Non completely solvable systems of complex first order PDEs

Non Completely Solvable Systems of Complex First Order PDE's

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

ABSTRACT- We revisit the lack of local solvability for homogeneous vector fields with smooth complex valued coefficients, in the spirit of Nirenberg's three dimen- sional example. First we provide a short expository proof, in the case ofCR dimension one, with arbitrary CR codimension. Next we pass to Lorenzian structures with anyCRcodimension1and CRdimension2. Several dif- ferent approaches are presented. Finally we discuss the connection with the absence of the Poincare lemma and the failure of localCRembeddability, and present a global example.

MATHEMATICSSUBJECTCLASSIFICATION(2010). Primary: 35F05 Secondary: 32V05, 14M15, 17B20, 57T20

KEYWORDS. Complex vector fields,CRmanifolds

Introduction

History and motivation

Hans Lewy in [30] and Louis Niremberg in [36] gave two fundamental results in the theory of linear partial differential equations. The first showed that a non homogeneous equation for a first order partial differ- ential operator with complex valued real analytic coefficients, but C¹- smooth right hand side, may, in general, have no local weak solution. The

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

E-mail: [email protected]

(**) Indirizzo dell'A.: Dipartimento di Matematica, II UniversitaÁ di Roma ``Tor Vergata'', Via della Ricerca Scientifica, 00133 Roma (Italy).

E-mail: [email protected]

second, that a homogeneous equation for a first order partial differential with complex valued smooth coefficients may have no non constant weak local solutions. Both results were formulated and proved for partial dif- ferential operators inR³. A fuller understanding of [30] opened different directions of investigation (see e.g. [3, 4, 19, 37, 38]), especially from the two points of view of p.d.e. theory and of the analysis ofCRmanifolds.

Nirenberg's example was especially relevant to the problem of embedding CR manifold into complex manifolds. From this point of view, there have been two types of results. The Nirenberg example means that pseudoconvex three dimensional CR hypersurfaces cannot be locally CR-embedded. However the existence of sufficiently many independent solutions of the tangential Cauchy-Riemann equations was shown to hold for pseudoconvex higher dimensional CRhypersurfaces (see e.g. [1, 9, 12, 26, 27, 28]), and some general results were also ob- tained in terms of Lie algebras of vector fields (see e.g. [5, 17]). In the opposite direction, the counterexample of [36] was extended to CR hypersurfaces with degenerate or non degenerate Lorentzian signature (see e.g. [14, 15, 20, 22, 23].

The results above were all obtained for the case ofCRhypersurfaces.

For higher codimension, a crucial invariant is the scalar Levi form, which is parametrized by the characteristic codirections of the tangential Cauchy- Riemann complex. The first result in higher codimension on the absence of the Poincare lemma at the placeqwhen some non degenerate scalar Levi form hasqpositive eigenvalues was first proved in [3]. In [18] this result was extended to some cases where the scalar Levi form is allowed to de- generate. Much less is known about the CR-embedding of manifolds of higherCRcodimension. In [32] some results of [22] are extended under some supplementary conditions of the Cauchy-Riemann distribution. We also cite some partial results in [10, 11, 16].

Here we want to reconsider some of these questions, also in the more general framework of general distributions of complex vector fields of [2].

Contents of the paper

Let M be a smooth paracompact manifold of dimension m, and let L1;. . .;Lnbe smooth complex vector fields onM. In local coordinates each L_j can be written as

L_j ˆL_j(x;D)ˆX^m

iˆ1

a_j;i(x) @

@x_i; (0:1)

with coefficients aj;i which are assumed to be complex valued and C¹- smooth. We are interested in considering local solutions of the homo- geneous system

L_juˆ0; forjˆ1;. . .;n:

(0:2)

Whenn>1, since every local distribution solutionuof (0.2) also satisfies [Lj1;Lj2]uˆ(Lj1Lj2 Lj2Lj1)uˆ0; . . .; [Lj1;[Lj2;[. . .;Ljr]]]uˆ0 for all sequencej1;j2;. . .;jrwith 1j1;j2;. . .;jrn, it is not restrictive to require thatL₁;. . .;L_n satisfy the formal Cartan integrability conditions, i.e. that all commutators [L_j1;L_j2] are linear combinations, with smooth coefficients, ofL₁;. . .;L_n.

When this condition is satisfied, and L1;. . .;Ln define linearly in- dependent tangent vectors on a neighborhoodU0of a pointp02M, there are at mostm nsolutionsu₁;. . .;u_{m n}of (0.2), withdu₁(p₀);. . .;du_{m n}(p₀) linearly independent. In fact this is always the case when the L_j's have coefficients that are real analytic in some coordinate neighborhood ofp₀. Nirenberg's result in [36] shows that in general this is not true in theC¹case ifnˆ1,mˆ3. A small perturbation of a vector field for which (0.2) has two analytically independent solutions changes to a vector field for which all local solutions of (0.2) are constant.

In § 1 we show how this result extends to the case wherenˆ1, butmis allowed to be any integer larger or equal to 3. Namely, we show that the smooth complex vector fields for which (0.2) admits non locally constant solutions near some point of M form a small nowhere dense set of first Baire category in the FreÂchet space of complex vector fields onM.

This generalization of [36] was already given in [32], and our main goal is to extend in fact the results of [22] to the case of higherCRcodimension.

We show that in general, given a smooth manifoldMand any locallyCR- embeddable LorentzianCRstructure onM, and a pointp02M, there is a new LorentzianCRstructure, which is defined on a neighborhood ofp₀in M, and agrees to infinite order with the original one atp₀, which is not locallyCR-embeddable. We also show that the corresponding system (0.2) is not completely integrable in the classC¹.

In § 2 we collect the notions on CR manifolds that will be employed throughout the rest of the paper. In § 3, § 4, § 5, § 6 we prove the analog of the result of § 1 for overdetermined systems by adaptations of the argu- ments therein. The results are weaker than those obtained for a scalar p.d.e.

