<< ñòð. 2(âñåãî 9)ÑÎÄÅÐÆÀÍÈÅ >>
So we find, summing up

V -+ R defined by
'rheorem 3.3 The map p : V X

pew, c) = E(w- I Â«1Â»
0

tlt:fines a bi-invariant metric on V. Moreover the map

Lipschitz continuous with Lipschitz constant equal to 1:
1.11

v.
II.cmark 3.4 The map H -+ WH extendJ to the completions C --+ Since for
˜. ˜ 2 the map H --+ 'II H can be considered as a map Ck --+ C k - I J the above theorem

.t4!t.Ol1JS philosophically that this map extends to a map Co --+ C-I. So in some sense
l' d(˜fines a C-I-topology.
22 Hofer: Symplectic capacities

Remark 3.5 The topology on V is very weak. It does for example not even imply
pointwise convergence. For example we can have

in (V,p) and

˜ 00.
as k -+ 00 for 1 ::; p It would be interesting to have a good model for the
completion V.

Remark 3.6 The displacement capacity can be considered - to a certain extent - as
a nonlinear extension of the 2-dimensional Lebesgue measure to higher dimensions.
To see this, note that bounded simply connected domain S with smooth boundary is
symplectomorphic to B 2 (R) for some R. Hence

= -rrR2 = = meas (S).
deS) = d(B 2 (RÂ» meas (B 2 (RÂ»

Next we consider whether there are other invariants of a set S C R 2 n which
are related to the displacement capacity. Let us assume S is a connected compact
smooth hypersurface in R 2n â€¢ The canonical line boundle Â£s ˜ S is defined by

.c s = {(x,e) E TS J w(e,1J) = 0, for all,., E TxS}.

We denote by Ls(x) the leaf through xES and call it a characteristic on S.
Of particular interest are the closed characteristics. We denote the set of closed
characteristics of S by V(S). We give Â£s an orientation by choosing a section e
of Â£ s -+ S such that w( e(x), n(xÂ» > 0, where x -+ n( x) is the outward pointing
normal vector field. Recall that by Alexander duality R 2 n \ S has precisely two
components.
It is an open problem if V(S) =F 4> for every compact hypersurface. By a result
of C. Viterbo, [8], we have that V(S) =F <P if S is of contact type, [9].
Now let S be a compact connected smooth hypersurface with the orientation of
Â£ s ˜ S just defined. If P E 'D( S) we have

and therefore P inherits on orient!ttion from Â£s. Define the action of P by

where D is a disk in R2n, with aD = P and compatible orientation. Clearly A(P)
is well defined.
We have the following strong squeezing theorem which follows immediately from
results in [10].
Hofer: Symplectic capacities 23

'I'heorem 3.7 (Strong Squeezing) Assume S is a smooth compact hypersurface
IUHtnding a convez domain C. Then

I'11(B 2n (r)) C C
sup {1rr 2 for some '11 E V}

IP
˜ inf {A(P) E V(S)}

inf {1rr 2 I '11(C) C z2n(r)
< for some '11 E V}

Jt˜I'1' C = B2n(1) we recover Gromov's squeezing theorem.

It might be possible that we have in fact equality for the three expressions. This
1M fitill an open problem.
The following theorem which is a consequence of results in [7], gives a nice a
In'iori estimate for th˜ displacement energy.

'I˜heorem 3.8 Assume S i3 a smooth compact hyper8urface bounding a convez do ..
C. Then
1u(tin
d(C) ˜ inf {A(P) IP E V(S)}.

Note that for ellipsoids we have equality. That symplectic capacities are related
t.C) closed characteristics on hypersurfaces is no accident and has been detected in
{2]. To make it plausible consider the following:
Let C be a subset of R2n and define its capacity as the infimum of the outer
IIU˜fl,sure of the projections of all symplectic images of C onto the (Zl = (Xl, Yl))-
{I( )ordinate-plane.

This defines a symplectic capacity, as an easy consequence of Gromov's squeezing
f.ll(˜orem. Assume S = BC is a compact hypersurface. Suppose also that the projec-
tioll onto the zl-coordinate has minimal area. In order to decr˜ase this projection-
IU'('a. further (which is impossib˜e by our assumption) one would like to take a Hamil-
f,{)uian H which is increasing on the leaves of S, because the associated Hamiltonian
vc'ctor field pushes S into C. Of course such an H does not exist globally, but on
r˜uhregions of S. If some region contains a closed characteristic it cannot be the do-
Illn.in of a Hamiltonian which is increasing on leaves. So closed characteristics occur
h,H obstructions against decreasing the measure of certain projections by pushing

4"(tl't.ain subregions of S into C.
We already motivated the oÂ·ccurence of periodic Hamiltonian trajectories in the
utudy of symplectic capacities. The next construction makes extensive use of peri-
tHlie solutions of Hamiltonian Systems and gives a whole sequence of independent
.Â·,q>a,cities.
Hofer: Symplectic capacities
24

Periodic Hamiltonian Trajectories and Capacities
In [11] I. Ekeland and the author studied periodic solutions of Hamiltonian
Systems on a prescribed energy surface. In [11] they associated to convex energy
surfaces S symplectic invariants and to their periodic trajectories three independent
invariants and showed that these three quantities are not independent. A little bit
later C. Viterbo, [8J, proved the R2n-Weinstein conjecture, [9]. Viterbo's key in-
gredient in his proof, meanwhile called Viterbo's trick, was simplified and exploited
by E. Zehnder and the author, [11], in the study of periodic Hamiltonian trajecto-
ries on a prescribed energy surface. Roughly speaking one associates to a bounded
open set U C R 2n a variational problem in the loop-space of R2n, which gives
useful information about the Hamiltonian flow on the energy surface au. These
developments were the starting point for the following construction of symplectic
capacities.
We define a particular class of Hamiltonian systems on R 2n, which we denote
by 'H, as follows:
R belongs to 'Ii if
A Hamiltonian H : R2n -+

u == O. Moreover H(z) ˜ 0
HI There exists an open nonempty set U such that H 1

for all z E R 2 n.

rt?r' Z, and H(z) = a I z
H2 There exist numbers p,a > 0, such that a > 2
a
1r, 1

for all I z I˜ p.

