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11hllowing Gromov (see [Gr) we say that an almost complex manifold is tame if
exists a symplectic structw"e w on X such that the form weT, JT), T E T(X)
Uu'rc
iN positive definite.
Nute that any Stein (i.e. affine) cOlnplex manifold, J(ahlerian manifold or (genuine)
(,oluplex manifold with pseudoconvex boundary is tame. Any symplectic manifold
,uhnits an ahnost cOlnplex structure ta.med by the symplectic structure.
All ahnost complex manifold )(, J is called holomorphically a8pherical if there are
tlo holomorphic embeddings Cpl -+ (X, J).
˜. FILLING BY HOLOMORPHIC DISCS
Iia 1969 E. Bishop (see [Bi]) discovered that a punctured neighborhood of an elliptic
point admits a (unique!) foliation by circles spQnninp; holomorphic discs. By simple
topological reasons these discs have to be di of a Levi-flat
--!.--Q
50 Eliashberg: Filling by holomorphic discs and its applications
3-manifold. Many attempts were made to globalize Bishop's result: How far from
the elliptic point can the family of circles spanning holomorphic discs be developed?
The first serious success in this direction was obtained by E. Bedford and B. Gaveau
in 1982 (see [BG]). They considered the following problem: Given a 2-sphere 8
embedded in C 2 with exactly two complex points (which are in this case necessarily
elliptic), does there exist a Levi-flat 3-ball in C 2 bounded by the sphere 8? Or
equivalently, does the complement of complex points in S admit a foliation by
circles spanning holomorphic discs?
Bedford and Gaveau gave a positive answer to this question under two additional
assumptions:
n c C2 ; .˜
-8 is contained in the pseudoconvex boundary of a domain :"˜ !
.˜I
-orthogonal projection of 8 onto R3 C C 2 is an embedding.
В·f
While the first assumption looks very important and cannot be completely removed
(see Section 9 below) the second is absolutely unnecessary and even the original
-:˜
Bedford-Gaveau's proof can be easily modified to work without this assumption . !˜
(see [EI2]). The first c0111plete proof of the result belongs to M. Gromov (see [Gr]). 1;
':˜
The key point in Bedford-Gaveau's proof is Lipshitz estimates for boundaries of :;,˜
holomorphic discs while Gromov uses the more general compactness theorem which
he proves in the same paper.
In our joint paper with V. M. Harlamov we related the technique of filling by
holomorphic discs with topological problems and, in particular, with Bennequin's ',˜
).˜
.,y(
theorem [Be] \vhich was just proved. We showed that if one could prove a filling i}
.;/
result for spheres which do not belong to a pseudo-convex boundary, that would ",
imply very strong corollaries in low-dimensional topology (see Problem 10.7 below).
Unfortunately (or fortunately?) the result in that generality is false (see Section 9 1
below) but the problem is far from being understood.
Since that time there were two major breakthroughs which sufficiently extended
possibilities of the method. In 1985 Gromov wrote his famous "Pseudoholomorphic
curves in symplectic manifolds" \vhere he showed that holomorphic curves and,
in particular, holomorphic discs can be successfully used not only in complex but
Eliashberg: Filling by holomorphic discs and its applications 51
".tHO in tame almost complex manifolds. Recently E. Bedford and W. Klingenberg
(Are [BK]) found a teccnique for developing the family of holomorphic discs in the
I..'('sence of hyperbolic complex points. That allowed them to prove the result about
til(' filling of a 2-sphere in C 2 without any additional assumptions (besides pseudo-
,В·cHlvexity).
the next section I formulate \vithout proofs a summary of results concerning
III
I'..,˜ularity and compactness properties for fa.milies of holomorphic discs. The main
"rrntlts (Theorems 3.4.1 and 3.4.2) can be proved within the ideology of [BG], [BK]
luul [Gr] but are not straightforward corollaries of results of these papers. The gen-
"I,tl.li˜a,tionof [BI(] for an almost complex case requires a fine analysis of singularities
ur .T-holomorphic curves which was recently done by D. McDuff (see [McD2]).
:i. COMPACTNESS AND REGULARITY PROPERTIES FOR FAMI-
L114˜S OF HOLOMORPHIC DISCS
:t.1 'P-admissible sets
By Morse function we mean a function with nondegenerate critical points and
H
puil'wise different critical values. By an almost Morse function we mean a function
wl,i('h has either non-degenerate or birth-death type critical points and at most
dlluhle critical values.
L.,t I( be a 3-manifold with the boundary 8I{ = M and 'P : K --+ R be a function
Wh()H(˜ restriction on /vI is ahnost Morse. For a closed subset A C K we denote by
ii' A the intersection A n 8I( and by 8"A the rest of its boundary: a˜˜=:A.
A closed subset A C I( is called tp-admissible if the following conditions AI-A5 are
˜"t.iHfi(\d:
A1. 'rhe function 'P does not have critical points in A.
A:J.. lutA is a union of some components of sets {x E IntI<, e < c.p(x) < e'} where
are critical values of <.p l/VI. in particular, <p is constant on components of 8"A.
' \ ,,I
/\:1. For any e E R all components of the level set A c = {x E Int A, <p(x) = c} are
ftilUpJy connected.
A-1. For any critical value c of'P 11\1 the intesection of closures in K of sets {<p(x) <
'I .,. > C, x E ..4} is x E A}.
C A} and {tp(x) open-closed subset of {tp =
all C,
Eliashberg: Filling by holomorphic discs and its applications
52
a"A
contains a critical point of the function ˜ 1M and
A5. Any component C of
it is either not simply connected or it does not cover the whole component of the I
1<, Ie}.
level set {x E <p( x) = 'P
HAC K is ep-admissible then one can form a graph GA,cp identifying each compo-
nent of level sets {x E A, ep(x) = c}, c E R, to a point. Vertices of G correspond to
local maxima and minima of 'P 1M and to components of aliA.
..
Let 1r be a projection A -+ GA.",. For a subgraph r c GA,cp we will denote by Ar
the inverse image 1r- 1 (r) and by Mr the intersection Ar n M = ()'Ar. Note that
= r. r
if the set Ar is ˜-admissible then G Ar.cp We will say in this case that is an
admi8sible subgraph of G.
The function 'P factorizes to a function <j; : G A,cp ˜ R with <p = 1r 0 tj;. The function
rp is non-degenerate on each edge of GA,cp and, therefore, defines an orientation of
the graph GA,'P.
3.2 Partial and maximal fillings
Let M now be an oriented closed 2-surface in an almost complex manifold X, J.
By a partial filling of M by holomorphic discs we mean a quintuple
, = (1<, cp, A, rt, F) which consists of:
.J
-a 3-manifold J( bounded by M, ;
1M' .˜
-a function <p : I( --+ R with an almost Morse restriction <p
.˜
-a cp-admissible set A , ' .˜
-a family 1ВЈ of complex structures on levels Ae = {x E Int A, <p(x) = c} which .˜
˜
depends smoothly on c E R,
I
-a smooth map F : A -+ X which is holomorphic on levels Ae , c E H, and coincides
with the inclusion 8'A ˜ )( on a'A = An M. /˜
'˜
Note that for any component ˜ of Ae , c E R, the image F(˜) is a holomorphiej
disc in X and its boundary is contained in M. For any partial filling ( all critical
points of ˜ 11\/ from A are, necessarily, complex points of M and elliptic points '
correspond to local maxima and minima of <p 1M' We will require, in addition,
that local maxima and minima of c.p 1M correspond to negative and positive elliptic I
points, respectively. With this choice of the orientation of M the orientation of any
Eliashberg: Filling by holomorphic discs and its applications 53
˜ C,
x E A'A = MnA} x E {)' A},
{<p(x) = {cp(x)
as the boundary of the set
.В·ul've C,
.l (: R, coincides with its orientation as the boundary of a corresponding holomorphic
F(˜).
.tiNe
A partial filling is called maximal if all elliptic points of M belong to A and are,
UH˜rcfore, local maxima and minima of cp 1M. A partial filling is called generic if
V' IM is a Morse function.
WtВ· Hay that M can be filled by holomorphic discs if it admits a maximal filling with
Ii. Note that in this case]( must be a handlebody bounded by the surface M.
" .::=
II. l>articular, if M is diffeolnorphic to S2 and can be filled by holomorphic discs
t.hf'1l it bounds a 3-ball in X.
:'.4 Existence of the Inaxilnal filling
'I'IIEOREM 3.4.1. Let X, J be a tame almost complex manifold, n c X be a domain
1U't.!I. a smooth J .. con1Jex boundary on and Mean be a closed 2-surface. Then M
2
1'1&71. be C -approximated by a surface AI which admits a unique generic maximal
Jilting by holomorphic discs.
n
3.4.2. Let X, J and 0, be as above. Suppose that, in addition, is
'I'UI':()REM
1&f1lcnnorphically aspherical. Let M be a closed 2-surface and ˜ be an embedding
an. Then the embedding ˜ can be C 2 -perturbed in such a way that
AI x I ---.
If
surface ˜(M x t), tEl, admit3 a maximal fiZling by holomorphic discs.
tJfl.fh
x t), tEl, can be filled by holomorphic discs then there exist a
fllld˜h ,˜urface ˜(M
Ilft'U.tllebody I{ and an embedding ˜ : !{ x I ---. n such that ˜ coincides with q, on
n is
AI x I and for any tEl the image 4В»(M x t) C Levi-flat.
:1." Graph of a filling
With any partial filling ( = (/(, tp, ..4 , H" F) we can associate the graph GA,<p (see
:1.1). If ( is a maxhnal filling of M then \ve ",{ill write GM instead of GA,cp. Vertices
ttl' (,t 1\1 correspond to all elliptic and some hyperbolic points of M. Elliptic points
un' ('ud-points of GM. The orientation of GA1 has been chosen in such a way that
ftcll1:(˜H start at negative elliptic points and end at positive ones. H M is filled by
JIt.ltHllorphic discs then all end-points of GAl correspond to elliptic points of M.
If this is not so then any connected cOlnponent of GM must contain an end-point
˜
j
Eliashberg: Filling by holomorphic discs and its applications
54 ':˜
,˜
1
which is a hyperbolic point of M. Note that edges can start at positive hyperbolic
.˜
points and end at negative ones.
˜
1
3.6 Cutting A out of M
l
Let ( = (1(, '1', A, 11" F) be a partial filling of M and G = GA,tp be the graph of
the filling. The boundary of A is a piecewise smooth surface which consists of two ˜
˜j,˜
parts: 8A = (f Au{)" A. It can be sInoothed in such a way that the new surface MA ˜.
.
is filled by holomorphic discs. The graph GM A is isomorphic to G. End-points of ':','1;,
G which correspond to hyperbolic points of M are replaced in GMA by end-points
:˜::::s;:nsd;˜l::::::::;˜LLEDBY HOLOMORPHIC DIscsl
n de- .˜
Through the rest of the paper )[, J denotes a tame almost complex manifold,
notes a domain in X \vith a smooth J-convex boundary and M denotes a connectedJ1 ',
2-surface (sometinles closed, sOluetilnes \vith a boundary) which is contained in an.
