A Note On Weinstein's Conjecture

0.1 MB·English

by  Renyi Ma

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0 A Note On Weinstein Conjecture∗
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Renyi Ma
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Department of Mathmatics

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2 Tsinghua University

] Beijing, 100084
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People’s Republic of China
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Abstract
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2 In this article, we give new proofs on the some cases on Weinstein
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conjecture and get some new results on Weinstein conjecture.
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0 Keywords Symplectic geometry, J-holomorphic curves, Periodic orbit.
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2000MR Subject Classification 32Q65,53D35,53D12
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/ 1 Introduction and results
h
t
a
m Let Σ be a smooth closed oriented manifold of dimension 2n−1. A contact
: formonΣisa1−formsuchthatλ∧(dλ)n−1 isavolumeformonΣ.Associated
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i to λ there is the so-called Reed vectorfield X defined by i λ ≡ 1, i dλ ≡
X λ Xλ Xλ
0. The integral curve of X is called characteritcs. There is a well-known
r λ
a conjecture raised by Weinstein in [17] which concerned the close Reeb orbit
in a contact manifold.
Conjecture(see[17]). If (Σ,λ) is a close simply connected contact man-
ifold with contact form λ of dimension 2n−1, then there is a close charac-
teristics.
Let (M,ω) be a symplectic manifold and h(t,x)(= h (x)) a compactly
t
supported smooth function on M ×[0,1]. Assume that the segment [0,1] is
∗Project 19871044Supported by NSF
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endowed with time coordinate t. For every function h define the (time −
dependent) Hamiltonian vector field X by the equation:
ht
dh (η) = ω(η,X ) for every η ∈ TM (1.1)
t ht
The flow gt generated by the field X is called Hamiltonian flow and its
h ht
time one map g1 is called Hamiltonian diffeomorphism. Now assume that
h
H be a time independent smooth function on M and X its induced vector
H
field.
Theorem 1.1 Let (M,ω) be an exact symplectic manifold convex at infinity
or with bounded geometry. Let (Σ,λ) be a contact manifold of induced type in
M with induced contact form λ, i.e., there exists a vector field X transversal
to Σ such that L ω = ω and λ = i ω, X its Reeb vector field. If there exists
X X λ
a Hamiltonian diffeomorphism h such that h(Σ)∩Σ = ∅, then there exists at
least one close characteritics on Σ
Corollary 1.1 Let (M,ω) be an exact symplectic manifold which is convex
at infinity or has bounded geometry(see[6]). M ×C be a symplectic manifold
with symplectic form ω⊕σ, here (C,σ) standard symplectic plane. Let r > 0
0
be a fixed number and B (0) ⊂ C the closed ball with radius r . If (Σ,λ) be a
r0 0
contact manifold of induced type in M ×B (0) with induced contact form λ,
r0
i.e., there exists a vector field X transversal to Σ such that L (ω⊕σ) = ω⊕σ
X