Denote by F the loop space of R2n, i.e. the space of all smooth maps R/Z -+
R n. We put 51 := R/Z and define ˜H : F --+ R for H E 1-l by
2

1
f f H(z)dt.
1 1
2' < -Jz,z > dt -
epH(Z) =
o 0

Here J is given by

(˜ ˜1 ) o
J=

(˜ ˜1)
o 0

The critical points of ˜ H are precisely the I-periodic solutions z of the differ-
ential equation -Jz = H'(z), where H' is the gradient of H with respect to the
z
standard inner product < ',' >. Alternatively we have of course = XH(Z), since
XH = JH' is the Hamiltonian vector field associated to H. Our aim is to find
infinitely many universal minmax-characterisations for critical points of ˜H. For
25
Hofer: Symplectic capacities

thiH we take the completion F of F with respect to the Hi-norm on F. This is
cl(˜nned as follows: Every element in F can be written as

E exp (21rtkJ)Zk, Zk E R 2n
Z=
keZ

< 00 for every s
L: I k
˜lI(˜h that ˜
2
1 1 Zk 1
8
O. We define a norm by

IIzU˜. =1 1 +21r Elk II Zk
2 2
Zo ,
1
keZ

n.llel denote the completion of (F, II Â· II) by F. F has an orthogonal splitting F =
1"- EB FO EI7 F+ by cutting a Fourier series into pieces k < 0, k = 0 and k > O. The
1
l I(J z, z} dt extends to a quadratic form z
z a(z) on F, with a gradient
IIln.p t-+ t-+

tt' I!;iven by Â°
= -z- + z+, = z- + zo + z+ E F.
a'(z) z
Moreover one can show that F embeds compactly into every LP([O, 1]; R2n) for
1 ˜ p < 00. We have a natural SI-action on F by phase-shift, i.e. r E R/Z acts
via
= z(t + r).
(r * z)(t)
W(˜ define a distinguished subgroup B of homeo(F) the homeomorphism group of
/1' by saying h E B if h : F --+ F is a homeomorphism admitting the representation

+ zO + exp (i+(Z))Z+ + K(z),
h(z) = exp (i-(Z))Z-
,+,,- :F ˜ R are continuous and SI-invariant, mapping bounded sets in F
where
illt,o bounded sets in R. Moreover K : F --+ F is continuous and Sl--equivariant and
IIUtpS bounded sets into precompact sets. It is easily verified that usual composition
turns B into a group. Next we need a pseudoindex theory in the sense of Benci,
113], associated to the Fadell-Rabinowitz index, [14], in order to measure the size
c)f Sl-invariant sets. This goes as follows: Given a paracompact SI-space X we
IHlild a free 5 1-space X X Soo by letting SI act through the diagonal action. Here
,,",'00 = US 2n - l , with S2n-l C R 2 n ˜ en. Taking the quotient with respect to 8 1
we obtain a principal 5 1 -bundle

,rile classifying map
˜
f: (X x SOO)/S' Cpoo
il)<luces a homomorphism

Alexander-Spanier-Cohomology with rational coefficients. Here R S l is the well-
ill
kllown Borel construction of a SI-equivariant cohomology theory. We know that
26 Hofer: Symplectic capacities

=
Q[t], the generator t being of degree 2. We define the index of X,
H(CpOO)
denoted by a(X) to be the largest number k such that

We put a(X) = 00 if f*(t k - 1 ) =F 0 for all k and a(0) = o. The a-index is the
well-known Fadell-Rabinowitz construction of a Sl-index theory. Next we define
ec
for an 5 1-invariant set F an index ind(e) by

where S+ is the unit sphere in F+. The key topological result is the following:

Lemma 3.9 Let X be an 5 1 -invariant linear subspace of F+ of dimension 2k.
Then

For a proof see [3].

Finally we define for H E 1i

= inf {sup <P H( e) lee F is 5 1-invariant and ind (e) ˜ k}.
Ck,H

By construction, it is clear that

o< cl,H ˜ C2,H ˜ â€¢â€¢. ˜ +00.
For a bounded set 5 c R 2n we put

OJ.
1i(5) = {H E 1i I H I U(S) ==

where U(S) denotes an open neighbourhood of S depending on H. For a bounded
set S let
Ck(S) = inf Ck H
HE'H(S) ,

For an unbounded set T let

IS c
ck(T) = SUp{Ck(S) T bounded}.

The first result proved in [3] is

Theorem 3.10 For every k E {I, 2, ... } the map defines a symplectic capacity.
Ck
Moreover
Cl˜C2˜.Â· â€¢â€¢
'27
Hofer: Symplectic capacities

= (rt, ... , rn)
The Ck have been computed in [3] for certain sets. For r with
< rl ˜ r2 ... ::; r n, define an ellipsoid E(r) by
()

For such a r define with N* = {1, 2, ...} :

Ie}.
j E {I, ... , n}, 1rr;1 ˜ ˜
= in! {r > 0 Icard{(l,j) 11 E N*, r}
dk(r)

= (1,1, ... ,1) we have
For example for r

where [*] denotes the integer part. Moreover observe that

lim dkÂ«1,n, ... ,n)) = k7r.
n-..oo

It. has been proved in [3] that

'I'heorem 3.11 The following equations hold:

Ilere all sets are equipped with the standard structure (1. The capacities Ck enjoy
n. useful representation property explained as follows. Call a compact connected
Hlllooth hypersurface S to be of restricted contact type provided there exists a
veetor field TJ on R2n, such that

'1 is transversal to S, L,,(1 = (1.

s:
W(˜ have, denoting by B s the bounded component of R 2n \

'I'heorem 3.12 Let S be as just described. Then there exists a sequence (Pk ) C
'/)(5) and a sequence nk E N* such that

Finally let us give an application to an embedding problem:

'rheorem 3.13 Assume B 2 (1) x ... x B 2 (1) admits a symplectic embedding into
"2n(r). Then r 2:: ,;n.
28 Hofer: Symplectic capacities

Let W: B2(1) x ... X B2(1) ˜ B 2 n(r) be the symplectic embedding. We
PROOF:
find, see [2], a symplectic map ˜ E 'D, such that

where ˜ = ˜ s and 0 < C < 1 was given. Hence, taking the n-th capacity we obtain

n1rc 2 :5 7T'r 2 â€¢
Since C < 1 was arbitrary we deduce r ˜ ..;n. We had to introduce the map ˜ since
the C/c are by construction only invariant under symplectomorphisms in V. â€¢

A capacity on Symp2n and the Weinstein
4
conjecture
Here we present a more general construction for a symplectic capacity on arbitrary
symplectic manifolds. Assume (M,w) E Symp2n. We denote by H(M) .the set of
all autonomous Hamiltonian Systems H : M -+ R such that

I U == o.
HI There exists a non-empty open set U C M \ 8M such that H
H2 There exists a compact subset K C M \ 8M and a constant m(H) > a such
that
o˜ H(x) ˜ m(H) for x E M
and
H(x)=m(H) forxEM\K.

