Chekanov torus and Gelfand–Zeitlin torus in S2 × S2

Yoosik Kim

Abstract. The Chekanov torus was the first known exotic torus, a monotone Lagrangian torus thatis not Hamiltonian isotopic to the standard monotone Lagrangian torus. We explore the relationshipbetween the Chekanov torus in S2 × S2 and a monotone Lagrangian torus that had been introducedbefore Chekanov’s construction [Che96]. We prove that the monotone Lagrangian torus fiber in a certainGelfand–Zeitlin system is Hamiltonian isotopic to the Chekanov torus in S2 × S2.

1. Introduction

A symplectic manifold (X,ω) is called monotone if the class of the symplectic form ω andthe first Chern class of the tangent bundle TX in H2(X;R) are positively proportional, i.e.[ω] = r · c1(TX) for a positive real number r > 0. Each Lagrangian submanifold L of X carriestwo group homomorphisms:

• Iω : π2(X,L)→ R is given by the symplectic area,• Iµ : π2(X,L)→ Z is given by the Maslov index.

A Lagrangian submanifold L is said to be monotone if there exists a positive real number s > 0such that Iω(β) = s ·Iµ(β) for every β ∈ π2(X,L). This constant s is called the monotonicity ofL. The notion of a monotone Lagrangian submanifold was introduced by Y.-G. Oh to constructits Floer cohomology in [Oh93a, Oh93b].

As an attempt to classify the monotone Lagrangians in a symplectic manifold, it has beenan interesting problem to construct monotone Lagrangian tori that are not related by anyHamiltonian isotopy. In the symplectic vector space R2n, Chekanov [Che96] constructed amonotone Lagrangian torus that is not Hamiltonian isotopic to any standard product torus.Moreover, by embedding a suitable subset of R2n symplectically, one can produce a monotoneLagrangian torus by transporting a Chekanov exotic torus of R2n into a closed symplecticmanifold including the complex projective space or the product of projective lines. Such anembedded monotone Lagrangian torus is also called a Chekanov torus. A monotone torus notHamiltonian isotopic to the standard one is called an exotic torus. The Chekanov torus is thefirst known exotic monotone Lagrangian torus. Recently, there have been exciting developmentsconstructing infinitely many exotic Lagrangian tori, see [Via16, Aur15, Via17] for instance.

In this article, we are concerned with the Chekanov torus in CP 1 ×CP 1. To begin with, webriefly recall several ways of constructing the Chekanov torus in CP 1 × CP 1 ' S2 × S2.

• (Chekanov–Schlenk torus TCS in [CS10]) Let D(√

2) := {ζ ∈ C | |ζ| <√

2} be an opendisk in the complex plane with radius

√2 and let H(

√2) := {ζ ∈ D(

√2) | Im(ζ) > 0} be

an open half-disk. Choose any simple closed curve Γ in H(√

2) which bounds the regionhaving the area π/2. The product space carries the Hamiltonian S1-action given by

e√−1θ · (ζ1, ζ2) −→

(e−√−1θζ1, e

√−1θζ2

).

Consider the product space CP 1×CP 1 of projective planes equipped with the productsymplectic form 2(ωFS ⊕ ωFS) where ([v0 : v1], [w0 : w1]) is its homogeneous coordinate.

1

arX

iv:2

109.

0143

5v1

[m

ath.

SG]

3 S

ep 2

021

2 YOOSIK KIM

Recall that 2ωFS = 12ωstd and the symplectic area of CP 1 measured by 2ωFS is 2π. We

denote the diagonal map by

(1) ∆D : D(√

2) −→ D(√

2)× D(√

2).

Also, the composition of the inclusion H(√

2) → D(√

2) and the diagonal map ∆D isdenoted by

(2) ∆H : H(√

2) −→ D(√

2) −→ D(√

2)× D(√

2).

