fact, in the hypersurface case its assumptions imply that M is of kind 2, hence strictly pseudoconcave.

We expect the result to extend to arbitrary finite kind. In [HN1], the weak identity principle for CR functions (coincidence on open sets im- plies coincidence everywhere) was shown for essentially pseudoconcave CR manifolds. For those CR manifolds covered by the assumptions of Theorem 1.1 our resultimmediately yields the strong identity principle

(coincidence of Taylor coefficients at some point yields coincidence in a neighborhood).

Theorem 1.1 is proved in Sections 2-4. The rest of the article is de- voted to applications, extensions and examples. In Section 5 we observe that CR manifolds as in Theorem 1.1 enjoy the local extension property E introduced in [HN2]. The results in [HN2], [HN3] yield far reaching global consequences for fields of CR meromorphic functions on such manifolds. Here CR meromorphic functions are functions which are lo- cally representable as fractions of CR functions. An alternative approach to CR meromorphic mappings originates from work of Harvey and Lawson [HL]. The idea is to require the graph to look like a CR manifold with appropriate singularities. In general, extension of such mappings is complicated. Based on [MP2], we prove in Section 6 that such CR mer- omorphic functions extend meromorphically from manifolds with prop- erty E to full ambient neighborhoods and are in particular representable as local quotients. In Section 7 we present several classes of homo- geneous CR manifolds to which all the local and global results indicated above apply. These were discovered in a much broader context (see [AMN]). For the reader's comfort, we give a reasonably self-contained presentation.

2. Preliminaries.

We will use some standard facts about the bundlesG₁G₂. . .: If G_kˆG_k‡1 then allG_j,jk, are equal (the proof is an application of the Jacobi identity). This means in particular that Gk is integrable in the sense of Frobenius. Moreover the map associating to smooth sections X2G(U;G1),Y 2G(U;G_k), the section [X;Y] modG_k 2G(U;G_k‡1=G_k) is tensorial, i.e. [X;Y](p) modG_k;pdepends only onX(p) and Y(p).

Letus now have a closer look atG₂. First we note that independently of concavity G2;p=HpM is spanned as a real vector space by the image Cpˆ fLp(X):X2HpMg of the vector valued Levi form. Indeed, Lˆspan_RCpis contained inG2;p=HpM by definition. On the other hand, polarization shows that LCˆspan_CfL_p(X;Y):X;Y 2H_pMg. Since the imaginary part of L_p(X;Y) is essentially [X;Y], we obtain G_2;p=H_pML. The above is equivalent to the fact thatG_2;pis spanned by the preimage ofCpunder the canonical projectionTpM!TpM=HpM.

A simple but crucial observation is that trace pseudoconcavity allows us to replace linear spans by convex hulls.

LEMMA 2.1. Assume that M is trace pseudoconcave at p2M. Then G2;p=HpM is the convex hull of Cpˆ fLp(X):X2HpMg.

PROOF. If the lemma fails, there is a nonzero linear functionaljon G_2;p=H_pMsuch thatC_p fj0g. We may extendjto an element ofH⁰_pM.

SinceG2;p=HpMis the linear span ofCp, there is anX2Cpwithj(X)>0.

AnyY2HpMwithLp(Y)ˆXsatisfiesLj(Y)>0. But this implies thatLj

has also some negative eigenvalue, in contradiction toCp fj0g p To keep track of directions of extension, we will use an analogue of the analytic wave front set, denoted byWF_u. It is defined for CR distributions uvia the FBI transform in [S], see also [T2]. ForUopen inM, letCR(U) denote the space of continuous CR functions defined onU. We do noteven have to recall the definition of WF_u, since the following basic properties will suffice for our purposes:

(a) Letube a CR distribution defined onUM. ThenWFuis a cone, closed in the pointed characteristic bundleH⁰Uno(odenoting the zero section).

(b) WF_u\H⁰_pMˆ ; holds if and only the CR distribution uextends holomorphically to an ambient neighborhood ofp.

(c) Let u2CR(U). If CR extension from U holds at(p;X), p2U, X2TpMnHpM, then for anyj2WFu\H⁰_pMwe havej(X)0.

In (c) we use the following terminology: We say thatCR extension from U holds at(p;X),p2U,X2T_pMnH_pM, if there is aC²-smooth (dimM‡1)- dimensional CR manifoldM~ attached toUalong someU-neighborhoodU⁰of p such that (i) for a representative ofX,JX points intoM~ and (ii) every u2CR(U) has a continuous extensions toM~ [U⁰which is CR onM. Ne-~ glecting the dependence onU, we will sometimes call (p;X) or justX, a di- rection of CR extension. For (a), (b), see [S], whereas (c) is observed in [T2].

Theorem 1.1 is a consequence of the following more precise result which does notrequire kind 3.

THEOREM2.2. Let M be a smooth generic CR manifold inCⁿ. Assume that on an open set UM, M is trace pseudoconcave and that G₂, G₃are bundles. Then for every continuous CR function u defined on U, we have WF_uG^?₃.

