Monotone Lagrangian twist tori
based on
Y. Chekanov, F. Schlenk, Electron. Res.
Announc. Math. Sci. 17 (2010)
August 30, 2014
L ⊂ (M, ω) Lagrangian torus
there are two “symplectic” homomorphisms:
Z
[D] 7→
ω
σL : π2 (M, L) → ,
R
Z
µL : π2 (M, L) → ,
D
[D] 7→ Maslov index of D
L ⊂ (M, ω) Lagrangian torus
there are two “symplectic” homomorphisms:
Z
[D] 7→
ω
σL : π2 (M, L) → ,
R
Z
µL : π2 (M, L) → ,
D
[D] 7→ Maslov index of D
L monotone :⇔
σL (A) = const µL (A)
Meaning: ω-size of D on L
❀ “symmetry” of L
≈
for all A ∈ π2 (M, L)
twisting of ω along D
Example.
Torus fibers:
×
µ
The goal
Construction of different monotone Lagrangian tori in
R2n , CPn ,
S2 × · · · × S2
The goal
Construction of different monotone Lagrangian tori in
R2n , CPn ,
1. Construction
2. Classification
3. Non-displaceability
S2 × · · · × S2
Construction
Fix k ∈
N
S(k) := sector in
R2
Choose a curve γ ⊂ S(k):
2π
k+1
1
Twist torus
Define
Θk =
(
k+1
X
i α1
i αk+1
√1
e
γ(t),
.
.
.
,
e
γ(t)
αj = 0
k+1
What is this ?
j=1
)
Take γ as above;
turn it into the complete diagonal
∆ = {(z, . . . , z)} ⊂
γ∆ (t) = √
Ck+1 :
1
γ(t), . . . , γ(t)
k +1
Take γ as above;
turn it into the complete diagonal
∆ = {(z, . . . , z)} ⊂
γ∆ (t) = √
Usual Tk+1 -action on
Ck+1 :
1
γ(t), . . . , γ(t)
k +1
Ck+1 :
(z1 , . . . , zk+1 ) 7→ ei α1 z1 , . . . , ei αk+1 zk+1
Take γ as above;
turn it into the complete diagonal
∆ = {(z, . . . , z)} ⊂
γ∆ (t) = √
Usual Tk+1 -action on
Ck+1 :
1
γ(t), . . . , γ(t)
k +1
Ck+1 :
(z1 , . . . , zk+1 ) 7→ ei α1 z1 , . . . , ei αk+1 zk+1
Take the subtorus
k+1
X
k
i α1
i αk+1 T0 =
e , ..., e
αj = 0
j=1
Θk = Tk0 · γ∆
Θk is
• a torus
Θk is
• a torus
• Lagrangian
Θk is
• a torus
• Lagrangian
• monotone
Θk is
• a torus
• Lagrangian
• monotone
• embedded:
Figure: Θ5 ∩ ∆.
Note :
Θk ⊂ D 2 (1 + ε) × · · · × D 2 (1 + ε) =: D 2(k+1) (1 + ε)
Note :
Θk ⊂ D 2 (1 + ε) × · · · × D 2 (1 + ε) =: D 2(k+1) (1 + ε)
For k = 1:
1
Θ1 : Chekanov ’96, Eliashberg–Polterovich ’97
Let’s twist again !
Let’s twist again !
Iterate this !
Fix k.
For z ∈ S(k) define
(
)
k+1
X
1
Θk (z) = √
αj = 0
ei α1 z, . . . , ei αk+1 z k +1
j=1
isotropic k-torus in
Ck+1
Recall Θk1 ⊂ D 2(k1 +1) (1 + ε)
Recall Θk1 ⊂ D 2(k1 +1) (1 + ε)
Let’s twist Θk1 at (e.g.!) the first coordinate z1 by Tk02 :
take symplectic embedding ψ : D 2 (1 + ε) ֒→ S(k2 ):
ψ
1+ε
1+ε
Recall Θk1 ⊂ D 2(k1 +1) (1 + ε)
Let’s twist Θk1 at (e.g.!) the first coordinate z1 by Tk02 :
take symplectic embedding ψ : D 2 (1 + ε) ֒→ S(k2 ):
ψ
1+ε
1+ε
k2
k1
Θ (Θ ) :=
n
Θk2 (z1 ) × (z2 , . . . , zk1 +1 ) |
(z1 , z2 , . . . , zk1 +1 ) ∈ (ψ × idk1 ) (Θk1 )
o
twist at the ℓ2 th coordinate ❀ Θkℓ22 (Θk1 )
Note: Θk1 (S 1 (1 + ε)) = Θk
twist at the ℓ2 th coordinate ❀ Θkℓ22 (Θk1 )
Note: Θk1 (S 1 (1 + ε)) = Θk
Twist again ...
