A NNALI DELLA
S CUOLA N ORMALE S UPERIORE DI P ISA Classe di Scienze
O LLE S TORMARK
On the theorem of Frobenius for complex vector fields
Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 4
esérie, tome 9, n
o1 (1982), p. 57-90
<http://www.numdam.org/item?id=ASNSP_1982_4_9_1_57_0>
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Complex
OLLE STORMARK
Generalizing
some results ofLewy,
Andreotti and Hillproved
in[1]
that any
system
of first orderhomogeneous
linearpartial
differential equa- tions withcomplex
valued realanalytic
coefficientslocally
may be reduced to a realpart, consisting
of the de Rhamsystem,
and acomplex part,
whichconsists of the
tangential Cauchy-Riemann equations aM
on ageneric locally
closed real
analytic
submanifold .lVl of some Cn. Then thequestion
ariseswhether the
j.-equations
can besimplified
in theirturn,
for instance in such a way thatthey
arecomposed
ofequations
ofCauchy-Riemann
andLewy type. Geometrically
this would mean that ~f admitscomplex
folia-tions and
Lewy
foliations. In this paper wepresent
a method forfinding
such and related
foliations,
which is based on Vessiot’stheory
for vectorfield
systems.
1. - Some theorems of Frobenius
type.
Let
be m real
linearly independent
vector fields on an open set U inRn,
andlet Y be the vector field
system generated by
these. If we areonly
interestedin solutions
... , xn)
of thesystem
we may as well assume
(by maybe having
added same brackets andthereby
increased
m)
that Y iscomplete, i.e.,
closed under Lie brackets.Pervenuto alla Redazione il 14 Novembre 1980.
Then the classical theorem of Frobenius says that it is
possible
to find localcoordinates
(xi ,
...,~)
such that the vector fieldsform a basis for Y.
With these
coordinates, (2)
goes over intoi.e., u(x’ )
is a function of alone.Or,
in otherwords,
to(1)
there is associated a local foliation with the leavesx~ + 1 = cm + 1, ... , xn = cn (where
the ci ,i = m -E- 1, ... , n,
are con-stants),
such that the solutions of(2)
are constantalong
theleaves,
butmay vary
arbitrarily
from one leaf to another. In such a situation we say that the variables ..., areprincipal,
while~+i?
...,x~
areparametric.
Another way of
stating
the Frobenius theorem(in
a way so that it canbe
generalized
to thecomplex case)
isby saying
that(1) locally
admitsn-m
functionally independent
invariants~m+1, ..., ~n,
i.e. functionssatisfying (2)
and the conditionNamely,
if one intro-duces the
$i
and m otherfunctions tj,
such that d iA... n dtm n d,+1 n ... / dn 0,
as new
coordinates,
then theXk
areexpressed by
means of the derivations8j8t; only,
and Gauss eliminationyields (3).
Thusthe ~i , i
= m+ 1,...,
11, appear asparametric
variables.In this paper we want to consider what
happens
when the vector fieldsin
(1)
are allowed to becomplex
valued. To see what difference thismakes,
we first look at the case of
just
one vectorfield,
sayP(x)
=P’(x) + iP"(x),
where P’ and
P"
are real vector fields defined on U c R’. If P’ andP"
arelinearly independent
inU,
P attaches to eachpoint x
E U two differentdirections
(instead
of one, as in the realcase),
whichtogether
span a 2-dimen- sionalplane
inTx( U) (where
denotes the realtangent
space of a(real
orcomplex)
manifold ~ at thepoint x).
In the most favourable casethese
planes
fittogether
to form a foliation of U with 2-dimensional leaves.By
the real Frobenius theorem this occurs if andonly
if the vector fieldsystem T spanned by
P’ andP"
iscomplete. Moreover,
it mayhappen
that these leaves can be
given
the structure ofcomplex
1-dimensionalmanifolds,
so that P appears as theCauchy-Riemann equations
on eachleaf. Then the solutions of Pu = 0 are
holomorphic along
all leaves(instead
ofbeing constant,
as in the realcase),
but arearbitrary
in theparametric
variables.However,
if J is notcomplete,
it is natural to consider the vector field;system 5" generated by P’, P’~
and[P’, P"].