Theorem 2.2 will be proved in the next two sections. It implies Theo- rem 1.1 in the following way: Kind 3 means thatG^?₃ is the zero bundle near

p0. By (b) a continuous CR function u extends holomorphically to an ambientneighborhood of everypcontained in some neighborhoodU⁰ofp0

inM. By a standard gluing argument one obtains extension to an ambient neighborhoodV⁰ ofU, which may a priori depend onu. Note that holo- morphic extension in particular shows thatu2 C¹(U⁰). Now a Baire ca- tegory argument as in [HN2] yields extension to a neighborhood of V whose size only depends onU. This proves the theorem for continuous CR functions.

Ifuis a CR distribution, we may use a method from [BT1], [T], to re- presentitnearp₀asuˆD^k_Mf, wherefis a continuous CR function. Herek is a sufficiently large integer, andD_M is a variantof the Laplace operator which is defined in an ambientneighborhood ofp0 and restricts nicely to M. If~f is a holomorphic extension off, then the various properties ofD_M imply thatD^k_M~f is the desired extension ofu. Hence Theorem 1.1 follows from Theorem 2.2.

3. Proof plan for Theorem 2.2.

Here we will prove Theorem 2.2 modulo some more technical results on CR extension which are postponed to the following section. Pick some u2CR(U).

STEP1. -WF_uG^?₂. This will follow from trace pseudoconcavity. It is a consequence of the following lemma which holds without constant rank assumptions.

LEMMA 3.1. Assume that MCⁿ is trace pseudoconcave at p2M.

Then for every continuous CR function u defined near p we have WF_u\H⁰_pMG^?_2;p.

PROOF. Letj2WFu\H⁰_pM. We will considerjboth as a functional acting on TpM and TpM=HpM. By [Tu2], every element X2Cp can be approximated by directions of CR extensions X_j2TpM=HpM. From property(c)and continuity we getj(X)0. Itfollows thatjis nonnegative on the convex hull of C_p. Since this convex hull is the vector space

G_2;p=H_pM,jvanishes onG_2;p p

Notice that the lemma together with property (b) already imply ex- tension to a full neighborhood for strictly pseudoconcave CR manifolds.

STEP 2. -WFuG^?₃. By Step 1 it suffices to show that a covectorj₀ that annihilatesG2butnotG3is notcontained inWFu. Letp02Udenote the base point to which j₀ projects. It is our aim to show j₀2= WFu by constructing an appropriate CR extension and applying property (c).

Since the imageC_p0of the vector valued Levi form spansG_2;p0=H_p0M(see the remarks before Lemma 2.1), we may select vectorsX1;. . .;Xk 2Hp0M such that the vectorsY~jˆ LM;p0(Xj),jˆ1;. . .;k, form a basis ofG2;p0=Hp0M.

Extending theX_jsmoothly to CR vector fields defined nearp0, we obtain a local basisY~_j(p)ˆ L_M;p(X_j) ofG₂=HM. SetY_jˆ[JX_j;X_j], and choose a local basisZ₁;. . .;Z_2mofHM. Then theZ_iform together with the theY_ja local basis ofG₂.

Firstwe claim thatG3is spanned in some neighborhood ofp0by theZi, Y_j, together with the brackets [Z_i;Y_j]. Indeed, by definitionG3;pis spanned byG_2;pand vectors of the form [Z;Y](p) whereZ2 G_1;p,Y2 G_2;pforpnear p₀. Aroundpwe may writeZˆP

z_iZ_i,Y ˆP~z_iZ_i‡P

y_jY_j, with smooth coefficientsz_i,~z_i,y_j. This yields

[Z;Y]ˆX

z_iy_j[Z_i;Y_j]‡R;

whereRis a germ inG_2;p. This proves the claim.

In the sequel, we will only need the following consequence: Since j₀ does notannihilateG3;p0, there arei0,j0such thatj₀([Zi0;Yj0](p0))6ˆ0. For notational convenience we will write from now onY ˆY_j0,ZˆZ_i0.

The following proposition, which will be proved in Section 4, yields CR extension at (p₀;Y(p₀)) realized by a CR manifold to whichY is complex tangentin a neighborhoodofp0inM.

PROPOSITION3.2. Let MCⁿ be a smooth generic CR manifold of CR dimension m and codimensiond. Letp₀ 2M and letU be an open neighborhood of p₀ in M. Let X be a smooth CR vector field onU with L_M;p0(X)6ˆ0. Then there is a localC⁴-smooth generic CR manifoldM~ of dimensiondimM‡1with the following properties:

(a) M\M is a neighborhood of p~ ₀ in U andMnM has two connected~ componentsM~ .

(b) The distribution on M\M spanned by HM and~ [JX;X]coincides with(HM~ \TM)j_M\M~.

(c) Denoting byM~^‡ the side into which J[JX;X](p₀) points, we have CR extension from U toM~^‡.

Sincej₀annihilatesHp0M~ \Tp0M, there is a unique extension~j₀2H⁰_p0M.~

We claim thatLM;~~j0 has eigenvalues of both signs. Indeed, bothZandY may be extended toC³-smooth CR vector fields onM~ which we denote by the same symbols. ForYwe use here thatYiscomplextangent toM~ along Mby Proposition 3.2. The fact thatj₀annihilates the image of the Levi form of M implies LM;~ ~j0(Z(p₀))ˆ L_M;j0(Z(p₀))ˆ0. Since ~j₀([Z;Y](p₀))6ˆ0 is essentially the imaginary part of the sesquilinear Levi form,Zdoes notlie in the kernel of LM;~ ~j0. Hence LM;~ ~j0() takes both positive and negative values in every neighborhood ofZ(p0) inHp0M. The claim follows.~

LetX^‡,X be positive and negative eigenvectors ofLM;~~j0, respectively.