❀ many monotone Lagrangian tori in
R2n :
Θkℓ := Θkℓmm · · · Θkℓ22 Θkℓ11 ⊂ D k1 +···+km +1 (1 + ε)
=: primitive twist torus
Taking products =: twist torus
Graphical representation
S1 ≈ γ ⊂
R2 :
R4 :
Θ4 ⊂ R10 :
Θ1 ⊂
Θ11 Θ1 :
Θ23 Θ11 Θ2 :
Θ2 × Θ1 Θ1 :
4
TCliff
:
II. Classification
up to what ?
II. Classification
up to what ?
1. Lagrangian isotopy (❀ all the same: product tori)
II. Classification
up to what ?
1. Lagrangian isotopy (❀ all the same: product tori)
2. global symplectomorphism
3. Hamiltonian isotopy
II. Classification
up to what ?
1. Lagrangian isotopy (❀ all the same: product tori)
2. global symplectomorphism
3. Hamiltonian isotopy
For twist tori:
In all our results 2. ≡ 3.
In
R2n : get
only n symplectic equivalence classes :
T n,
(as in Chekanov ’96)
Θ1 ,
Θ2 , . . . , Θn−1
In
R2n : get
only n symplectic equivalence classes :
T n,
Θ1 ,
Θ2 , . . . , Θn−1
(as in Chekanov ’96)
Now : Put Θkℓ into closed monotone manifolds
(such as
CPn, ×n S 2).
Can this be done in a monotone way ?
In
R2n : get
only n symplectic equivalence classes :
T n,
Θ1 ,
Θ2 , . . . , Θn−1
(as in Chekanov ’96)
Now : Put Θkℓ into closed monotone manifolds
(such as
CPn, ×n S 2).
Can this be done in a monotone way ?
Yes, one can !
Recall
Θkℓ ⊂ D 2 (1 + ε) × · · · × D 2 (1 + ε)
⊂ D 2 (2) × · · · × D 2 (2)
⊂ S 2 (2) × · · · × S 2 (2)
where D 2 (2) ≡ S 2 (2) \ ∞
All twist tori are monotone in ×n S 2 (2)
Recall
Θkℓ ⊂ D 2 (1 + ε) × · · · × D 2 (1 + ε)
⊂ D 2 (2) × · · · × D 2 (2)
⊂ S 2 (2) × · · · × S 2 (2)
where D 2 (2) ≡ S 2 (2) \ ∞
All twist tori are monotone in ×n S 2 (2)
Also Θkℓ ⊂ B 2n (n + 1) ⊂
CPn (n + 1)
F1 , F2 : rooted forests
F1 ∼ F2 ⇐⇒ ∃ a homeomorphism F1 → F2
mapping roots to roots
F1 , F2 : rooted forests
F1 ∼ F2 ⇐⇒ ∃ a homeomorphism F1 → F2
mapping roots to roots
Note:
F1 ∼ F2
=⇒
Θ(F1 ) ≈ Θ(F2)
in
CPn
and ×n S 2
Theorem 1 Let Θ, Θ′ be two twist tori in ×n S 2 .
Then
Θ ≈ Θ′ ⇐⇒ F (Θ) ∼ F (Θ′ )
Theorem 1 Let Θ, Θ′ be two twist tori in ×n S 2 .
Then
Θ ≈ Θ′ ⇐⇒ F (Θ) ∼ F (Θ′ )
The same is true in
CPn for n ≤ 7.
Theorem 1 Let Θ, Θ′ be two twist tori in ×n S 2 .
Then
Θ ≈ Θ′ ⇐⇒ F (Θ) ∼ F (Θ′ )
The same is true in
Note:
CPn for n ≤ 7.
# { ample rooted trees } =
Sloan Sequence A000669 ≈ (3.692)n
Proof.