˜
Given a symplectic embedding '1J : M N we define a morphism

W. : H(M) -+ H( N)
by
W- 1 (x)
H if x E im ('11)
0

(w.H)(x) =
{
if x E N \ im (w)
m(H)
We call a Hamiltonian H E H(M) admissible provided every T-periodic solution
of x = XH(x) for T E [0,1] is constant. We denote by ?te(M) the collection of all
admissible Hamiltonians in H(M). Clearly w.Ha(M) C Ha(N) and H a(AI) f:. Â¢ as
one easily verifies. We define c : Symp 2n --+ r by

= sup{m(H) IH
(M,w) ?ta(M,w)}.
E
c
29
Hofer: Symplectic capacities

c is a covariant functor and c (M, a Â· w) =1 a 1Â·c (M, w) for a =f o. The
(a(˜nrly
clif-Hculty lies in proving axiom (N) which is done in [10]. Some other results listed
h<'low are taken from [15,16].

r
'I'heorem 4.1 c: S ymp2n -+ is a symplectic capacity and

Moreover for every every convex bounded domain C with smooth boundary S = 8C
111t˜ have
c (C,u) = inf{A(P) 1 P E V(S)}
nnd additionally for every compact symplectic manifold (N, w) with w 11r2(N) = 0

= 7rr 2
B 2k (r),w EB u)
c (N X

l)1'o'vided k 2 1. Also for the standard Cpn:

The above capacity is very useful in dealing with the Weinstein conjecture, [9].
114't us recall some notations. Consider a compact (2n - 1)-dimensional manifold S
rqllipped with a I-form ,\ such that ,\ A (d,\)n-l is a volume. One calls ,\ a contact
fcn'rn.
The manifold S carries a natural line bundle defined by

Id'\(x)(e,'fJ) = 0 forall 'fJ E T˜S}.
Â£5 = {(x,e) E TS

Â£s c TS and since '\A(d,\)n-I is a volume A(X, e) i: afor (x, e) E Â£5, if e=F O.
1I(˜nce
I)(˜note by V(S) the closed integral curves for the (integrable) distribution Â£5 --+ S.
Weinstein conjectured that 1>(8) =1= Â¢ at least if HI(S; R) = O. Given (5,'\) we
dt'fine a germ of a symplectic manifold structure on (-˜,e) X S by w = d(e t ,\)
where (t, x) E (-e:, e:) x S. We consider S as the subset {OJ x S in (-â‚¬, â‚¬) X S. First
W(˜ have the following trivial observation.

I)roposition 4.2 Assume the previously defined capacity of (-e, e) x S for some
t
r '> a is finite. Then V(S) Â¢.

==
Fix a smooth map <p : (-e, e) ˜ R such that cp 0 near zero and
I) ROO F:
(f1(S) = m > 0 for S close to e or -e. Define a Hamiltonian

)
H E 1i ( (-â‚¬, â‚¬) S, d (et,\)
X
Hofer: Symplectic capacities
30

by
is} X s.
H(x) = ep(6) for x E
We pick m in such a way that m > c( (-e,e) X S).
Hence H is not admissible and has a non-constant periodic solution x of period
T E [0,1]. We have
x(R) {6} S
C X

for some 6 and
d
˜ dt (prs
t x (t) )
0

˜
is a non-vanishing section of Â£s 8. Hence it parametrises a closed characteristic.
â€¢
Let us say that a compact smooth hypersurface S in a symplectic manifold
(M,w) is of contact type provided there exists a I-form A on S such that
dA = i*w
where i : S ˜ M is the inclusion and A(X, e) 1= 0 for non-zero elements in Â£s. It
is not difficult to show that Â«-e,e) X S,d(etAÂ» admits for some small e > 0 a
symplectic embedding into (M,w) mapping {O} X 8 into S.

Definition 4.3 Let (8, A) be a compact (2n-l)-dimensional manifold equipped with
a contact form. We call (S, A) embeddable into the symplectic manifold (M, w) E
Symp2n provided there exists an embedding <p : S ˜ M such that

4>*w = dA.

.As a corollary we obtain

Theorem 4.4 Let (S, A) be described as before and assume (S, A) is embeddable
into N X B2k(R) or CP'''', where (N,w) is a compact symplectic manifold with
w I 1r2(N) = O. Then V(S) 1= 4>. (Here of course we talk about codimension
l-embeddings).

The Weinstein conjecture is one of the key conjectures in the Existence theory
for periodic solutions of Hamiltonian Systems. For more details we refer the reader
to [8,9,10,12J.

Capacities and Instanton HOInology
5
In this section we describe some results presented at the symplectic year at MSRI,
Berkeley. We restrict our˜elves to the case of open sets in R2n. The construction
Hofer: Symplectic capacities 31

cÂ°n.n be done in much greater generality. For this we refer the reader to a forthcoming
I)n.per, [17], in which a theory, called symplectology, is developed (Symplectology:
Sylnplectic Homology). For the following we require some familarity with Floer-
hornology for example as described in [18,19,20]. We consider as in 3.3 the family
()f I-Iamiltonians 11. For H E 11 we have the map ˜H : F ˜ R as introduced in 3.3.

'1. '1 induces a map, still denoted by ˜H,

Itoughly speaking we study the L 2-gradient flow on (F x Soo)/ Sl. In practice we
have to replace Soo by a sufficiently large compact approximation S2n-l and to show
that the following construction stabilizes. From now on we shall simply ignore any
tpehnical difficulty in order to present the idea. For a generic H E 1i the map .p H
luts finitely many critical points. For numbers c ˜ d we consider the free abelian
˜roup /\˜(H) defined by
/\˜(H) = ffixeaZx
where G is the set of all x with:

˜#(x) = 0
{ f) H( x) E (c, d].

˜H
We can associate to critical points of a relative Morse index J.t E Z, which can
h(˜ understood in terms of a Maslov class for example as in [21]. So /\˜(H) has a
nn.tural Z-grading. We define a boundary operator {) : /\˜(H) ˜ /\˜(H) by

E
8y= #<x,y>x
J1.(x)=J1.(y)-l

where # < x, Y > denotes the number of trajectories of the flow equation " = epi!(,)
I'tlnning from x to y. Here these orbits carry natural orientations so that we actually
(fount their signed number. We have {)2 = 0 and define a homology group Ig(H).
'I'he choice of an inner product and a positive almost complex structure with respect
t.o the symplectic form is involved in studying " = epk(,). However it turns out
that Ig(H 0 \J!) :k Ig(H) for all \II E 1), where 1) consists of all symplectomorphisms
(Â»)>tained as time-I-maps for compactly supported time-dependent Hamiltonians.
We define an ordering on 11 by

H >!{ *=> There exists '1J E 1) such that
{
H(z) ˜ K 0 \II(z) for all z E R2n.
-

'I'aking a homotopy from !{0'1J to H which is increasing, say L s such that L s = !{0'1J
ror s :::; -1 and L s = H for s ˜ 1 and :sLs(x) ˜ 0 we can study the flow equation

" = ˜˜s(1'),
Hofer: Symplectic capacities
32

where one looks for solutions running from critical points of ˜ 1(0\11 to critical points
of c1> H. Studying this problem for critical points of the same Morse index one obtains
an induced morphism
1:(K) ˜ 1:(H).
One defines for a bounded subset 5 of R2n using 11(5) as defined in 3.3

1:(S) = l˜ 1:(H)

where the direct limit is taken over all H with H E H(S). Recall that H(S) is
partially ordered. We also note that we have for c ::; d ˜ e morphisms

1:(H) ˜ 1˜(H),
which induce

e by
We define a terminal object

e := lim ItOO(H)withH E 11(*).
It turns out that e := Z[t] is the polynominal ring in one variable of degree 2
(recall that we have a Z-grading ).
Assume '!I(S) C T for some WE V, where Sand T are bounded sets. Then we
obtain a natural map
1:(T) ˜ 1:(8)
We define for an unbounded set U the group 1:(U) to be the inverse limit of the
Ig(8) for S running over the bounded subsets of U. The family Ig, for c :5 d, of
functors is the symplectology. Using our terminal object e we have induced maps

Ig˜e,

c T, '11 E V and c :5 d
such that if w(S)

e

19(T)
'\,
e
Taking the second diagram we see that the image of the map 19(T) ˜ e is increasing
18
- - ? e we define a non-decreasing sequence
in d. Studying the images of the maps
33
Hofer: Symplectic capacities

{d k } as follows. The sequence {d k } consists of all points where the rank of the
ilnage changes. Here we repeat a point of discontinuity according to its multiplicity,
namely the net change of the rank. The first diagram shows that the numbers d k
are monotonic invariants, Le. if w(S) c T then dk(S) ˜ d!c(T). This Instanton-
homology approach to symplectic capacities shows that symplectic capacities can
he understood as numbers, where a certain classifying map into Z[t] changes its
ra.nge.