If this iscomplete,
onegets
a foliation of U with 3-dimensional
leaves,
so that P actsonly
within each leaf. Then it mayhappen
that each leaf can be imbedded in acomplex
2-dimensional manifold in such a way that the restriction to any leaf of a solution
u(x)
to Pu = 0 is theboundary
value of aholomorphic
function(in
twocomplex variables).
And if 5" is not
complete,
one adds more Lie brackets until acomplete system
isobtained, whereupon
the Frobenius theoremgives
a foliation.Let us now consider
systems
ofcomplex
valued C°° vector fields-
P’(x) +
k =1,
... , m, whereP’
andPk
are real. Thesimplest (nonreal)
case that occurs is handledby Nirenberg’s complex
Frobeniustheorem
(see [10]) :
THEOREM 1. Let 5’ be the vector
field system generated by P,, ... , Pm
onan open
neighbourhood of
0 inltn,
be the vectorfield system generated by P1, ... , Pm .
Assume that T has constant rank m and has constant rank m+
r(where rmin (m, n - m)) . Then, if
and
it is
possible
to introduce new coordinat.es rj , yj( j
=1,
...,r),
tk(k
_=-1,
..., ..., m -r)
and Sz(1 == 1,
..., n - m -r) f or
Rn such that 5’ isgenerated by
the
vectorfields
’
and
,where zj = Xj
+
Here zj and tk
areprincipal variables,
whilethe 8z
areparametric.
Andthe foliation associated to T has leaves of the form On the acts as the de Rham
system,
and on the Cr-factor S forms theCauchy-Riemann system.
In the
general
case it is natural to preserve the condition[T, ~’,
,since this
changes nothing
as far as the solutions of 5u = 0 are concerned.Under this
assumption
Andreotti and Hill haveproved
thefollowing generalization
of the real Frobenius theorem(see [1]):
THEOREM 2.
Suppose
that thecomplete
vectorfield system
{f is realanalytic.
Then
locally
one canf ind
n - mfunctionally independent complex
valued realanalytic
invariantsC1(Xl,
...,xn), ... , Cn-m(x1,
...,Xn) f or J ,
where m = rank T.We remark that
Nirenberg
has shown in[11]
that this theorem is notalways
true in the 000category.
For this reason(and
forothers,
that willemerge later
on)
we assume from now on that all vector fieldsconsidered,
as well as the manifolds
they
live on, are realanalytic.
However,
we do not want to make such a restrictiveassumption
on thesolutions of the
system
may be a distribution or a
hyperfunction
for instance. But since we are interested infoliations,
it is natural to suppose that the solutions u may be restricted to all leaves underconsideration,
and this we do from now on.Now, just
as in the real case, it is natural to use the invariants of J as new coordinates in order to reduce J to canonical form. Moreprecisely,
assume that T is
complete,
that rank T = m and thatJ }
= m+ r (where
on an open set U ~: Then thefollowing
hasbeen
proved by
Andreotti-Hill in[1]:
THEOREM 3. Consider the map
~‘ :
U ->- Cn-mdefined by
x F-*(’1 (X),
...,".y~-w(~))y
and alifting
1p:of ~~
such that thediagram
commutes.
Then
for
eachpoint
x E U there is anneighbourhood
o-)of
x suckthat M:=
C((o)
is ageneric locally
closed realanalytic submanifold of
Cn-mof
real dimensionequal
to n - m+
r, and such that N:=1p(w)
isdiffeo- morphic
to w, is an open set in MxRm-r.The
system (4)
on w is thenequivalent
to thesystem
on N c where the induced
Ca2cchy-Riemann system on
If and the de Rhamsystem
on an open set inAs
usual,
functions u on Msatisfying ~M~
= 0 are called CR functions.We see that when the
system (4)
is written in the form of(5),
it issplitted
into a realpart, namely
the de Rhamsystem,
and acomplex part.
Since the real
part
is more or less trivial(at
leastlocally),
we consideronly
the
complex part dM
in thefollowing.
The
J,,,-equations
are ingeneral
not aspleasant
as the de Rham orCauchy-
Riemann
equations.
Forinstance, 3~ gives
rise to acomplex
on if in anatural way,
corresponding
to the de Rham and Dolbeault sequences on Rn and C"respectively.
But it turns out that thecohomology
groups of thiscomplex
are ingeneral
notlocally trivial,
as one can see from[2]
or[14].
And this fact
gives
anexplanation
of the famous counterexample
ofLewy
(see [2], part 1, § 5).