For M := S 2 × S 2 ,
that is TCliff =: T 6≈ Θ
Proof.
For M := S 2 × S 2 ,
that is TCliff =: T 6≈ Θ
Problem: All classical invariants at T, Θ are the same !
Proof.
For M := S 2 × S 2 ,
that is TCliff =: T 6≈ Θ
Problem: All classical invariants at T, Θ are the same !
Example. Displacement energy
R
For H : [0, 1] × M → define
Z 1
kHk =
max H(t, x) − min H(t, x) dt.
0
x∈M
x∈M
For A ⊂ M define
n
o
e (A) = inf kHk | ΦH (A) ∩ A = ∅
H∈H
Proof.
For M := S 2 × S 2 ,
that is TCliff =: T 6≈ Θ
Problem: All classical invariants at T, Θ are the same !
Example. Displacement energy
R
For H : [0, 1] × M → define
Z 1
kHk =
max H(t, x) − min H(t, x) dt.
0
x∈M
x∈M
For A ⊂ M define
n
o
e (A) = inf kHk | ΦH (A) ∩ A = ∅
H∈H
Then e(T) = ∞ = e(Θ)
Versal deformations
Idea (Chekanov ’96): Compute at nearby tori :
Ham
Assume ψ : T −→ Θ:
T
L
ψ
Θ
Weinstein: locally
R
{ Lagrangian tori near L } / Ham = H 1 (L; )
Choose identifications
R
replacements
H 1 (T; ) ∼
=
R2,
R
H 1 (Θ; ) ∼
=
Since e : L → [0, ∞] is Ham-invariant:
R2
eT
(ψ|T )∗
◦
[0, ∞]
R2
eΘ
R2
Choose identifications
R
replacements
H 1 (T; ) ∼
=
R2,
R
H 1 (Θ; ) ∼
=
Since e : L → [0, ∞] is Ham-invariant:
R2
eT
(ψ|T )∗
◦
[0, ∞]
Hence: The function germs eT , eΘ : (
R
are GL(2; ) related
R2
R2
eΘ
R2, 0) → [0, ∞]
Computation of eT
y
µ
×
Neighbourhood of T in L
≡
x
neighbourhood of (0, 0) in ✷
Computation of eT
y
µ
×
Neighbourhood of T in L
≡
x
neighbourhood of (0, 0) in ✷
For (x, y ) 6= (0, 0):
eT (x, y ) = e(Tx,y )
(∗)
= min 1 − |x|, 1 − |y |
= 1 + min ±x, ±y
Computation of eT
y
µ
×
Neighbourhood of T in L
≡
x
neighbourhood of (0, 0) in ✷
For (x, y ) 6= (0, 0):
eT (x, y ) = e(Tx,y )
(∗)
= min 1 − |x|, 1 − |y |
= 1 + min ±x, ±y
minimum of four (and not less!) linear functionals
≤ in (∗) is clear
≤ in (∗) is clear
≥ : look at
D0 :=
D
u : ( , S 1 ) → (M, T) | u is J0 -holomorphic
and non-constant
≤ in (∗) is clear
≥ : look at
D
u : ( , S 1 ) → (M, T) | u is J0 -holomorphic
and non-constant
nR
o
Chekanov ’98: min u(D) ω ≤ e(T)
D0 :=
D0
≤ in (∗) is clear
≥ : look at
D
u : ( , S 1 ) → (M, T) | u is J0 -holomorphic
and non-constant
nR
o
Chekanov ’98: min u(D) ω ≤ e(T)
D0 :=
D0
Clearly
min
D0
nR
ω
u(D)
o
= min 1 − |x|, 1 − |y |
Computation of eΘ
Recall
Θ =
where
1 √ e2π i α γ(t), e−2π i α γ(t)
2
γ
1
Θ is invariant under the S 1 -action
2π i α
−2π i α
(u, v ) 7→ e
u, e
v
Θ is invariant under the S 1 -action
2π i α
−2π i α
(u, v ) 7→ e
u, e
v
and
µ(Θ) =
π |γ(t)|2 , |γ(t)|2
=: σ
2
1
σ
1
2
Z
Z
Basis of H 1 (Θ; ) ∼
= H1 (Θ; ):
class of γ∆ ⊂ ∆,
class of an S 1 -orbit