We have a symplectic embedding

(3) ρ :(D(√

2)× D(√

2), (ωstd ⊕ ωstd)|D(√

2)×D(√

2)

)−→

(CP 1 × CP 1, 2(ωFS ⊕ ωFS)

)such that the image of ρ is the intersection of v0 6= 0 and w0 6= 0. Chekanov and Schlenkin [CS10] constructed the monotone Lagrangian torus TCS as follows:

(4) TCS :={ρ(e−√−1θζ, e

√−1θζ

)∈ CP 1 × CP 1 | θ ∈ [0, 2π], ζ ∈ Γ

}.

• (Entov–Polterovich torus TEP in [EP09]) Let us regard the product space S2×S2 as anembedded submanifold of R3×R3 equipped with 1

2(ωstd⊕ωstd). Entov and Polterovichin [EP09] constructed a monotone Lagrangian torus as follows:

TEP :={

(a,b) ∈ S2 × S2 | (a + b) · e1 = 0,a · b = −1/2}

where e1 = (1, 0, 0).• (Fukaya–Oh–Ohta–Ono torus TFOOO in [FOOO12]) Start with the symplectic toric orb-

ifold associated to the triangle whose vertices are (0, 0), (0, 1), and (2, 0) as the momentpolytope. By replacing a neighborhood of the singular point with the Milnor fiber toobtain a symplectic manifold isomorphic to S2 × S2. Through this process, one obtainsa semi-toric system Φsemi whose image is the above polytope. Fukaya–Oh–Ohta–Onomonotone Lagrangian torus is located at the center (1/2, 1/2), that is,

TFOOO := Φ−1semi(1/2, 1/2).

• (Albers–Frauenfelder torus TAF in [AF08]) Let ∆ be the diagonal of S2 × S2. Choose asymplectomorphism

υ : (S2 × S2)\∆ −→ D∗1S2

where D∗1S2 is the open unit disk bundle. Albers–Frauenfelder monotone torus is defined

byTAF := υ−1

({(p,q) ∈ D∗1S2 | |p| = 1/2, (p× q) · e1 = 0

}).

• (Biran–Cornea torus TBC in [BC12]) Let us start with the diagonal ∆ of CP 1 ×CP 1, acomplex hypersurface of CP 1 × CP 1. Set

P∆ ={

(a,b) ∈ R3 × R3 : |a| = |b| = 1, a · b = 0}.

with the circle action

e√−1θ · (a,b) = (a, (cos θ)b + (sin θ)(a× b)).

Identifying ∆ with S2, we then have the principal S1-bundle π : P∆ → ∆ given byπ([(a,b)]) = a, whose connection 1-from for P∆ is defined by α(a,b)(p,q) = q · (a× b).According to a work of Biran in [Bir01], there is a symplectomorphism from

D√2(P∆) =P∆ ×D(

√2)

(e√−1θ · (a,b), ζ) ∼ ((a,b), e

√−1θ · ζ)

CHEKANOV TORUS AND GELFAND–ZEITLIN TORUS 3

adorned with a certain symplectic form defined by using the connection 1-form to (S2×S2)\∆ where ∆ is the anti-diagonal.

For each monotone Lagrangian submanifold L in ∆ and each r with 0 < r <√

2, wehave the submanifold

L(r) ={

[((a,b), ζ)] ∈ D√2(P∆) | |ζ| = r, π([((a,b), ζ)]) ∈ L}.

According to [BC12], there is a unique value r such that the embedding of L(r) becomes

a monotone Lagrangian submanifold in S2 × S2. In this case, we choose a great circleof S2 ' ∆ for L and one for r. The choice leads to Brian–Cornea monotone Lagrangiantorus

TBC = L(1)

in S2 × S2. For an explicit description for the embedding and the embedded monotoneLagrangian, the reader is referred to [OU16, Section 4].