In fact, our constructions involve perturbations of an original system which, to keep formal integrability, employ either functions that are constant with

respect to some variables, or, in § 5, special morphisms ofCRmanifolds, and, in the more special cases of § 6.7, analytic objects, calledCR-divisors. In general, we obtain new overdetermined systems which are only defined in small coordinate neighborhoods. In § 6.8 we prove that we can globally de- fine a newCRstructure on the Lorentzian real quadricQinCPⁿwhich is not locallyCR-embeddable at all points of a hyperplane section.

In § 5 we also describe theCRcomplexes and show in § 6.2 how the technique used in the rest of the paper can be also employed to give proofs of the non validity of the Poincare lemma different from those of [3, 4, 18].

1. Homogeneous equations with no nontrivial solutions

In this section we prove a generalization to dimensions 3 of a re- markable theorem of Nirenberg about local homogeneous solutions to a single homogeneous linear partial differential equation having smooth variable complex coefficients ([36], see also [32]). Our construction is close to the one in [22, Corollary p. 238] and the argument that completes the proof of Theorem 1.1 akin to [23] (compare p. 469 there and our (1.3)). Thus the methods in these section are not original, but we found it convenient to explain the general original construction in a simpler case, before ex- tending and adapting it to discuss more general situations.

Here and in the following sections,Mwill denote a smooth paracompact real manifold of dimensionm.

We denote byX^C(M) the FreÂchet space of allC¹ complex vector fields onM. Note thatX^C(M) includes also real vector fields onM. WhenMis an open set inR^m, eachL2X^C(M) can be written as

Lˆa₁(x) @

@x1‡a₂(x) @

@x2‡ ‡a_m(x) @

@xm;

wherexˆ(x1;x2;. . .;xm), and the coefficientsaj(x)2 C¹(M) are (in gen- eral) complex valued.

THEOREM1.1. Let M be a smooth manifold of dimension m3. Then the setEof L2X^C(M)for which there exists a non empty open subset U of M, ane>0, and a solution u2 C^1‡e(U)of Luˆ0on U with du(p)6ˆ0for at least one p2U, is a nowhere dense set of first (thin) Baire category.

In other words: the set of allLonMhaving the property that any u with HoÈlder continuous first derivatives, which is a local solution toLuˆ0,

in any neighborhood of any point, must be constant, is a dense set of the second (thick) Baire category.

First we prove a Lemma.

LEMMA 1.2. Let M be a smooth Riemannian manifold of dimension m3, and let L₀2X^C(M). Then for every point p₀ 2M, h2N, ande>0 we can find L2X^C(M)with

kL L0k_h;M5e on M;

(1:1) such that

u2 C¹(U); U^open3p₀; Luˆ0on Uˆ)du(p₀)ˆ0:

(1:2)

PROOF. We can argue on a small coordinate patchVaboutp₀, and then, substitutingL₀by another vector fieldL₀sufficiently close in theh-norm, we can assume that the coefficients ofL₀are real analytic in the coordinates inV, and that

L0(p); L0(p); [L0;L0](p) are linearly independent inCTpM;8p2V:

Letkˆm 2. By the real analyticity assumption, using the Cauchy-Ko- walevski theorem, and by shrinkingVif needed, we can findk‡1 complex valued real analyticz₀;z₁;. . .;z_monVwith

L₀z_iˆ0; foriˆ0;. . .;k; dz₀^dz₀^dz₁^ ^dz_k6ˆ0 onV:

Let xiˆRezi, yiˆImzi. We can also arrange that x0;y0;x1;. . .;xk are real coordinates inV centered atp0, and that y_i(p0)ˆ0, dy_i(p0)ˆ0 for iˆ1;. . .;k. This preparation yields a local CR-embedding of V as aCR submanifold ofCRdimension 1 andCRcodimensionkinC^k‡1, given by

y_iˆh_i(z0;x) foriˆ1;. . .;k;

After multiplication by a nowhere zero function, we can take L0 of the form

L₀ˆ @

@z0‡X^k

iˆ1

a_i @

@xi; a_i2 C¹(V):

The condition that [L₀;L₀](p₀)6ˆ0 implies that@²h_i=@z₀@z₀6ˆ0 atp₀ for some index i. Moreover, we note that we obtain new solutions of the homogeneous equationL0uˆ0 by taking foruany holomorphic function of z0;z1;. . .;z_k. This allows us to use biholomorphic transformations to obtain that

the real Hessian ofh1(z0;x) is positive definite inV : ()

In this way, the setsVrˆ fp2VjImz15rg, forr>0, form a fundamental system of open neighborhoods ofp0inM. Set

M_t ˆ fp2Vjz₁ˆtg; fort2C:

Then, by (),M₀ˆ fp₀g, and there is an open connected neighborhoodvof 0 inCand a smooth real curve Imtˆf(Ret) inv, passing through 0, with the properties

Mt Vif t2v; (i)

M_t ˆ ; if Imt5f(Ret);

(ii)

M_t ˆ fa pointg if Imt ˆf(Ret);

(iii)

M_t 'S^k if Imt >f(Ret):

(iv)

LetfD_ngbe a sequence of pairwise disjoint closed discs in v^‡ˆ ft2vjImt>f(Ret)g;

with centersand radii converging to 0 forn! 1. For a suitabler0>0, all sets v^‡_r ˆ ft2vjf(Ret)5Imt5rg

are connected, for 05r5r₀. Set v⁰_rˆv^‡_r n[

n

Dn; V⁰_rˆ fp2Vjz1(p)2v⁰_rg:

Letube aC¹solution ofL0(u)ˆ0 onV⁰_r, for some 05r5r0. For eacht2v⁰_r we define

F(t)ˆ Z

Mt

u dz₀^dz₂^ ^dz_k:

We claim thatFis holomorphic inv⁰_r. Let indeedkbe an arbitrary smooth simple closed curve inv_r. Then S

t2kM_tis the boundary of a domainN_kinV, that is diffeomorphic to the Cartesian product of a 2-disc and a (k 1)-ball, andI

k

F(t)dtˆ I

k

dt Z

Mt

u dz0^dz2^ ^dz_k ˆ Z

@Nk

u dz0^dz1^dz2^ ^dz_k

ˆ Z

Nt

du^dz₀^dz₁^ ^dz_kˆ0;

because (see [29])

du^dz0^dz1^ ^dz_k ˆ(L0u)dz0^dz0^dz1^ ^dz_k ˆ0:

By Morera's theorem,F is holomorphic onv⁰_r. Moreover,Fextends to a continuous function on the closure ofv⁰_rinv\ fImt5rg, that equals 0 for Imtˆf(Ret) because of (iii). It follows thatF(t)ˆ0 fort2v⁰_r.