Since all vectors inH_p0Mare null vectors ofLM;~~j0, we may assume, after replacingX by an appropriate complex multiple if necessary, that X both point into M~^‡. But now the following proposition, whose proof is postponed to Section 4, yields for M two directions of CR extension Y2T_p0M=H_p0Mwithj₀(Y^‡)>0 andj₀(Y )50. This impliesj₀6ˆWF_u by property(c). Hence the proof of Theorem 2.2, and thus also of Theorem 1.1, will be complete as soon as we have shown the following.

PROPOSITION3.3. LetM,M~,M~ andp₀ 2M be as above. Assume that there is~j2H_p⁰₀M~ andX2H_p0M~nT_p0M pointing intoM~^‡such that L_M;~~j(X)>0. Then there is a directionZ2T_p0M=H_p0M of CR extension fromU satisfying~j(Z)>0.

4. CR extension.

In this section we will prove Propositions 3.2 and 3.3. First we recall some basic material on Bishop discs introduced in the seminal paper [B]. Consider a generic CR manifold MCⁿ of CR dimension m and codimension d. A C-smooth analytic disc is a mapping A(z)ˆ(Z(z);W(z))2 C(D;Cⁿ)\ O(D;Cⁿ) whereDˆ fz2C:jzj51g. We say that A is attached to M if A(S¹)M.

For jmj 1 let Tm denote the Hilbert transform of a function U:S¹!Rto its harmonic conjugate T_mU, normalized by the condition that the harmonic extension ofT_mUvanishes atzˆm. Itis known thatT_m is a continuous linear operator on the HoÈlder spacesC^k;a(S¹;R) ifk2N0, 05a51. We shall use the same notation for the Hilbert transform applied componentwise to vector valued functions ofC^k;a(S¹;R^d).

We work in coordinates

z₁ˆx₁‡iy₁;. . .;z_mˆx_m‡iy_m;w₁ˆu₁‡iv₁;. . .;w_dˆu_d‡iv_d (1)

centered at the origin in whichMis locally given as a graphvˆh(z;u) with h(0)ˆ0,dh(0)ˆ0. The Bishop equation is the nonlinear functional equa- tion

Uˆ T_m(h(Z;U))‡u:

(2)

Here UˆU(z) is the unknown function mapping the unit circle S¹ˆ fjzj ˆ1g CtoR^d, whereasZˆZ(z) is the boundary value of a given holomorphic function fromDtoC^mandu2R^dis a prescribed vector. It is known that the Bishop equation can be solved inC^k;a(S¹;R^d) provided the dataZ(z) areC^k;a-small andjujis small (see [MP4] for detailed information).

The solution U corresponds to a unique analytic disc A:D!Cⁿ whose restriction to S¹ is (Z(z);U‡ih(Z(z);U(z))). More precisely, A is holo- morphic onDandC^k;a-smooth up toS¹. By construction,Ais attached toM.

PROOF OFPROPOSITION3.2. - The construction ofM~^‡ will be a refine- mentof [HT, proof of Theorem 9.1]. We may assume p₀ˆ0 and choose coordinates (z;w) as in (1). After appropriate rotations and dilations in z andw, we may furthermore assumeL_M;0 @

@x1

ˆ @

@u1 modH₀M. We will constructM~^‡as the union of real curvesg_p(s),s2[0;e₁), having their initial points atg_p(0)ˆp2Mand such that the segmentsg_p((0;e₁)) foliateM~^‡.

Let us start by the construction ofg₀. Following [HT], we firstsimplify the defining equations by removing some of the pure terms. After a change of thew₁-variable

w₁7!w₁ i X^m

j;kˆ1

@²h₁

@zj@zk(0)z_jz_k‡X^d

jˆ1

@²h₁

@z1@uj(0)z₁w_j‡X^d

j;kˆ1

@²h₁

@uj@uk(0)w_jw_k

!

; we have

h₁(z;u)ˆ X^m

j;kˆ1

a_j;kz_jz_k‡O₃(z;u);

witha1;1ˆ1. In particular, we still haveLM;0

@

@x1

ˆ @

@u1.

For a smalle₁>0, to be specified later, and a parameter 0t5pe₁, we letUtbe the solutions of the parameter-dependent Bishop equation

U_tˆ T₀(h(W_t;U_t)); whereW_tˆ(tz;0. . .;0):

(3)

Since the data are smooth, we can solve this in the HoÈlder spaces C^k;a(S¹;R^d) (k1, 05a51) fore₁ sufficiently small. Let theA_t(z) be t he corresponding holomorphic disc and setg₀(s)ˆA 

ps(0). One reads off from

(3) that the curve g₀ starts at the origin and runs in the space iR^d_v ˆ fzˆu₁ˆ. . .ˆu_nˆ0g. Itis shown in [HT] that(i)dg₀

ds(0) is a po- sitive multiple of @

@v1and that (ii)g₀is on [0;e₁) as smooth as we please ife₁is sufficiently small.