I
For purposes of our applications nothing changes with a small C 2 -perturbation. So
we will speak about the filling of a stuface !vI Ineaning that it can be done after,
'j
probably, some small C 2 -perturbation.
'˜i
an different/rom 52 satisfies the inequality
4.1. Any closed sur/ace A1 C
THEOREM
X(M) ::; -lc(M)1 .
If M is diffeomorphic to 52 then it can be filled by holomorphic discs. In particular, '
n.
it bounds an embedded ball B 3 C :1
iJ
Proof. IT M is not 52 but can be filled by holomorphic discs then c(M) = 0 and the ˜l.˜ ,
inequality is autonlatically satisfied. Hence it is enough to consider'the case when :1
1
M cannot be filled by holomorphic discs. Let ( = (K, cp, A, 'H., F) be a maximal
filling of M. Then a'A conta.ins no closed components. Cutting A out of M we get :t
a surface M' = AlA which can be filled by holomorphic discs. The graph GM' of
this filling coincides with G A1. Let us denote by k+ and k_ numbers of positive and
negative hyperbolic points of M \vhich have been substituted by elliptic points of
MA. Then
eВ±(Al') = eВ±(1\l) + k=f
hВ±(Al) 2 hВ±(J..l') + kВ±
Eliashberg: Filling by holomorphic discs and its applications 55
Note that if bo = bo(M') is the number of components of M' then k+ + k_ ˜ boo
Now we have c(M') = 0 because M' is filled by holomorphic discs. Therefore,
qllalities 1.4.1 imply that
11
11.'uee,
dВ±(M) = eВ±(M) - hВ±(M)
+ hВ±(l\1') - hВ±(M)
˜ bo - k˜
˜˜=F - kВ± = b - (k - + k+) ::; 0
::; bo -
But
1
dВ±(A1) = 2(X(l\1) В± c(M))
which implies the required inequa.lity
X(1\1) ˜ -lc(1\I)1
Q.E.D.
that Theorem 4.1 provides an obstruction for a 4-manifold to carry a structure
Not.e
flf a tame almost complex Inanifold ,vith the J-convex boundary. For example,
( ˜()HOLLARY 4.2 S2 x D 2 has no tame almost complex structure with the J -convex
bO'l/,'fI,dary.
Illdc\ed, a sphere S2 x p,p E 8D2 does not hound any ball in S2 x D 2 which
4.1.
Hltradicts
fie
A eontact structure on a 3-n1anifold V is called fillable if it can arise as the dis-
eof complex tangencies to the J-convex boundary V = an of a domain
tribution
X, J. Note that the first Chern class Cl(X)
\l ill a tame aln10st cOlnplex Inanifold
e.
Je'stricted to V coincides with the Euler class e(e) of the bundle Let us denote
Ily Fill(V) the subset of H 2 (V) which consists of classes e(e) for all fillable contact
eon
!1t.1't1(˜tures V.
56 Eliashberg: Filling by holomorphic discs and its applications
4.3 The set Fill(V) is finite.
COROLLARY
E H 2 (V) can be realized by a smooth surface MeV.
Proof. Any homology class J.L ;J
According to 4.1 we have le(e)[Jlll = te(M)1 ˜ -X(M). Thus e(e) can take only :˜
finite many values on any homology class from H 2 (V).
on be a 2-surface with the boundary:::
Now we consider the relative case. Let AI C
eof
aM complex tangencies to an. Let us choose ˜
transversal to the distribution
the orientation of M which induces the positive orientation of 8M (see 1.2). ;
equivalently, X(M) $ e(M).
4.4. If lv/ is a.'.f above then d_ $ 0
THEOREM OT,
This is a generalized Bennequin's theorenl. In [Be] he proved the inequality for the ,I
= C2 вЂў
n is a ,˜
case when round ball in .'\.
˜˜
"1
Mwhich arises as the boundary'
Proof. Let us double M: consider a closed surface
an.
of a tubular neighborhood of M in We can consider the oriented surface M to)
be contained in Iv! as a half of M. Let G= Gif be the graph of the maximal filling 'j
va
be the set of those vertices of G\vhich correspond to negative elliptic ˜
of M. Let
points of M which belong to A([. Let Go be the least admissible (see 3.1) subgraph ':˜
of Gwhich contains Vo. Let Ao = AGo and Afo = Mao be parts of A and Mwhich ˜
correspond to Go (see 3.1). Note that 1\10 is contained in IntM. Indeed, if it is not:
so then there exists a hololnorphic disc 6. whose boundary {)˜ C M is tangent to '
a6. ;
8M at a point p E 8A1. By our choice of orientation of M the orientation of
(as the boundary of the hololnorphic disc ˜) at p coincides with the orientation
of oM as the boundary of AI. But the complex orientation of a˜ is opposite to :
at::. induced by the orientation of the set {ep ::; c} (see 3.2). This
the orientation of
M we get
c l11t AI. Cutting Ao out of
contradiction proves that M o a sunace
MAo
Mo = (see 3.6). Then, arguing as in 4.1, we get
o= c(Mo) = d+(Mo) - d-(Mo), d+(Alo) + d_(Mo) = X(Mo)
= e_(M)
e_(Mo)
h-(Mo) ˜ h_(M) - bo(Ma)
Eliashberg: Filling by holomorphic discs and its applications
- 1 ---- ..-.
e_(M) = e_(Mo) = 2X(Mo) + h_(Mo)
1 ..-.
'2 X(Mo) + h_(M) - bo(Mo)
$
= h_(M) - bl(˜O) ::5 h_(M)
o. Q.E.D.
which means that d_(M) = e_(.1\l) - h_(M) :5
=L
4.5. Let Mean be a 8urface with a Legendrian boundary aM
(˜()ROLLARY
It be the homology class of M in H 2 (an; L). Then
tВЈ11.d
X(M) ;5 tb(Llp) -lr(Llp)lВ·
prove 4.5 we can consequently approximate M by a surface whose boundary is
'('0
eand apply 4.4 and 1.4.2 ( see[Be] for similar
n. negative or positive transversal to
uol'guments).
IS. MANIFOLDS BOUNDED BY A J-CONVEX SPHERE
'l'he theorem of this section indicates that the topology of the J-convex boundary
itllposes very strong restrictions on the topology of the domain.
5.1. let X and n be as above and suppose, in addition, that n is holo-
'I'IIEOREM
1norphically aspherical. If an is diffeomorphic to S3 then n i8 diffeomorphic to the
"nIl B 4 вЂў Moreover, n admits a J -con'vex function which is con8tant on an and has
(::J;(J,ctly one critical point-the minim'lI,m-in O.
Take a Morse function h: an ˜ R with exactly two critical points. All non-
1"1'0of.
c"riticallevels of h are diffeomorphic to S2. By a small C 2 -perturbation of 8W near
critical points of h we can arrange that level-sets of h which are close to critical
IH)ints are filled by holomorphic discs. This defines a foliation of a neighborhood
of critical points in n by Levi-flat 3-balls. By an additional C 2 -perturbation of
as! we can arrange (via 3.4.2) that all other levels also admit maximal filling by
lu)lornorphic discs. But all these levels are diffeomorphic to 8 2 and, therefore,
j
'˜
58 Eliashberg: Filling by holomorphic discs and its applications
I˜I
according to Theorem 4.1 they are actually filled by holomorphic discs. Hence ,.i
1
an
each 2-sphere {h = c} c bounds a Levi-flat 3-ball Be. The family of these
balls depends smoothly on c E R (see 3.4.2) and are pairwise disjoint. Indeed, if ˜
;˜
nBc' =F 0 for c :/= c' then there exist holomorphic discs ˜c and ˜e' with aA e E :.%
Be
В·iВ·˜
aBe, at1 c E aBel and ˜e n ˜Cl =F 0. The intersection index of two holomorphic ::˜
l
˜e' is positive (see [McD2] for the non-integrable case) but it has to be :J,
discs A c and
zero because their boundaries aA c c aB˜ and. aA c: C aBcl bo˜d discs in disjoint,)
n IS folIated by LeVI-fiat balls Be and, ;:!
spheres aBc and aBel. Hence the domaIn
J
therefore, ˜eomorphic.to B4. To prove the .second part of the .theorem, denote by
H the functIon on 0 whIch extends h and whIch has balls He as Its level-sets. There jJ
exists a diffeomorphism c.p: n - n which is fixed at the boundary an and such that :˜
,!J.ll
all level-sets of the function HI = H 0 t.p are J-convex. As it was explained in 1.1 ".j
there exists a function i: R ˜ R such that the composition i 0 HI is a J-convex ˜˜
function. Let M = maxo H2, m = minn H2. Let H3 be a J-convex function which ,˜
is defined in a tubular neighborhood U :> an and which is constant on an and au.:˜;
˜˜
We can arrange that H a Ian> M and H 3 launn< m. Now extend somehow H a on.˜
n \ U with maxH3 Ifl\U< m and let H 4 = max(H3 , H 2 ). Smoothing H 4 we get the!
required J-convex function H s : n ˜ R. :˜
QED :;J
... ˜
:˜
D. McDuff's study of blowing down in tame almost complex manifolds allows us to:˜;j
j
understand what is going on when n is not holomorphically aspherical.
,{˜
,:rQ
5.2. Let X, J be a tame symplectic manifold and 0 c X be a domain::˜
THEOREM
В·˜:˜S
bounded by a J .. convex 3..sphere. Then fOT an almost complex structure J' which i,,˜i˜
',1j:;
Coo -close to J the manifold (Q, JI) is a 4..ball up to blowing up a few points. In;:˜˜
.˜
particular, n is diffeomorphic to B4#kCP2.
.˜
,;;;,
ietl
Proof. According to [McDl] there exists an almost complex structure J' which
';˜
COO-close to J and such that all elnbedded J-holomorphic spheres in n are disjoint,!˜
non-singular and have self-intersection number (-1). These spheres can be blown:1
down to give a non-singular hololnorphically aspherical almost complex manifold:˜
0' with the same boundary an. Thus we can apply 5.1 to conclude that Q' is:˜
diffeomorphic to B4 and, therefore, n is diffeomorphic to B 4 #kCP2.
Eliashberg: Filling by holomorphic discs and its applications 59
eon
5.3. Let X, J and n be as in 5.2. Then the contact structure
(;OROLLARY
O!! (formed by complex tangencies to an) is isomorphic to the tJtandard contact
.' trhlJ,cture on S3 (i. e. 3tructure induced on a unit sphere in C 2 .)