The relation between the above monotone tori had been explored. In [FOOO12], Fukaya,Oh, Ohta, and Ono discussed the relation between TFOOO and TAF. Gadbled [Gad13] showedthat the TBC and TCS are Hamiltonian isotopic. Oakley and Usher ultimately proved that allfive monotone Lagrangian tori are Hamiltonian isotopic to each other by constructing detailedand explicit symplectomorphisms in [OU16].

Theorem 1 (Theorem 1.1 in [OU16]). The five monotone Lagrangian tori TCS, TEP, TFOOO,TAF, and TBC listed above are Hamiltonian isotopic to each other.

The main goal of this article is to add one more monotone Lagrangian torus to the abovelist. This torus had been constructed even before the Chekanov’s construction of “twist tori” in[Che96]. Specifically, we shall prove that the monotone Gelfand–Zeitlin (GZ) Lagrangian torusfiber in the orthogonal Grassmannian OG(1,C4) is Hamiltonian isotopic to one (and hence each)of the above listed tori.

Let us recall the quickest way of constructing the monotone GZ Lagrangian torus.

• (Gelfand–Zeitlin torus TGZ in [GS83b]) Consider the set O of (4 × 4) skew-symmetricmatrices with prescribed four eigenvalues ±λ

√−1 and 0 with multiplicity two for some

real number λ. In other words, the space O is the orbit of the block diagonal matrix

(5)

0 λ 0 0−λ 0 0 00 0 0 00 0 0 0

under the adjoint SO(4)-action. It carries the Kirillov–Kostant–Souriau (KKS) sym-plectic form ωKKS, which is determined by λ. cf. Remark 2 below. To match up withthe convention on the symplectic form 2(ωFS ⊕ ωFS) on CP 1 × CP 1, we take λ = 1.

For a matrix A ∈ O, let A(k) be the leading principal submatrix of the size (k × k).

Since A is skew-symmetric, A(1) is the zero matrix. In order to define the desired GZtorus, we impose the following conditions on A(2) and A(3).(1) The eigenvalues of A(2) are required to be 0 with multiplicity two.

(2) The eigenvalues of A(3) are required to be ±12

√−1, and 0.

The conditions (1) and (2) determine the GZ Lagrangian torus TGZ in O. That is,

(6) TGZ ={A ∈ O

∣∣∣ A(2) = O and the eigenvalues of A(3) are ±12

√−1, 0

}.

4 YOOSIK KIM

Remark 2. When defining the KKS form ωKKS, the symplectic form is sometimes normalized bydividing the form by 2π so that the class of the symplectic form agrees with an integral multipleof the Chern class. In this article, we do not normalize the KKS form to facilitate comparisonof the KKS form with other symplectic forms. Namely, it is defined by

ωKKS (adξ1(A), adξ2(A)) = 〈A, [ξ1, ξ2]〉where A ∈ O and adξi(A) is a tangent vector at A of O for i = 1, 2.

Additional explanation on TGZ and O is in order. The real special orthogonal group SO(4) :=SO(4;R) transitively acts on O by conjugation. A choice of an element of O gives rise toa diffeomorphism O ' SO(4)/S(O(2)×O(2)). Therefore, the orbit O is diffeomorphic to theorthogonal Grassmannian OG(1,C4), which is the space of isotropic subspaces of C4 with respectto a non-degenerate symmetric bilinear form. Thus, OG(1,C4) can be regarded as a quadrichypersurface. As any two smooth quadric hypersurfaces are symplectomorphic, the quadrichypersurface is symplectomorphic to the Segre variety

{[z] ∈ CP 3 | z0z1 − z2z3 = 0

}. Hence,

the space O is regarded as the product CP 1 × CP 1 of two complex projective planes.The min-max principle, a system of inequalities between eigenvalues of submatrices of a

skew-symmetric matrix, says that the eigenvalues of A(2) and A(3) are of the form:

• the eigenvalues of A(2) are ±λ(2)√−1.