However,
in the situation of Theorem 1 thej,,,-equations
are easy tounderstand,
because then if islocally
of the formCr,
where Ran---ris a
parameter
space,and aM equals
theCauchy-Riemann system
on thefactor Cr. Thus = 0 means in this case that it is
holomorphic
on each leaf of the formCr ,
and isarbitrary
in theparametric
variables.In the
general
case it canhappen
that M has a foliation where the leaves areperhaps
notcomplex manifolds,
but where each leaf can be im- bedded in acomplex
manifold with some(real)
codimensionk,
such thatthe restriction of a CR function u to each leaf is a
boundary
value of aholomorphic
function. Inparticular,
if k =1(in
which case the foliation is said to be ahypersurface foliation),
we then have a lot of informationconcerning
the restrictions of tt to the different leaves thanks to the work in[2]
for instance. The purpose of this paper is to indicate how foliations such as these maybe
found.Since we shall have to deal a lot with vector field
systems
in thesequel,
we introduce the
following
convenient notation: If.X~ , ... , ~YN
are realanalytic
vector fields defined on a realanalytic
manifoldM,
then(Xl,
...,XN)
denotes the vector field
system generated by X1, ... , (as
a module over thering
of realanalytic
functions onM).
2. -
Examples.
Before
proceeding further,
wepresent
someexamples
which show howone in an intrinsic manner can find foliations related to the
tangential Cauchy-
Riemann
equations im.
Since the foliations areeventually given by
thereal Frobenius
theorem,
we willmainly
work with real vector fields.EXAMPLE 1. Define the real 4-dimensional submanifold .NI of C3
by
Then a
holomorphic
vector field on C3 istangent
to ifal = 0 and
-f --
= 0. The choice a2 = Z3 and a3= - ~~, gives
thevector field
where .
and J is the
complex
structure of Thesystem
= 0 then consists of thesingle equation Z2~
= 0.Calculations show that
and that
J[JX, X]
commutes withX,
JX and[JX, X].
Hence
(X, JX)
is notcomplete,
so there is nocomplex
foliation of -31.However., (X, JX, [JX, X])
iscomplete,
and it iseasily
seen that the cor-responding
foliation of JI has the leavesi. e. ,
ri is aparametric
variable.But also
(X, JX, [JX, X], J[JX, X])
iscomplete,
andgives
a foliationof
C3%(z
E C3: Z2 = =01.
Since this vector fieldsystem
is invariant underJ,
the leaves inherit thecomplex
structure J fromC3,
and henceare
complex
manifolds. Infact, they
areThus forms a
hypersurface
foliation of M.Now the Bochner theorem shows that if u is a CR function on M then for each is the
boundary
value of aholomorphic function ùx,
x defined onC3: Z, = X,, IZ212 2 -r- z3 2
1-(In case u.,,,
isa
hyperfunction
we refer to[12]
or[14]
for the existence ofUX1.)
The idea behind the considerations above is the
following:
If(X, JX)
were a
complete system,
if would have acomplex
1-dimensional foliation.Since this is not the case, we first add
[JX, X]
so as toget
thecomplete
system (X, JX, [JX, X]),
whichgives
the foliationHowever,
(X, JX, [JX, X])
is not invariant underJ,
so weenlarge
thissystem
to(X, JX, [JX, X], J[JX, X]),
which is bothcomplete
and invariant under J.Hence it
gives
rise to a foliation withcomplex leaves,
which is an«
analytic
continuation » ofREMARK 1. On
J[JX, X].
Let M be a realhypersurface
defined in aneighbourhood
of theorigin
in C2by
a realequation ~)
= 0 such thatç(0)
= 0 andgrad ç(0)
~ 0. We assume that the coordinates are chosen such that istangent
to thehyperplane
at0, i.e.,
IoplêJY1(0)
= =0,
and =grad ç(0)
~0, where zk
= xk+
=1,
2.Up
tomultiplication
with afunction,
there isjust
oneholomorphic
vector field which is
tangent
toM, namely,
where
Then
XIM
andgenerate
the module of CR vector fields on .lVl(i.e.,
the vector fields Y onM,
such that also J Y istangent
toM).
Straightforward
calculations now show thatwhere
ô2cplozl
is the Levi form of M at 0.Hence the
following
statements areequivalent
at 0:1) J[JX, X]1,,,
has anonvanishing
normalcomponent;
2)
the Levi form of M isnonvanishing;
3)
if isstrictly pseudoconvex.