/',"oo!. Perturbation of J does not change the contact structure. So using 5.2 and
tвЂў. l we can get a ball n' bounded by an and a J-convex function H on n whic˜
1M (˜onstant on 8n = an' and has exactly one (non-degenerate) critical point: the
luillimum. All non-singular level sets of Hare J-convex and diffeomorphic to 8 3.
IIrllee all of them have contact structure and, therefore, all these contact structures
urn iHomorphic (see 1.2). But near the critical point the almost complex structure
."Ul he made integrable and level sets can be made biholomorphically equivalent to
Ulf' round sphere in C 2 and, therefore, the contact structure on all level-sets is the
Mtn.ll<!ard one.
o. I(ILLING OF ELLIPTIC COMPLEX POINTS
surface At c X has hyperbolic and elliptic points p, q E M of the same sign
If Il.
t.1U\1l one can easily CO-perturb M along a path connecting p and q to kill these 2
IJuilll.s (see [HE]). If M is contained in a J-convex boundary an of a domain n c X
an.
t.lU'1i it would be itnportant to be able to kill p a.nd q by an isotopy of M in8ide
tl'laiH is impossible to do in a neighborhood of a path connecting p and q (it would
11tlllt.l'H.dict the sY111plectic rigidity) but as it follows fronl Theorem 6.1 below this is
pu,mihle when enough room is provided.
VVC' Htart with the following definition. Let G be the graph of a maximal filling
Itt' A'I. An elementary contraction C, of G is the removal of an edge I with ends
whi.1b eorrespond to elliptic and hyperbolic points of the same sign. The following
11"'01'('111 which will be proved in [EI3] sho\vs that any elementary contraction can
lUI nВ·ali7.cd by an isotopy of !vI inside an.
Mean, be a closed surface, G be a graph of a generic
6.1. Let M,
'I'IIEOHEM
1'" "lzt/,I filling and I be an edge of G with ends which correspond to elliptic and
'''JI,,В·.,.!Jolic point3 of the ..˜ame sign. Let All be a part of M which is covered by
,.,,,, "tln1'i c., of holo m01В·phic discs 'which correspond to points of 1. Then there ezist8
an an, t E [0, 1], ..fntch that h o = id, h t is fixed outside a
" ,h'l'cot()py ht. : -+
Eliashberg: Filling by holomorphic discs and its applications
60
neighborhood of M, for all t E [0,1] and the surface h 1 (M) admits a maximal filling
whose graph is the result of the contraction C, of G. In particular hI (M) has two '
complex points less than M.
Now we apply 6.1 to prove
6.2. Let M be a closed surface in aQ. Suppose that X(M) = -lc(M)1
THEOREM
(comp. 5.1). Then M either can be filled by holomorphic discs (and it has to be ,(
diffeomorphic to the torus T 2 in this case) or is isotopic in an to a surface M' ,˜
,l
without elliptic points. In particular if M is a torus then either it bounds a Levi-flat
solid torus in n or it is isotopic in an to a totally real torus. .J
,˜
;˜
Proof. Suppose that :AI cannot be filled by holomorphic discs and let GM be the .;˜
graph of a maximal filling ( = (1(,!.p, A, 11., F). In view of 6.1 it is enough to provel
. :˜
that the graph G can be exhausted by a sequence of elementary contractIons. In ;˜
'i)
other words, that we can always find an edge whose ends correspond to elliptic and ',:˜
hyperboli˜ points of the same sign. Let MA be a result of cutting A out ˜f.M (see ]
3.6). As In the proof of 4.1, let us denote by k+ and k_ numbers of posItIve and .˜
negative hyperbolic points of M which have been substituted by elliptic points of I!
.˜
;˜
MA. Then we have
.˜
+ k_
eВ±(MA) = eВ±(.I\1") + k˜, hВ±(fl.1) 2:: hВ±(MA) + kВ±, k+ ;:::: bo(MA ) (*) 7:'
'1
o. Then X(M) = -c(M) ˜
Let us choose the orientation of M such that c(M) 2:
0, d_(M) = !(X(M}-c(M)) = X(M) . j
= !(X(Al)+c(Al)) =
and, therefore, d+(Al)
e_(.J\,I) = h_(M) + X(M).
or e+(M) = h+(IvI), Suppose that GM admits no)!
elementary contractions. This means that there is no edge whose ends are elliptic ˜
and hyperbolic points of the same sign \vhich implies inequalities
Then
+ h_(M) ,
˜
h+(MA ) X(M)
˜
h_(MA) h+(IVI) and
+ h+(M) + h_(M)
+ h_(MA) 2
h+(MA) x(1\.f)
Eliashberg: Filling by holomorphic discs and its applications 61
or
the other hand, the equality c(MA) = 0 implies that
()n
Now we get
1
"2 X(MA)
h+(Al) + k_ - h+(MA) = ,
1
+ X(1\l) + k+
h_(M) - h_(MA) = 2X(MA) .
'rile first equality and (*) give
,u1d therefore,
1
+ k_ = '2 X(MA)
bo(MA) = k+ В·
But then the second equality and (**) imply that
= o. But then MA is empty which is impossible if M has
",lid, therefore, bo(1\IA)
(illiptic points.
Q.E.D.
6.3. Let M be a 2-s'u1jace in an 'with the Legendrian boundary L = aM
'('IIEOREM
tВЈnd p, be the homology class of Al in H 2 ( an, L). Suppose that M is in normal form
7tt:(LT aA! and either M = D2 or
X(1\I) = -tb(LIJt) - Jr(LIp,)I.
there exists an isotopy of }vI in an which i3 fixed near L and moves M to a
'I'h(˜n
.Iu/.1jace M without interior elliptic point3.
an
which contains M and let ( = (1<, cp, A, 1t, F)
I'7В·0o!. Let M' be any closed surface in
I)(˜ its maximal filling by hololnorphic discs. Let r be the minimal admissible (see
:1.1) subgraph of the graph of ( which contains all interior elliptic points of M.
Eliashberg: Filling by holomorphic discs and its applications
62
Let :Alr = 8'Ar = M' n Ar be the corresponding part of M'. I claim that Mr is
contained in IntM. Indeed, if it is not then there must exist a holomorphic disc ˜ '!˜
whose boundary a˜ is tangent to L = 8M. But that is impossible either in the .˜
totally real part of L or in a complex elliptic point which belong to L. The rest of:˜
the proof repeats the proof of 6.1 \vhere, of course we need to use the formula from :˜
,˜˜
1.4.2 insteed of 1.4.1. J
.\
Q.E.D..}
. 'Iii
:.˜
˜
,
.˜t
7. LEGENDRIAN KNOTS .:˜
';)
'iJI
:ii
eo
be the standard contact structure ,˜
2
Let S3 be a boundary of a. unit ball in C and
on S3 (formed by complex tangencies). By a standard Legendrian curve in S3 .˜
we will mean an intersection L o of S3 with a Lagrangian plane P in C 2 or any ':˜1
Legendrian curve \vhich is Legendrian isotopic to this intersection. Note that if J˜
L o is a standard Legendrian curve in S3 then r(L o) = 0 and tb(L o) = -1. Any'::;˜
contact 3-manifold M, eis locally isolTIorphic to S3 \ p (Darboux chart) with the ˜ ;˜
'.,˜
standard contact structure. 'Ve will call a Legendrian curve L C M standard if it ': :˜ ˜
is Legendrian isotopic to a Legendrian cW've \vhich is standard in a Darboux chart...',˜
"J:
With-˜
The curve L o bounds in S3 a hemisphere V o which is the intersection of S3
a hyperplane which contains the Lagrangian plane P. The hemisphere V o has no;'1
interior complex points and has exactly two elliptic points on the boundary Lo.;˜
That implies that leaves of the characteristic foliation (V o)eo are Legendrian arcs,';˜f˜
which connect the t,vo elliptic points on the boundary. Note that using these arcs \f
ВЈ':1'
one can easily construct a Legel1dl'ian isotopy in an arbitrary small neighborhood"˜:!:
I вЂў.!J.
of V o into a small standard Legcndrian curve. ':';\;
n be an di8C'\;;˜
7.1. let ..Y , J and a.s in 6.3 and V C be an embedded
-I.':
THEOREM
= L. = 0 and tb(OV) =
bounded by a Legendrian curve OV Suppose that r(OV)
:?˜
L is standard.
Then the Legendrian C'ltrve
Proof. The disc V can be deformed (without cha.nging the boundary) to a disc 1)'
xCV') = -tb(L) - r(L) and ,i˜
which is in norlnal for111 near L. The equality 1 =
Theorem 6.3 guarantee a possibility of finding a disc V" which coincides with V'
near the boundary and which does not have interior elliptic points. But according "
˜ i˜
Eliashberg: Filling by holomorphic discs and its applications 63
'De
= o.
to 1.4.2 we have dВ± Therefore, has no interior complex points at all and
two elliptic complex points p and q at the boundary. Then the characteristic
llltH
eVil on V" is a foliation by Legendrian arcs connecting p and q.
fc)lintion But this
huplies that the pair V", {)T)" is contactomorphic to 'Do, L o and, therefore, L can
IJr eontracted to a small standard Legendrian curve through a Legendrian isotopy
ill an arbitrary small neighborhood of V".
Q.E.D.
II. l>rilne decomposition of l11anifolds with a pseudo-convex boundary
111 t.his section we consider talne ahnost complex manifolds il, J with a J-convex
an to be defined up to a deforlnation of an which keeps it J-convex.
bouudary
lH VPll a Levi-flat embedding h : (B, aB) --+ (0, an) of a 3- ball B one can extend it
tu 'Ul embedding H : B x [-1,1] --+ n with HIBxO = h, H(8B x [-1,1]) C an and
˜tlcth that balls H(B x 1) and H(B x (-1) are J-concave (with the orientation wich
,˜ illcluced 011 the boundary of the dOlnain H(B x [-1,1]). The boundary of the
tt.ullplement domain n \ H(Bx] -1, 1[) can be canonically smoothed to become J-
.˜cJllvex. We say that the new Inanifold 0' is the result of cutting 0 along a Levi-flat
,,,,11 h( B).