Z
Z
Basis of H 1 (Θ; ) ∼
= H1 (Θ; ):
class of an S 1 -orbit
class of γ∆ ⊂ ∆,
(s, t) near (0, 0) ❀ Θs,t
Equivariant Weinstein: can choose Θs,t S 1 -invariant
Then
µ(Θs,t ) =: σs,t k σ
2
σ
σs,t
2
Meaning of deformation parameters:
1+s =
t =
area of Θs,t ∩ ∆
action of any S 1 -orbit
Meaning of deformation parameters:
1+s =
t =
Claim:
For t 6= 0:
area of Θs,t ∩ ∆
action of any S 1 -orbit
Θs,t ∼
= T 1 + s, 1 + s + |t|
Meaning of deformation parameters:
1+s =
t =
Claim:
For t 6= 0:
With this:
e Θs,t
area of Θs,t ∩ ∆
action of any S 1 -orbit
Θs,t ∼
= T 1 + s, 1 + s + |t|
= 1 + min {±s, ±(s + |t|)}
= 1 + min {s, −s − |t|)}
= 1 + min {s, −s ± t}
minimum of three (and not less!) linear functionals
Proof of Claim:
Take Tx,y = T(1 + x, 1 + y ) with (x, y ) ∈ ℓs,t
2
ℓs,t
y
σs,t
x
2
Proof of Claim:
Take Tx,y = T(1 + x, 1 + y ) with (x, y ) ∈ ℓs,t
2
ℓs,t
y
σs,t
x
2
Note: Tx,y is also S 1 -invariant
Its orbits have action (1 + x) − (1 + y ) = x − y
Thus
t = x −y
Σs,t := µ−1 (ℓs,t )
since t 6= 0:
Σs,t /S 1 is a smooth disc D
Σs,t := µ−1 (ℓs,t )
since t 6= 0:
Σs,t /S 1 is a smooth disc D
cT := Tx,y /S 1 ,
cΘ := Θs,t /S 1
smooth curves in D, of areas
min{1 + x, 1 + y },
1+s
Σs,t := µ−1 (ℓs,t )
since t 6= 0:
Σs,t /S 1 is a smooth disc D
cT := Tx,y /S 1 ,
cΘ := Θs,t /S 1
smooth curves in D, of areas
min{1 + x, 1 + y },
1+s
Thus: Choose {x, y } such that t = x − y and
s = min{x, y }
that is,
{x, y } = {s, s + |t|}
Now take Hamiltonian isotopy in D with cT ❀ cΘ
lifts to Hamiltonian isotopy in Maff with Ts,t ❀ Θs,t
Now take Hamiltonian isotopy in D with cT ❀ cΘ
lifts to Hamiltonian isotopy in Maff with Ts,t ❀ Θs,t
Σs,0 /S 1
Θs,0 /S 1 6≈ Tx,x /S 1
❀
Σs,t /S 1
Θs,t /S 1 ≈ Tx,y /S 1
Back to
R2n !
Twist tori in
R2n are
Tn ,
≈ to one of
Θ1 ,
Θ2 , . . . , Θn−1
Since: there is room to “untwist under the Θ-operation”
Inversion trick
Inversion trick
Note
◦
◦
◦
Θkℓ ⊂ D(2) × · · · × D(2) =: D 2n
Inversion trick
Note
◦
◦
◦
Θkℓ ⊂ D(2) × · · · × D(2) =: D 2n
◦
◦
Choose symplectomorphism ψ : D(2) → D(2)
turning inside out (“like z 7→ 1/z”)
Inversion trick
Note
◦
◦
◦
Θkℓ ⊂ D(2) × · · · × D(2) =: D 2n
◦
◦
Choose symplectomorphism ψ : D(2) → D(2)
turning inside out (“like z 7→ 1/z”)
Apply ψ to some coordinates ❀ inverted twist torus
◦
Ψ Θkℓ ⊂ D 2n (2) ⊂ 2n
R
Inversion trick
Note
◦
◦
◦
Θkℓ ⊂ D(2) × · · · × D(2) =: D 2n
◦
◦
Choose symplectomorphism ψ : D(2) → D(2)
turning inside out (“like z 7→ 1/z”)
Apply ψ to some coordinates ❀ inverted twist torus
◦
Ψ Θkℓ ⊂ D 2n (2) ⊂ 2n
R
often: 6≈ any twist torus !