• the eigenvalues of A(3) are ±λ(3)√−1, 0.

Each real-valued function determined by λ(•) generates a Hamiltonian circle action (on an opendense subset of O) by the result of Guillemin–Sternberg [GS83a]. Indeed, the fiber TGZ is a freeHamiltonian T 2-orbit so that it is a Lagrangian torus. Employing the technique of the gradientHamiltonian disks in [CK19], the torus TGZ was shown to be monotone.

Lemma 3 (Proposition 3.7 in [Kim21]). The above Gelfand–Zeitlin torus fiber TGZ is monotone.

The main theorem of this article is stated below.

Theorem 4. There is a symplectomorphism from O to CP 1 ×CP 1 taking the Gelfand–Zeitlintorus TGZ to the Chekanov–Schlenk torus TCS. Consequently, all monotone Lagrangian toriTCS, TEP, TFOOO, TAF, TBC, and TGZ are Hamiltonian isotopic to each other.

There had been decisive clues that TGZ is Hamiltonian isotopic to the Chekanov torus inliteratures. First, Nishinou–Nohara–Ueda [NNU10] described and constructed the toric degen-eration of the Gelfand–Zeitlin system. The image of the system on the quadric hypersurfaceof complex dimension two agrees with the moment polytope of CP (1, 1, 2), which is one of thestarting points of this work. Second, the disk potential of the GZ torus fiber TGZ computed in[Kim21] agrees with that of TCS in [Aur07, CS10] and that of TFOOO in [FOOO12] up to somecoordinate changes. Theorem 4 claims a stronger relationship between TGZ and TCS.

2. Proof of Theorem 4

Consider the product space CP 1×CP 1 of complex planes equipped with the product symplecticform 2(ωFS ⊕ ωFS) where ωFS is the Fubini–Study form on CP 1. It has the S1-action given by

(7) ([v0 : v1], [w0 : w1]) 7→([v0 : e−

√−1θv1

],[w0 : e

√−1θw1

]).

The product space CP 1 × CP 1 is embedded into CP 3 via the Segre embedding

σ : CP 1 × CP 1 −→ CP 3 ([v0 : v1], [w0 : w1]) 7→ [v0w1 : v1w0 : v0w0 : v1w1].

CHEKANOV TORUS AND GELFAND–ZEITLIN TORUS 5

The image under the Segre map σ is a hypersurface of CP 3 defined by z0z1 − z2z3 = 0 where[z] := [z0 : z1 : z2 : z3] is the homogeneous coordinates for CP 3. The hypersurface is denoted by

(8) Q′ :={

[z] ∈ CP 3 | z0z1 − z2z3 = 0}.

Let S1 act on CP 3 as follows.

(9) [z0 : z1 : z2 : z3] 7→[e√−1θz0 : e−

√−1θz1 : z2 : z3

].

Adorning CP 3 with twice Fubini–Study form 2ωFS, the S1-action becomes Hamiltonian. Setting‖z‖2 := |z0|2 + |z1|2 + |z2|2 + |z3|2, we choose a moment map µS1 : CP 3 → R with respect tothe symplectic form 2ωFS for the action (9) as

(10) µS1(z) =|z0|2 − |z1|2

‖z‖2.

Then the map σ is S1-equivariant. The embedded variety Q′ is equipped with the Kahler forminherited from (CP 3, 2ωFS). We denote the form restricted to Q′ ⊂ CP 3 by ωQ′ .

To show that the map σ preserves the symplectic form, it suffices to show that it is a sym-plectomorphism on an affine chart, an open dense subset of CP 1 × CP 1. A straightforwardcomputation shows that σ preserves the symplectic form.

Proposition 5. The Segre embedding σ is an S1-equivariant symplectomorphism.