References
[1] M. Gromov: Psettdoholomorphic curves in symplectic manifolds, Inv. Math.
, 1985 , 82, 307-347.

[2] I. Ekeland, H. Hofer: Symplectic topology and Hamiltonian Dynamics, Math.
Zeit. , 1989, 200, 355-378.

[3] I. Ekeland, H. Hofer: Symplectic topology and Hamiltonian Dynamics II,
Math. Zeit. , 1990, to appear.

[4] M. Gromov: Partial differential relations, Springer, Ergebnisse der Mathe-
matik, 1986.

[5] Y. Eliashberg: Funct. Anal.
A Theorem on the structure of Wave Fronts,
Appl., 1987, 21, 65-72.

[6] M. Gromov: Proc. of the ICM at
Soft and Hard Symplectic Geometry,
Berkeley 1986, 1987, 81-98.

[7] H. Hofer: On the topological properties of symplectic maps, Proc. Royal Soc.
of Edinburgh, special volume on the occasion of J. Hale's 60th birthday.

[8] C. Viterbo: A proof of the Weinstein conjecture in R'n., Ann. Inst. Henri
Poincare, Analyse non lineare, 1987, 4, 337-357.

[9] A. Weinstein: On the hypotheses of Rabinowitz's periodic orbit theorems, J.
Diff. Eq. , 1979 , 33,353-358.

[10] H. Hofer, E. Zehnder: A new Capacity for symplectic Manifolds, to appear in
the proceedings of a conference on the occasion of J .Moser's 60th birthday.

Ekeland, H. Hofer: Convex Hamiltonian Energy Surface3 and their periodic
r11] I.
Trajectories, Comm. Math. Phy. , 1987 , 113, 419-469.

/12] H. Hofer, E. Zehnder: Periodic Solution3 on Hyper3urface3 and a result by C.
Viterbo, Inv. Math. , 1987 , 90, 1-9.
34 Hofer: Symplectic capacities

[13] V. Benci: On the critical Point Theory for indefinite Functionals in the Pres-
ence of Symmetries, Transactions Am. Math. Soc. , 1982 , 274 , 533-572.

[14] E. Fadell, P. Rabinowitz: Generalized cohomological index theories for Lie
Group Actions with an Application to Bifurcation Questions for Hamiltonian
systems, Inv. Math. , 1978 , 45, 139-173.

C', Math.
[15] A. Floer, H. Hofer, C. Viterbo: The Weinstein Conjecture in P X
Zeit., 1990, to appear.
[16] H. Hofer, C. Viterbo: The Weinstein Conjecture in the Presence of holomor-
phic Spheres, in preparation.

[17] A. Floer, H. Hofer: in preparation.

[18] D. Salamon: Morse theory, the Conley Index and the Floer Homology, Bull.
of the London Math. Soc., to appear.

[19] D. McDuff: Elliptic Methods in symplectic geometry, Lecture notes.

[20] A. Floer: Morse theory for Lagrangian intersection theory, J. DifF. Geom., 1988,
28, 513-547.

[21] D. Salamon, E. Zehnder: Floer Homology, the Maslo1J Index and periodic orbits
of Hamiltonian Equations, preprint , Warwick, 1989 .

[22] Y. Eliashberg, H. Hofer: Towards the definition of a symplectic boundary, in
preparation.
TIle Nonlinear Maslov index
A.I˜. GIVENTAL
LfiUill Institute for Physics and Chemistry, Moscow

I will present here a nonlinear generalisation of the Maslov-Amold index concept [l],and
it to deduce the following theorem .
liM"

TUIÂ·;OREM (GIVENTAL).
llt1t Rpn-l C cpn-l be the fixed-point set of the standard anti-holomozphic involution
rtf cpn-l. Then if f : cpn-l --+ cpn-I is a map which can be deformed to the identity
tlll'ough a Hamiltonian isotopy, the image f(Rpn-l) intersects Rpn-l in at least n points.

= 2 ; it is evident that the equatorial circle in the 2-sphere
1\ ... a simple example, take n
1

IlltÂ·(˜ts
any area bisecting circle at least twice.
'l'hiR theorem is a typical fact of symplectic topology; similar to results proved by Conley-
˜rhnder and Floer. We shall see, I hope, that the nonlinear Maslov index provides a
IU\tllral and convenient language to formulate "Arnold- type" conjectures on symplectic
nx(˜d points, or Langrangian intersections (see [2]).

'I'he linear Maslov index.
fly the linear Maslov index we mean the only homotopy invariant of loops in the Lagrange-
(: ..n.ssman manifold An, the space of n-dimensional Lagrangian linear subspaces of R 2 n.
Il(ofore generalising this notion it is convenient to projectivise it.
L.˜t en be complex n-space , endowed. with its standard symplectic structure. The real
pl'ojectivization, Rp2n-l has a standard contact structure. A point p in Rp2n-l is a
IÂ·t˜nlline in en ; its' skew-orthogonal complement is a hyperplane containing this line. In
))('ojective space we get a hyperplane through p , and the tangent space of this defines
t.ll(˜ element of the contact structure at p. With thi˜ contact structure, the Legendrian
IU'ojective subspaces of Rpn-l are exactly the projectivisation˜of Lagrangian subspaces
or en (if a Lagrangian subspace contains a line then it is contai˜ed in the skew-orthogonal
complement of the line, Le. it is tangent everywhere to the contact structure). Thus An is
t.he manifold of projective Legendrian subspaces in Rp2n-l. It is a homogeneous space of
the the group Gn = Sp(2n, R)/ Â± 1, and its' subgroup H n = U(n)/ Â± 1. A linear Maslov
iudex m('Y) E Z of a loop 'Y ,in any of these three spaces, is just its' homotopy class under
the canonical identification ([1])
.:˜
,˜.˜˜

:;1˜
Givental: The nonlinear Maslov index
36

Legendre-Grassmann manifolds. ;0
We shall deal with the following infinite-dimensional manifolds: ,j