The curvesg_pwill be obtained by varying the initial point in this con- struction. To this end we first produce a family of coordinates (zp;wp) centered atpsuch thatTpˆ0Mˆ fvpˆ0g. Clearly this can be achieved by an affine linear change of coordinates Fp which depends smoothly on p, wherep ranges in some M-neighborhood of the origin. Thus (z_p;w_p) are holomorphic forpfixed, butare only smooth inp. Next we rotate and dilate in such a way thatdF_p(X) transforms to @

@x₁ at the origin and such that L_M;0 @

@x₁ ˆ @

@u₁. Then we modify (z_p;w_p) again as before in order to simplify second-order terms. Clearly all this can be done by a family of local biholomorphisms depending smoothly on the parameter p. In these last coordinates, which we still denote by (z_p;w_p), we construct a curve~g_p(s), 0s5e1, verbatim by the same construction as above. Thus~g_p(s) starts at the origin in (zp;wp)-space, which corresponds to the pointp, and its time derivative atsˆ0 is a positive multiple of @

@v1. The desired curveg_p with g_p(0)ˆpis then obtained by reversing the coordinate transformations. The nature of the process implies thatdg_p

ds (0) is a positive multiple ofX(p).

We claim thatM~^‡[Wˆ S

p2W;0s5e1

g_p(s) is a manifold with boundary W attached to M along W, provided W is a sufficiently small open neighborhood of the origin in M, and e1 is sufficiently small. Actually, optimal regularity results for the Bishop equation yield that the solution, which depends on a finite dimensional set of parameters, has arbitrarily small loss of smoothness with respect to the parameters ([Tu2], see also [MP4]). Thus we have that data dependingC^k;a-smoothly on all variables and parameters lead toC^{k;a 0}-smooth solutions. As the smoothness insis as good as needed and we can assumekas large as we please, the mapping (p;s)7!g_p(s) is as smooth as we please. First, the inverse function theorem implies the claim aboutM~^‡[W. Second, we may extend (p;s)7!g_p(s) t o W( h;e₁) for some 05h1, and obtain an extended manifold M~ ˆ S

p2W; h5s5e1

g_p(s) as required in (a). The before mentioned fact that dg_p

ds (0) is a positive multiple ofX(p) completes the proof of (b).

The proof of (c) is standard. One chooses anM-neighborhoodW⁰U ofp0so small that everyu2CR(U) can be uniformly approximated onW⁰ by holomorphic polynomialsP_j, using the Baouendi-Treves approximation theorem [BT1]. Restricting domains again, one restricts the above con- struction using only discs attached to W⁰. Then the maximum modulus principle applied to the discs implies that the Pj converge uniformly on M~^‡[W to a continuous function u‡, which is CR onM~^‡. The proof of

Proposition 3.2 is complete p

PROOF OFPROPOSITION3.3. - Let us first sketch the geometrical idea.

SinceY ˆ L_{M;p~ 0}(X) is not complex tangent toM, itis classical (see [HT])~ that appropriateC^2;a-small discs attached nearp0toM~ whosez-coordinates are parallel to CX are nontangent toM~ along their boundaries, sticking outalong directions which are approximatelyYmoduloH_p0M. This would~ be enough in order to extend CR functions defined on all ofM. In the case~ at hand, we are only allowed to use discs attached to M~^‡[W. We will construct a family of discs whose boundaries touch M quadratically in exactly one point and obtain CR extension to a (dimM‡1)-dimensional manifold (distinct fromM~^‡) contained in the union of the discs. Since es- sentially the same construction is explained in great detail in [MP3, Section 5], it will suffice to give a concise review of what has to be done.

We will first construct a single disc attached to M~^‡[W whose boundary touchesMatp0. Choose local coordinates

z₁;. . .;z_m‡1;w₁ˆu₁‡iv₁;. . .;w_{d 1}ˆu_{d 1}‡iv_{d 1}

centered at p0 such that M~ is locally given as a graph vˆ~h(z;u), with h(0)~ ˆ0, d~h(0)ˆ0. After convenient rotations, we may assume that Xˆ @

@x₁,LM;0~ Xˆ @

@u₁ modH0M~ andT0MˆiRy1C^m_z2;...;zm‡1R^d_u1;...;u¹ _{d 1} (by multiplying by somez2S¹, we have rotatedXso thatJX2T0M)). For technical reasons, we also arrange that

h(z~ 1;0;. . .;0;u1. . .;ud 1)ˆcjz1j²‡O(j(z;u)j³); c>0;

(4)

by eliminating pure terms of second order.

ForZ_r(z)ˆ(r(1 z);0;. . .;0), 05r1, let

Ar(z)ˆ(Zr(z);Ur(z)‡iVr(z))ˆ(Zr(z);Ur(z)‡ih(Zr(z);Ur(z))) be the analytic disc obtained by solving the Bishop equation

U_rˆ T₁(h(Z_r;U_r)):

(5)

Note thatAr(1)ˆ0. The crucial pointis thatr(1 z) lies in the right z1- halfplane and touches the imaginary axis quadratically at the origin, and that the curvature of its boundary at the origin becomes large forr!0.