Furthermore,
the local Bochner theorem(see [12]
or[14]
for thehyper-
function
case)
shows that in this situation each CR function on if has ananalytic
continuation to a one-sidedneighbourhood
of if in C2 at 0. Andby (6),
thisneighbourhood
lies on that side of M into whichJ[JX, .~Y’]~M points.
Let co be a
neighbourhood
of 0 in M. Then(6)
alsoimplies
that thefollowing
statements areequivalent:
1)
the Levi form of .lVl vanishesidentically
in w;2)
the normalcomponent
ofJ[JX,
vanishesidentically
in oi;3)
is a CR vectorfield;
4)
co has acomplex
foliation.Here
3) implies 4),
since3)
shows that[JX, X]
is a linear combination of ,X and JX on OJ, so that(X, JX)1.
iscomplete,
and this vector fieldsystem
isobviously
closed under J.Conversely,
if c~ has acomplex
folia-tion,
this is associated to(X,
which must therefore becomplete.
Thus
[JX,
is a linear combination ofXI.
and and hence isa OR vector field.
REMARK 1’. One-sided continuation and the heat Let M be a
strictly pseudoconvex hypersurface
defined in an open set U inC2 ,
where Uis so small that the vector fields
X, JX, [J.X, X]
andJ[JX, X], given
inthe above
remark,
are defined andlinearly independent
in all of U.Set Y : =
J[JX, X]
and Z : =X - iJX,
W := Y - iJ Y. Then Z and 1V form a basis for theholomorphic
vector fields on U. Hence a functionis
holomorphic
in U if andonly
ifin
U,
thatis,
and
Thus
(7)
isequivalent
to thesystem
of realequations
Now let
f
be a C°° OR function onM,
and consider thefollowing Cauchy
problem
in U:Of course, this
problem
is notwell-posed,
butfollowing [6]
we can for-mulate it in such a way that we at least obtain a formal solution.
Namely, by (8),
This
equation
remains true withreplaced by F2,
and henceNow
Y/M
has anonvanishing
normalcomponent,
whileXIM
andare vector fields which are
tangent
to M. Thusand an iteration of this
equation
shows that F isuniquely
determined ina formal
neighbourhood
of in U.If
J[JX, X]
commutes with the vector fieldsystem (X, JX, X])
and if the latter is
complete (as
was the case inexample 1),
we can makethis
argument
moreexplicit
in thefollowing
way(we give
the calculations in somedetail,
sincethey
indicate ageneral procedure
forfinding
suitablelocal coordinates associated to
complete
vector fieldsystems):
Let
$1 , ~2
and~3
be realfunctionally independent
invariants ofX],
and
let
be a real invariant of(X, [JX, .X]).
Then
The
commutativity assumption
shows thatthat
is,
Analogously,
and
Since
X,
JX and[JX, ~]
arelinearly independent,
this shows that= 0 for i =
1, 2,
3. Hence ai,bi
and Ci are functionsof $ only,
while honly depends
on q. Since M is anintegral
manifold of(X, JX, [JX,
the functions =
1, 2, 3,
form asystem
of local coordinates for M.Setting
Since t is
independent of ~,
tis
constant on M. Hence we can choose theintegration
constant in(11)
so that t = 0 on M. With these new co-ordinates, (10)
goes over intowhere
L1ç
= XoX+
JXoJX is a second order differentialoperator
in the~-variables.
Hence theCauchy problem (9)
has the formal solutionAs
(12)
resembles the heatequation,
onesuspects
that(9)
will havean actual solution when t > 0
(and small)
and this is confirmedby
the localBochner theorem.
Perhaps
this observation(besides
the usual discarguments)
may
give
an intuitiveexplanation
of the one-sidedness ofanalytic
continua-tion from
pseudoconvex hypersurfaces.
EXAMPLE 2. The real 4-dimensional manifold M in C3 is defined
by
Then there is
(up
tomultiplication
with afunction) only
oneholomorphic
vector field which is
tangent
toM, namely
were
Furthermore,
and
J[JX, X]
commutes withX,
JX and[JX, X].
The vector fields
X,
JX and[JX, X]
arelinearly independent
on Msave on the
exceptional
setHence
-Mo
has acomplex foliation,
which one also seesdirectly:
On the
complete
vector fieldsystem (X, JX, [JX, X])
definesa foliation with real 3-dimensional leaves
To
get
ananalytic
continuation >> of we add onJ[JX, Xi
so as to
get
thecomplete
J-invariantsystem (X, JX, [JX, X], J[JX, X]).