˜
(1) C!5 n- the identity component of the contactomorphism group of Rp2n-l.
(2) ..e n- the space of all embedded Legendrian submanifolds of the contact manifol˜
Rp2n-l which can be obtained from the standard Rpn-l C Rp2n-1 by a Legen˜
drian isotopy. We call ..en the Legendre- Grassmann manifold. It is a homogeneou˜
:'˜˜
space of C!S n .
(3) i)n the identity component of the subgroup of (!5n consisting of transformation˜
'˜˜
which preserve the standard (U(n)-invariant) contact I-form no on Rp2n-l.
'!!
One may consider the contact form ao as a pre-quantisation connection on the square˜
Hopf bundle Rp2n-l ˜ cpn-l , having structural group the circle T = {e it } / Â± 1. Thu˜
J)n is a central T-extension of the identity component of the symplectomorphism group o˜
cpn-l. We call fJ n a quantomorphism group because it realises, at the group level, th˜
:˜˜
Poisson bracket extension of the Lie algebra of Hamiltonian vector fields: ..;,

,j
o-+ R coo(cpn-l) sym(cpn-l) O.
-+ -+ -+
˜!
The finite dimensional manifold An is naturally contained in the infinite dimensional spacJ
C!Sn. Similarly, Q5 n contains G n and .JJ n contains H n . The weak statement about thes4
space.s is that the linea:r Maslov index can be. exte.nded to the loops in ˜hese infinite dil.'l

menslonal spaces, that IS we have a commutatIve dIagram of homomorphIsms:

cw
1t"t(A n ) ˜ Z
˜
1/ 1
J
˜˜
1t"1 (..en)
'1
The stronger statement is that Arnold's geometrical definition ([1]) of the linear Maslov;j
˜
index can also be extended to these infinite dimensional spaces.

Discriminants.
To extend Arnold's geometrical definition we define a discriminant, a subspace ˜ c .cn.i
Let us mark a point in ..en, for example the projectivized imaginary subspace iRn c cn..,:˜
The discriminant ˜ consists of all Legendrian submanifolds which intersect this marked:l
one. Two Legendrian subspaces are , in general, linked so ˜ is a hypersurface in ..en:˜
(with singularities). We also define a discriminant in C!Sn to consist of contactomorphisms'˜
9 which fix a point of Rp2n-l, and fix the contact I-form at that point; i.e. g(x) =f
x and g*(ao}Jx = aol x , for some x in Rp2n-l. Analogously we define the discriminant in'i
1)n to consist of quantomorphisms which have fixed points (in fact, circles of fixed points, ˜;:
fibres of the bundle Rp2n-l -+ cpn-l).

1.
THEOREM
Givental: The nonlinear Maslov index 37

l1:n('l. of these three discriminants admits a co-orientation such that their intersection num-
1.,Â·,..-; with oriented loops define commuting homomorphisms

!:!
7I"t(A n ) Z
1rl(Gn)
1rt(Hn) ---t
---t
---t

1
1 1
1rl(˜n)
1rt(.f)n) 1rt(Â£n)
---t ---t

m=int. index1 ml ml

Z
Z
Z

f.lxt.(˜llding the linear Maslov indices.
'fIlii,; theorem yields two corollaries, as we shall explain below.
1.
Â«!onOLLARY
'1("" PI, P2 be two points in ..en, i.e. embedded Legendrian submanifolds diffeomorphic to
II,I)U-l. The projections of PI and P2 to cpn-l have at least n intersection points in
C˜1)1I-1.

2, (KLEINER-OH).
(!()Il.OLLARY
'Il/lt! standard Rpn-l C cpn-l has the least volume among all its images under Hamilto-
";11.11 isotopies (and, more generally, among all projections of Legendrian submanifolds in
).
'˜n
Ilere the volume is measured with respect to the standard U(n)-invariant Riemannian
1I1r't.ric on cpn-I. A simple illustration of Corollary 2 is furnished by the theorem of
Poincare which asserts that the equator in the 2-sphere has least length among all area-
ltiHt'eting curves. The proof of Corollary 2 is based on integral geometry. The volume of a
L"'˜l'angian submanifold is proportional to the average number of its intersections with the
',l'H.llslates of the standard Rpn-l by U(n). This number is not less than n, by Corollary
I , and equals n if the Lagrangian submanifold is standard.

'I'he Morse Inequality.
It˜n("h of the manifolds .en, (!Sn, jjn is modelled on a space of smooth functions. For example,
II IH'ighbourhood of any Legendrian submanifold L in a contact manifold is contactomorphic

I.â€¢â€¢ t.he I-jet space JI L of functions on L (that is, J1 L == RxT* L, with the contact structure
tltt == pdq). A Legendrian submanifold GI-close to L is represented by a Legendrian section
of .IlL, i.e. by the graph of a smooth function f on L , with u = f(q) , p = dqf. The
".'ction meets the zero section L at a critical point of f with zero critical value. Thus the
diseriminant ˜ ,near the marked point, looks like the hypersurlace in GCXJ(L) of functions
with singular zero level. Consider, for example the space of polynomials in one va.riable in
I,ll,.ec of GOO (L) in which the analogue of ˜ is the subset of polynomials with multiple roots.
F(H' polynomials of degree 4 this subvariety has the form of the "swallow tail" singularity
Givental: The nonlinear Maslov index
38

.;r.
,'I:'

f

Diagram 1

in 3-space, depicted below, and this is in fact the general model for codimension-one ';
singularities in the discriminants. '0,;.

To co-orient the discriminant 6 C COO(L) we introduce, for every Coo function I on L, al˜
topological space ."

= {q E Llf(q) ˜ o},)
LJ

and an integer bl = b*(LI, Z/2) - the sum of the Betti numbers of LI' with Z/2 co_O)I˜
efficients. As I moves in the complement of 6 the boundary 1- 1 (0) remains smooth, and;:i.'
b, is unchanged. When I crosses ˜ at a nonsingular point a Morse bifurcation ocurs: a':˜
new cell is glued to Lf and bl changes by either +1 or -1. We co-orient 6 in the direction 0.:;
in which bf increases.
Now we give an interpretation of the total Morse inequality in our intersection-index terms. ,;
Let us consider the flow I ˜ I +t in COO(L). The number, ", of intersections with 6 of :',
the orbit of a Morse function I on L is, by the definition of 6 , equal to the number of
critical levels of f. On the other hand the intersection index of the orbit with 6 is equal
to
Givental: The nonlinear Maslov index 39

= b.(L : Z/2).
bf+oo - b/- oo
Thus the total Morse inequa1ity,˜ ˜ b.(L; Z/2), follows from the fact that an intersection
index is never more than an intersection number.
An analogous argument is used to deduce Corollary 1 from Theorem 1, at least in the case
when the intersections are transverse. Intersections of ˜ with a T-orbit through L E ..en
eorespond to the intersections of the image of L in cpn-I with the standard Rpn-I ,just
hy the definition of˜. But the intersection index, being homotopy invariant, is equal to
t,he linear Maslov index of T- orbits in An , that is, to n.
The Calabi-Weinstein Invariant.
'rhe Calabi-Weinstein invariant([9]) is the homomorphism

which is defined by the Lie algebra homomorphism

r
w: coo(cpn-l) -+ R : h 1-+ hdi-t,
}cpn-t
- integration with respect to normalised Liouville measure. In fact such a homomorphism
'U1 is defined for any quantomorphism group.