First, it is proved in [MP3, 5.5] that for r>0 small, we have A_r(Dnf1g)M~^‡ and thatA_r(S¹) touchesMquadratically at 0. There is a geometric estimate for admissibler which is stable under C^2;a-small de- formations of~h. Second, the usual classical argument (see [HT]) based on (4) shows that

@V_r

@l

zˆ1ˆr(c⁰;0;. . .;0)‡o(r);

(6)

with somec⁰>0 independentofr(herezˆl‡ih).

Let us take for granted, for the moment, that we can construct for a small fixedra (dimM‡1)-dimensional manifoldM^^‡attached toM, and containing the imageA_r((1 e;1)) of the segment (1 e;1), such that CR functions extend from M toM^^‡. Then (6) yields for the normalized out- going direction at the origin

@v_r

@l(1).@v_r

@l (1)!(c⁰;0;. . .;0); asr!0:

But this implies that we can approximate the element defined by @

@v1 in T₀M=H₀M by directions of CR extensions. Since ~j @

@v1

>0 we have found a direction of extensionZas desired. Now since a sufficiently small r>0 has been fixed, we drop it from the notation and writeA.

Itremains(i) to construct the manifoldM^^‡ and (ii) to establish CR extension fromMtoM^^‡. The method for getting(i)is very similar to that of the proof of Proposition 3.2. We construct a family of local holomorphic coordinates

z1;p;. . .;zm‡1;p;w1;p;. . .;w_{d 1;p}

coinciding with the above coordinates forpˆ0 and satisfying the following properties:

a) (zp;wp) dependC³-smoothly on the parameterp, which ranges in a smallM-neighborhoodW⁰⁰of the origin.

b) Forpfixed, (z_p;w_p) are holomorphic and centered atp, and we have T0MˆiRy1;pC^m_{z2;p;...;zm‡1;p}R^d_u1;p¹_{;...;ud 1;p}.

We apply the above construction with dependence on the parameterp.

This yields discsApattached toM~^‡[W⁰and touchingMquadratically at A_p(1)ˆp. Using regularity results for the Bishop equation ([Tu2], also

[MP4]), we see thatAp(z) dependsC^2;1=2-smoothly onpandz2D. After a further shrinking of W⁰⁰ and with a smaller e, we obtain a manifold M^^‡[W⁰⁰ˆ S

p2W⁰⁰Ap((1 e;1]) as required in(i).

To show(ii)we just have to remember from the construction ofM~^‡that everyu2CR(U) can be approximated by holomorphic polynomialsPjwhich converge uniformly onW⁰[M~^‡. Hence thePjalso converge onM^^‡[W⁰⁰to a continuous CR function extendingu. This yields(ii)and completes the proofs of Proposition 3.3, and also of Theorems 1.1 and 2.2 p

5. Applications to CR meromorphic functions.

One of our motivations is to find concrete applications for the Siegel- type theorems proved in [HN2], [HN3]. In these papers, far reaching global consequences for the field of CR meromorphic functions are proved for CR manifolds satisfying a local extension property E. For aC¹-smooth generic CR submanifold M of a complex manifold N, property E means that the canonical restriction mappingO_N;p! CR_M;pis surjective for ev- eryp2M. HereONandCRM denote the sheaf of germs of holomorphic functions onNand the sheaf of germs ofC¹-smooth CR functions on M, respectively, andO_N;p,CR_M;pare their stalks atp. We obtain immediately that a manifold satisfying the assumptions of Theorem 1.1 in a coordinate neighborhood of each of its points has property E. Hence we obtain all results proved in [HN2], [HN3] by carrying the local situation studied here to general manifolds.

Actually the main results in [HN2], [HN3] concern CR meromorphic functions rather than CR functions. Similarly as ordinary meromorphic functions, we define CR meromorphic functions in the usual sense on UMas functions which are defined on a dense open subset ofUand can be represented neareverypointp2Uas the quotientp=q ofC¹-smooth CR functionsp,q, where qdoes not vanish identically on any nonempty open subset. IfMhas property E, every CR meromorphic function onUis the restriction of a meromorphic function defined on some ambient neighborhood ofUinN.

Let M be a smooth compact locally embeddable CR manifold of CR dimensionmand CR codimensiond, which at each point satisfies the hy- potheses of Theorem 1.1. Then the fieldK(M) of CR meromorphic func- tions onMhas transcendence degreekm‡d. Iff₁;. . .;f_kis a maximal setof algebraically independentCR meromorphic functions on M, then

K(M) is a simple finite algebraic extension of the fieldC(f1;. . .;fk) of ra- tional functions of thef1;f2;. . .;fk. Assuming thatMis connected, there is also an equivalence between the algebraic dependence over C, and the analytic dependence, of a finite set of CR meromorphic functions inK(M).

WhenMhas a projective embedding there is an analogue of Chow's the- orem, andK(M) is isomorphic to the fieldR(Y) of rational functions on an irreducible projective algebraic varietyY, andMhas a CR embedding in reg Y. For details, and further applications and remarks, see [HN2], [HN3].