For eachn2N, letQnˆ fp2Vjz1(p)2Dng. We fix smooth functions c_iinV, such thatc_idz0^dz0^ ^dz_k is, for eachiˆ0;1;. . .;k, a non- negative real regular measure, with

suppc_iˆ[¹

jˆ0

Q_i‡j(k‡1) and such that, for

LˆL0‡c₀ @

@z0‡X^k

iˆ1

c_i @

@xi

we have kL L0k_h5e. Assume now that u2 C¹(Vr) satisfies Luˆ0.

Hence, for allnsufficiently large,Qn Vr, and 0ˆ

I

@Dn

dt Z

Mt

udz0^dz2^ ^dzkˆ Z

@Q_n

u dz0^dz1^ ^dzk

ˆ Z

Qn

(L0u)dz0^dz0^dz1^ ^dzkˆ Z

Qn

((L0 L)u)dz0^dz0^dz1^ ^dzk

implies, by the mean value theorem, that, for all large i2N, there are pointsp_i;p⁰_i2Q_jsuch that, for largej,

Re@u(p_j(1‡k))

@z0 ˆ0; Im@u(p⁰_j(1‡k))

@z0 ˆ0;

Re@u(p_i‡j(k‡1))

@x_i ˆ0; Im@u(p⁰_i‡j(k‡1))

@x_i ˆ0; foriˆ1;. . .;k:

8>

>>

<

>>

>:

By passing to the limit, asp_j!p₀, we obtain that

@u(p₀)

@z0 ˆ0; @u(p₀)

@xi ˆ0; foriˆ1;. . .;k:

Together withLu(p₀)ˆ0, this yieldsdu(p₀)ˆ0. p PROOF OFTHEOREM1.1. We fix a Riemannian metric onM, so that we can compute the length of vectors and covectors and the C^h-norms of

functions defined on subsets ofM. Then we have seminorms which endow X(M) with a FreÂchet space topology, and we may discuss Baire category.

LetfUng_n2Nbe a countable basis of non empty open subsets ofM, and for eachn2Nfix a pointp_n2U_n. Forh2Nwe defineE(n;h) to be the clo- sure inX^C(M) of the set ofLsuch that

9u2 C^1‡1h(U_n) with

Luˆ0 onUn; kuk_1‡¹

h;Un h;

jdu(p_n)j 1 h: 8>

><

>>

: (1:3)

The setE(n;h) has an empty interior. This can be proved by contradiction. If someE(n;h) had an interior point, by Lemma 1.2 it would contain an interior pointLsatisfying (1.2) withp₀ˆp_n. By definition, there is a sequencefL_jg_j2N withL_j!LinX^C(M) forj! 1such that for eachj, there isu_j 2 C^1‡1h(U_n) withL_ju_jˆ0 onU_n,ku_jk_1‡1

h;Un handjdu(p_n)j 1

h. By the Ascoli-ArzelaÁ theorem, passing to a subsequence we can assume thatu_j !u2 C^1‡h1(U_n), uniformly with their first derivatives on every compact neighborhood ofp_nin U_n. ThenLuˆ0 onU_n, andjdu(p_n)j 1

h>0 contradicts (1.2).

Therefore the union S

n;hE(n;h) is a countable union of closed subsets having empty interior, hence of first Baire category. Then alsoEis of first Baire category, because ES

n;hE(n;h). This completes the proof of the

Theorem. p

2. Involutive systems andCRmanifolds

To extend the result of § 1 to overdetermined systems of homogeneous first order p.d.e.'s, we will develop ideas from [14, 15, 20]. In this section we begin by describing the general framework. In the following,Mwill denote aC¹-smooth manifold of real dimensionm.

2.1 ±Generalized complex distributions and CR structures

Letzbe a generalized distribution of smooth complex vector fields onM.

This means thatzdefines, for each open subsetUofMa,C¹(U) submodule ofX^C(U), in such a way that the assignmentU!z(U) is a sheaf:

(1) IfU^openV^openM, thenZj_U 2z(U) for allZ2z(V);

(2) IffU_ngis a family of open subsets ofM, a smooth complex vector

fieldZ, defined on S

n Un, belongs toz S

n Un

if and only ifZj_Un2z(Un) for alln.

Our main interest in the sequel will be focused on thelocalsolutions to the homogeneous system

Zuˆ0; 8Z2z:

(2:1)

It is therefore natural to assume in the following that zisinvolutive, or formally integrable. This means that

[Z₁;Z₂]2z(U); 8Z₁;Z₂2z(U); 8U^openM:

(2:2)

Sincezis a fine sheaf, every germZ_(p) ofzat a pointp2M is the re- striction of a global Z2z(M). Thus we can for simplicity utilize global sectionsZ2z(M) in most of the discussion below.

For each pointp2M, we consider the set

ZpMˆ fZ(p)jZ2z(M)g CTpM:

(2:3)

If the dimension of theC-linear spaceZpMis constant, we say thatzis a distributionof complex vector fields.

DEFINITION2.1. A CRstructureon M is the datum of an involutive distributionzof smooth complex vector fields withz\zˆ0.

The constant dimension n ofZpM is its CR dimension, and kˆm 2n its CR codimension. We call the pair(n;k)the type of the CR manifold M.

In the case wherezis aCRstructure onM, we write sometimesT^0;1M forZM.

When M is a real smooth submanifold of a complex manifold X, we consider onMthe generalized distribution

z(U)ˆ fZ2X^C(U)jZp2T^0;1_p X; 8p2Ug;

whereT^0;1Xis the bundle of anti-holomorphic complex tangent vectors toX.

Thenzis involutive andz\zˆ0. Whenzhas constant rank,zdefines aCRstructure onM, for which we say thatMis aCR-submanifoldofX.