(Hvctu two tame almost complex rnanifolds 0 1 , Jl and O2 , J2 with pseudo-convex
IItt1I1Hlaries one ca.n canonica.lly define the boundary connected sum 0 1 #0 2 , J 1 #J2
ant #80,2.
f'˜ It t.alne almost cOlnplex 111anifold \vith the pseudo-convex boundary
tl'llc' manifold can be characterized by the property that cutting it along a Levi-
IH'W
""t. ball we can get the disjoint union of original manifolds n1 , J1 and O2 , J2 вЂў
I,,'t. lt˜ eall a manifold n, J prime if it cannot be presented as the boundary connected
12 1#n 2 ,J1 #J2 of two tame allnost complex manifolds with pseudo-convex
lttlill
t."Huda.ries where ill and n2 are not diffeomorphic to the 3-ball B. We say that
\',.J iH almost prime if nt and Q2 are not difFeolnorphic to B#kCp2.
r1I1';() It E tvl n, J 1]Jith a J -convex boundary is almost prime if and
8.1. A ma nifoltl
an is a p1'ime 3-manifold.
u"iJl if 1:t.'J boundary
64 Eliashberg: Filling by holomorphic discs and its applications
9. EXAMPLE OF A FAILURE OF THE FILLING PROPERTY
n Jut B 3 , t E] - 1, 1[, are holomorphic;
F n restricted to open discs V t = Pt
Eliashberg: Filling by holomorphic discs and its applications 65
Suppose that F n with these properties exists for all n = 1,.... First of all the
tuutlysis of the local behavior of hololnorphic discs near a given disc (see, for eXaIn-
pl(˜, [Gr]) shows that F n 152: 8 2 --. ˜n(82) has to be an immersion and, therefore,
elnbedding. Then the boundary of a holomorphic disc Fn(V t ), t e] - 1,1[, di-
hВ·1l
vicles the sphere <Pn(S2) into two henlispheres S˜t and S;,t. Let w be the standard
˜Ylnplectic form in C 2 вЂў Then, according to Stokes' theorem,
J Jwi
= I
w= $ Area S:.t $ Area rt'n(S2) В·
Area Fn(Vt )
S˜,t
Fn ('Dr )
'rllis means that a.reas of all hololllorphic discs are uniformly bounded and according
to Gromov's cOlnpactness theoreln (see [Gr]) for each t E] - 1,1[ the sequence
II˜, Iv&, n = 1, ... , has a converging subsequence. (It is easy to check that cusp-
.1'В·˜(˜llerations cannot occur.) But then each point of the immersed sphere <,0(82 ) C
l 2
H has to belong to a hololnorphic disc. But it is not so for points from
CC
.,,('Do ) C cp(S2). Q.E.D.
10. OPEN QUESTIONS AND CONJECTURES
10.1 Which prime 3-manifolds (LrC pseudo-convex boundaries of compact domains
tit 1.lJ,',ne almo..qt complex manifold˜?
Wr' will call such 3-Inanifolds fillable. No one example of a nonfillable manifold
i˜1(llown. Note that without the condition "talne" all 3-manifolds are, obviously,
UIII1.I)le.
111. I.A If one drops the word "almost" in 10.1 does it change the class of fillable
:J uULnifolds ˜
N. If,f' that results of [Ell] ra.ise the hope tha.t the clas of fillable (in both senses)
:1 Illu.nifolds is pretty large.
10.2 How many different (lВЈP to blowing up) 4-manifolds can have the same 3-
'" fUI.7!()ld as a pseudo-convex boundary f
'I'll(' sphere S3 has the unique filling (see 5.1) but it is possible that the same M 3
1)(' filled by a few different 4-ll1al1ifolds (for example, the lense space L(4, 1) can
.. " II
I..В· liJhВ·<t at least by t\VO different lunnifolds, see [MeDl]).
66 Eliashberg: Filling by holomorphic discs and its applications
10.3 How many different fillable structures can arise on the same 3-manifolds?
Conjecture: A fillable contact structure on a 3-manifold is uniquely determined by
its Euler class (see Section 4). Together with 4.3 it would imply the existence of
only finite different fillable contact structures on a given 3-manifold. Theorem 5.3
proves the conjecture for 1'.1 = 8 3 вЂў
an
n be a compact domain with the J-convex boundary
10.4 Let in a tame
almost complex manifold X, J. Does n admit a J -convex function h : n --+ R which
an
f The answer is kno\vn to be negative if one does not allow certain
is constant on
degeneration of h.
Conjecture: There exists a s'ubset E clot n which is a union of J -holomorphic
n admits a J -convez function <p : n --+
curves and Levi-flat hypersurfaces such that
;˜
and J -con'vex in n\ E. If J is integrable than this is true '.'˜
IR which is constant on E
::˜
(even without Levi-flat components of E) according to a Grauert-Rossi's theorem. ,Ii
;˜
10.5 Let L be a Legendrian knot in the standard contact 53. Let top(L) be L itself
considered as a topological knot, 7"( L) and tb( L) be rotation number and Thurston- '˜
Bennequin's invariant of L (see 1.3). Does the triple (top(L), r(L), tb(LВ») define L ;'˜
J
up to a Legendrian isotopy?
˜j
I think that the answer is positive \vhen top(L) is trivial (compare 7.1) and negative ˜:˜
˜˜
in the general case.
'$
˜
10.6 J-convex h-cobordism problem. Let 11 be a domain in a tame almost .вЂў
j
complex (X, J) which is diffeomorphic to M x I for a 3-manifold M. Let h : n -+
\.(
n admit a ':i˜:
1. Does
I be a J-convex function such that h IMxO= 0, h IMxl=
!
nondegenerate J-convex function with this property? .;!,.t,
˜i;
t˜
10.7 Prove the conjecture from [HE] which gives 4-dimensional generalization of ˜
n be a domain with J-convex boundary in a .:˜
Bennequin's inequality (see 4.2). Let
nwhose boundary is contained
tame (almost) complex (X, J) and ˜1 be a surface in
in an and is transversal to the conta.ct structure eon an. Let us orient M in such a
aM.
way that it induces the positive (see 1.2) orienta.tion on the transversal Then
:5 c(lvI).
˜0
d_(M) or, equivalently, X(lvI)
Eliashberg: Filling by holomorphic discs and its applications 67
it. is shown in section 9, the procedure of filling by holomorphic discs which
an (see 4.2) fails in the general case but one
c
)ves the inequality for the case 1\1
hope that the result is, nevertheless, true.
1
REFERENCES
D. Bennequin, Entrelacements et equations de Pfaff, Asterique 107-108 (1983),
83-161.
E. Bishop, Differentiable Inallifolds in cOlnplex Euclidean space, Duke Math. J., 32
(1965), 1-22.
E. Bedford and B. Gaveau, Envelopes of holomorphy of certain 2-spheres in C 2 ,
Amer. Journal of Math., 105 (1983), 975-1009.
E. Bedford and W. I(lingehbcl'g, On the envelope of holomorphy of a 2-sphere in
C 2 , preprint, 1989.
Ya. Eliashberg, Topological cha.ra.cterization of Stein ma.nifolds of dimension> 2,
preprint, 1989, to appear in International Journal of Math.
Va. Eliashberg, Three lectures synlplectic topology, Proceedings of the Confer..
011
t:nce on Diff. Geometry in Gala Gonone, 1988.
Va. Eliashberg, !(illing of elliptic cOlnplex points and the Legendrian isotopy, in
I>reparation.
M. Gromov, Pseudo-hololnorphic curves in sYlnplectic ma.nifolds, In'll. Math., 82
(1985),307-347.
V. M. Harlanl0v and Va. Eliashberg, On the nUlllber of complex points of a real
Proc(˜etlings
surface in a compex sluface, of Leningrad Int. Topology Conference,
1082, 143-148.
I). McDuff, The structure of rational nnd ruled sYlIlplectic manifolds, preprint, 1989.
(). Mcduff, The loca.l behavior of holo1l10rphic curves in alnlost complex 4-manifolds,
preprint, 1989.
PART 2
JONES/WITTEN THEORY
71
Introduction
The discovery by Vaughan Jones of a new polynomial invariant of links in the
a-sphere was an important breakthrough which has lead to the introduction of a
whole range of new techniques in 3-dimensional topology. The original Jones invari-
hut, a Laurent polynomial in one variable, was obtained via a "braid" description
c)f a link, utilising the remarkable properties of some representations of the braid
KI'OUP which arose in the theory of Von Neumann algebras. Early developments
were largely combinatorial, leading to alternative definitions of the invariant and
tC) generalisations, including a 2-variable polynomial which specialises to both the
.'()lles polynomial and the classical Alexander polynomial after appropriate substi-
ttlt,ions. The new invariants are comparatively easy to calculate and have had many
.'C)11Crete applications but for some time no really satisfactory conceptual definition
c)f the invariants was known: one not relying on special combinatorial presentations
c.r u.link. It was not clear, for example, whether such invariants could be defined for
links in other 3-manifolds. While there were many intriguing connections between
t.ll.˜ Jones theory and statistical mechanics, for example through the Yang-Baxter
f'cl'lations and the newly developed theory of quantum groups, it was a major prob-
ItВ·lll to find the correct geometrical setting for the Jones theory. We refer to [L] for
˜\ Httrvey of this phase of the theory.
his lecture at the International Congress of Mathematical Physicists in Swansea,
III
.J uly 1988, Witten proposed a scheme which largely resolved this problem. He
Mhowed that the invariants should be obtained from a quantum field theory with
14n.˜rangian involving the Chern-Simons invariant of connections. This scheme pro-
vi.led a truly natural definition of the invariants, and indeed allowed considerable
"Pll(˜ralisations, to links in arbitrary 3-manifolds. Taking, in particular, the empty
llllk, Witten obtained new invariants of closed 3-manifolds. The challenge in this
h,pproach arose from the notorious difficulties of quantum field theory, in attach-
illS( real meaning to the functional integrals over the space of connections which
w..r.˜ involved. It is, to a mathematician, a striking fact that, despite these serious
foundational difficulties, Witten's theory made concrete predictions which could be
vt˜l'ified on a more elementary level. The new invariants appear to have great poten-
Un.l, Hnd it seems possible that they might even be able to detect counter examples
t,CJ t.he Poincare conjecture, should any exist.
'n) develop the quantum field theory of the Chern-Simons function on a more rigor-
basis, Witten has also given a Hamiltonian approach, which unites these ideas
CUlM
wit.h conformal field theory in two dimensions. This approach starts by quantising
n ct('rtain symplectic manifold naturally associated to a closed surface E, the moduli
f.1IHl.Ce of representations of ?rI(E) in a given compact Lie group. This leads to the
.'C )Ilst.ruction of a projectively fiat connection on a certain bundle over the moduli
NIUl.(В·(˜ of complex structures on E, whose monodromy defines a representation of the
u,,,pping class group. More generally, one should take a Riemann surface with a set
of luarked points (corresponding to a link in a 3-manifold), and in the case when
t.I,вЂў. surface is the Riemann sphere one gets representations of the braid group.
I
Introduction
72
Meanwhile Reshetikhin and Turaev have defined new invariants, from a more com- :˜
binatorial and algebraic point of view, which it seems should agree with the Witten 1˜
invariants. They use a Dehn surgery description of a 3-manifold, and obtain a J
manifold invariant from a link invariant of Jones type. This approach by-passes the .˜
analytical difficulties in the quantum field theory, although the conceptual geometric :;i
:1
picture is less clear.