6. CR meromorphic mappings according to Harvey and Lawson An alternative notion of CR meromorphic functions and mappings was suggested by Harvey and Lawson in the context of the complex Plateau problem and studied in [HL], [DH], [DS], [MP1], [MP2]. The following definitions appear in [HL] for hypersurfaces and in [DS] for CR manifolds of arbitrary codimension. LetMbe a smooth generic CR submanifold of a complex manifold N of CR dimension m and codi- mension d, and let X be an arbitrary complex manifold. Then a CR meromorphic mapping F in the sense of Harvey-Lawson of an open UM with values in X is given by a triple (F;DF;GF) with the fol- lowing properties:

(a) D_F is an open dense subsetofU, (b) F:DF!Xis aC¹-smooth CR mapping,

(c) the closure of the graph of F in UX equals GF and is a local scarred CR cycle of CR dimension m and dimension 2m‡d in NX.

In (c) we mean thatGF is of locally finite (dimM)-dimensional Haus- dorff measure and contains a closed subset s (the scar set) of (dimM)- dimensional Hausdorff measure zero such that (i)G_Fnsis aC¹-smooth CR manifold of same dimension and CR dimension as M and (ii) in a neigh- borhood of every (p;x)2GF, integration overGFnsyields aclosedcurrent (see [HL], [DS], [MP1] for full details). IfX equalsP¹, the complex pro- jective line, we also speak of CR meromorphic functions in the sense of Harvey-Lawson.

Meromorphic extension of these CR meromorphic mappings is tech- nically complicated because of the presence of the scar set. Actually it requires a certain machinery to derive the counterpart of Theorem 1.1 for

CR meromorphic functions in the sense of Harvey-Lawson. We can prove this for all manifolds with property E.

THEOREM6.1. Let M be a smooth generic CR submanifold of a complex manifold N with property E. Then every CR meromorphic function F in the sense of Harvey-Lawson, defined on an open set UM and with values in P¹has a meromorphic extensionF to a neighborhood V of U in N. More~ precisely, the graph GF~ VP¹ of F satisfies~ GF~\(UP¹)ˆG_F. In particular, F is a CR meromorphic function in the usual sense.

A CR manifoldM is calledminimal at a point p2M (in the sense of Tumanov) if there is no germ of a CR manifold NM of the same CR dimension asMand of lower dimension thanMcontainingp.

LEMMA 6.2. Let M be a smooth generic CR manifold in Cⁿ with property E. Then M is minimal at every point p2M.

PROOF. Letus assume thatMis notminimal atp0. By [BR] there is a smooth CR functionudefined on an open neighborhoodU⁰Uofp₀which does not extend holomorphically to any open wedge attached toMnearp₀,

in contradiction to property E. p

PROOF OF THEOREM 6.1. - Let F be a P¹-valued CR meromorphic function in the sense of Harvey-Lawson defined on UM. Firstwe construct a local extension of F to an ambient neighborhood of a given p₀2U. Because of Lemma 6.2, [MP2, Theorem 1.2] gives meromorphic extension to an open wedgeWattached to a neighborhoodU⁰ofp0inM.

More precisely, there is an open truncated coneCCⁿwith vertex at the origin and a meromorphic function F~ onW ˆU⁰‡C which attainsF as continuous boundary value onD_F.

From property E and a Baire category argument (see [HN2]), it follows that smooth CR functions on U⁰ extend holomorphically to a uniform ambientneighborhoodV⁰of U⁰. Since the envelope of meromorphy coin- cides with the envelope of holomorphy for domains inCⁿ, functions which are meromorphic on an arbitrarily thin neighborhood of U⁰ extend mer- omorphically to V⁰. For fixed c2C, the rigid translates U_e⁰ˆU⁰‡ fecg approachU⁰fore#0. To obtain the desired extension to a neighborhood of p0, itsuffices to choosee2(0;1) so small thatp02V⁰‡ fecgand to extend F~ toV⁰‡ fecg.

Now a standard gluing argument yields a meromorphic function F~ which is defined on a neighborhoodV ofUinCⁿand coincides withFon D_F. It remains to prove thatGF~ \(UP¹)ˆG_F. Near points inD_Fthis is obvious. From(b)in the definition of CR meromorphic mappings and the corresponding (well known) property of meromorphic functions, we deduce GF GF~\(UP¹). Assume that there is (p0;z₀)2GF~\(UP¹)nGF. This is obviously impossible if p0 is a pointnear which F~ is a smooth mapping. Hence it remains to consider the case in which p0 lies in the indeterminacy setSF~ ˆ fp:fpg P¹GF~g.

Following [DS], we also consider the indeterminacy setS_FofFdefined by

S_Fˆ fp2U:fpg P¹G_Fg:

Our assumption onp0means thatp02=SF. Letabe a biholomorphism ofP¹ mappingz₀to1. It is observed in [DS] that the set-valued functionaF naturally induces a CR distributiongof order one on a neighborhoodU_p0of p₀ in M. AsM is minimal in p₀, Tumanov's theorem [Tu1] and the usual extension techniques for CR distributions yield holomorphic extension to an open wedge attached toMatp0(assuminggas boundary value in the weak sense). Then an argumentwith approach manifolds as above yields a holomorphic extension~gto a full neighborhood ofp₀. Observing thataF and~gare smooth and coincide at points ofD_F, we obtain that the equality F~ ˆa ¹~g holds near p0. In particular, F~ is smooth near p0, in contra- diction top02SF~. The proof of Theorem 6.1 is complete p REMARK6.3. It requires only little extra work to derive a corresponding result for CR meromorphic mappings with values in a projective manifoldX.