Let 1a 1. A complex CR-immersion of class C^a of M is a C^a- smooth immersion f:M!X of M into a complex manifold X with df(Z_pM)T^0;1_f(p)Xfor allp2U.

For any open setUofMwe set

oM(U)ˆ fu2 C¹(U)jZuˆ0; 8Z2z(M)g:

(2:4)

The assignmentU^open!oM(U) defines a sheaf of rings of germs of com- plex valued differentiable functions onM.

2.2 ±The differential ideal and complete integrability Let V_Mˆ L

0pmV^p_M be the sheaf of germs of alternated smooth dif- ferential forms onM. We associate tozthedifferential ideal

J_MˆM

p1

J^p_M with J^p_Mˆ fh2V^p_Mjhj_z(M)ˆ0g:

(2:5)

This is a graded ideal sheaf ofV_M, generated by its elements of degree 1.

Being interested in the local solutions to (2.1), we can assume thatJ_Mis completeand thatzis thecharacteristic systemofJ_M, i.e. that

z(U)ˆ fZ2X^C(M)jZcJ_M(U) J_M(U)g

ˆ fZ2X^C(U)jh(Z)ˆ0; 8h2 J¹_M(U)g; 8U^openM:

Ifzis a distribution, it is the characteristic system of its differential ideal.

The pointwise evaluation of the elements ofJ¹_Myields in this case a smooth subbundleZ⁰MofCTM, given by

(2:6) Z⁰MˆG

p2M

Z⁰_pM; with Z⁰_pMˆ fh2CT_pMjh(Z)ˆ08Z2z(M)g:

In general, (2.6) defines a subset of the complexified tangent bundle ofM.

DEFINITION2.2. Let zbe a generalized distribution of smooth com- plex vector fields on M and p₀2M. We say thatzis completely integrable at p0if

8h2Z⁰_p0M 9u2oM(p0) with du(p0)ˆh:

(2:7)

This means that (2.1) has at p0 the largest number of differentially independent local solutions that is permitted by the rank ofz.

2.3 ±The case of CR manifolds

Letzbe aCRstructure of type (n;k) onM. Complete integrability at p₀2Mis equivalent to the existence of a complexCR-immersion of class C¹ of an open neighborhoodUofp₀intoC^n‡k.

The question of the regularity of complex CR-immersions seems in general a rather delicate open problem (see e.g. [31]). Note that anyC¹- immersion is in factC¹-smooth whenMsatisfies suitable pseudo-concavity assumptions (see [2]).

For C¹-smooth complex local CR-immersions we introduce a special notation.

DEFINITION2.3. A CR-chart on M is the datum of an open subset U and of n‡k smooth CR functions z₁;. . .;z_n‡k2o_M(U)\ C¹(U), such that

dz₁(p)^ ^dz_n‡k(p)6ˆ0; 8p2U:

Clearly f(p)ˆ(z1(p);. . .;z_n‡k(p)) provides in this case a C¹-smooth CR-immersion ofUinC^n‡k.

The functionsz₁;. . .;z_n‡k of aCR-chart are not independent complex coordinates when k>0. For each pointp₀ of U there are indeedk real valued functionsr₁;. . .;r_k, defined andC¹on an open neighborhoodGof f(p0) in C^n‡k, with r_i(z1;. . .;z_n‡k)ˆ0 on a neighborhood of p0, and

@r₁(f(p₀))^ ^@r_k(f(p₀))6ˆ0.

DEFINITION2.4. We say that a CR manifold M is locally CR-embed- dable if the open subsets U of its CR-charts make a covering.

Locally CR-embeddable CR manifolds can be abstractly defined as ringed spaces, using the structure sheafo¹_M ˆoM\c¹of the germs of its smoothCRfunctions.

LEMMA2.5. Let M be a CR manifold of type(n;k)and p₀2M. Then we can find an open neighborhood U of p0in M and a newCR structure on M which is locally CR-embeddable and agrees to infinite order with the original one at p0.

PROOF. L etzbe theCRstructure onM. It suffices to consider smooth functions z1;. . .;zn which are defined on a neighborhood of p0, satisfy Zz_jˆ0¹atp0, and havedz1(p0)^ ^dzn(p0)6ˆ0. To prove the existence of such functions, we observe that it is always possible to find a smooth coordinate chart (U;x₁;. . .;x_m) centered atp₀such thatzis generated in Uby vector fields of the form

Z_iˆ @

@xi‡i @

@xi‡n‡ X^m

jˆn‡1

a_j(x) @

@xj; witha_j(x)ˆO(jxj):

Let Liˆ @

@x_i‡i @

@x_i‡n, andRiˆ P^m

jˆn‡1aj(x) @

@x_j.

We denote bymthe maximal ideal of the local ringCffx1;. . .;xmggof formal power series ofx1;. . .;xm. We obtain formal power series solution to (2.1) by constructing by recurrence sequencesff_hg_h0Cffx1;. . .;xmgg which solve the equations

f_h2m^h;

Ljf12m; forjˆ1;. . .;n;

L_jf_h‡1‡R_jf_h2m^h‡1; forjˆ1;. . .;n:

8>

>>

<

>>

>: ()

We observe that, takingf1equal toxi‡ixi‡n foriˆ1;. . .;n, or tox2n‡i, for iˆ1;. . .;k, we obtain n independent solutions of L_if1ˆ0 for 1in.

Assume now thatd1 andf_d2m^dsatisfies L_if_d‡R_if_{d 1}2m^d; for 1in:

The integrability conditions yield [Z_i;Z_j]ˆ0 for 1i;jn. Hence we obtain

0ˆ[Z_i;Z_j]f_dˆ L_iR_jf_d‡L_jR_if_d‡[R_i;R_j]f_d: ()

We haveRifd2m^d, and hence there is a polynomialgi;d2C[x1;. . .;xm], homogeneous of degree d, such that Rifd gi;d2m^d‡1. Since [R_i;R_j]f_d2m^d‡1, we obtain from () that L_ig_j;dˆL_jg_i;d for all 1i;jn and therefore there is a polynomial f_d‡12C[x₁;. . .;x_m], homogeneous of degree d‡1, such that L_if_d‡1ˆg_i;d for iˆ1;. . .;n.