We were fortunate to have a number of lectures in Durham on these recent develop-.˜˜
ments, leading to the four articles below. The paper of Hitchin and Jeffrey's notes ;˜˜ ..˜
on Witten's three lectures concentrate on the geometric quantisation of the repre- .:˜
sentation spaces for surfaces; the paper of Kirby and Melvin gives the definition of .:˜
the Reshetikhin-Turaev invariants and evaluates them in some special cases. Jef- :.t
frey's notes also give Witten's explanation of how the Reshetikhin-Turaev formulae i˜
should be obtained in the Chern-Simons picture. The lecture of Atiyah lies closer to . ˜˜
the earlier work in Jones theory, involving the Hecke algebras; the paper describes .:˜
new topological constructions of braid group representations, due to Lawrence. .::1
11
:˜
[L] Lickorish, W.B.R. Polynomials for links Bull. London Math. Soc. 20 (1988)
1
˜˜˜8
,˜
n
:f1
New Results in Chern-Simons
Theory
Lectures by
Edward Witten
Institute for Advanced Study, Princeton, NJ, US.A. 08540
Notes by
Lisa Jeffreyl
Mathematical Institute, 24-29 St. Giles, Oxford OXl 3LB, UK
In these lectures, I will describe some aspects of an approach to the Jones polynomial
of knots, and its generalizations, that is based on a three-dimensional quantum
Y-ang-Mills theory in which the usual Yang-Mills Lagrangian is replaced by a Chern-
Simons action. This approach gives a manifestly three-dimensional approach to the
subject, but some of the key aspects of the story, involving the Feynman path
integral, are somewhat beyond the reach of present rigorous understanding. In
the first two lectures, I will describe aspects of the subject that can be developed
rigorously at present. This basically consists of a gauge theory approach to the
.lones representations of the braid group and their generalizations. In the last
lecture, I will describe the more ambitious Feynman path integral approach, which
is an essential part of the way that physicists actually think about problems such
as this one, and which gives the most far-reaching results.
'fhe first two lectures describe joint work with S. Della Pietra and S. Axelrod. In
this work, methods of symplectic geometry and canonical quantization are used to
associate a vector space 'HE to every oriented surface 1:. In this approach, we will
have to pick a complex structure J on ˜ as an auxiliary tool in constructing the 'Hr"
a.nd much of the effort will go into constructing a projectively flat. connection whose
lSupported ill part by all NSF Graduate Fellowship.
Witten: New results in Chern-Simons theory
74
existence shows that in a suitable sense, the 1-lt are independent of J. There are
several other approaches to this connection, due to Tsuchiya, Dena, and Yamada,
Segal, and Hitchin; Hitchin's approach, which is based on algebraic geometry, has
been presented at this meeting.
Once one has understood the association :E -+ 1-lt, one can use it to construct
topological invariants of three-manifolds (and, more generally, of links in three..
manifolds). To give a simple example, since the association :E -+ 'fit is natural,
a diffeomorphism f of ˜ gives rise to an endomorphism 9/ of 'Ht. If one forms a
three-manifold M f by taking the mapping cylinder of f (that is, by multiplying :E
by a unit interval and gluing the ends using f), then the invariant one associates to
M J is the trace of q>!.
The first lecture will be devoted to background about canonical quantization. In
the second lecture, we will focus more specifically on the problem of actual interest
- quantization of the moduli space .M of representations of the fundamental group
of a surface in a compact Lie group G. The third lecture describes more concretely
how these constructions lead to link invariants. A detailed description of these
results can be found in [2] and [15], along with more extensive references.
1. GEOMETRIC QUANTIZATION
(a) The prototype case: affine space
We begin with a standard affine symplectic space A = 1R 2n with coordinates :c 1 , вЂўвЂўвЂў ,
:c 2n and a symplectic form W = Wij d:c i A d:c j (Wij = -Wji is a real antisymmetric
matrix such that W is nondegenerate.) We may regard w as a transformation T ˜ T*
(T is the tangent space to A) by sending a tangent vector to its interior product
with w. Then the inverse w- 1 : T" -+ T will be denoted wii : thus we have
We examine the Heisenberg Lie algebra (essentially just given by the Poisson bra-
cket) given by {:c 1 , вЂўвЂўвЂў :c 2n } with the commutation relations
(1)
The quantization procedure for A constructs an irreducible unitary Hilbert space
representation of this algebra. In fact it gives a representation of the Heisenberg.
group, an extension by U(l) of the group of affine translations of 1R 2n вЂў By a theorem
75
Witten: New results in Chern-Simons theory
of Stone and von Neumann, there is a unique such representation up to isomor-
phism: moreover, the isomorphism between any two such representations is unique
up to multiplication by 8 1 C C. SO any two such representations are canonically
isomorphic up to a projective multiplier.
To construct representations of the Heisenberg group (first a rather uninteresting
reducible one and then an irreducible subrepresentation of it), we fix a unitary line
bundle ВЈ over A with a connection D such that the curvature D 2 = -iw. (Such
a bundle exists whenever w /21r represents an integral cohomology class - here it it
the trivial one!) The isomorphism classes of such bundles are given by H 1 (A, U(l)),
which is 0; so ВЈ is unique up to isomorphism. The isomorphism between two such
bundles is unique up to a constant gauge transformation, which must be of modulus
1 to preserve the metric.
r
We define 11,0 = L 2 (A, ВЈ). The Coo functions on A act naturally on 1-ВЈ0 as follows.
Given 4> E COO(A), the Hamiltonian function associated to it is VВў = w- 1 (dВўВ». Then
Вў acts on rio via an operator
(2)
(This represents a lifting to ВЈ of the vector field VВў, which preserves the symplectic
structure: VВў thus acts on the space of sections of ВЈ.) The map Вў 1--+ U</J is a
homomorphism of algebras, i.e.,
so this gives a representation of the Lie algebra of all Hamiltonian vector fields,
including those, the infinitesimal translations, comprising the Heisenberg algebra.
To get an irreducible subrepresentation of this, we pick a translationally invariant
complex structure J on A ( i.e., a constant endomorphism of T whose square is -1 )
such that:
(i) w is compatible ˜ith J in the sense that w(JX, JY) = w(X, Y) , or in other
words, in the complex structure J, w a form of type (1,1).
> 0 if X # 0).
(ii) w is positive with respect to J (i.e., w(X, J X)
2:i dz i 1\ dzi вЂў
Thus there are holomorphic linear functions zi such that w = i
Let us now consider the line bundle ([" D) which was introduced and fixed before
picking J. It acquires a new structure once J is picked. Indeed, as D has curvature
of type (1,1) with respect to J, D and J combine to give ВЈ a holomorphic structure.
76 Witten: New results in Chern-Simons theory
.˜
.˜
(One defines a lJ operator on [, as DO,l; the condition that this actually defines a ,i
:˜
holomorphic structure is [P = (DD)O,2 =101В°,2 = 0.)
",!
The line bundle ВЈ, can be given a particularly simple description once J is picked.
Let [,0 be the trivial holomorphic line bundle on A with the Hermitian metric (for
"p E coo(.Co) )
:i
'˜t
::˜
':˜
.˜˜
and the connection compa.tible with this metric a.nd the standard holomorphic struc-
"˜
ture" The curvature of this connection is
:i
˜
:˜
,'t
it
,
.˜˜
This is the formula that characterizes ВЈ', so ВЈ, is isomorphic to [,0, by an isomorphism ˜
that is unique up to a projective ambiguity" ::˜
::.˜
Now that J has been picked, the Heisenberg algebra that we are seeking to represent :˜
.˜
can be written
˜\
(3) ,˜;1
[Zi, zi]
˜˜
=
[zi, Z3] (1
,'˜
'i
We can represent this algebra. on the subspace ,˜
i
':j
of our original Hilbert space consisting L2 sections that are also holomorphic" This .˜
representation p is j
:˜
')˜
p(zi) "1/J = zi"p (4)
1
p(zi) "1/J = 8˜' 1/J ,:1
˜
,˜
The commutation relations are preserved, as ˜
y
8В· .. .8
-.(zJ"p) = S'3"p + Z3_."p"
8z" 8z'
To verify unitarity, note that
But
77
Witten: New results in Chern-Simons theory
so upon integrating by parts and using the holomorphicity of "p, we find that this
equals
Jd"zd"z,Вў exp(- :˜:>kzl')ix= (.,p,p(z')X)'
Ie
This representation of the Lie algebra in fact exponentiates to a representation of the
(complexified ) Heisenberg group (see for instance the discussion in [8], p. 188). It
is irreducible, because holomorphic functions can be approximated by polynomials.
Thus we have found the unique irreducible representation of the Heisenberg group
described by the Stone-von Neumann theorem.
We now want to let J vary, fixing its key properties (translation invariance and the
fact that w is positive of type (1,1)). The space T of all complex structures on
A with these properties is the Siegel upper half plane of complex symmetric n X n
matrices with positive imaginary parts. Over T we get a Hilbert space bundle 1-ВЈ
whose fibre over JET is /1,J. The uniqueness theorem for representations of the
Heisenberg group implies that there is a natural way to identify the fibres of this
bundle, i.e., 'H, has a natural projectively flat connection.
We would like to identify this connection explicitly. To this end, we observe that if
xi ˜ ii is any representation of (1), then the objects
themselves obey a Lie algebra, which can be worked out explicitly by expanding
the commutators, i.e.,
[Dij, D lel ] = _i(Dilw jle + Dikwil + D lj w i1c + Dkjw il )
This is just the Lie algebra of Sp(2n, R), if we identify A with R2n ( it is given by
the algebra of homogeneous quadratic polynomials under Poisson bracket). Thus,
the Lie algebra of Sp(2n, R) acts in any representation of the Heisenberg algebra..
Actually, the group Sp(2n, R) acts on the Heisenberg algebra by outer automor..
phisms, and so conjugates the representation ?-lJ of the Heisenberg group to another
representation. By the uniqueness of this irreducible representation, Sp(2n, R) must
act at least projectively on fiJ. The Dij give the action explicitly at the Lie algebra
level.
The zi act on /1,J as zeroth order differential operators, while the zi act as first order
operators. The Dij are thus at most second order differential operators. The Dij,
which generate vector fields that act transitively on T, essentially define a connec-
tion on 'H, and this connection is given by a second order differential operator since
the D's have this property. (Actually, the full group of symplectic diffeomorphisms
of A has a natural 'loprequantum" action on the big Hilbert space 'H,o. This gives
78 Witten: New results in Chern-Simons theory
a representation of the Lie algebra of Sp(2n,R) by first order operators Dij '. The
connection is really the difference between the D ij and the D ij '.)