1. The Calabi-Weinstein invariant is proportional to the Maslov index:
PROPOSITION
'Hore precisely m = (n/1r)w.
The homomorphism w takes values in (1r/n)Z C R.
COROLLARY.

If Sjn were compact one could prove Proposition 1 by integral geometry arguments.
'rhere exists a (CO, locally exact, adjoint invariant) I-form M on j)n which represents the
('ohomology class m. The value of M on a tangent vector h E Tqil n is

:E
Mq(h) = Â±h(x),
q(z)=x

where q denotes the underlying symplectomorphism, and signs are defined by the co-
()l'ientation of a at its' intersection points with the T-coset through q. IT we could average
t.his form over all translations of M we would obtain a left and right invariant form, which
Hhould clearly be proportional to W.
Generating functions.
'[0 prove the existence of the nonlinear Maslov index we mark in Rp2n-l another La-
˜rangian subspace L o = Proj(Rn) C cn. Given an ambient Hamiltonian isotopy h t of
Itp2n-l ( 0 ˜ t ˜ 1) , we factor hI into a large number N of small isotopies, and construct
a function

f : Rp2D-l ˜ R , D = nNe
Givental: The nonlinear Maslov index
40

This function is a kind of finite approximation to the action function of the isotopy;;:
'˜.˜
lifted to en homogeneously. It is chosen to have the property that

is non-singular {:} hl(Lo) E ˜.
1- 1 (0) .
Then we define the relative Maslov index of the path 'Y in .en formed by L t = ht(Lo) to bal
me,) = b.(f- 1 (R+); Z/2) - D, and co-orient ˜ in the increasing direction of the relativ˜˜;
index when the end of the path crosses˜. A crucial point in the proof of the validity of thi˜
definition is the additivity of the relative index. H 1'0 and 1'1 are, respectively, a loop and ˜j
path in.e n which are subdivided into No and N 1 parts, then m(,O,l) = m(,o) + m(,l)?˜
Moreover, if I is the function associated to the composite 1'0,)1, and fl is associated t˜
1'1, then the space f-I(R+) is cohomologicallyequivalent to the Thom space of the s˜1
of Do = nNo copies of the Mobius line bundle over 1- 1 (R+). In particular, the twq˜i
spaces fol(RÂ±) are cohomologically equivalent to RPd:l:-l, where the "inertia indicesn˜
dÂ± = b*(f;I(RÂ±); Z/2) are complementary (d+ + d_ = 2Do ) , and differ from the middl˜1
dimension by twice the Maslov index of the l o o p : : ˜

= d+ - Do = Do - d_. .::˜
m(/o)
.f.˜

A similar method can be used to co-orient the discriminant in Q5n. We decribe the gen;j
erating function for this case explicitly. Let hI = hN 0 â€¢â€¢ â€¢ 0 hI be a decomposition of th˜˜

isotopy hlinto N small parts - lifted into C n homogeneously. Then we take

t
f = Proj (Q - H) : (Cn)N R,
-+

where Q is the quadratic generating function of the cyclic permutation in (C<n ))N whicil
0t
II.
map˜ (Xl,.;;, XN) to (X2.' .â€¢. ' X˜, Xl) , and = EJ1f:l Hi is the generating function
t?e ve˜t?r tra˜sformatl0n whIch maps a.poInt (Xl, ... ,XN) to (h1Xl, ... ,hNXN). ThlÂ·˜.˜ â€¢.'"
SIngular!tIes of f 1 (0) correspond to fixed hnes of h I . .:'
For the group Sln, the lifted isotopy preserves the unit ball in en and the generatin'" I

function, being eit-invariant, is naturally defined on a complex projective space: :1 I

,:!.. ˜

= n[N/ 2 ] . " 1
f : Cp2D-l R,D
-+
:'"
generating functions proves the consistenc˜
A topological comparison of all of these
01
statements of Theorem 1. Besides, the "signature" (d+,d_) of the generating function
a l?"p (in ˜n for ex˜ple) i.s actually a signature of t˜e ?enerating fu˜etion at ˜he criticaj!
pomts. The ch˜ge.m t˜e slg?at˜re of the second vanatIon of the ,actIon functlOnal.alon1
a path of Hamtltoman Isotoples IS known as the Conley-Zehnder Index ([3]), so thIS last˜˜

observation yields the corollary:
::˜˜Â·I

PROPOSITION 2.
The Maslov index ofa loop in <!5 n is equal to the Conley-Zehnder index of the corresponding:˜

trajectory of Hamiltonian isotopies of en.
˜

According to the "Sturm theory " the variation in the signature is equal to the linear i

Maslov index of the Hamiltonian flow, linearised along its trajectory. This gives
Givental: The nonlinear Maslov index 41

3.
IlUOPOSITION
1t/u˜ lJonlinear Maslov index of a loop ( in <!Sn ) is equal to the linear Maslov index of loops
( ,,, G n ) formed by linearising Bows along the the trajectories in en.
w(˜ use this result, applied to the subgroup J)n , to prove Proposition 1. Averaging over
All t,ra.jectories, one finds that the nonlinear Maslov index is equal to the average value
UV˜I' the ball of the Laplacian divgradHt of the Hamiltonian. For homogeneous H t this is
......n.l to an average of H t over the unit sphere, which proves Proposi˜ion 1.
'J'"lis.
1\. c'cnnplete the proof of our Lagrange intersection theorem for the non- transverse case one
Ut˜f-(IH to modify the definition of the discriminant. Let us define the cohomologicallengtk,
I( X), of a subset X of a real projective space to be the least degree in which the restriction
ur t,Il(˜ generator of the cohomology of the projective space vanishes on X. The length is
'U4Htotone with respect to inclusion. Let us denote by r , and call a tail the subset of the
tlbwl'illlinant ˜ where the length of the positive subset f-l(R+) changes - rather than the
n' 1 f,ti Hum we considered before. The tail is a hypersurface with no topological singularities

III UollY finite codimension (see the diagram). Its' co-orientation by the length-increasing
tUnÂ·(Â·t,ion coincides with the positive T-co-orientation (monotonicity) and makes r into a
1 c'oc˜ycle cohomologous to ˜ (additivity). Thus the inequality between the intersection
fluu"ber and index, applied to r and T-orbits, gives estimates for geometrically different
'˜np˜l'u.ngian intersections, or fixed points of symplectomorphisms.

r

Diagram 2

3 [5]).
(FORTUNE-WEINSTEIN
( Â·OUOLLARY
:\ "y l11uTliltonian transforrnA,tjoll of cpn-l at least n different fixed points.
JUtS
\1
Givental: The nonlinear Maslov index
42 '''f˜!'
;':("
;.˜.:t