Note that in the general case we can only expectG_FG_F~ \(UX).

7. Homogeneous examples

Atfirstglance, itmay seem hard to find examples of CR manifolds of kind 3 satisfying the conditions of Theorem 1.1. However, the theory of homogeneous CR manifolds provides many of them in a very natural way (see [MN1], [MN2], [MN3], [MN4], [AMN]). We give a concise description of a class of such homogeneous CR manifolds, and refer to [AMN] for more details.

LetG^Cbe a complex connected semisimple Lie group, with Lie algebra g^C, and Ga connected real form ofG^C, with Lie algebrag. Fix a Cartan

subgroup Hof G, which is maximally noncompact, that is a Cartan sub- group such that a maximal compact torus inHhas minimal dimension, and denote byhandh^Cthe Lie algebras ofHand of its complexificationH^C. In the set of roots R ˆ R(g^C;h^C) choose a subset R^‡ of positive roots, adapted tog(cf. [AMN], Proposition 6.1), and letBbe the corresponding set of positive simple roots. To any subsetF Bwe associate the parabolic subalgebra and subgroup

q_Fˆh^C‡ X

a2R^‡

g^C_a ‡ X

a2 R‡

(suppa)\Fˆ ;

g^C_a;

Q_FˆNorm_GC(q_F):

Hereg^C_a is the eigenspace ing^Cof a roota, and suppais the support of a root ainB. The groupQFhas Lie algebraq_F.

The group G^C acts via the adjoint representation on g^C and on linear subspaces ofg^C. Fix a subsetF Band letdbe the dimension ofq_F. The orbit

Yˆ fAd_g^C(g)(q_F)jg2G^Cg Grd(g^C)

through q_F in the Grassmannian ofd-planes ing^C is the flag manifold of parabolic subalgebras ofg^Cconjugate by an inner automorphism toq_F. Itis a smooth irreducible projective subvariety of Grd(g^C), isomorphic toG^C=QF. The orbit

Mˆ fAd_gC(g)(q_F)jg2Gg Gr_d(g^C)

of Gthrough q_Fis a smooth generic CR submanifold ofY which, by our choice of the Cartan subgroup and of the system of positive roots, is com- pact, and is called the minimal orbit ofGinY.

If all local CR functions near a pointpofMextend to a full neighbor- hood ofpinY, then the pair (M;Y) has property E of [HN2],hence the field K(M) of CR meromorphic functions is isomorphic to the field R(Y) of rational functions on Y, because Y is the smallest projective variety containingM(see [HN2]).

EXAMPLE7.1. IdentifyC⁶, with the standard basisfe_jg_1j6, with the quaternionic vector spaceH³by setting, forl2C,

jle_{2j 1}ˆle_2j; jle_2jˆ le_{2j 1}: Consider the complex flag manifold

Yˆ`1 `3`5C<sup>6</sup>jdim`2j 1ˆ2j 1; 1j3 :

ThenY is a compact13-dimensional complex manifold, homogeneous for the action ofG^CˆSL(6;C). Near the point

oˆ he₁i_C;he₁;e₂;e₃i_C;he₁;e₂;e₃;e₄;e₅i_C

it admits a holomorphic chart given by the nonconstant entries of the matrix

Aˆ

1 0 0 0 0

z₁ 1 0 0 0 z₂ 0 1 0 0 z₃ z₆ z₉ 1 0 z₄ z₇ z₁₀ 0 1 z5 z8 z11 z12 z13

0 BB BB BB

@

1 CC CC CC A :

LetMbe the real submanifold ofY given by

Mˆ f(`<sub>1</sub>; `₃; `<sub>5</sub>)2YjH`₁`<sub>3</sub>; H`₃`₅)g:

ThenMis homogeneous for the action of the real formGˆSL(3;H) ofG, and itis a compactreal-analytic homogeneous generic CR submanifold of Y. Denoting byA_jthej-th column ofA, near the pointothe manifoldMis defined by the system of equations

rk(A1;A2;A3;jA1)ˆ3;

rk(A1;A2;A3;A4;A5;jA1;jA2;jA3)ˆ5;

which, in the coordinatesfz_jg_1j13, are

z2 z6‡z1z3‡z3z9 z1z1z6 z1z2z9ˆ0;

z4 z8‡z1z5‡z3z11 z1z1z8 z1z2z10ˆ0;

z10 z12‡z9z11‡z11z13 z9z9z12 z9z10z13ˆ0;

z5‡z7 z1z4 z3z10‡z1z1z7‡z1z2z10ˆ0;