Also via
CPn :
Θkℓ ⊂ B 2n (n + 1) ⊂
CPn
is disjoint from all coordinated hyperplanes
Also via
CPn :
Θkℓ ⊂ B 2n (n + 1) ⊂
CPn
is disjoint from all coordinated hyperplanes
Remove different hyperplane ❀ inverted twist torus in
B 2n ⊂ 2n
R
Also via
CPn :
Θkℓ ⊂ B 2n (n + 1) ⊂
CPn
is disjoint from all coordinated hyperplanes
Remove different hyperplane ❀ inverted twist torus in
B 2n ⊂ 2n
R
Example
Get two new tori in
R6 by inverting ΘΘ and Θ2 in CP3
For n large:
#
inverted twist tori in
R2n
≥ 2n
Proof: again by versal deformations
no use of J-holomorphic curves
use only e S 1 (a1 ) × · · · × S 1 (an ) = min ai
III. Non-displaceability
A ⊂ (M, ω) displaceable if ∃ Φ ∈ Ham(M, ω) such that
Φ(A) ∩ A = ∅
Theorem 2.
Twist tori in ×n S 2 and
CP n are not displaceable.
Pearl homology
“Proof”: for Θ ⊂ S 2 × S 2 = M
Pearl homology
“Proof”: for Θ ⊂ S 2 × S 2 = M
We use pearl cohomology (Biran–Cornea 2008–2010)
f:Θ→
Λ :=
R
perfect Morse function
Z2[H2(M, Θ)]
group ring
HP∗ (Θ) cohomology of C ∗ (f ) ⊗ Λ, dP
where
C ∗ (f ) =
Z2 ⊕ (Z2 ⊕ Z2) ⊕ Z2
degree of x ⊗ D = |x| + µ(D)
dP :
Λ-linear of degree +1, counting
x
σ
Θ
y
dP is of the form
dP = d0 + d2 + d4 + . . .
where
dk : C ∗ (f ) ⊗ Λ → C ∗−k+1 (f ) ⊗ Λ
dP is of the form
dP = d0 + d2 + d4 + . . .
where
dk : C ∗ (f ) ⊗ Λ → C ∗−k+1 (f ) ⊗ Λ
f perfect =⇒ d0 = 0
can identify C ∗ (f ) ≡ H ∗ (Θ)
dim Θ = 2 =⇒ dk = 0 for k ≥ 4
dP = d2 : H ∗ (Θ) ⊗ Λ → H ∗−1 (Θ) ⊗ Λ
HP∗ (Θ) 6= 0 =⇒ HF∗ (Θ) 6= 0 =⇒ Θ non-displaceable
Thus: need to show HP∗ (Θ) 6= 0
HP∗ (Θ) 6= 0 =⇒ HF∗ (Θ) 6= 0 =⇒ Θ non-displaceable
Thus: need to show HP∗ (Θ) 6= 0
σ1 , . . . , σℓ : J-holomorphic with µ = 2 and boundary on Θ
P
Note d2 = ℓk=1 dσk
HP∗ (Θ) 6= 0 =⇒ HF∗ (Θ) 6= 0 =⇒ Θ non-displaceable
Thus: need to show HP∗ (Θ) 6= 0
σ1 , . . . , σℓ : J-holomorphic with µ = 2 and boundary on Θ
P
Note d2 = ℓk=1 dσk
Lemma 1 dσk α = [σk ] ι[∂σk ] α
Hence let’s find the Maslov 2 discs !
Basis of H2 (M, Θ) [Dγ ], [Dτ ], [S1 ], [S2 ]
Fix v ∈ Θ
Basis of H2 (M, Θ) [Dγ ], [Dτ ], [S1 ], [S2 ]
Fix v ∈ Θ
Lemma 2. In each of the five classes
[Dγ ]
−[Dγ ] −[Dτ ] + [S1 ]
−[Dγ ]
+ [S1 ]
−[Dγ ]
+ [S2 ]
−[Dγ ] +[Dτ ]
+ [S2 ]
D D
∃! J0 -holomorphic disc σk : ( , ∂ ) → (M, Θ) passing
through v . They are regular.