.:.:Â·˜ Â·i
!˜;;˜
-;˜˜
The asymptotical Maslov index.
Let a be a contact I-form for the standard contact structure on Rp2n-l; so a = hao foti;˜
some positive function h. The characteristic vector field VOl is defined byf˜
,...˜

vEkerda, a(v)=I,
,.,
;;J!˜
â€¢..

and this vector field generates the characteristic flow of contactomorphisms. Let us fiJGt˜
on such a flow gt or, more generally, a nonautonomous flow of a time periodic characteristi˜˜
field. Let L E .Â£n and let mL.,.(t) (respectively metÂ») be the intersection index of the patlij
(g1" L) ( respectively (g1") , 0 ˜ T ::; t, with the tail r c .en (respectively C!5 n). Â·t˜

2.
THEOREM
'1.˜
The following limits exist and are equal :

= lim m(t)/t = lim m L ( t ) / t . 1
m
The asymptotical Maslov index m is monotone and homogenerous of degree -1 o˜
the space of contaxct I-forms; it is continuous in CO-nonn and does not change und˜
conjugations in ˜ n . ; ; ˜
'.:-'i
For example, a positive quadratic Hamiltonian in en generates a projective characteristi˜
flow on Rp2n-l. Its nonlinear asymptotical Maslov index is equal to the linear one; th61
:,
sum of frequencies m = LWi. For example ao generates the Hopf flow with m = n/1r. .˜˜
':1˜
COROLLARY 1.
˜m˜
n/(1rmax. h) n/(1rInin. h ) . j
COROLLARY 2 (VITERBO). Any closed I-form on

3
COROLLARY (GIVENTAL-GINZBURG).1
The nu˜ber #of chords (i.e. characteristics starting and finishiD? on the ˜ame Legendria.4
jl
submanlfold L) of length less than or equal to t grows at least bnearly wlth t :

˜ mt - ˜ n[t/1rmax h]- c o n s t . ( L ) . "
#(L, t) const.(L)
)'ti
Remarks.
(1) For the quantomorphism group tl n one can also define the asymptotical Calabi!
Weinstein index. If a characteristic flow of a contact I-form hao (where h is ˜
positive function on cpn-l) is precompact in j)n then its Calabi-Weinstein inde˜1
coincides with its asymptotical Maslov index, in accordance with Proposition 1: ˜\J
.i.˜
1
1
m=(n/1r)jh- â€¢

(2) In general the asymptotical Maslov index is equal, just by the definition, to the:'
density of the length spectrum of closed characteristics and seems to coincide with \˜˜
the density of the capacity spectrum of the domain {r 2 h \$ I} C en which has been?
43
Givental: The nonlinear Maslov index

defined by Ekeland-Hofer and Floer-Hofer ([7]). When applied to a precompact
How in tÂ»n this turns (i) into the Duutermaat.. Heckmann integration formula:

(see [9]).
(3) Our proof of Theorem 2 is mainly based on the monotonicity of cohomologicallength
and the qUGsiatlditivity of generating functions : a homogeneous generating function
for the flow {gT'} , T ˜ t 1 + t 2 , coincides with the direct sum of such functions for
T ˜ t 1 and T ˜ t 2 after their restriction to a subspace of low codimension (that is,
low compared with large values of tl and t2).
Let us note finally that our intersection-theoretic interpretation of Arnold's fixed point
..11<1 Lagrange intersection problems can be extended to some other symplectic and contact
IUltuifolds using direct methods of Floor homology ([4]), instead of singular homology and
Morse theory.

REFERENCES

I. Arnold,V.I., On a characteristic class entering the quantisation conditions, Funct. Anal. Appl. 1
(1967), 1-6.
2. Arnold,V.I., First ,tel's in symplectic topology, Russian Math. Surveys 41:6 (1986), 1-21.
:1. Conley,C. and Zehnder,E., Mor,e type indez theory for flows and periodic solutions of Hamiltonian
equations, Comm.Pure Appl. Math. (1984),207-253.
4. Floer,A., Morse Theory for Lagrangian intersections, J. Differential Geometry 28 (1988), 513-547.
I). Fortune,B. and Weinstein,A., A symplecticfized point theorem for comple:t projecti.1Je space, Bul1.Amer.
Math. Soc.12 (1985).
n. Givental,A., Periodic maps in symplectic topology, Funct. Anal. Appl. 23:4 (Russsian) (1989),37-52.
7. Hofer,H., In this volume (1989).
K. Viterbo,C., Inter,ections de sous variites Lagrangienne" functionelle, d'action et andice des sys-
temes Hamiltoniens, Bull.Math. Soc France 115 (1987),361-90.
O. Weinstein,A., Oohomology of symplectomorphism groups and cntical1Jalues of Hamiltonians, Preprint,
Berkel (1˜88).
Filling by Holomorphic Discs and its Applica-
tions
YAKOV ELIASHBERG

Stanford University

The survey is devoted to application of the technique of filling by holomorphic discs
to different symplectic and complex analytic problems.

1. COMPLEX AND SYMPLECTIC RECOLLECTIONS

1.1 J -Convexity
Let X, J be an almost complex manifold of the real dimension 4 and E be an
oriented hypersurface in X of the real codimension 1. Each tangent plane Tz(E),
ex c Tz(E) which we will callÂ°a complez
x E E, contains a unique complex line
tangency to E at x. The complex tangency is canonically oriented and, therefore,
eon
cooriented. Hence the tangent plane distribution E can be defined by an
equation a = 0 where the I-fonnii is unique up to multiplication by a positive
function. The 2-form da Ie is definea up to the multiplication by the same positive
factor. We say that E is J -convez (or pseudo-convex) if da(T, JT) > 0 for any
non-zero vector T E ˜:t, X E E. We use the word "pseudo-convex" when the almost
complex structure J is not specified.

An important property of a J-convex hypersurface E is that it cannot be touched
inside (according to the canonical coorientation of E) by a J-holomorphic curve.
In particular, if n is a domain in X bounded by a smooth J -convex boundary ao
then all interior points of a J-holomorphic curve C c X with 8C c an belong to
IntO. Moreover, C is transversal to on in all regular points of its boundary ac.

A hypersurface E is called Levi-flat if do: le= o. According to Frobenius' theorem
eintegrates
this means that to a. foliation of E by J-holomorphic curves.
46 Eliashberg: Filling by holomorphic discs and its applications

A function c.p on X is called J ..convex or pseudoconvex if it is strictly subharmonic
on any J-holomorphic curve in X. All level-sets {'P = c} of a J-convex function 'P
are J-convex if we oriente them as boudaries of domains {t.p ::; c}. Conversely, any
function with this property can be made J-convex by a reparametrization: <p = h0'P
˜
for a diffeomorphism h : R R.