The series P

fd of the terms of a sequence ffdg solving () is a formal power series solution of (2.1).

In particular, we can find solutionsfz1g;. . .;fzng 2Cffx1;. . .;xmggto (2.1) with dfz_ig(0)ˆdx_i(0)‡idx_i‡n(0) for iˆ1;. . .;n and dfz_ig(0)ˆ dx_n‡i(0) for iˆn‡1;. . .;n. It suffices then to take smooth functions z₁;. . .;z_nhaving Taylor seriesfz₁g;. . .;fz_ngat 0. p

2.4 ±Characteristic bundle and Levi form[33]

The underlying real distribution and the characteristic bundle ofzare:

H ˆ fReZjZ2zg; i:e H(U)ˆ fReZjZ2z(U)g; 8U^openM;

(2:8)

H⁰Mˆ fj2TMjj(X)ˆ0; 8X2 H(M)g:

(2:9)

To each characteristic covectorj₀2H⁰_p0Mwe associate a Hermitian sym- metric form onZp0M, by

L_j0(Z1;Z2)ˆij₀([Z1;Z2]); 8Z1;Z22z(M):

(2:10)

In fact a straightforward verification shows that the value of the right hand side of (2.10) only depends onZ₁(p₀);Z₂(p₀)2Z_p0M.

Moreover,Lj0(Z1;Z2)ˆ0 if one of the two vector fields is real valued on a neighborhood ofp0. ThusLj0defines a Hermitian symmetric form on the quotient of Zp0M by the subspace Np0Mˆ fZ(p0)jZ2z(M)\z(M)g, consisting of the values at p₀ of the complex multiples of the real vector fields inz(M). Set

Z_p0MˆZ_p0M=N_p0M:

(2:11)

Ifj₀2H⁰_p0M, then (2.10) defines a Hermitian symmetric formL_j0onZ_p0M, that we call theLevi formofzatj₀.

DEFINITION 2.6. Let p₀2M and j₀2H⁰_p0M. We say that z is q- pseudoconvex at j₀ ifL_j0 is nondegenerate and has exactly q positive eigenvalues on Zp0M.

Ifzis1-pseudoconvex at somej₀2H⁰_p0M, we say thatzisLorentzian at p0.

Ifz(M) is generated by a single vector fieldLnearp₀, the condition of being Lorentzian atp0means thatL(p0),L( p0), and [L;L]( p0) are linearly independent inCTp0M.

2.5 ±Reduction of complete integrability to the case of CR manifolds When NpM has constant dimension on a neighborhoodU ofp02M, then the real vector fields inz(U) define an involutive distributionnof real vector fields on U. By the Frobenius theorem, there is an open neighborhood Wof p₀ inUand a smooth fibrationp:W!Bof W such thatBis a smooth manifold and the fibers ofpare integral submanifolds of n. One easily proves

LEMMA2.7. There is a CR structurez⁰on B such that for every p2W we haveo_M;(p) ˆpo_B;(p(p)), andzis completely integrable at p2W if and only ifz⁰is completely integrable atp(p).

3. Involutive systems which are not completely integrable atp₀ In this section, we give a weak generalization of the results of § 1 to Lorentzian CR manifolds M with arbitrary CR-codimension k1 and CR-dimension n2. We recall that mˆdim_RMˆ2n‡k, and we set nˆn‡k.

We closely follow the arguments of § 1.

Assume thatMis locallyCR-embeddable and Lorentzian atp0. Then there is a real valued functionr, defined on a full generic complex neigh- borhood ofp₀, and vanishing onM, whose complex Hessian has signature (1;n 1) onH_p0M. By adding a real valued functionc, vanishing to the second order onM, we obtain a new functionr‡c, vanishing onM, with

@@(r ‡c)ˆ@@r onHp0M, while the signature of the full complex Hessian ofr‡chas been arbitrarily adjusted in the transversal directions toM.

Then there is aCR-chart (U;z₁;. . .;z_n) centered atp₀, withdzⁱ(p₀) real for iˆn‡1;. . .;n, and

Imz_n‡z_nz_n‡X^{n 1}

iˆ1

z_jz_jˆX^{n 1}

iˆn

z_jz_j‡O(jzj³) onU:

(3:1)

By shrinking, we getⁿP¹

iˆnzizi1

2ImznonU.

We consider the mapp:U3p!wˆ(z₁(p);. . .;z_{n 1}(p);z_n(p))2Cⁿ. By a further shrinking, we can assume that there is an open ballBCⁿ, centered at 0, such that

- p(U)ˆv, with Bnvstrictly convex, and@v\Bsmooth;

- if Imt0, thenfw2BjImw_nˆtg v;

- for allw2vthe setM_wˆp ¹(w) is diffeomorphic to the sphereS^k; - forw2@v\Bthe setMw ˆp ¹(w) is a point.

As in § 1, we have:

LEMMA3.1. If u2oM(U), then F(w)ˆ

Z

Mw

u dzn^ ^dzn 1ˆ0; 8w2v:

(3:2)

PROOF. We prove first thatFis holomorphic onv.

Fix any polycylinder DˆD1 Dn in v, with D_iˆ ft2Cj jt t_ij e_ig. For 1jn we set@_j(D)ˆ fw2Dj jw_j t_jj ˆe_jg,

g_j ˆ @

@w_jc(dw1^ ^dwn) and consider the integral I

@jD

F(w)dw1^ ^dwn^g_jˆ I

@jD

dw1^ ^dwn^g_j Z

Mw

u dzn^ ^dzn 1:

LetNiˆp ¹(@iD) andNˆp ¹(D). We have@NˆPⁿ

iˆ1Ni. Moreover, the formu dz1^ ^dzn^pg_jis zero onN_ifori6ˆj. Thus we obtain:

