$C^0$-rigidity of characteristics in symplectic geometry

0.21 MB·

by  Emmanuel Opshtein

0
C
-rigidity of
hara
teristi
s in symple
ti
geometry.
Emmanuel Opshtein.
9
0
0
2
Abstra
t

n C0
a The paper
on
erns a -rigidity result for the
har
teristi
folia-
J
tions in symple
ti
geometry. A symple
ti
homeomorphism (in the

3 sense of Eliashberg-Gromov) whi
h preserves a smooth hypersurfa
e
2
also preserves its
hara
teristi
foliation.

]
G
S Introdu
tion
h. C0
t GromovandEliashbergshowedthata -limitofsymple
ti
di(cid:27)eomorphisms
a
whi
h is itself a di(cid:27)eomorphism is symple
ti
([7, 5℄, see also [8℄). This
m C0
rigidity result leadsto the de(cid:28)nition of symple
ti
homeomorphisms (the -
[
limitsofsymple
ti
di(cid:27)eomorphismswhi
harehomeomorphisms),andshows

1
that they de(cid:28)ne a proper subset of volume preserving homeomorphisms in
v 4
9 dimension greater than . It also raises the question of the survival of the
3 symple
ti
invariants to this limit pro
ess. Whi
h
lassi
al invariants of
6 C0
symple
ti
geometry remain invariants of this maybe softer -symple
ti
3
. geometry? This paper shows that the
hara
teristi
foliation is one of them.
1
0 S S′
9 Theor(eMm,ω1). L(Met′,ωa′n)d be smooth hypersurfa
es of some symplMe
ti
maMn-′
0 ifolds , . Any symple
ti
homeomorphism between and
S S′ S S′
:
v whi
h sends to transports the
hara
teristi
foliation of to that of .
i
X
The
hara
teristi
foliation is a symple
ti
invariant of a given hypersur-
r S
a fa
e ,whi
h
anbede(cid:28)nedastheintegral foliationofthe(onedimensional)
S
null spa
e of the restri
tion of the symple
ti
form to . This de(cid:28)nition
S
is intrinsi
ally smooth sin
e it involves the tangent spa
es of . But the
roles of this foliation in symple
ti
geometry are many. In parti
ular, one
of its rather folklori
property
on
erns non-removable interse
tion : if two
smoothly bounded open sets interse
t exa
tly on their boundaries, and if
no symple
ti
perturbation
an separate them, then the boundaries share a

ommon
losed invariant subset of the
hara
teristi
foliation [9, 10, 12, 13℄.
Thispaper pro
eeds from the remark that this property have ameaning also
in the
ontinuous
ategory, so de(cid:28)ning this foliation in
ontinuous terms is

on
eivable.
1
An appli
ation of this theorem is a weak answer to a question by Eliash-
berg and Hofer about the symple
ti

hara
terization of a hypersurfa
e by
the open set it bounds : Under whi
h
onditions the existen
e of a symple
-
tomorphism between two smoothly bounded open sets in symple
ti
manifolds
imply that their boundaries are symple
tomorphi
also [4℄ ? Some results
are known [2, 1℄, but nothing when the sets are standard balls. Theorem 1
allows a partial answer in this
ase.
U R4
Theorem 2. Let be asmoothly bounded open set in . Assume thatthere
B4(1) U
is a symple
tomorphism between and whi
h extends
ontinuously to
S3 ∂U ∂U S3
a homeomorphism between and . Then is symple
tomorphi
to .
The paper is organized as follows. We (cid:28)rst de(cid:28)ne symple
ti
hammers
and explain their roles : theorem 1 pro
eeds from a lo
alization of their
a
tionsalong
hara
teristi
s(se
tion1). Thislo
alizationisprovedinse
tion
2. We then present the appli
ation in the last se
tion.
Aknowledgements. I wish to thank Leonid Polterovi
h for making me
aware of a serious mistake in the (cid:28)rst version of the proof.
1 Symple
ti
hammers.
S M B
Let be a hypersurfa
e in a symple
ti
manifold . We say that is a
S
small ball
entered on if iMt is a symple
ti
Re2mn−b1ed:d=inRg×ofCann−e1u
⊂lidCenan baSll