To find these classes first compute candidate classes :
[Dγ ] [Dτ ]
S1 × 0
0
0 × S2
[S1 ] [S2 ]
−1
0
1
0
0
1
S1 × ∞
0
0
1
1
0
∞ × S2
0
0
1
0
z1 z2 = w 2
1
0
1
1
µ
2
0
4
4
A
b2 ≥ aτ
b2 ≥ 0
b1 ≥ −aτ
b1 ≥ 0
aγ + b1 + b2 ≥ 0
1 = aγ + 2(b1 + b2 )
Rest of the proof: Linear Algebra !
Want HP0 (Θ) 6= 0 i.e. 1 ∈
/ d2 (C 1 (Θ) ⊗ Λ)
Rest of the proof: Linear Algebra !
Want HP0 (Θ) 6= 0 i.e. 1 ∈
/ d2 (C 1 (Θ) ⊗ Λ)
Abbreviate R := [Dγ ], T := [Dτ ]
Z
With Lemma 1 + 2 and working over 2 :
im d2 = R + R −1 (T −1 S1 + S1 + S2 + TS2 ), R −1 T −1 S1 + TS2
1∈?
Rest of the proof: Linear Algebra !
Want HP0 (Θ) 6= 0 i.e. 1 ∈
/ d2 (C 1 (Θ) ⊗ Λ)
Abbreviate R := [Dγ ], T := [Dτ ]
Z
With Lemma 1 + 2 and working over 2 :
im d2 = R + R −1 (T −1 S1 + S1 + S2 + TS2 ), R −1 T −1 S1 + TS2
1∈?
Specialize !
R 7→ R, T 7→ R, S2 7→ R, S1 7→ 1
Then ϕ(im d2 ) = hR 2 + R + 1i does not contain
1 ∈ 2 [R, R −1 ]
Z
hence im d2 does not contain 1 either.
Other constructions of exotic tori
Different constructions of an exotic torus in S 2 × S 2 :
ΘBC
(Biran–Cornea)
ΘEP
(Entov–Polterovich)
ΘFOOO
ΘAF
(Fukaya–Oh–Ohta–Ono)
(Albers–Frauenfelder)
Sometimes equivalence is hard to show (!)
Agnès Gadbled 2010:
Θ ∼
= ΘBC ∼
= ΘEP ∼
= ΘFOOO ∼
= ΘAF
New developments motivated by mirror symmetry
Renato Vianna 2013:
There is a third monotone torus in
CP2
Denis Auroux 2014 arXiv:1407.3725 :
There are infinitely many monotone tori in
And in
CP3 or S 2 × S 2 × S 2 ???
R2n , 2n ≥ 6
Auroux’ construction in
R6
For n ≥ 0 take
hn (z, w ) = c z n + c −1 w − 1,
c ≥ 10
and define hypersurface
Xn :=
(x, y , z, w ) ∈
C4 | xy = hn (z, w )
pr
∼
=
C3(x, y , z)
Auroux’ construction in
R6
For n ≥ 0 take
hn (z, w ) = c z n + c −1 w − 1,
c ≥ 10
and define hypersurface
Xn :=
(x, y , z, w ) ∈
C4 | xy = hn (z, w )
pr
∼
=
C3(x, y , z)
Kähler form on Xn :
ωXn
i
i
i
i
= dz ∧ d z̄ + dw ∧ d w̄ + ε dx ∧ d x̄ + dy ∧ d ȳ
2
2
2
2
Xn is invariant under S 1 -action
e i α (x, y , z, w ) = e i α x, e − i α y , z, w
with moment map
µXn = επ |x|2 − |y |2
Xn is invariant under S 1 -action
e i α (x, y , z, w ) = e i α x, e − i α y , z, w
with moment map
µXn = επ |x|2 − |y |2
reduced space:
1
Xn,red = µ−1
Xn (0)/S
pr
∼
=
C2(z, w )
⊃ T2
Xn is invariant under S 1 -action
e i α (x, y , z, w ) = e i α x, e − i α y , z, w
with moment map
µXn = επ |x|2 − |y |2
reduced space:
1
Xn,red = µ−1
Xn (0)/S
pr
∼
=
C2(z, w )
Tn := pr−1 (T 2 ) ⊂
⊃ T2
Xn
are all different: there are 2n + 1 Maslov-two discs on Tn