8>

><

>>

:

thusMhas CR dimension 5 and codimension 8. The holomorphic tangent spaceT^1;0_o Madmits the basisf@=@zjgjˆ1;3;9;11;13. The space of Levi forms has real dimension 6, and in the basisf@=@zjgjˆ1;3;9;11;13itis the space of Her- mitian symmetric matrices of the form

0 a 0 0 0

a 0 a b 0 0 a 0 g 0 0 b g 0 g 0 0 0 g 0 0

BB BB

@

1 CC CC

A; a;b;g2C;

as the fourth equation does not contribute to the Levi form. In some

characteristic codirections the Levi forms are zero, but in the remaining characteristic codirections, the Levi forms all have signature (1;1) or sig- nature (2;2), andMis trace pseudoconcave. MoreoverMhas kind 3, indeed G₂ has real dimension 16 and G₃ is the whole tangent space of M. By Theorem 1.1 germs of CR distributions onMare real-analytic and extend to germs of holomorphic functions onY.

With the notation of [AMN],Mis the minimal orbit of the simple Lie group of type A II5associated to the parabolic subalgebraqˆq_fa1;a3;a5g.

EXAMPLE7.2. The example above can be generalized to the following pairs of complex flag manifolds, homogeneous for the action of SL(2n;C), and compact generic CR submanifolds, homogeneous for the action of SL(n;H):

Yˆ

`2j1 1`2j2 1`2j3 1C<sup>2n</sup>jdim`2jk 1ˆ2jk 1;1k3 ; Mˆ (`2j<sub>1</sub> 1; `2j₂ 1; `2j<sub>3</sub> 1)2Y dim (`2j_k 1\H`2j_k 1)ˆ2jk 2; 1k3

H`2jk 1`2jk‡1 1; 1k2

( )

; for 1ˆj15j25j3 n:

8>

>>

>>

<

>>

>>

>:

In this case the complex dimension ofY, the CR dimension ofM, and the CR codimension ofMare:

dim_CYˆ(2n 1)‡2(j₂ 1)(2n 2j₂‡1)‡2(j₃ j₂)(2n 2j₃‡1);

dimCRMˆ2n‡2j3 7;

codim_CRMˆ4 (n j3)(j3 1)‡(j3 j2)(j2 1)‡1 : The space of Levi forms ofMhas dimension

(2j2 3)(2j3 2j2‡1)‡(2j3 2j2 1)(2n 2j3‡2j2 1)‡

‡(2n 2j3)(2j3 2j2);

and all of them, in a suitable basis (as described in [AMN]), have all diagonal entries equal to zero. HenceMis trace pseudoconcave, and has not kind 2.

Itcan be checked thatMhas kind 3, thus Msatisfies the hypotheses of Theorem 1.1 and germs of CR distributions extend holomorphically to a full neighborhood.

With the notation of [AMN] M is the minimal orbit of the simple Lie group of t ype A II_{2n 1} associated to the parabolic subalgebra qˆq_fa1;a2j

2 1;a2j3 1g. By duality, a completely analogous statement holds for the case 1j15j25j3ˆn.

EXAMPLE 7.3. Similar to the previous one is the case of pairs of complex flag manifolds, homogeneous for the action of SL(2n;C), and compact generic CR submanifolds, homogeneous for the action of SL(n;H):

Yˆ

`<sub>2j1 1</sub>`_{2j2 1}C²ⁿ jdim`_{2jk 1}ˆ2j_k 1; kˆ1;2 ;

Mˆ (`2j1 1; `2j2 1)2Y dim (`2jk 1\H`2jk 1)ˆ2jk 2; kˆ1;2 H`2j1 1`2j2 1

; for 15j15j25n:

8>

>>

><

>>

>>

:

Also in this case,Mhas kind 3 and is trace pseudoconcave.

EXAMPLE7.4. LetG^C be the connected and simply connected simple complex Lie group of type F4, and let g^C its Lie algebra. Fix a Cartan subalgebrah^Cofg^Cand a system of simple rootsB ˆ fa_jg_1j4of the root system R ˆ R(g^C;h^C) (we use the root numbering scheme of Bourbaki, see [AMN], Appendix). Letfv_jg_1j4 be the set of fundamental weights dual to Band letV ˆV_v2 be the fundamental representation ofG^C with highestweightv₂. LetY be theG^C-orbit, inP(V), of the highest weight root space. With the notation of [AMN],Yis the complex flag manifold of G^Cconsisting of parabolic subalgebras ofg^Cthat are conjugate toq_fa2g.

InsideY consider the minimal orbitMof the real form ofG^C of type F II (also denoted byF_{4( 20)}). Itis a CR manifold of CR dimension 9 and CR codimension 11. By [AMN], Theorem 9.1, itis of finite type, and by [AMN], Theorem 13.5, it is trace pseudoconcave. Direct computation shows thatG2 has codimension 4 andG3 is the whole tangent space ofM, thusMhas kind 3. Hence also in this case we have holomorphic extension for germs of CR distributions.

Acknowledgments. The second and the fourth author would like to thank the Mathematisches Institut der Humboldt-UniversitaÈtzu Berlin, and Professor JuÈrgen Leiterer in particular, for their kind hospitality.

Furthermore we thank Anna Siano for information about the literature.

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Manoscritto pervenuto in redazione il 17 marzo 2009.