entered at the origin into whi
h sends to .
S S
+
Su
h a ball is dis
onne
ted by into two
omponents denoted by and
S S
−
. By a
lassi
al result, any point of is the
enter of su
h a ball. Fix also
M
a metri
on in order to refer to small sets.
x y S∩B ε
De(cid:28)nition 1.1. Given two points , on and a (small)positive real ,
ε x y B
an -symple
ti
hammer between and with support in is a
ontinuous
Φ t ∈ [0,1]
t
path of symple
ti
homeomorphisms ( ) with
ommon supports in
B U (x) U (y)
ε ε
, and for whi
h there exist two open sets and
ontained in the
ε x y
-balls around and respe
tively su
h that :
Φ =
0
1. Id,
Φ (z) ∈ S t ∈]0,1] z ∈S ∩U (x)
t + ε
2. for all and ,
Φ (z) ∈ S t ∈]0,1] z ∈S ∩U (y)
t − ε
3. for all and ,
Φ (z) ∈ S t ∈ [0,1] z ∈ S\ U (x)∪U (y)
t ε ε
4. for all and .
(cid:0) (cid:1)
A smooth hammer will refer to a smooth isotopy of smooth symple
tomor-
phisms verifying the four
onditions above.
Φ S
t
In other terms, preserves the hypersurfa
e ex
ept for two bumps in
opposite sides (a symmetry is ne
essary in view of the volume preservation).
One
an easily
onstru
t examples of symple
ti
hammers.
2
y
Φt
x
Φt(x)
x,y ∈ B ∩S
Proposition 1.2. If lie in the same
hara
teristi
, there exist
ε x y ε > 0
-symple
ti
hammers between and for all .
Proof : Sin
e all hypersurfa
es are lo
ally syRm2pn−le1
t=i
a{lly tzhe=sa0m}e,⊂itCins
1
enough to produ
e a symple
ti
hammer for Im
p = 0 q = (1/2,0,...,0) x = z
1 1
between the points and . Putting Re ,
y = z r = |z |
1 1 i i
Im and ,
onsider a Hamiltonian of the following type.
H(z ,...,z ) := χ(y )ρ(x )Πn f(r ).
1 n 1 1 i=2 i
χ ρ f
If , and are the bell fun
tions represented in (cid:28)gure 1, and maybe
H
multiplying by a small
onstxant inyorder to slow the (cid:29)ow down produ
e(cid:3)s
a symple
ti
hammer between and .
1+ε ΦtH
1−ε
ε
χ(y1) x T y
−εf(ri)
−ε ε ρ(x1) 12−ε 21+ε
H
Figure 1: The Hamiltonian (cid:29)ow of in the proof of proposition 1.2.
Proposition1.2
aneasilybereversedinthesmooth
ategory: twopoints
S ∩B
lie in the same
hara
teristi
leaf of if and only if there exist smooth
symple
ti
hammers between them. It is less obvious, but still true that
all the symple
ti
hammers also meet this
onstraint. Theorem 1 obviously
follows be
ause the
lass of symple
ti
hammers is preserved by symple
ti
homeomorphisms.
S B S
Proposition 1.3. A hypersurfa
e and a small ball
entered on being
ε x,y ∈ B ∩S B
given, there exists an -hammer between with support in for
ε x y
all small if and only if and are on the same
hara
teristi
leaf.
3
M,M′,S,S′ Φ
Proof of theorem 1 (assuming proposition 1.3) : Let and be
M M′
as in theorem 1, and put any metri
on and . Consider two points
x,y ∈S B = {B }
α
whi
h lie in the same
hara
teristi
. Consider a
overing
S Φ
of by small balls (in the above sense) whose images by are
ontained
B′ S′ (x ) x y
α i i≤N
in small balls
entered on . Let be a
hain between and
x = x x = y x x
0 N i i+1
(that is , ) su
h that and are always in a same ball
(ε)
B ε Φ B x
i t i i
. Then there exist -hammers with supports in between and
x ε Φ◦Φ(ε)◦Φ−1 δ(ε)
i+1 t
for all . The isotopies de(cid:28)ne
ontinuous -symple
ti
B′ Φ(x ) Φ(x ) δ(ε)
i i i+1
hammers with support in between and , where goes
ε Φ(x ) Φ(x )
i i+1
to zero with . TherefoΦre(xb)y proΦpo(syi)tion 1.3, and are on th(cid:3)e
same
hara
teristi
, so and are also on the same
hara
teristi
.
2 Proof of proposition 1.3
The idea is the following. Sin
e preserving a foliation is a lo
al property,
and sin
e all hypersurfa
es are lo
ally the same in the symple
ti
world,
we
ould translate the non-preservation of one
hara
teristi
by a symple
-
ti
homeomorphism to the existen
e of a lo
al, hen
e universal obje
t (a
hammer between points on distin
t
hara
teristi
s) whi
h would exist on all
hypersurfa
es. These
ontinuous hammers would allow to break interse
-
tions between open sets as long as these interse
tions only
onsist of one