The rank of this is
equal
to 4 when y, ~0,
and the associated foliation has the leavesSince
J[JX,
has anonvanishing
normalcomponent,
the localBochner theorem
gives
ananalytic
continuation of CR functions onL~3
into on that side of into which
J[JX, points.
So if it is a OR function on
M, u
has thefollowing properties:
i)
u isarbitrary
in theparametric
variable y,;ii)
for fixedy,, =A 0,
is theboundary
value of aholomorphic
function defined in a one-sided
neighbourhood
of inis
holomorphic along
the leaves in thecomplex
foliation ofifo’
Moreover,
theexceptional
case ~3 = 0 is related to the fact that theanalytic
continuation of u from intochanges
side at theparameter
value y3 = 02013cf.
figure
1.Figure
1EXAMPLE 3. We now consider a
simple example
where there are twolinearly independent holomorphic
vector fieldstangent
to M.Let -I’ Then is
tangent
The choices
respectively,
ygive
theholomorphic
vector fieldsand
Here
and
Then
(X, JX, [JX, X])
and(X, JX, [JX, X], J[JX, X])
arecomplete
and
give
foliations andrespectively,
whereSimilarly, (Y, JY, [JY, Y])
and(Y, JY, [JY, Y], J[JY, Y])
are com-plete.
The associated foliations are andrespectively,
withHere IZ21
= 1 is anexceptional
case in the same way as y3 = 0 was in thepreceding example.
REMARK 2. Foliations
of non degenerate hypersurfaces.
The lastexample
can be
generalized
toarbitrary nondegenerate hypersurfaces
in Cnby using
nonintrinsic methods
(which, however,
do notgenerally
work when the codimension isbigger
thanone).
In
fact,
let be such ahypersurface passing through
theorigin
in Cn.It follows from
[4],
sections13,
14 and18,
that it ispossible
to introduceso called normal coordinates near
0,
such that M is the zero set of a functionwhere
z’= ..., z~_1)
and~E~~~-i ,
with êj==±1?
is thesignature
of theLevi form of M at 0.
From
(14)
it followsimmediately
that the foliations =1,
..., 1,/;-1,
wherehave the
property
that for each is a real 3-dimen- sionalstrictly pseudoconvex hypersurface
in thecomplex
2-dimensional manifoldL~
near 0. Hence weget
n -1pairs
of foliationsk =
1,
...,n -1,
where eachpair
is of the same kind as those consideredin the
examples
above.REMARK 3. On
misleading examples.
Theexamples
discussed here aresomewhat
misleading
in the sense that the considered vector fields have been defined on all ofCn,
and notonly
on the real submanifold Me C".This
depends
on the fact that theholomorphic
vector fields Z have been definedby requiring
thatZ(qJi)
= 0 for a set ofdefining equations fggi
=ol
for
M, i.e.,
Z istangent
to all manifolds =constant}
as well. But then the definition of Z outside of Mdepends
on the choice of the functions qJi, and hence is ratherarbitrary.
When
considering J,,,-equations arising
as in Theorem3,
there is ingeneral
no obvious
privileged
extension of the vector fields inquestion
outside of M.As a consequence, the
complex
leavesLt
will nolonger present themselves,
but
require
some construction.3. - The
theory
of Vessiot.In order to find
complete
vector fieldsystems
of the kind that we areinterested
in,
we will use thetheory
of Vessiotgiven
in[15].
For the con-venience of the
reader,
and in order to establishterminology,
wegive
arough
sketch of thattheory
in this section. We work in the realanalytic category throughout,
and all functions and vector fields are real valued.The
problems
to which the Vessiottheory
can beapplied
are of thefollowing
kind: For agiven integer
p and agiven noncomplete
vector fieldsystem:F
defined on an open set U in one wants to find a foliation of U withp-dimensional leaves,
such that all leaves are invariant under the action of plinearly independent
vector fields from 5-.By
Frobenius this isequivalent
tofinding
acomplete
sub vector fieldsystem 9
of Y of rank p.By
Gauss elimination andby renumbering
the indices if necessary,we may suppose that Y has a basis of the form
i.e.,
the basis is in resolved form(with respect
toalax,)
in Vessiot’s termi-nology. Assuming + [Y, Y]
has the basis ... ,Xm ; Zi ,
7... ~Zjy
there are structure functions
... , xn ) )
such thatBy
above we look for a sub vector fieldsystem
of Y with a basissuch that
shows that
so that the
problem
offinding Yi satisfying (16)
can be divided into twosubproblems (where i, j
=1, ... , p ) :
which is a
problem
in linearalgebra,
y andwhich is a
system
ofpartial
differentialequations
of a ratherspecial type.