I

@jD

F(w)dw₁^ ^dw_n^g_jˆ Z

Nj

u dz₁^ ^dz_n^pg_j

ˆXⁿ

iˆ1

Z

Ni

u dz1^ dzn^pg_jˆ Z

N

du^dz1^ dzn^pg_jˆ0

because du2 hdz1;. . .;dzni by the assumption that u2oM(U). This equality, valid for all closed polycylinderDinvand all 1jn, implies that Fis holomorphic inv. ClearlyF(w)!0 when w!@v\B, because M_w0 is a point for w₀2@v\B, and hence Fˆ0 on v by Holmgren's uniqueness theorem, since@has constant coefficients inCⁿ. p

Letcbe a smooth function with compact support inC, and set

^ c(t)ˆ 1

2pi

Z Z c(z)dz^dz z t : Then@^c

@t ˆcand therefore

c^](z;t)ˆzc(t)dt‡^c(t)dzˆ@(zc(t))^ is a@-closed form in C², with

dc^]ˆ@c(t)^

@t dt^dz‡z@c(t)

@t dt^dtˆdt^ @

@tc^]:

LEMMA3.2. Let U⁰U. Ifc_i, for iˆ1;. . .;n 1, are smooth functions of a complex variablet, withjc_ijsufficiently small. Then

u1ˆdz1‡c^]₁(zn;zn); . . .; un 1ˆdzn 1‡c^]_{n 1}(zn;zn); unˆdzn

(3:3)

generate the involutive ideal sheafJ⁰_Mof a CR structure of type(n;k)on U⁰. PROOF. The ideal sheaf is generated on U by dz₁;. . .;dz_n. After shrinking, we can assume that dz₁;. . .;dz_n, dz₁;. . .;dz_n are linearly independent onU.

Thus, by takingjc_ijsufficiently small, we may keepu1;. . .;un;u1;. . .;un

linearly independent in any neighborhoodU⁰ofp0withU⁰U. Moreover, sincedc^]_i(zn;zn)^dznˆ0, for 1i5n, we obtain

(du_i)^u1^ ^unˆdc^]_i(zn;zn)^u1^ ^un 1^dznˆ0; 8iˆ1;. . .;n 1:

This shows that the ideal sheafJ⁰_Mgenerated byu1;. . .;unis involutive and defines aCRstructure of type (n;k) on U⁰. p

Let us fix a sequence of distinct complex numbersft_jg, such that Imt_j>0 for allj; t_j !0; fwnˆt_jg \v6ˆ ;for all j:

For eachjwe choose an open diskD_j inC, centered att_j, and such that Dj\S

i6ˆj

Diˆ ;. Provided thetj's are sufficiently close to 0, for eachjwe can fix a pointw^(j)2v, withw⁽_n^j) ˆt_j, andw^(j)!0, and take the functionsc_jin Lemma 3.2 in such a way that

suppc_iˆ[¹

jˆ0

D_i‡jn; for iˆ1;. . .;n;

c_i‡jnˆ Z

Ai‡jn

c^]_i(z_n;z_n)^dz_n^ ^dz_{n 1}^dz_n is real and > 0;

where (e1;. . .;enis the canonical basis ofCⁿ)

A_jˆp ¹(fw^(j)‡(t t_j)e_njt2D_jg):

Let u be aCR function on an open neighborhood V of p₀ in U for the structure defined by (3.3). This means that du_(p)2 J⁰_M(p) for all p2V.

SinceJMandJ⁰_Magree to infinite order outsideS

j p ¹(fwjwn2Djg, and S

jfw2vjw_n2D_jgdoes not disconnectv, by the argument of Lemma 3.1 we have (4.2) for all w in the complement inp(V)nS

jfw2vjw_n2D_jg.

Thus we obtain 0ˆ

I

t2@Dj

dt Z

M_w(j)‡ten

udz_n^ ^dz_{n 1}ˆ Z

@Aj

u dz_n^ ^dz_n

ˆ Z

Aj

du^dz_n^ ^dz_n:

This yields

Z

Ai‡jn

@u

@zi c^]_i^dz_n^ ^dz_nˆ0;

where, to compute @u

@zi, we consider anyC¹-extension ofuas a function of the complex variablesz₁;. . .;z_nfor which@u ˆ0 at all points ofU. Taking the limit, we observe that

c_i‡jn¹ Z

Ai‡jn

@u

@zi c^]_i^dz_n^ ^dz_n !@u(p₀)

@zi ˆ)@u(p₀)

@zi ˆ0 8iˆ1;. . .;n 1;

which, together with (2.1) shows thatdu(p₀)2Cdz_n(p₀).

We have proved:

THEOREM3.3. Let M be a CR manifold of type(n;k)and assume that M is Lorentzian at a point p₀. Then we can find a new CR structure of type(n;k)on a neighborhood U of p₀, which agrees with the original one to infinite order at p0, and a real codirectionh₀2T_p0M such that, ifzis the distribution of(0;1)-vector fields for this newstructure, all solutions u2 C¹ on a neighborhood of p₀ to the homogeneous system (2.1)satisfy du(p₀)2Ch₀.

PROOF. Indeed, using Lemma 2.5 we can always reduce to the case in

whichMis locally embeddable atp₀. p

COROLLARY3.4. We can find a newCRstructure of type(n; k)onU, which agrees with the original one to infinite order atp0, and which is not CR-embeddable atp0.

4. Involutive systems whose solutions are critical atp₀

In this section we improve the result of the previous section in the case of a LorentzianCRmanifold of the hypersurface type.

A proof of Theorem 4.3 is contained in [24], where, however, the result is not correctly stated (see Remark 4.4 below). However, our proof here is different and outlines, in the simpler case of the hypersurface-type, the argument used later for higherCRcodimension.

We assume that M has CR-dimension n2 and CR-codimension 1, and is Lorentzian and locally embeddable at p02M. We have mˆdim_RMˆn‡2 and we setnˆn‡1,

We can fix aCR-chart (U;z₁;. . .;z_n) centered atp₀, with Imzn‡Xⁿ

iˆ2

ziziˆz1z1‡O(jzj³) onU:

(4:1)

By shrinking, we get that z1z11

2Imzn on U. Consider the map p:U3p!wˆ(z2(p);. . .;zn(p))2Cⁿ. By a further shrinking, we can assume that there is an open ballBCⁿ, centered at 0, such that

- p(U)ˆv, with Bnvstrictly convex, and@v\Bsmooth;

- if Imt0, thenfw2BjImwnˆtg v;

- for allw2vthe setMwˆp ¹(w) is diffeomorphic to the circleS¹; - forw2@v\Bthe setM_w ˆp ¹(w) is a point.