hara
teristi
. But some su
h interse
tions are known to be non-removable :
the most famous one being the interse
tion between the
omplement of the
Z(1) B2n(1)

ylinder and the
losed ball .
x
Lemma 2.1. If Spr2onp−o1si⊂tioCnn1.3 does not hold, theεn for any point of the
eu
lidean sphere and for any positive , there exists a
ontinu-
ε x y
ous -symple
ti
hammer between and a point whi
h does not lie in the
x

hara
teristi

ir
le passing through .
Proof : Assume that proposition 1.3 does not hold. Then there exists a
S p,q ∈ S ∩B
small ball
entered on a hypersurSfa
∩eB, two points Φ(ε) := (Φ(εw))hi
h are
t t∈[0,1]
not in the same
hara
teristi
of and a family of
ε B p q
-symple
ti
hammers with supports in between and . By de(cid:28)nition
Ψ B
1
of a small ballB, th⊂ereCnis a symple
ti
di(cid:27)eomorΨph(Sis∩mB)=bRet2wn−ee1n∩B and an
1 1
eu
lidean ball aroundtheoriginwith . Then
Ψ Φ(ε) ε Ψ (p) Ψ (q)
1 1 1
takes to an -hammer between and whi
h are not on
the same
hara
teristi
. By use of translation and res
aling, we
an assume
Ψ (p) B 0
1 1
that is the origin and is as small a neighbourhood of as wished.
x ∈ S2n−1 B
1
Now given the point Ψ :B, a−n→d iCfn is sΨma(lBl en∩ouRg2hn,−t1h)e⊂reSe2xni−st1s
2 1 2 1
Ψa s(y0m)p=le
xti
di(cid:27)eomorphism with R2n−1 ∩B ,
2 1
and su
h that di(cid:27)erent
hara
teristi
s of are sent by
Ψ S2n−1∩Ψ (B ) S2n−1
2 2 1
not only to di(cid:27)erent
hara
teristi
s of but even of
Ψ B
2 1
(this means that we do not allow to "bend" so asto take two di(cid:27)erent
4
hara
teristi
s to two di(cid:27)erent segments of a same
hara
teristi

ir
le of
S2n−1 Φ(ε)
). The
ontinuous symple
ti
isotopies obtained by transporting
Ψ ◦Ψ ε x y := Ψ (Ψ (q))
2 1 2 1
by are -hammers betweenx and the point whi
h(cid:3)is
not on the
hara
teristi
through .
U ⊂ Z(1) := B2(1) × Cn−1
Lemma 2.2. Any bounded open set whose
∂U S1×{·} ∂Z(1)
boundary does not
ontain a
hara
teristi

ir
le of
an
∂Z(1) Z(1)
be symple
ti
ally displa
ed from to the interior of .
∂Z(1)
Proof : Re
all that the
hara
teristi
(cid:29)ow of
an be oriented by the
JN N
∂veZ
(t1o)r (cid:28)eldJ where the ve
tor is the outwardCnnormal ve
tor (cid:28)eld to
and is the standard
omplex stru
ture on . Now observe that if
K := ∂U∩∂Z(1)
the
ompa
t set H doeCs nnot
ontain any
hara
teristi

ir
le,
there exists a smooth fun
tion on whi
h de
reases along the
hara
-
K
teristi
(cid:29)ow on a neighbourhood of [14℄. The
orresponding Hamiltonian
ve
tor (cid:28)eld points toward the inside of the
ylinder on this neighbourhood
K
of be
ause
g(X (x),N(x)) = ω(X (x),JN(x)) = dH(JN(x)) < 0,
H H
U Z(1) H (cid:3)
so is driven inside by the (cid:29)ow of for small enough times.
B2n(1)
Letus
omeba
ktoproposition1.3. Consider asanopensetlying
Z(1) S2n−1 Z(1)
in . Its boundary meets along pre
isely one
hara
teristi
∂Z(1)