The connection can be described explicitly as follows. Let 6 be the trivial connection
on bundle A x 'H o, regarded as a trivial Hilbert space bundle over A. 6 does not
respect the subspace rtJ C 1-{,o. A connection which does respect it can be described
by adding to the trivial connection a certain second-order differential operator. In
describing it, note that the Siegel upper half plane has a natural complex structure,
so the connection decomposes into (1,0) and (0,1) parts. Let T denote the tangent
space to A; then a tangent vector to T corresponds to a deformation of the complex
structure of A which is given explicitly by an endomorphism 6J : T ˜ T (which
obeys J 6J + 6J J = 0). Associated with such an endomorphism is the object
6J 0 w- 1 E T В® T, i.e.,. 6J 0 w- 1 : T* ˜ T, which in fact lies in 8ym2 T; indeed,
it lies in Sym2 T 1 ,O в‚¬I:) Sym2 TO,1. The (2,0) part of this (which is the (1,0) part of
aJ with respect to the complex structure on T ) will be denoted aJij. Then the
connection on 'Ii can be written:
V O,l
= SO,1
\71,0
= Sl,O - 0 (5)
t
= -˜ i,j=l Dz"
D.SJij˜
0
DZ3
4
tzi
is the covariant derivative acting on sections of ВЈ. As promised, this
where
connection is given by a second-order differential operator on sections of ВЈ'. In more
invariant notation we have
стр. 3


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Персидская казачья бригада в 1882-1885 гг
.
Поездка генерал-лейтенанта Генерального штаба А.Н. Куропаткина в Персию в 1895 гг.
Ксения Кривошеина (Франция).
Возвращение на родину: к 70-летию гибели матери Марии (Скобцовой)
Владимир Кружков (Россия).
Кто же он – генерал Карбышев
?
Советский остров в Вене
.
Д. П. Татищев: взлет и «падение» необыкновенного дипломата
Алла Новикова-Строганова (Россия).
"Слово, которое проповедуем". К 120-летию памяти Н.С. Лескова
Геннадий Литвинцев (Россия).
Пастырь добрый в чужой стране. Памяти святителя Ионы, Маньчжурского чудотворца
Людмила Хухтиниеми. (Финляндия).
Фельдмаршал Маннергейм, Императорская семья и Анна Александровна Танеева (монахиня Мария)
Павел Густерин (Россия).
Политика Ивана III на Востоке
Светлана Баконина (Россия).
Первый опыт духовного руководства русскими паломниками (из дневника священника, сопровождавшего богомольцев на Святую Землю в 1912 году)
Новая рубрика!
Протоиерей Георгий Митрофанов.
"Трагедия России: «запретные» темы истории XX века в церковной проповеди и публицистике"
Полемика, отзывы
Наши авторы
Страничка
памяти протоиерея Василия Ермакова
АДРЕСА ДОБРЫХ ДЕЛ:
Спасем от гибели храм, построенный Александром Суворовым!
Творческие мастерские православной службы помощи "Милосердие"
Заказ икон и картин из бисера
Московской службе «Инватакси» - нужна помощь
Электронный журнал "Россия в красках"
Вышел
летний номер
№ 51
журнала "Россия в красках"
Архив номеров
Проекты ПНПО "Россия в красках":
Публикация из архивов:
А. П. Ладинский. Путешествие в Палестину. 1937 г.
Русское Александровское подворье близ храма Воскресения. Путь к Александровскому подворью. Извлечение из издания ИППО 1901 г.
Храм Воскресения Господня в Иерусалиме и окружающие его святыни
Журнал "Русское Возрождение"
Закончена работа над созданием
первой серии цикла фильмов о Святой Земле
.
Фонд для фильмов о Святой Земле
Палестина под властью христианских императоров. (326-636 г.г.)
По Альфонсу Курэ. С.Петербург. 1894 год.
X. Свв. Савва и Феодосий
.
XI. Св. Савва и император Анастасий. До 528 г.
Раритетный сборник стихов из архивов "России в красках".
С. Пономарев. Из Палестинских впечатлений 1873-74 гг.
Павел Платонов.
К 10-тысячелетию Иерихона - самого древнего города на земле
Славьте Христа
добрыми делами!
Рекомендуем:
Наши баннеры для обмена ссылками
Почтовый ящик
интернет-портала "Россия в красках"
Наш сайт о паломничестве на Святую Землю
Новости
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12 июня 2007
В Азове открыт барельеф выдающемуся военачальнику Головину
Федор Головин был сподвижником Петра I, блестящим военачальником и государственным деятелем. Человек, чье имя связано с величием России. Его память увековичили в Азове.
далее..
11 июня 2007, 00:01
Архиепископ Берлинский Марк совершил Божественную литургию в возрожденном храме Всех святых, на земле Российской просиявших в Горненском монастыре в Иерусалиме
10 июня 2007 года в Иерусалиме была совершена Божественная литургия на праздник Всех святых, на земле Российской просиявших, в возрожденном храме в Горненском монастыре.
далее..
11 июня 2007
В храме Русской Православной Церкви Заграницей в Брюсселе был отслужен совместный благодарственный молебен по случаю восстановления единства Русской Церкви
Богослужение единой Русской Церкви состоялось в воскресенье 10 июня 2007 года, в праздник Всех святых, в земле Российской просиявших, в храме Воскресения Христова Русской Православной Церкви Заграницей в Брюсселе.
далее..
11 июня 2007
Утварь из походного храма императорской семьи передана храму в Мадриде
Глава Российского Императорского дома Мария Владимировна Романова передала в воскресенье общине храма Русской Православной Церкви в Мадриде утварь из походного храма императорской семьи - ларец со священными сосудами и антиминсом.
далее..
11 июня 2007
В Австрии открыт барельеф Тыртышникова "Россия. Повесть временных лет
Торжественная церемония открытия барельефа "Россия. Повесть временных лет" работы молодого российского скульптора Андрея Тыртышникова прошла в воскресенье в небольшом австрийском городке Кайзерштайнбрух (земля Бургенланд).
далее..
10 июня 2007
Германо-российский фестиваль проходит в Берлине
Посетители имели возможность увидеть различные концертно-развлекательные мероприятия, проходившие сразу на нескольких сценах, посетить живописный средневековый городок, отведать блюда русской и немецкой кухни, принять участие в спортивных конкурсах на площадке знаменитого футбольного клуба Schalke 04, познакомиться с деятельностью многочисленных фирм и организаций, участвующих в германо-российских обменах по самым различным направлениям.
далее..
10 июня 2007
Год русского языка в Мали
В рамках проведения Года русского языка Посольством России в Мали проводится целый ряд культурных мероприятий.
далее..
10 июня 2007
Конференции российских соотечественников
В конце мая по всему миру прошли конференции российских соотечественников.
далее..
10 июня 2007
Крестный ход памяти Михаила Романова пройдет в Перми
Впервые 12 июня в Перми состоится крестный ход, посвященный памяти великого князя Михаила Романова.
далее..
10 июня 2007
Государственный музей истории религии проводит выставку «Неизвестные работы известных мастеров»
В год своего 75-летия Государственный музей истории религии (ГМИР) открывает выставку, которая предоставляет зрителям уникальную возможность познакомиться с неизвестными ранее страницами истории русской художественной культуры конца XVIII — начала XIX вв. и преподносит ряд сюрпризов.
далее..
09 июня 2007
Патриотическая акция "Российский триколор" посвящена Дню России
"Единая Россия" призывает россиян прикрепить ленточку с российским триколором к одежде или на автомашину, показав тем самым свой патриотизм и причастность к нашему общему дому - России", - говорится в сообщении.
далее..
09 июня 2007
Объявлены лауреаты литературной премии "Единорог и лев
"Что объединяет британскую и российскую культуру? Наша любовь к слову", - сказал на церемонии посол Великобритании в России Энтони Рассел Брентон.
далее..
09 июня 2007
Памяти Палестинского Общества
Юбилейные торжества, посвященные 125-летию Императорского Православного Палестинского Общества (ИППО), пройдут 15 июня 2007 в Санкт-Петербурге
далее..
09 июня 2007
Выставка, посвященная Ордин-Нащокину, откроется в Пскове
"Государственного деятеля такого масштаба в России не было ни в допетровскую эпоху, ни при Петре I. Недаром иностранцы называли Ордин-Нащокина "русским Ришелье", - отметил сотрудник комитета.
далее..
09 июня 2007
В Новгороде открылась выставка, посвященная княгине Елизавете Романовой
Выставка "От милосердия к святости", посвященная основательнице московской Марфо-Мариинской обители милосердия, преподобномученице, великой княгине Елизавете Федоровне Романовой, открылась в пятницу в Государственном музее художественной культуры Новгородской земли на территории Десятинного монастыря Великого Новгорода.
далее..
08 июня 2007
В Брюсселе прошла конференция "Соотечественники в Бельгии: вопросы самоорганизации, интеграции, культурной идентичности"
6 июня 2007 года в Брюсселе прошла конференция "Соотечественники в Бельгии: вопросы самоорганизации, интеграции, культурной идентичности", организованная Посольством России в Бельгии, Брюссельско-Бельгийской архиепископией Русской Православной Церкви и Координационным советом ассоциаций соотечественников.
далее..
08 июня 2007
В Петербурге открылся XI Международный экономический форум
С 8 по 11 июня северная столица России станет местом встречи деловой элиты мира. Петербургский форум в этом году стал частью Всемирного экономического форума в Давосе.
далее..
08 июня 2007
Российские художники принимают участие в 52-й Биеннале современного искусства в Венеции
52-я Биеннале современного искусства - одно из главных событий в сегодняшнем арт-мире. Россию представляют несколько художников, работы которых выставлены в историческом здании, построенном в 1914 году по проекту Алексея Щусева и отремонтированном этой весной.
далее..
08 июня 2007
В греческом городе Трикала проходит фотовыставка РИА Новости
Около 40 широкоформатных фотографий рассказывают о народных обычаях России, знакомят с традиционной русской Масленицей, татарским праздником Сабантуй, праздниками адыгейцев, калмыков, чукчей, цыган, народов Дагестана и других национальностей России.
далее..
08 июня 2007
Учрежден специальный приз Конкурса имени Чайковского "Золотой рубанок"
Приз представляет собой произведение ювелирного искусства из золота с бриллиантами.
далее..
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Всего: 4242
[
версия для печати
]
© 2004 – 2015 Educational Orthodox Society «Russia in colors» in Jerusalem
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markdown:
### Cọ bình sữa Dr.Brown
Hotline:Bán lẻ : 0915 092 097
Bán buôn : 0983 092 097
Bán buôn : 0936 092 097
|
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Tên sản phẩm
|
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Cọ bình sữa Dr.Brown
|
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Xuất xứ
|
|
Mỹ
|
|
Kích thước
|
|
27 x 5 cm
|
|
Chất liệu
|
|
Sợi PP
**được sử dụng cho bình sữa cổ thường hoặc cổ rộng, giúp làm sạch dễ dàng các bình sữa và đầu ty silicone mà không làm xước hay hỏng. Sản phẩm không dùng các độc tố hoá học, cũng như các hợp chất gây hại cho sức khoẻ. Cọ bình sữa Dr.Brown ** **Cọ bình sữa Dr.Brown có nhiều ưu điểm:**
· Làm từ nhựa PP an toàn, không chứa BPA.