To solve
(I),
start with anarbitrary
and determinethe coefficients in from
For fixed
1,
..., m, this is asystem
of linearequations
in the a,,.We suppose that the coefficients
alY
are chosen such that the rank of(Ii)
is
maximal,
in which casethe a,,
are said to begeneric.
If this rankequals
ythen m - q1 of the
a28, ~
=1,
..., m, are leftarbitrary.
~ m
Then we
try
to find a suitableYg =,I a,,,(x) - X,, by solving
a=1
with
Y2
thegeneral
solution of(11).
Also here we choosea1y
and the a2a leftarbitrary
in ageneric
way, so that therank q~
of(I,),
as asystem
oflinear
equations
for is maximal.And then we continue in this way until at last determined from
where
again aly’
...,a. - i, y ~ 7 - - 7 - .. ~
m, are chosengenerically,
so that therank of is maximal.
If all the
systems (h)-(Ip_1)
have nontrivialsolutions,
we say that Y is involutive oforder p,
y and the vector fieldsYl, ... , Yp
found in this way form an involution ofp-th
order. In this case we continue to solve(II).
Suppose
that with a suitable choice of local coordinates one wants"Oil ..., xp to be
functionally independent
on thesought-for integral
mani-folds. Then a basis for ( j’ =
(Yi,
...,YJ
can be written in the formwhere, according
to what was doneabove,
all Vl(X arearbitrary (though generic),
y of the v2a arearbitrary,
y and so on. Because of the resolved form ofV,,
thepartial
differentialequations
forvka (x) corresponding
to
(II)
can now be solvedby
a clever inductionargument using
theCauchy-
Kowalewski theorem
repeatedly.
This determines the up to the fol-lowing
arbitrariness:The whole
theory
isreally
invariant under coordinatechanges, except
for the last
steps involving
theCauchy-Kowalewski
theorem. And it isprecisely
these whichgive
thecurious-looking
result above.However,
ithas been shown in
[9], chapter I,
that in such a resultonly
the arbitrariness in thebiggest
number of variables isreally
invariant(of
course this wasknown
already
to Cartan see e.g. letter XXXVII in[3]).
So if ra is thefirst nonzero
integer
in the sequence rl , r2 , ..., rp , we say that the solutiondepends
on « raarbitrary
functions of n - a+
1 variables)>(see [9]
for theprecise meaning
ofthis).
’
The solutions found in this way
(with generic
are calledregular.
Then there may also exist
singular
solutionscorresponding
tonongeneric vka(x).
However,
each of these can under rathergeneral
circumstances-as is in- dicated in[15],
andproved
in[9], chapter III,
for the dual case of exterior differentialsystems-be regularized >> by
means of a finite number ofprolongations,
and then be obtainedby
thepreceding
method. We refer to[15],
section15,
for this.4. -
Complex
submanifolds of Cn and the Cartan-Kihlertheory.
The
examples
in section 2 show that thefollowing problem
is reasonable:Given a real submanifold M of Cn we want to find a
family (where
tbelongs
to someparameter space)
ofcomplex
manifolds defined nearM,
so that is a
nondegenerate hypersurface
inLt
for each t.One way to find
(ii)
isby
firstconstructing
a suitable foliation(Li)
of lVl
by
means of the Vessiottheory,
and thenextending
eachE,
to acomplex
manifold. It is the latter
step
that we shall describe in this section.Suppose
that ~ is acomplex
m-dimensional submanifold of Cn . Thenlocally
one can findcomplex
coordinates Zl’...’ Zm, 7 z~+1, ... , zn for Cn sothat ~ is defined
by
where the functions
f k (zl , ... , zm )
areholomorphic.