ir
le of :
S2n−1∩∂Z(1) = {|z |= 1, z′ = 0} ⊂ B2 (1)×Cn−1.
1 z1 z′
Assume then by
ontradi
tion that proposition 1.3 does not hold. Then by
ε Φ (1,0) ∈ S2n−1 ∩
t
lemma 2.1, there exists -hammers between the point
∂Z(1) y ∈ S2n−1∩Z(1) ε ε
and an interior point for arbitrarily small . If is
smallenough, and slowingthe(cid:29)ow ofthehammer down enough (
onsidering
Φ Φ U B(1) Φ
at t 1
in pla
e of ), the image of by is therefore an open set of
Z(1) ∂Z(1) S1 ×{0}
whose boundary interse
tion with is the
ir
le minus
(1,0)
a small neighbourhood of whi
h was taken inside the ball - hen
e
inside the
ylinder - by our hammer. By lemma 2.2, there exists therefore
K Φ1 (U) Z(1)
K
a smooth Hamiltonian su
h that is relatively
ompa
t in .
Φ Φ Φ1 ◦ Φ (B(1))
t t K 1
Taking a good enough smooth approximation of , is
Z(1)
still relatively
ompa
t in . But this is in
eontradi
tion withe Gromov(cid:3)'s
non-squeezing theorem.
3 Symple
ti
geometry from the inside.
In this se
tion, we prove theorem 2. We (cid:28)rst prove that the
hara
teristi
s
U
of the sphere are sent to the
hara
teristi
s of the boundary of . The
next point is to see that the a
tion on these
hara
teristi
s
oin
ide on both
hypersurfa
es. The result follows.
5
3.1 A one-sided version of theorem 1.
In fa
t, theorem 1 holds also in a slightly more general framework.
U U′
Theorem 3. Let and be smoothly bounded open sets in symple
ti
U U′
manifolds. Any symple
ti
homeomorphism between and whi
h extends

ontinuously to a homeomorphism of their
losures transports the
hara
ter-
∂U ∂U′
isti
foliation of to that of .
The proof below is rather qui
k be
ause everything has already been ex-
plained. Exa
tly as for theorem 1, the point is to de(cid:28)ne a
onvenient notion
of symple
ti
hammer whi
h isinvariant by one-sided symple
ti
homeomor-
phisms. Note that de(cid:28)nition 1.1 has to be modi(cid:28)ed sin
e it involves both
sidesofthehypersurfa
ethroughthebumps. Thesolutionissimplytoforget
U
about the part of the hammer whi
h goes outside .
U
De(cid:28)nition 3.1 (One-sidedhammers). Let be a smoothly bounded open set
M B ∂U x,y ∈ B∩∂U ε
in , a smallball
entered in , . A one-sided -symple
ti
x y Φ :
t
hammer between and is a
ontinuous isotopy of homeomorphism
U\B (y) −→ U U\B (y)
ε ε
whi
h
an be uniformly approximated in by smooth
B
symple
ti
isotopies with
ommon supports in , and whi
h veri(cid:28)es also
properties 1), 2) and 4) of de(cid:28)nition 1.1.
This de(cid:28)nition a
tually provides a one-sided de(cid:28)nition of the
hara
teristi
s
be
ause of the following.
ε x y
Proposition 3.2. There exists one-sided -hammers between and for all
ε x y ∂U
small if and only if and belong to the same
hara
teristi
of .
Proof : The proof is very similar to the proof of proposition 1.3. On one
hand, sin
e the de(cid:28)nition of a one-sided hammer is a lo
al one, and sin
e
smooth hypersurfa
es have no lo
al invariants, the existen
e of a one-sided
∂U∩B
hammer between two pointsnot inthe same
hara
teristi
of ensures
the existen
e of su
h a hammer on the ellipsoid
|z′|2
E(1,2) := {|z1|2+ ≤ 1} ⊂ Z(1) ⊂ Cz ×Cz′
4
(1,0) C
0
between the point of the "least a
tion"
hara
teristi
and another
y E(1,2)\B (y)
ε
point not in this
hara
teristi
. This hammer isotops to an
U ⊂ Z(1) ∂Z(1)
openset whoseboundary
ontainsno
hara
teristi

ir
leof .
U ∂Z(1)
By a Hamiltonian (cid:29)ow,
an be symple
ti
ally displa
ed from inside
Z(1) ε y C E(1,2)\B (y)
0 ε
. For small enough, and sin
e does not belong to ,
1