·
**với lông mềm mại, cọ sạch mọi ngõ ngách của bình. Cọ bình sữa **
· Sản phẩm có thiết kế đặc biệt có thể tự đứng thẳng, không đổ kể cả khi bạn dựng trên mặt bàn bếp.
· Tay cầm chống trượt.
· Thiết kế vừa có đầu mút, vừa có các lông nhựa giúp làm sạch dễ dàng các bình sữa và đầu ty silicone mà không làm xước hay hỏng
**sữa. bình **
· Sản phẩm không dùng các độc tố hoá học, cũng như các hợp chất gây hại cho sức khoẻ.
· Có nhiều màu sắc để mẹ lựa chọn.
### Sản phẩm cùng tuổi
[Xem tất cả](http://shopquynhhoa.com.vn/cs0-cọ-bình-sũa-dr-brown.html)
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#### Giá :65,000đ
[Bánh ăn dặm Pigeon 6 tháng vị cà rốt, rau bina](http://shopquynhhoa.com.vn/cp950-banh-an-dam-pigeon-6-thang-vi-ca-rot-rau-bina.html) [Đặt hàng](http://shopquynhhoa.com.vn/gh950-banh-an-dam-pigeon-6-thang-vi-ca-rot-rau-bina.html)
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#### Giá :268,000đ
[Sữa nước Matilia bú vị socola, 200ml](http://shopquynhhoa.com.vn/cp946-sua-nuoc-matilia-bu-vi-socola-200ml.html) [Đặt hàng](http://shopquynhhoa.com.vn/gh946-sua-nuoc-matilia-bu-vi-socola-200ml.html)
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#### Giá :268,000đ
[Sữa nước Matilia bú vị caramel, 200ml](http://shopquynhhoa.com.vn/cp945-sua-nuoc-matilia-bu-vi-caramel-200ml.html) [Đặt hàng](http://shopquynhhoa.com.vn/gh945-sua-nuoc-matilia-bu-vi-caramel-200ml.html)
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#### Giá :268,000đ
[Sữa nước Matilia bú vị vani, 200ml](http://shopquynhhoa.com.vn/cp944-sua-nuoc-matilia-bu-vi-vani-200ml.html) [Đặt hàng](http://shopquynhhoa.com.vn/gh944-sua-nuoc-matilia-bu-vi-vani-200ml.html)
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#### Giá :50,000đ
[Váng sữa monte Pháp, vị socola, 4x55g](http://shopquynhhoa.com.vn/cp943-vang-sua-monte-phap-vi-socola-4x55g.html) [Đặt hàng](http://shopquynhhoa.com.vn/gh943-vang-sua-monte-phap-vi-socola-4x55g.html)
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#### Giá :50,000đ
[Váng sữa monte Pháp, vị vani, 4x55g](http://shopquynhhoa.com.vn/cp942-vang-sua-monte-phap-vi-vani-4x55g.html) [Đặt hàng](http://shopquynhhoa.com.vn/gh942-vang-sua-monte-phap-vi-vani-4x55g.html)
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#### Giá :Liên hệ
[Sữa chua Bledina 6 tháng vị quả mâm xôi, 6x60g](http://shopquynhhoa.com.vn/cp937-sua-chua-bledina-6-thang-vi-qua-mam-xoi-6x60g.html) [Đặt hàng](http://shopquynhhoa.com.vn/gh937-sua-chua-bledina-6-thang-vi-qua-mam-xoi-6x60g.html)
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#### Giá :Liên hệ
[Sữa chua Bledina 6 tháng vị mơ, 6x60g](http://shopquynhhoa.com.vn/cp936-sua-chua-bledina-6-thang-vi-mo-6x60g.html) [Đặt hàng](http://shopquynhhoa.com.vn/gh936-sua-chua-bledina-6-thang-vi-mo-6x60g.html)
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#### Giá :129,000đ
[Bột pha sữa Nestle, 6 tháng, vị cacao](http://shopquynhhoa.com.vn/cp934-bọt-pha-sũa-nestle-6-tháng-vị-cacao.html) [Đặt hàng](http://shopquynhhoa.com.vn/gh934-bọt-pha-sũa-nestle-6-tháng-vị-cacao.html)
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#### Giá :Liên hệ
[Sữa chua Bledina 6 tháng vị tự nhiên (6x60g)](http://shopquynhhoa.com.vn/cp925-sua-chua-bledina-6-thang-vi-tu-nhien-6x60g.html) [Đặt hàng](http://shopquynhhoa.com.vn/gh925-sua-chua-bledina-6-thang-vi-tu-nhien-6x60g.html)
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#### Giá :Liên hệ
[Sữa chua Bledina 6 tháng vị lê đào, 6x60g](http://shopquynhhoa.com.vn/cp924-sua-chua-bledina-6-thang-vi-le-dao-6x60g.html) [Đặt hàng](http://shopquynhhoa.com.vn/gh924-sua-chua-bledina-6-thang-vi-le-dao-6x60g.html)
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#### Giá :Liên hệ
[Sữa chua Bledina 6 tháng vị dâu, 6x60g](http://shopquynhhoa.com.vn/cp923-sua-chua-bledina-6-thang-vi-dau-6x60g.html) [Đặt hàng](http://shopquynhhoa.com.vn/gh923-sua-chua-bledina-6-thang-vi-dau-6x60g.html)
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#### Giá :Liên hệ
[Sữa chua Bledina 6 tháng vị chuối, 6x60g](http://shopquynhhoa.com.vn/cp922-sua-chua-bledina-6-thang-vi-chuoi-6x60g.html) [Đặt hàng](http://shopquynhhoa.com.vn/gh922-sua-chua-bledina-6-thang-vi-chuoi-6x60g.html)
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#### Giá :Liên hệ
[Ruốc cá hồi Nissui, 500g](http://shopquynhhoa.com.vn/cp921-ruoc-ca-hoi-nissui-500g.html) [Đặt hàng](http://shopquynhhoa.com.vn/gh921-ruoc-ca-hoi-nissui-500g.html)
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#### Giá :Liên hệ
[Phomai tròn Kraft Nhật vị việt quất](http://shopquynhhoa.com.vn/cp920-phomai-tron-kraft-nhat-vi-viet-quat.html) [Đặt hàng](http://shopquynhhoa.com.vn/gh920-phomai-tron-kraft-nhat-vi-viet-quat.html)
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#### Giá :Liên hệ
[Phomai tròm Kraft , Nhật vị chanh leo](http://shopquynhhoa.com.vn/cp919-phomai-trom-kraft-nhat-vi-chanh-leo.html) [Đặt hàng](http://shopquynhhoa.com.vn/gh919-phomai-trom-kraft-nhat-vi-chanh-leo.html)
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#### Giá :Liên hệ
[Cháo Wakodo 12 tháng vị thịt bò rim rau củ](http://shopquynhhoa.com.vn/cp911-cháo-wakodo-12-tháng-vị-thịt-bò-rim-rau-củ.html) [Đặt hàng](http://shopquynhhoa.com.vn/gh911-cháo-wakodo-12-tháng-vị-thịt-bò-rim-rau-củ.html)
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#### Giá :Liên hệ
[Cháo Wakodo 9 tháng vị gà, cá ngừ, nấm](http://shopquynhhoa.com.vn/cp908-cháo-wakodo-9-tháng-vị-gà-cá-ngù-nám.html) [Đặt hàng](http://shopquynhhoa.com.vn/gh908-cháo-wakodo-9-tháng-vị-gà-cá-ngù-nám.html)
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#### Giá :Liên hệ
[Cháo Wakodo 9 tháng vị cá trắng, rau củ](http://shopquynhhoa.com.vn/cp907-cháo-wakodo-9-tháng-vị-cá-tráng-rau-củ.html) [Đặt hàng](http://shopquynhhoa.com.vn/gh907-cháo-wakodo-9-tháng-vị-cá-tráng-rau-củ.html)
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#### Giá :Liên hệ
[Cháo Wakodo 7 tháng vị cá ngừ, rau](http://shopquynhhoa.com.vn/cp903-cháo-wakodo-7-tháng-vị-cá-ngù-rau.html) [Đặt hàng](http://shopquynhhoa.com.vn/gh903-cháo-wakodo-7-tháng-vị-cá-ngù-rau.html)
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#### Giá :Liên hệ
[Cháo Wakodo 7 tháng vị cá hồi, rau củ](http://shopquynhhoa.com.vn/cp902-cháo-wakodo-7-tháng-vị-cá-hòi-rau-củ.html) [Đặt hàng](http://shopquynhhoa.com.vn/gh902-cháo-wakodo-7-tháng-vị-cá-hòi-rau-củ.html)
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#### Giá :Liên hệ
[Cháo Wakodo 7 tháng vị cá bơn, trứng](http://shopquynhhoa.com.vn/cp901-cháo-wakodo-7-tháng-vị-cá-bon-trúng.html) [Đặt hàng](http://shopquynhhoa.com.vn/gh901-cháo-wakodo-7-tháng-vị-cá-bon-trúng.html)
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#### Giá :Liên hệ
[Cháo Wakodo 7 tháng vị gạo Nhật, gà](http://shopquynhhoa.com.vn/cp900-cháo-wakodo-7-tháng-vị-gạo-nhạt-gà.html) [Đặt hàng](http://shopquynhhoa.com.vn/gh900-cháo-wakodo-7-tháng-vị-gạo-nhạt-gà.html)
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#### Giá :Liên hệ
[Cháo Kewpie 9 tháng vị thịt gà và cây ngưu bàng](http://shopquynhhoa.com.vn/cp899-chao-kewpie-9-thang-vi-thit-ga-va-cay-nguu-bang.html) [Đặt hàng](http://shopquynhhoa.com.vn/gh899-chao-kewpie-9-thang-vi-thit-ga-va-cay-nguu-bang.html)
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#### Giá :Liên hệ
[Cháo Kewpie 9 tháng vị sukiyaki](http://shopquynhhoa.com.vn/cp898-chao-kewpie-9-thang-vi-sukiyaki.html) [Đặt hàng](http://shopquynhhoa.com.vn/gh898-chao-kewpie-9-thang-vi-sukiyaki.html)
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#### Giá :Liên hệ
[Cháo Kewpie 9 tháng vị rau củ và gan gà](http://shopquynhhoa.com.vn/cp897-chao-kewpie-9-thang-vi-rau-cu-va-gan-ga.html) [Đặt hàng](http://shopquynhhoa.com.vn/gh897-chao-kewpie-9-thang-vi-rau-cu-va-gan-ga.html)
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#### Giá :Liên hệ
[Cháo Kewpie 9 tháng ngũ sắc](http://shopquynhhoa.com.vn/cp896-chao-kewpie-9-thang-ngu-sac.html) [Đặt hàng](http://shopquynhhoa.com.vn/gh896-chao-kewpie-9-thang-ngu-sac.html)