Under the coordinate change

(11)

z0

z1

z2

z3

=

1√−1 0 0

1 −√−1 0 0

0 0 1√−1

0 0 −1√−1

x0

x1

x2

x3

,the variety Q′ in (8) maps into the Fermat hypersurface :

(12) Q ={

[x0 : x1 : x2 : x3] ∈ CP 3 | x20 + x2

1 + x22 + x2

3 = 0}.

The linear transformation (11) is denoted by Λ: Q → Q′.Let G := SO(2) = SO(2;R) ' diag (SO(2;R), I2) act on C4 linearly. The linear SO(2)-action

induces the action on CP 3 and the action on Q. Let us fix the following isomorphism

(13) ι : S1 −→ SO(2;R), e√−1θ 7→

[cos θ − sin θsin θ cos θ

].

The hypersurface Q in (12) admits the S1-action via (13). We then observe the following.

Lemma 6. The map Λ: Q → Q′ is S1-equivariant.

Proof. Using the identification (13) and the expression (11), one can directly verify the S1-equivalence of Λ. �

We now discuss symplectic forms on Q. Two adorned symplectic forms on Q are takeninto account. The first one is the pull-backed symplectic form Λ∗ωQ′ . The second one is thesymplectic form coming from the twice Fubini–Study form 2ωFS on the ambient space CP 3.Namely, we take the reduction CP 3 ' (C4 − {0}) // S1 where the S1-action is diagonal actionwith the stability condition ‖x‖ =

√2 so that CP 3 is equipped with twice Fubini–Study form

2ωFS. The Kahler form restricted to Q is denoted by ωQ.

6 YOOSIK KIM

The Fermat hypersurface Q carries two circle actions. First, the moment map of the corre-sponding S1-action onQ with respect to the first symplectic form Λ∗ωQ′ is obtained by replacingz with Λ(x) in (10) as follows.

(14) µS1(x) =

√−1 (x0x1 − x0x1)

‖x‖2.

Second, by taking the Killing form

〈ξ1, ξ2〉 = −1

2Tr(ξ1, ξ2),

we identify the Lie algebra g and its dual Lie algebra g∗. Under this identification, a momentmap of the above SO(2)-action on Q is then of the form

(15) µG : Q −→ g, µG(x) =

√−1

‖x‖2

[0 (x0x1 − x0x1)

−(x0x1 − x0x1) 0

].

Via the isomorphism (13), the SO(2)-action can be regarded as the S1-action. By the functo-riality of moment maps, a moment map of the S1-action can be chosen as

(16) (ι∗ ◦ µG)(x) =

√−1 (x0x1 − x0x1)

‖x‖2: Q −→ R.

The following lemma compares those two circle actions.

Lemma 7. The Hamiltonian function (14) on (Q,Λ∗ωQ′) and the Hamiltonian function (16)on (Q, ωQ) generate the same S1-action on Q.

Proof. We shall compare those two S1-actions onQ after passing them toQ′ via the isomorphismΛ in (11). Recall that the S1-action onQ′ corresponding to the first S1-action is described in (9).By Lemma 6, the S1-action on Q′ corresponding to the second S1-action is exactly the actionin (9) as desired. �

To interpolate these forms, we need an equivariant version of the Moser theorem, which isstated below.

Theorem 8 (Equivariant Moser Theorem). Let X be a compact symplectic manifold. Supposethat ω0 and ω1 are two symplectic forms in the same cohomology class, that is, [ω0] = [ω1] inH2(X;C). If ωt := (1− t)ω0 + tω1 is symplectic for each t ∈ [0, 1], then there exists an isotopy

φ : [0, 1]×X −→ X

such that φ∗tωt = ω0 for each t.If a compact Lie group G acts on X symplectically with respect to ωt for each t, then the map

φt is G-equivariant for each t.