ontains the ball of radius ,
ontradi
ting again Gromov's non-squeezin(cid:3)g
theorem.
6
3.2 Proof of theorem 2.
S3 = ∂B4(1)
Letus(cid:28)x the notatπion:sS.3O−n→ P1 , the
ωhara
teristi
foliation de(cid:28)nes
0
thePH1opf (cid:28)bration π . Moreover, if is the standardωareaRfo4rm
on with total area , the restri
tion of the symple
ti
form on to
S3 π∗ω
0
is .
U R4 f
Let be a smoothly bounded domain in , a symple
ti
di(cid:27)eomor-
U B4(1)
phism from to whi
h extends
ontinuously to a homeomorphism
f ∂U
of the boundaries. Then by theorem 3, sends the
hara
teristi
s of
S3 ∂U
to those of , so the
hara
teristi
foliation of is a topologi
al Hopf
Ψ : ∂U −→ S3
(cid:28)bration. By a work of Epstein [6℄, there is a di(cid:27)eomorphism
∂U S3
whi
h takes the
hara
teristi
s of to the Hopf
ir
les of . Sin
e the
Ψ ω S3
∗
rΨesωtri
tion of to vanishes along thωe HopPf1
ir
Ψlesωb=utπn∗eωver vanishes,
∗ 1 ∗ 1
is alωso the pull-Pb1a
k of an area form on : . We
laim
1
that the -area of is the expe
ted one :
ω = π.
(cid:12)(cid:12)ZP1 1(cid:12)(cid:12) (1)
(cid:12) (cid:12)
(cid:12) (cid:12)
Let us a

ept this as a fa
t for a momPe1nt. Then, after a possible orientation-
reversing
hanϕge:o(fP
1o,oωrd)in−a→tes(oPn1,ω,)there existsΦan arϕea-preservinπg dif-
1 0
feomorphism . Any lift of through is a
S3 π∗ω0 = ω|S3 Ψ∗ω
self-di(cid:27)eomorphism of whi
h pulls ba
k to .
(∂U,ω) Ψ // (S3,Ψ ω) Φ // (S3,π∗ω )
∗ 0
π π
(cid:15)(cid:15) (cid:15)(cid:15)
(P1,ω ) ϕ // (P1,ω )
1 0
(Φ◦Ψ)∗ω|S3 = ω|∂U
Hen
e, and theorem 2 follows by a
lassi
al argument
of standard neighbourhood (see [11℄).
In order to prove (1), observe that putting
ω = πr2,
(cid:12)Z 1(cid:12)
(cid:12) P1 (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
Φ
the above argument shows the existen
e of a symple
ti
di(cid:27)eomorphism
r
between a neighbourhood of the eu
lidean sphere of radius and a neigh-
∂U S3(r) ∂U g = f ◦ Φ
bourhood of whi
h takes to . The map de(cid:28)nes
V
r
therefore a symple
tomorphism between two one-sided neighbourhoods ,
V r = 1
1
(cid:3) of the eu
lidean spheres. The following lemma (cid:28)nally ensures that .
Lemma 3.3. There exists a symple
tomorphism between two open one-sided
S3(r) S3(R) r = R
neighbourhoods of eu
lidean spheres and if and only if .
7
It is straightforward in view of [3℄. We give however the argument for
the
onvenien
e of the reader. Noti
e that it would be obvious, should the
g S3(r)
map extend smoothly to (whi
h is pre
isely not the
ase) be
ause
S3(r) S3
the a
tion of the
hara
teristi
s on and should
oin
ide.
r ≤ R V V
r R
Proof : Assume that ,
all and the one-sided open neighbour-
g : V −→ V
R r
hoods of the two spheres and a symple
tomorphism. Noti
e
V S3(R) V
R R
thatsin
e issymple
ti
ally
onvexwithrespe
tto ,sois withre-
S3(r) V
r
spe
t to (meaning that there isa
ontra
ting ve
tor (cid:28)eld on (cid:29)owing
S3(r) V V
r R
away from ), so both and are
ontained inside the
orresponding
B4(R) X = − r ∂/∂r
i i
eu
lidean balls. On , the
ontra
ting ve
tor (cid:28)eld
ω λ = − r2dθ g X P
0 i i ∗
is -dual to the form . Its image is also a
ontra
t-
V ω Pg λ H1(V ) = 0 g λ
r ∗ 0 r ∗
ing ve
tor (cid:28)eld on , -dual to . Sin
e , extends to a
−ω B4(r) B4(r)
primitive of on . This extension provides with a
ontra
ting
X′ g X V
∗ r
ve
tor (cid:28)eld whi
h
oin
ides with on and whi
h is forward
om-
B4(r)
plete (its (cid:29)ow is de(cid:28)ned for all positive time) be
ause it points inside
g
near its boundary. Therefore
an be extended to a symple
ti
embedding
g : B4(R) −→ B4(r)
by the formula :
e g(p)= ΦtX′ ◦g Φ−Xt(p) , ∀p∈ B4(R), ∀t su
h that Φ−Xt(p)∈ VR.
(cid:0) (cid:1)
e
R ≤ r
B(cid:3)ut su
h an embedding is only possible if by volume
onsiderations.
Referen
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Emmanuel Opshtein,
Université de Strasbourg,
IRMA, UMR 7501
7 rue René-Des
artes
67084 Strasbourg Cedex
9