and λ = i (ω ⊕σ), X its Reeb vector field. Then there exists at least one
X λ
close characteristics.
Corollary 1.1was proved in [10] by Hofer-Viterb’s method(see[8]).
Corollary 1.2 Let M be any open manifold and (T∗M,dα) be its cotangent
bundle. Let (Σ,λ) be a close contact manifold of induced type in T∗M. there
exists at least one close characteristics on Σ.
Corollary 1.2 generalizes the results in [9, 15, 10]. The proof of Theorem1.1
is close as in [11].
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2 Lagrangian Non-squeezing
Let W be a Lagrangian submanifold in M, i.e., ω|W = 0.
Definition 2.1 Let
l(M,W,ω) = inf{| f∗ω| > 0|f : (D2,∂D2) → (M,W)}
ZD2
Theorem 2.1 ([12])Let (M,ω) be a closed compact symplectic manifold or
a manifold convex at infinity and M ×C be a symplectic manifold with sym-
plectic form ω⊕σ, here (C,σ) standard symplectic plane. Let 2πr2 < s(M,ω)
0
and B (0) ⊂ C the closed disk with radius r . If W is a close Lagrangian
r0 0
manifold in M ×B (0), then
r0
l(M,W,ω) < 2πr2
0
This can be considered as an Lagrangian version of Gromov’s symplectic
squeezing.
Corollary 2.1 (Gromov[6])Let (V′,ω′) be an exact symplectic manifold with
restricted contact boundary and ω′ = dα′. Let V′×C be a symplectic manifold
with symplectic form ω′⊕σ = dα = d(α′⊕α , here (C,σ) standard symplectic
0
plane. If W is a close exact Lagrangian submanifold, then l(V′×C,W,ω) ==
∞, i.e., there does notexist anyclose exactLagrangiansubmanifoldin V′×C.
Corollary 2.2 Let Ln be a close Lagrangian in R2n and L(R2n,Ln,ω) =
2πr2 > 0, then Ln can not be embedded in B (0) as a Lagrangian submani-
0 r0
fold.
3 Constructions of Lagrangian submanifolds
Let (Σ,λ) be a contact manifolds with contact form λ and X its Reeb vector
field, then X integrates to a Reeb flow η for t ∈ R1. Let
t
(V′,ω′) = ((R×Σ)×(R×Σ),d(eaλ)⊖d(ebλ))
and
L = {((0,σ),(0,σ))|(0,σ) ∈ R×Σ}.
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Let
′ ′
L = L×R,L = L×{s}.
s
Then define
′ ′ ′
G : L → V
′ ′ ′
G(l ) = G(((σ,0),(σ,0)),s) = ((0,σ),(0,η (σ))) (3.1)
s
Then
′ ′ ′
W = G(L) = {((0,σ),(0,η (σ)))|(0,σ) ∈ R×Σ,s ∈ R}
s
′ ′ ′
W = G(L ) = {((0,σ),(0,η (σ)))|(0,σ) ∈ R×Σ}
s s s
for fixed s ∈ R.
Lemma 3.1 There does not exist any Reeb closed orbit in (Σ,λ) if and only
′ ′ ′
if W ∩W is empty for s 6= s.
s s′
Proof. First if there exists a closed Reeb orbit in (Σ,λ), i.e., there exists
σ ∈ Σ, t > 0 such that σ = η (σ ), then ((0,σ ),(0,σ )) ∈ W′ ∩ W′ .
0 0 0 t0 0 0 0 0 t0
Second if there exists s 6= s′ such that W′ ∩W′ 6= ∅, i.e., there exists σ
0 0 s0 s′0 0
such that
((0,σ0),(0,ηs0(σ0)) = ((0,σ0),(0,ηs′0(σ0)),
then η(s0−s′0)(σ0) = σ0, i.e., ηt(σ0) is a closed Reeb orbit.
Lemma 3.2 If there does not exist any closed Reeb orbit in (Σ,λ) then
there exists a smooth Lagrangian injective immersion G′ : W′ → V′ with
′
G(((0,σ),(0,σ)),s) = ((0,σ),(0,η (σ))) such that
s
′ ′
G : L×(−s ,s ) → V (3.2)
s1,s2 1 2
is a regular exact Lagrangian embedding for any finite real number s , s ,
1 2
here we denote by W′(s ,s ) = G′ (L×(s ,s )).
1 2 s1,s2 1 2
Proof. One check
G′∗((eaλ−ebλ)) = λ−η(·,·)∗λ = λ−(η∗λ+i λds) = −ds (3.3)
s X
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since η∗λ = λ. This implies that G′ is an exact Lagrangian embedding, this