By repeating the proof of Lemma 3.1, we obtain LEMMA4.1. If u2o_M(U), then

F(w)ˆ I

Mw

u dz₁ˆ0; 8w2v:

(4:2) p

Since 2z₁z₁Imz_n on U, for any smooth function c of a complex variable t, with suppc fImt0g, the function z₁¹c(z_n) can be ex- tended to a smooth function on U, vanishing to infinite order on fz1ˆ0g \U.

LEMMA 4.2. If c_i, for iˆ1;. . .;n are smooth fuctions of a complex variablet, with support contained infImt0g, then

u1ˆdz1‡z₁¹c₁(zn)dzn; . . .; unˆdzn‡z₁¹c_n(zn)dzn

(4:3)

(the functions z₁¹c_i(zn)are putˆ0for z1ˆ0) generate the ideal sheafJ⁰_U0

of a CR structure of type(n;1)in a neighborhood U⁰of p0in U, which agrees to infinite order with the original one at p₀.

PROOF. By the condition on the supports, the functionsz₁¹c_i(z_n) are smooth onUand vanish to infinite forz1ˆ0, and in particular atp0. Thus u1;. . .;un,u1;. . .;unyield a basis ofCTpMforpin a suitable neighborhood U⁰ofp₀, and agree withdz₁;. . .;dzn;dz₁;. . .;dznto infinite order atp₀.

We have moreover

duiˆz₁¹@c_i(z_n)

@zn dzn^dzn z₁²c_i(zn)dz1^dzn: Hence

du_i^u₁^ ^unˆdu_i^dz₁^ ^dznˆ0

shows thatJ⁰_U0is involutive. The proof is complete. p Let us fix a sequence of distinct complex numbersft_jg, such that

Imt_j>0 for allj; t_j!0; fwnˆt_jg \v6ˆ ; for allj:

For eachjwe choose an open diskD_j in C, centered att_j, and such that D_j\S

i6ˆj

D_iˆ ;. Provided thet_j's are sufficiently close to 0, for eachjwe can fix a pointw^(j)2v, withw⁽n^j)ˆtj, andw^(j)!0, and take the functionsc_jin Lemma 4.2 in such a way that

suppc_iˆ[¹

jˆ0

D_i‡j(n‡1); for iˆ1;. . .;n;

c_i‡j(n‡1) ˆ Z

Ai‡j(n‡1)

z₁¹c_i(z_n)dz_n^dz₁^dz_n is real and > 0;

where

Aj ˆp ¹(fw^(j)‡(t tj)enjt2Djg):

Here we denoted bye1;. . .;enthe canonical basis ofCⁿ.

Letube aCRfunction on an open neighborhoodV ofp₀ inU⁰for the structure defined by (5.2). This means that du_(p)2 J⁰_U0(p) for all p2V.

SinceJ_MandJ⁰_U0agree to infinite order onp(U⁰) outsideS

i suppc_i(w), and this set does not disconnectU, by the argument of Lemma 3.1 we have (4.2) for allwin the complement inp(V) ofS

jfw2vjwn2Djg. Thus we obtain 0ˆ

I

t2@Dj

dt I

M_w(j)‡ten

u dz_nˆ Z

@Aj

u dz₁^dz_nˆ Z

Aj

du^dz₁^dz_n:

This yields

I_i‡j(n‡1)(u)ˆ Z

Ai‡j(n‡1)

@u

@ziz₁¹^dz_n^dz₁^dz_nˆ0;

(4:4)

where, to compute@u

@z_i, we consider anyC¹-extension ofuas a function of the complex variablesz1;. . .;znfor which@u ˆ0 at all points ofV. When j! 1, c_i‡j(n‡1)¹ I_i‡j(n‡1) !@u(p0)

@z_i . Hence, from (4.4) we obtain that

@u(p0)

@z_i ˆ0 for 1in, which, together with (2.1) shows that du(p0)ˆ0.

We have proved:

THEOREM4.3. If M is a CR manifold of type (n;1) and is Lor- entzian at p02M, then we can find a new CR structure of type (n;1) on an open neighborhood U of p₀ in M, which agrees with the original one to infinite order at p₀, such that, ifz is the distribution of (0;1)-vector fields for this newstructure, all solutions u2 C¹ on a neighborhood of p0 to the homogeneous system (2.1) satisfy du(p0)ˆ0.

PROOF. We can apply the discussion above after reducing, by Lemma 2.5, to the case in whichMis locallyCR-embeddable atp0. p REMARK4.4. The statement of Theorem 4.3 is weaker than that of Theorems I, II in [24], in which the perturbedCRstructure is claimed to exist globally. However, the final part of the argument there has a serious gap. Let us explain in what it consists. Let L1;. . .;Ln and L1;. . .;Ln be basis for the original and the perturbed structure, re- spectively, defined on a coordinate neighborhood (U;x), centered at a chosen point p2M. In [24, p. 290, (4.10)] new global structures are defined, which agree with the original one outside ofU, and having, in U, the basis L⁽ⁿ⁾_j ˆgnLj‡(1 gn)Lj, jˆ1;. . .;n. Here gn(x)ˆg(x=rn) for a scalar function g, having a star-shaped compact support in x(U), equal to 1 on a neighborhood of 0, and 05r_n51, r_n&0. The flaw consists in the fact that these new structures, which are perturbations with arbitrarily small compact support of the given one, fail to be formally integrable in 05gn51.

We think instead that, when n>1, there are geometrical ob- structions to the existence of perturbations with arbitrarily small compact supports and that in fact propagation should be a foremost important topic in further investigations on the deformations of CR structures.

5. The case of higher codimension

In this section we extend the result of Theorem 4.3 to someCRmani- folds withCRdimension andCRcodimension both greater than 1. To this aim we will first recall some results on weak unique continuation and next consider morphisms ofCRmanifolds.