Hence
son
A,
so thatfor all
(m -+-1)-tuples (j1’
...,By complex conjugation,
alsoNow let 3m be the differential ideal on
Cn r’-I
R2ngenerated by
or, in real
form, by
Then the
argument
above shows that ~ is anintegral
manifold of 3m with the real dimension 2m.Conversely,
suppose that JV’ is a realanalytic
submanifold of ~2n of real dimension2m,
such that JW is anintegral
manifold of 3m. Thenone can find local coordinates z~, ..., zn,
Zl’’’.’
zn in Cn such that,for certain
j1,
...,jp, k1,
...,kq with p -p ~ ==
2m.In
fact,
since X is anintegral
manifold of3-, necessarily p
= q = m.locally
onJf, where g,
is a
convergent
power series in the indicated variables. HenceBy hypothesis,
which
by (17)
shows thatogjlozkz
= 0 for l =1,
..., m.Thus gj
is aholomorphic
function in thevariables 3z,,,,
...,z,..
The fol-lowing
lemma is a consequence of all this:LEMMA. A real
analytic submanifold of
enof
real dimension 2m iscomplex analytic (of complex
dimensionm) if
andonly if
it is anintegral manifold of
thedifferential
ideal Jm.The
importance
of this lemma is that it makes itpossible
to constructcomplex
m-dimensional submanifolds of ~nstarting
from realanalytic integral
manifolds of m of real dimension 2 mby
means of the Cartan- Kählertheory (see
e.g.[8]
or[9]).
To be able to discuss
integral
elements of3m,
we first introduce a con-venient notation:
Let V be a real vector
subspace
ofTxCn ,
where x EC’,
and let Y1 bethe
orthogonal complement
of V in Then thecomplex
structureand the
projection together
define a.mapping Q :
= If a vector X c Vbelongs
toker Q,
thenalso JX E ker
Q
sinceJ(JX) = -
X E V. Sodimr ker Q
is an evennumber, equal
to2p
say. In this situation we say that V is of the formCP X R (where dimr
V =2p + q),
or that thecomplex
dimension of V isequal
to p.Then it is easy to see that the
integral
elements V c of the dif- ferential ideal 3m are of thefollowing
form:1)
any V with and2)
those V with(where
for which thecomplex
dimension of TT is at leastequal
to a.Furthermore,
in the latter case V is aregular integral
element of 3mif the
complex
dimension of V isequal to a,, i.e.,
if V is of the form Ca X Rm-a. .So if L is a real
analytic integral
manifold of 3m of real dimension m+
a, such thatTx(L)
is of the formCa X Rm-a
for somex E L,
then L can belocally
extended near x to acomplex
submanifoldL
of Cn ofcomplex
dimension m. And this extension is
unique,
as one can see for instanceby calculating
the characters of 3m(cf. [8],
pp.60-61).
In
particular,
consider a foliation of a real submanifold .M in Cn associated to acomplete
vector fieldsystem (X, JX, [JX, X]),
whereX,
JX and
[JX, .X]
arelinearly independent
vector fields on and J is thecomplex
structure of Cn. Then thetangent
spaces ofL,
are of the form C XR,
where the « C » is formedby
X andJX,
and the « R »corresponds
to
[JX, X].
Thus eachL,
canlocally
be extended to aunique complex
2-dimensional manifold
Lt,
such thatJ[JX, X] gives
one of the two normaldirections for
E,
inLt.
Moreover,
the Cartan-Kiihlertheory
is valid also in the presence ofparameters,
so that if(Li) depends
realanalytically on t,
will do so too.5. - The Levi module and its derivatives.
Now we want to consider the
tangential Cauchy-Riemann equations
on a submanifold 31 as in Theorem
3, i.e.,
we assume that .M~ is ageneric locally
closed realanalytic
submanifold of Cn of real dimension m, wheren m 2n. That if is
generic
means that for each x in if the ORtangent
space to .lVl
at x,
definedby
is of minimal
complex dimension, namely dimc
= m - n.Then
has the structure of a real
analytic
subbundle ofT(M),
and is called theOR bundle of M.
To avoid confusion about what is
regarded
as real and ascomplex,
wewill from now on
try
to considereverything
from a realpoint
of view.In
particular,
Cn is identified with thepair (R2n, J),
where J is anintegrable complex
structure.or
equivalently,
Then A is a module of vector fields
(the
so called CR vectorfields)
over thering
of realanalytic
functions on M. It is called the Levi module.-
The connection between A and the
j,,,-equations
is thefollowing:
anya.,,-operator Z
is of the form X+
iJX withX,
JX EA,
andconversely,
each X e A can be written as X =
Re Z,
withZ
aj,,,-operator.