s
proves Lemma 3.2.
Now we modify the above construction as follows:
′
F : L×R×R → (R×Σ)×(R×Σ)
′
F (((0,σ),(0,σ)),s,b) = ((0,σ),(b,η (σ))) (3.4)
s
Now we embed a elliptic curve E long along s−axis and thin along b−axis
such that E ⊂ [−s ,s ]×[0,ε]. We parametrize the E by t.
1 2
Lemma 3.3 If there does not exist any closed Reeb orbit in (Σ,λ), then
F : L×S1 → (R×Σ)×(R×Σ)
F(((0,σ),(0,σ)),t) = ((0,σ),(b(t),η (σ))) (3.5)
s(t)
is a compact Lagrangian submanifold. Moreover
l(V′,F(L×S1,d(eaλ−ebλ)) = area(E) (3.6)
Proof. We check that
F∗(eaλ⊖ebλ) = −eb(t)ds(t) (3.7)
So, F is a Lagrangian embedding.
If the circle C homotopic to C ⊂ L×s then we compute
1 0
F∗(eaλ−ebλ) = F∗(e0λ−e0λ) = 0. (3.8)
Z Z
C C1
since λ−λ|C = 0 due to C ⊂ L. If the circle C homotopic to C ⊂ l ×S1
1 1 1 0
then we compute
F∗(eaλ−ebλ) = (−)ebds = n(area(E)). (3.9)
Z Z
C C1
This proves the Lemma.
Gromov’s figure eight construction: First we note that the con-
struction of section 3.1 holds for any symplectic manifold. Now let (M,ω) be
an exact symplectic manifold with ω = dα. Let Σ = H−1(0) be a regular and
close smooth hypersurface in M. H is a time-independent Hamilton function.
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Set (V′,ω′) = (M×M,ω⊖ω). Iftheredoesnotexist anyclose orbitforX in
H
(Σ,X ), one can construct the Lagrangian submanifold L as in section 3.1,
H
let W′ = L. Let h = h(t,·) : M → M, 0 ≤ t ≤ 1 be a Hamiltonian isotopy
t
of M induced by hamilton fuction H such that h (Σ) ∩Σ = ∅, |H | ≤ C .
t 1 t 0
Let h¯ = (id,h ). Then F′ = h¯ : W′ → V′ be an isotopy of Lagrangian
t t t t
embeddings. As in [6], we can use symplectic figure eight trick invented by
Gromov to construct a Lagrangian submanifold W in V = V′ ×R2 through
′ ′
the Lagrange isotopy F in V , i.e., we have
Proposition 3.1 Let V′, W′ and F′ as above. Then there exists a weakly
exact Lagrangian embedding F : W′ ×S1 → V′ ×R2 with W = F(W′ ×S1)
is contained in M ×M ×B (0), here 4πR2 = 8C and
R 0
′ ′
l(V ,W,ω) = area(M ) = A(T). (3.10)
0
Proof. Similar to [6, 2.3B′].
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Example. Let M be an open manifold and (T∗M,p dq ) be the cotan-
i i
gent bundle of open manifold with the Liouville form p dq . Since M is open,
i i
there exists a function g : M → R without critical point. The transla-
tion by tTdg along the fibre gives a hamilton isotopy of T∗M : hT(q,p) =
t
(q,p+tTdg(q)), so for any given compact set K ⊂ T∗M, there exists T = T
K
such that hT(K)∩K = ∅.
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3.1 Proof on Theorem 1.1
Since (Σ,λ) be a close contact manifold of induced type in M with induced
contact form λ, then by the well known theorem that the neighbourhood
(U(Σ),ω) of Σ is symplectomorphic to ([−ε,ε]×Σ,deaλ) for small ε. So, by
Proposition3.1,wehave aclose LagrangiansubmanifoldF(L×S1) contained
in M ×M ×B (0). By Lagrangian squeezing theorem, i.e., Theorem 2.1, we
R
have
l((M ×M ×C),F(L×S1,ω ⊕ω) = area(E) ≤ 2πR2. (3.11)
If s −s large enough, area(E) > 2πR2. This is a contradiction. This con-
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tradiction shows there exists at least one close characteristics.
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