1.2 Contact Structure on a J-Convex Hypersurface
eon a J-convex E C X is completely nonintegrable (indeed if a is
The distribution
as in 1.1 then a A dO! > 0) and, by the definition, defines a contact structure on E.
There are no local invariants of contact stuctures where the word "local" may be
refered to the manifold as well as to the space of all contact structures on a given
manifold. In particular, a deformation of J which leaves the hypersurface J-convex
e.
does not change the underlying contact structure

to˜.
A curve / C E is called Legendrian if it is tangent An oriented transversal
eis called positive or negative according to the sign of the I-form a IT(1)
1 C E to
eto
(recall that we have chosen the I-form a with {a = O} = define the canonical
coorientation of the distribution e). Any Legendrian curve can be C<X>-approximated
e.
by a positive as well as negative transversal to

1.3 Invariants of Legendriall Curves
Let 1 be a Legendrian curve in a J-convex hypersurface E eX. Suppose that ,
is homological to 0 in E and fix a relative homology class f3 E H2(˜,1). Pushing
eI,
/ along a vector field transversal to in E one can compute the intersection
number of the perturbed curve /' with (3. This number tb(1 I (3) we will call the
Thurston-Bennequin's invariant. Note that in many cases (for instance, when :E is
a homology sphere) tb(l I (j) does not depend on (j. We will write tb(l) in these
cases or when the choice of /3 is clear from the context.

Now let M C E be any surface which is bounded by , and represents the class
e1M.
f3 E H 2 (E, ,,). Take any trivialization of the bundle Suppose that / is
oriented and let T be a tangent to the I vector field which defines the orientation.
e1M
Then the degree of T with respect to the chosen trivialization of depends only
on I and {3. We will denote it by 1"('Y I(3) and will call the rotation of, with respect
to p. Note that in contrast to tb('l I (3) the rotation number changes the sign with
Eliashberg: Filling by holomorphic discs and its applications 47

w˜en the choice of
the change of the orientation of "{. As above we will write r("{)
fJ is clear or irrelevant.

1.4 Complex Points of a Real Surface in all Almost Complex 4-M˜nifold
Let M be a real 2-surface in an almost complex 4-manifold X, J. A point p E M is
called complex if the tangent plane Tp(M) is a J-complex line in Tp(X). A generic
surface M c X has isolated complex points. If the surlace M is oriented then we
call a complex point p08itive if the orientation of Tp(M) coincides with its complex
orientation and negative in the other case. A surface without complex points is
called totally real.

If the surface M is contained in a J-convex hypersurface E c X then complex points
eof
of M are points where M is tangent to the distribution complex tangencies to
ewith
˜. Intersections of tangent planes to M form an (orientable) line field Me
on the totally real part of M. In the generic case the index of this field at complex
points is equal to Â±1. We will call a complex point p E M elliptic if the index of Me
at M is equal to +1 and hyperbolic in the other case. The field Me integrates to a
I-dimensional foliation (which we will still denote by Me) on M with singularities at
complex points of M. The foliation Me is called the characteristic foliation. Leaves
of the characteristic foliation are, by the definition, tangent to ˜ and, therefore,
Lcgendrian. The foliation Me has a focus type singularity near an elliptic point and
t.he standard hyperbolic singularity near a hyperbolic one.

n.E˜fARJ(. The notion of ellipticity and hyperbolicity of a complex point can be
(\xtended to a generic complex point of an arbitrary surface in X (and not necessar-
ily one which is contained in a pseudoconvex hypersurface). To do that, consider
n fibration Gr2(X) ˜ )[ over . .JC whose fiber at a point x E X is the Grassma-
Ilian of oriented 2-planes in Tx(X). Let CP+(X) and CP_(X) be subbundles of
˜
(ir'2(X) X which consist of conlplex lines \vith, respectively, complex or an-
ticomplex orientation. Note that CP+(_Â¥) and CP_(X) have codimension 2 in
(;'r2(X). For a surface !vI c X the inclusion map can be lifted to a Gaussian map
y : M ˜ Gr2(X), Points from g-l(CPÂ±(X)) are exactly positive and negative
cOlnplex points of M. Properly fixing a coorientation of CP+(X) U CP_(X) in
(ir2(X) (to agree \vith the special case \vhich we considered above) we say that a
48 Eliashberg: Filling by holomorphic discs and its applications

complex point p E M is elliptic (hyperbolic) if 9 is transversal to CPÂ±(X) at p and
the intersection index of CPÂ±(X) with geM) at the point g(p) is equal to +1( -1).

For an oriented surface M C X (possibly with boundary) we denote by eÂ±(M) and
hÂ±(M) numbers of positive or negative elliptic or hyperbolic interior points of M.
= eÂ±(M) -
Let d:i:(M) hÂ±(M).

If M is closed we denote by c(M) the value of th˜ first Chern class Cl(X) of X
on M and by lIeM) the normal Euler number of M in X. If 8M f: 0 let Tbe a
vector field along aM which is tangent to 8M and defines on aM the orientation
induced by the orientation of M. Let n be a vector field tangent to M and outward
transversal to 8M. Suppose that M has no complex points at 8M. Then vector
fields T and n are linearly independent over C and JT is transversal to M in X.
We denote by c(M) the obstruction to the extension of T and n on M as linearly
independent over C vector fields and by v(M) the obstruction to the extension of
J T on M as a transversal to J.1 vector field.

The proof of the following formula ,vhich relates invariants d:i:(M), c(M), v(M) and
the Euler characteristic X(M) of M is straightforward (see [HE] for the discussion).

c
1.4.1. Let M X be either closed or does not have complex poinu
PROPOSITION
on 8M. Then
1
dÂ±(M) = 2(X(M) + v(M) Â± c(M)) .

˜
If M is contained in a J-convex hypersurface C X then the absence of complex
points at 8M can be guaranteed by the condition that aM is transversal to the
distribution ˜. In this case II(M) = 0 and the formula 1.4.1 takes a simpler form

We will need also an analog of 1.4.1 for a sunace M C ax bounded by a Legendrian
curve 8M. IT, in addition, M has to be contained in a pseudo-convex hypersurface
E it is impossible, in general, to avoid complex points at 8M. Instead we want to
standardize them in the follo\ving sense. There exists an isotopy of M in E which
Eliashberg: Filling by holomorphic discs and its applications 49

aM
and such that all complex points at the boundary of the resulting
h.. fixed at
,.urfnce M' are elliptic and their signs alternates along aM' = aM. We say in this
that M' is in normal form near its Legendrian boundary aM'. It is easy to see
,Itt"('
1,III1,t. if M is in normal form near aM then the number of complex points on aM is
fJfllial to 2Itb(8M)I, where the Thurston-Bennequin's invariant tb(8M) is calculated
with respect to the homology class of M in H 2 (E, aM).

1.4.2. Suppo8e that a 8urface M C E c X i3 in normal form near
PUOPOSITION
.1... lJ(˜.gendrian boundary 8/\1. Then

1
dÂ±(M) = 2(X(M) + tb(1\l) Â± reMÂ»˜

WII,t:1'(˜ tb(}'!) and reM) are calculated with re8pect to the homology clas8 of M in
II˜(E, 8M).

'1'0 prove the equality one should approximate At! by surfaces M+ and M- bounded
positive and negative transversals, observe that c(MÂ±) = tb(8M)Â±r(M) (comp.
Ity
(lIe'J) and apply 1.4.1.

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