Consider Q together with two symplectic forms Λ∗ωQ′ and ωQ. Since two forms are Kahlerforms, each form ωt := (1−t)Λ∗ωQ′+tωQ in the linear interpolation is also symplectic. Since theS1-action is Hamiltonian (and hence symplectic) with respect to both ω0 and ω1, the S1-actionis symplectic with respect to ωt for each t ∈ [0, 1]. By Theorem 8, there exists an S1-equivariantisotopy φt : Q → Q such that φ∗t (ωt) = ω0. Set

Φ := φ1 ◦ Λ−1 ◦ σ : (CP 1 × CP 1, 2(ωFS ⊕ ωFS)) −→ (Q′, ωQ′) −→ (Q,Λ∗ωQ′) −→ (Q, ωQ).

Lemma 9. The map Φ is an S1-equivariant symplectomorphism

Proof. By Proposition 5, the Segre map σ is an S1-equivariant symplectomorphism. By Lemma 6,so is the second map Λ−1. It follows from Theorem 8 that φ1 is an S1-equivariant symplecto-morphism. Therefore, Lemma 9 is established. �

CHEKANOV TORUS AND GELFAND–ZEITLIN TORUS 7

Note that the quadric Q is the zero locus of the symmetric bilinear form B correspondingto the identity matrix I4 on CP 3. On the other hand, the vanishing condition imposes theisotropy condition on the bilinear form B on C4 = C〈x1, x2, x3, x4〉. Namely, Q is isomorphicto the orthogonal Grassmannian OG(1,C4), which consists of one dimensional subspaces Vof C4 satisfying B(v1, v2) = 0 for all v1, v2 ∈ V . The orthogonal Grassmannian OG(1,C4)is adorned with the complex structure from the identification with SO(4;C)/P where P is aparabolic subgroup of SO(4;C). We denote this isomorphism by % : OG(1,C4)→ Q as complexmanifolds.

Regarding G = SO(2) = SO(2;R) ' diag (SO(2), I2) as a subgroup of SO(4) = SO(4;R),recall that the quadric Q is acted by G in (15). The linear G-action on C4 induces the G-actionon OG(1,C4). Note that the map % : OG(1,C4)→ Q is G-equivariant.

The group SO(4) acts on the moduli space of one dimensional isotropic subspaces in C4

linearly and transitively. Also, the adjoint SO(4)-action on O is transitive. We then have

• SO(4)/S(O(2)×O(2))→ OG(1,C4)• O → SO(4)/S(O(2)×O(2)),

by choosing one flag and one element of O respectively. We then have a diffeomorphism Υ: O →OG(1,C4). The orbit O has the adjoint action of the subgroup G of SO(4). Regarding G =SO(2) ' diag (SO(2), I2) as a subgroup of SO(4), recall that the quadric Q is acted by G in (15).The linear G-action on C4 induces the G-action on OG(1,C4). The map Υ is G-equivariant.Recall that ωKKS is a Kahler form with respect to the complex structure from SO(4;C)/P .

The orbit O carries two Kahler forms (% ◦ Υ)∗ωQ and ωKKS. Each form ω′t := (1 − t)(% ◦Υ)∗ωQ+ tωKKS in the linear interpolation is also symplectic. Since the G-action is Hamiltonianwith respect to both ω′0 and ω′1, there exists an G-equivariant isotopy φ′t : O → O such that(φ′t)

∗(ω′t) = ω′0 again by Theorem 8. Set

Φ′ := φ′1 ◦Υ−1 ◦ %−1 : (Q, ωQ) −→ (OG(1,C4), %∗ωQ) −→ (O, (% ◦Υ)∗ωQ) −→ (O, ωKKS).

In summary, we have derived the following lemma.

Lemma 10. Under the identification (13), the map Φ′ is an S1-equivariant symplectomorphism.

We are now ready to transport the Gelfand–Zeitlin torus TGZ into CP 1×CP 1. By Lemma 9and Lemma 10, we obtain an S1-invariant Lagrangian torus (Φ′ ◦ Φ)−1(TGZ) in the productspace (CP 1 × CP 1, 2(ωFS ⊕ ωFS)). Let

(17) T ′GZ := (Φ′ ◦ Φ)−1(TGZ).