5.1 ±Minimal locally CR-embeddable CR manifolds and unique contin- uation

We recall that aCRsubmanifoldMisminimalatp02Mif there is no germ (N;p0) ofCRsubmanifold ofMatp0, having the sameCRdimension, but smallerCRcodimension. We have

LEMMA5.1. Assume that M is minimal and locally CR-embeddable at p02M.

Let (S;p0)be a germ of a CR submanifold of M, of type(0;n). Then a germ f 2o_M;(p0)of a CR function at p0, vanishing on(S;p0), is equal to0.

If M is minimal and locally CR-embeddable at all points, then the CR functions on M satisfy the weak unique continuation principle.

PROOF. In the first part of the proof, we can assume thatMis a generic CRsubmanifold of an open set inCⁿ. For any open neighborhoodUofp₀in M, there are an open neighborhoodU₀ofp₀inU, and an open wedgeWin Cⁿ, with edgeU₀, such that, the restrictionuj_U0 of anyu2o_M(U) is the boundary value of a holomorphic function~u, defined onW(see [43, 44, 6]).

Assume now thatu2oM(U) vanishes onS. Thenu~ˆ0 by the edge of the wedge theorem (see [39]), and thereforeuˆ0.

The last statement follows by unique continuation for holomorphic

functions on open subsets ofCⁿ. p

5.2 ±CR-maps with simple singularities

Let M;N be CR manifolds. A smooth map p:M!N is CR if dp(T^0;1M)T^0;1N. We say thatpis

- aCR-immersion if kerdpˆ0 anddp(T^0;1M)ˆdp(CTM)\T^0;1N;

- a CR-submersion if dp(T_pM)ˆT_p(p)N and dp(T_p^0;1M)ˆT^0;1_p(p)N, 8p2M;

- a localCR-diffeomorphism if it is at the same time aCR-immersion and aCR-submersion.

Next we consider critical points of someCR-maps.

Letk1 andp:M!NaCR-map, withMof type (n;k) andNof type (n;k 1).

Ifp₀2Mis not a critical point ofp, thenpis aCR-submersion nearp₀. Assume now thatp₀ is a critical point ofp, andq₀ˆp(p₀) the corre- sponding critical value. Then the rank ofdp(p₀) is less than 2n‡k 1.

Assume that it is exactly equal to 2n‡k 2. Then the dual map dp(p0):T_q0N!T_p0M is not injective, and has a 1-dimensional kernel.

DEFINITION5.2. If kerdp(p₀)\H⁰_q0Nˆ f0g, we say that phas at p₀ a CR-noncharacteristic singularity.

Assume that this is the case and fix 06ˆh₀2kerdp(p₀). Then there is h⁰₀, uniquely determined moduloH⁰_q0N, such thath₀‡ih⁰₀2Z⁰_q0N, and we obtain an elementj₀2H⁰_p0M, with 06ˆj₀ˆdp(p₀)(h⁰₀).

DEFINITION5.3. If we can chooseh⁰₀in such a way thatLj₀has1positive and n 1 negative eigenvalues, we say thatp has aLorentzianCR-non characteristic singularityat p₀.

Assume now that M and N are locally CR-embeddable at p0, q0, respectively, and thatLj₀has 1 positive and (n 1) negative eigenvalues.

We setnˆn‡k. We can chooseCR-charts (U;z₁;. . .;z_n) ofM, centered at p₀, and (W;w₂;. . .;w_n) of N, centered at q₀, with p(U)W and zjˆpwj forjˆ2;. . .;n, such thatij₀ˆdzn(p0), and

Imz_nˆh(z) on U; withh(z)ˆz₁z₁ Xⁿ

iˆ2

z_iz_i‡O(jzj³):

LEMMA 5.4. Let Dˆ fp2Ujz1(p)ˆ0g. Then, there is an open neighborhood U⁰ of p0 in U, an open neighborhood v of q0 in N, and an open domain v in v, w ith q02@v , such that

(1) v p(U⁰)v,p(D\U⁰)@vandvis strictly pseudoconcave at q0;

(2) p:U⁰!N is proper and, for q2p(U⁰),p ¹(q)is either a point or is diffeomorphic to a circle.

PROOF. ProvidedU is sufficiently small, the restriction ofptoDis a smooth diffeomorphism of D onto a closed hypersurface p(D) in an

open neighborhood v of q0 inN. By further shrinking, we can assume thatAnp(D) consists of two connected componentsv‡ andv and that v p(U).

Sincev ˆ fImw_n‡Pⁿ

iˆ2w_iw_i‡O(jwj³)>0gnearq₀, we haveq₀2@v andv strictly pseudoconcave atq₀. Moreover, by takingUsmall, we can assume that

Imz_n‡3 2

Xⁿ

iˆ2

z_iz_i1

2z₁z₁ onU;

and therefore we obtain an U⁰ satisfying (1) and (2) by setting U⁰ˆU\p ¹(v) for a smaller neighborhoodvofq0inN. p

5.3 ±Perturbation of the CR structure of M

We keep the notation of § 5.2, and we shall assume that (1) and (2) of Lemma 5.4 hold true withU⁰ˆU.

LEMMA5.5. Assume that N is a minimal CR manifold. If u is a CR function on a connected open neighborhood V of p₀in U, then

g(q):ˆ

I

p¹(q)

u dz1ˆ0; 8q2p(V):

(5:1)

PROOF. First we note thatWˆp(V)[(vnp(U)) is a neighborhood of q₀inN. The functiong, equal to the left hand side of (5.1) forw2p(V) and 0 onWnp(V) is continuous, because the fiberp ¹(q) shrinks to a point when q!@p(V)\v. Sincep(V) is connected and its connected component inW contains an open subset where gˆ0, our contents follows by the weak unique continuation principle (see Lemma 5.1) if we show that gis aCR function onW. To this aim, it suffices to show that

Z

N

dg^hˆ0; 8h2V^2n‡k₀ ²(W)\ J^n‡k_N ¹(W);

whereV₀(W) means smooth exterior forms with compact support inW. We note that ph2V^2n‡k₀ ²(V)\ J^n‡k_M ¹(W), because the map p is CR and proper. Thus we obtain

Z

N

dg^hˆ Z

M

du^dz₁^phˆ0;