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#### Giá :Liên hệ
[Cháo Kewpie 9 tháng vị cá hồi và rau củ](http://shopquynhhoa.com.vn/cp895-chao-kewpie-9-thang-vi-ca-hoi-va-rau-cu.html) [Đặt hàng](http://shopquynhhoa.com.vn/gh895-chao-kewpie-9-thang-vi-ca-hoi-va-rau-cu.html)
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#### Giá :Liên hệ
[Cháo Kewpie 9 tháng ninh ngũ vị](http://shopquynhhoa.com.vn/cp894-chao-kewpie-9-thang-ninh-ngu-vi.html) [Đặt hàng](http://shopquynhhoa.com.vn/gh894-chao-kewpie-9-thang-ninh-ngu-vi.html)
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#### Giá :119,000đ
[Bột ăn dặm Gerber yến mạch vị chuối, dâu tây, 227g](http://shopquynhhoa.com.vn/cp893-bot-an-dam-gerber-yen-mach-vi-chuoi-dau-tay-227g.html) [Đặt hàng](http://shopquynhhoa.com.vn/gh893-bot-an-dam-gerber-yen-mach-vi-chuoi-dau-tay-227g.html)
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#### Giá :119,000đ
[Bột ăn dặm Gerber yến mạch, chuối, 227g](http://shopquynhhoa.com.vn/cp892-bot-an-dam-gerber-yen-mach-chuoi-227g.html) [Đặt hàng](http://shopquynhhoa.com.vn/gh892-bot-an-dam-gerber-yen-mach-chuoi-227g.html)
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#### Giá :119,000đ
[Bột ăn dặm Gerber vị yến mạch, 227g](http://shopquynhhoa.com.vn/cp891-bot-an-dam-gerber-vi-yen-mach-227g.html) [Đặt hàng](http://shopquynhhoa.com.vn/gh891-bot-an-dam-gerber-vi-yen-mach-227g.html)
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#### Giá :119,000đ
[Bột ăn dặm Gerber vị lúa mỳ, táo, việt quất](http://shopquynhhoa.com.vn/cp890-bọt-an-dạm-gerber-vị-lua-my-tao-viet-quat.html) [Đặt hàng](http://shopquynhhoa.com.vn/gh890-bọt-an-dạm-gerber-vị-lua-my-tao-viet-quat.html)
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#### Giá :119,000đ
[Bột ăn dặm Gerber vị yến mạch, táo, quế, 227g](http://shopquynhhoa.com.vn/cp889-bot-an-dam-gerber-vi-yén-mạch-tao-que-227g.html) [Đặt hàng](http://shopquynhhoa.com.vn/gh889-bot-an-dam-gerber-vi-yén-mạch-tao-que-227g.html)
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#### Giá :119,000đ
[Bột ăn dặm Gerber vị ngũ cốc, táo, khoai lang](http://shopquynhhoa.com.vn/cp888-bọt-an-dạm-gerber-vị-ngu-coc-tao-khoai-lang.html) [Đặt hàng](http://shopquynhhoa.com.vn/gh888-bọt-an-dạm-gerber-vị-ngu-coc-tao-khoai-lang.html)
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#### Giá :119,000đ
[Bột ăn dặm Gerber vị gạo, táo, chuối, 227g](http://shopquynhhoa.com.vn/cp887-bot-an-dam-gerber-vi-gao-tao-chuoi-227g.html) [Đặt hàng](http://shopquynhhoa.com.vn/gh887-bot-an-dam-gerber-vi-gao-tao-chuoi-227g.html)
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#### Giá :119,000đ
[Bột ăn dặm Gerber vị gạo, 227g](http://shopquynhhoa.com.vn/cp886-bot-an-dam-gerber-vi-gao-227g.html) [Đặt hàng](http://shopquynhhoa.com.vn/gh886-bot-an-dam-gerber-vi-gao-227g.html)
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#### Giá :119,000đ
[Bột ăn dặm Gerber vị ngũ cốc, dâu, mâm xôi, 227g](http://shopquynhhoa.com.vn/cp885-bọt-an-dạm-gerber-vị-ngu-coc-dau-mam-xoi-227g.html) [Đặt hàng](http://shopquynhhoa.com.vn/gh885-bọt-an-dạm-gerber-vị-ngu-coc-dau-mam-xoi-227g.html)
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#### Giá :119,000đ
[Bột ăn dặm Gerber vị yến mạch, đào, táo, 227g](http://shopquynhhoa.com.vn/cp883-bọt-an-dạm-gerber-vị-yen-mach-dao-tao-227g.html) [Đặt hàng](http://shopquynhhoa.com.vn/gh883-bọt-an-dạm-gerber-vị-yen-mach-dao-tao-227g.html)

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NĂM SINH HOT
sim nam sinh 1985
sim nam sinh 1986
sim nam sinh 1987
sim nam sinh 1988
sim nam sinh 1989
sim nam sinh 1990
sim nam sinh 1991
sim nam sinh 1992
sim nam sinh 1993
sim nam sinh 1994
sim nam sinh 1995
Yêu Cầu
(Quý khách chú ý) Những đơn đặt hàng đầy đủ thông tin cần thiết (*) và được viết bằng tiếng việt có dấu là những đơn đặt hàng chính xác.
0968.16.6789
0943.000.666
HỆ THỐNG WEBSITE
http://simnamsinh.com.vn
http://nganhangsodep.vn
http://simtaybac.vn
TRUNG TÂM SIM SỐ ĐẸP - SIM NĂM SINH LỚN NHẤT VIỆT NAM
Hotline : 0943.000.666 - 0968.16.6789 Email: simnamsinh.com.vn@gmail.com
Địa chỉ: 147 Nguyễn Trãi - Thanh Xuân - Hà Nội

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##
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markdown:
-
[THÔNG BÁO TUYỂN ADMIN DIỄN ĐÀN 2013](http://spkt.net/threads/thong-bao-tuyen-02-admin-cho-4rum.32345/)
[Tìm kiếm nhà trọ - Ở ghép](http://www.spkt.net/forums/thong-tin-nha-tro.138/)
[THÔNG BÁO BÁN ÁO SPKT.NET CHO THÀNH VIÊN DIỄN ĐÀN](http://spkt.net/threads/thong-bao-ban-ao-dien-dan-spkt-2013.27570/)
**HÃY TÌM KIẾM Ở ĐÂY TRƯỚC KHI ĐẶT CÂU HỎI**{xen:phrase loading}
# Những giai thoại hay nhất về tình yêu và cuộc sống (Tiếng Việt)
Thảo luận trong '
[Tổng hợp](forums/tong-hop.89/)' bắt đầu bởi [svsp](members/svsp.38727/), [6 Tháng một 2011](threads/nhung-giai-thoai-hay-nhat-ve-tinh-yeu-va-cuoc-song-tieng-viet.11854/).


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markdown:
Theo bác sĩ Phòng khám phụ khoa Thiên Hòa thì có rất nhiều
** mà chị em không để ý tới. Vậy, những nguyên nhân đó là gì? Cùng tìm hiểu về nguyên nhân gây vô sinh nữ qua những thông tin được chia sẻ trong bài viết dưới đây. nguyên nhân gây vô sinh nữ**
**Những nguyên nhân gây vô sinh nữ:**
– Kinh nguyệt không đều: là tình trạng rối loạn nội tiết, ảnh hưởng trực tiếp tới việc phát triển và phóng noãn của trứng làm cản trở trực tiếp tới quá trình thụ thai.
– Bị các bệnh viêm nhiễm phụ khoa: khí hư ra nhiều,
[, viêm cổ tử cung, bệnh viêm loét cổ tử cung và các bệnh lây truyền qua đường tình dục. Đây là nguyên nhân chủ yếu gây vô sinh ở nữ giới. Các bệnh viêm nhiễm phụ khoa thông thường là do vệ sinh không đúng cách, quan hệ tình dục thô bạo… bệnh nấm âm đạo](http://suckhoephukhoa.net/viem-nhiem-phu-khoa/nam-am-dao/)
– Các bệnh lý bất thường: đau bụng kinh dữ dội, đau thắt lưng và đau vùng bụng dưới có thể dẫn tới viêm vùng chậu,
**bệnh viêm cổ tử cung**, viêm buồng trứng, ung thư buồng trứng, lạc nội mạc tử cung, ung thư tử cung làm ảnh hưởng nghiêm trọng tới chất lượng trứng gây vô sinh.
– Trong độ tuổi tiền mãn kinh: Thông thường từ 55 tuổi trở đi mới là độ tuổi mãn kinh, tuy nhiên cũng có những chị em mãn kinh từ 40 tuổi, thậm chí có thể còn sớm hơn.
– Tình trạng viêm nhiễm lâu dài dẫn tới ứ dịch, tắc vòi trứng làm cản trở tinh trùng bơi đến gặp trứng. Đây cũng là một nguyên nhân dẫn tới tình trạng vô sinh ở nữ giới ngày càng nhiều vì nhiều khi chị em không biết mình đang bị viêm nhiễm và bị viêm nhiễm từ khi nào.
– Do tử cung dị dạng, vòi trứng gấp khúc làm trứng không thể thụ tinh hoặc không thể làm tổ.
– Do ảnh hưởng của các loại thuốc: Một trong những nguyên nhân gây vô sinh nữ là do sử dụng các loại thuốc tránh thai, thuốc kháng sinh hoặc các loại thuốc giảm cân làm thay đổi nội tiết tố nữ trong cơ thể dẫn đến rối loạn quá trình rụng trứng.
– Bệnh u xơ tử cung sẽ làm hạn chế quá trình hình thành phôi dẫn tới trứng khó phát triển và gây trở ngại trong quá trình cấy phôi. Đây là 1 nguyên nhân làm cho phụ nữ ở tuổi 30-40 khó có con.
– Sử dụng quá nhiều các chất kích thích làm trì hoãn quá trình rụng trứng. Các chuyên gia đã chứng minh, sử dụng chất kích thích làm giảm khả năng sinh sản, thay đổi nội tiết tố, là tác nhân gây ra tình trạng vô sinh ở nữ ngày càng nhiều.
Để giảm thiểu tối đa nguy cơ vô sinh, chị em cần phải đến ngay các
[ thường xuyên để phát hiện sớm nhất những bệnh viêm nhiễm phụ khoa, những dị tật ở tử cung, buồng trứng… để có hướng điều trị bệnh sớm nhất. Ngoài ra, chị em nên bổ sung các loại vitamin cần thiết đặc biệt là vitamin E để có một sức khỏe tốt, tăng khả năng thụ thai. phòng khám phụ khoa uy tín](http://suckhoephukhoa.net/)


markdown: Một thương hiệu thời trang và thể thao cao cấp từ Mỹ. Hơn 100 năm qua, New Balance luôn tìm hiểu nhu cầu của những vận động viên để đầu tư, thiết kế những sản phẩm phù hợp nhất

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