By
definition A is invariant under the action of J.However,
ingeneral A
is not closed under Lie brackets. This makes it natural to consider the derivatives of A
(cf. [15]):
A’-c
A" C ... CT(liT)),
we assume(by shrinking
.M if ne-cessary)
that all(as
well as all other vector fieldsystems
on if to be considered in thesequel)
are of constant rank on M. Since rankT( )))
== dim if = m is
finite,
y there is aninteger
c such that allA (k)
areequal
for
k ~ c.
Then is acomplete
vector fieldsystem,
which is called the Levialgebra
of .112(because
it is a Liealgebra
under the bracketoperation).
The number e : = rank is sometimes called the excess dimen- sion of M. Since rank A =
2 ( m - n )
and rankA(’) rank
= m,we see that e 2n - m.
The
simplest
case occurs if c = 0. Then A iscomplete,
and thus definesa foliation of 31.
Moreover,
since A is invariant underJ,
the restriction of J to the leaves of that foliation willprovide
them with anintegrable complex
structure. Thus A defines acomplex
foliation of M.(This
isreally just
aspecial
case of Theorem1.)
CR functions on M are then characterizedby being holomorphic along
thecomplex leaves,
butarbitrary
in the para- metric variables.Suppose
now that c > 0. Then also e >0,
and rankA(c)
=2(m - n) + + e m.
Ife 2n - m,
rank~~~’ C m,
and thecomplete
vector fieldsystem A(C)
defines a nontrivial foliation ofM,
where theparameter
space
R2n-m-e.
Since the invariants of are also invariants ofA,
CR functions may vary
quite arbitrarily
from one leaf toanother,
andonly
their behaviour
along
the leavesBt
is restricted.Consider now a fixed leaf
B t ,
and letx E B be arbitrary. By
definition of A and~~~’, Tx(Bt)
will be of the formHence Bt
is aregular
integral
manifold of the differential ideal3
on Cn.By
the Cartan-Kahler
theory
we can thus extendB, locally
to aunique complex
manifoldBt
of
complex
dimension m - n+
e(which
is less than ii, if e 2n -m).
Since no derivatives with
respect
to theparametric
variables occur inA and
~~~’,
we can restrict these toBt,
so thatthey
form vector fieldsystems
and there. ThenAt
is the Levi module ofBt
consideredas a real submanifold of and =
T(Bt)). Clearly
the excessdimension of
Bt
isequal
to e.Now from a
deep
result due to Hunt and Wells(see [7])
it follows that iff
is a C1 CR function onBt,
thenf
has aunique
local extension to aCR function
I
defined on a CR manifoldU, c!3,
i with dimUt
= dimBt -~-
e =2(m - n + e)
andUt:2 Bt. Consequently
dimUt
=dim-F3t,
so thatUt
is in fact an open subset ofBe,
andf
isholomorphic
onHence we have obtained the
following
THEOREM. M has a
foliation
with leavesBt of
real dimension2(m - n) -E- e (where
e c 2n - m;if
e = 2n - m, thefoliation
is trivial in the sense that there isonly
oneleaf,
Mitself).
EachBt
has aunique
local ex-tension to a
complex (m -
n+ e)-dimensional mani f otd Bt. If
u is a CRfunc-
tion on
M,
then u isarbitrary
in theparametric
variables t.If
Ut:= E EC’(Bt),
then Ut has acunique
local extensiorc to aholomorphic function i, defined
on an open setUt
t inPt.
If e
(which
isequal
to the real codimension ofBt
in isbig,
ut is notexpected
to inherit so many of theanalytic properties
of4t-t. However, by making
local extensions of ut intoBt
andgluing
thesetogether by
meansof the
unique
continuation theorem forholomorphic
functions on open subsets ofBt,
we have at least:COROLLARY. C’ OR
functions
on M have theunique
continuationproperty along
the leavesBt.
In favourable cases there may exist subfoliations of such that the
corresponding
e issmaller,
and whichgive
further information about the C.R functions on M. The existence of such foliations will beinvestigated
in the
following
sections.6. -
Complex
foliations.As remarked
before,
when A iscomplete
the leaves of thecorresponding
foliation have a
complex
structure. But also when A is notcomplete
theremay exist