The next proposition compares T ′GZ with TCS in the same space (CP 1×CP 1, 2(ωFS⊕ωFS)).

Proposition 11. Two monotone Lagrangian tori T ′GZ and TCS are Hamiltonian isotopic.

Let us do preliminary work for verifying Proposition 11.Let U be an affine chart of CP 1 × CP 1 defined as

(18) U := (CP 1 × CP 1)\{v0w0 = 0} = CP 1\({v0 = 0})× CP 1\({w0 = 0}) ' C× C.Recall that U is exactly the image of ρ in (3). The Chekanov–Schlenk monotone torus TCS iscontained in the level set µ−1

S1 (0) where the moment map µS1 is in (10). Note that the S1-action

generated by µS1 induces that on U ∩ µ−1S1 (0).

A fundamental domain of the S1-action on U ∩ µ−1S1 (0) can be chosen as the image of

(19) F(√

2) := {ζ ∈ C | Im(ζ) > 0, 0 < |ζ| <√

2} ∪ {ζ ∈ C | Im(ζ) = 0, 0 ≤ ζ <√

2}

8 YOOSIK KIM

under the composition ρ ◦ ∆H of (2) and (3) as depicted in Figure 1. Since each pair of theantipodal points ζ and −ζ in D(

√2) := {ζ ∈ C | 0 ≤ |ζ| <

√2} lies on the same S1-orbit,

F(√

2) can be regarded as D(√

2) via the map

Θ: F(√

2) −→ D(√

2),(re√−1θ 7→ re2

√−1θ).

Figure 1. The fundamental domain F(√

2) in (19)

Lemma 12. Suppose that a symplectic manifold (X,ω) admits a Hamiltonian free S1-action.Choose a moment map µS1 of the S1-action. Let L be an S1-invariant connected Lagrangiansubmanifold. Then the Lagrangian submanifold L is contained in the set µ−1

S1 (r) for some r ∈R ' t∗.

Proof. For each point x ∈ X, we know that

ker(dµS1,x

)= {v ∈ TxX | ωx(v, w) = 0 for every w ∈ TxOx}

where Ox is the orbit through x, see [MS17, Lemma 5.2.5]. Since S1 acts freely on the levelset µ−1

S1 (r), each point x ∈ µ−1S1 (r) is regular and hence r ∈ R is a regular value of µS1 . The

dimension counting yields that ker(dµS1,x

)= Tx

(µ−1S1 (r)

). Since L is Lagrangian, we have

(20) TxL ⊂ {v ∈ TxX | ωx(v, w) = 0 for every w ∈ TxOx} = Tx(µ−1S1 (r)

)where r := µS1(x).

We claim that the image of a connected submanifold L under µS1 is a singleton set. Supposeon the contrary that the image of L contains an interval having positive measure. By Sard’sTheorem, there exist x ∈ L and v ∈ TxL such that dµS1,x(v) is non-zero. Then a non-zerovector w ∈ TxOx satisfies ωx(v, w) 6= 0, which contradicts to (20). �

Corollary 13. The monotone S1-invariant Lagrangian torus T ′GZ in (17) is contained in the

zero level set µ−1S1 (0) where µS1 is (10).

Proof. Observe that S1 acts on the complement of the four points

([1 : 0], [1 : 0]), ([0 : 1], [1 : 0]), ([1 : 0], [0 : 1]), and ([0 : 1], [0 : 1])

in CP 1 × CP 1 freely. Applying Lemma 12 to the complement, the torus T ′GZ is contained in

µ−1S1 (r) for some r ∈ R.We claim that T ′GZ is contained in the level set of 0, that is, r = 0. By Lemma 3, TGZ is

monotone and so is T ′GZ in CP 1×CP 1. Take a point in T ′GZ and choose an S1-invariant almostcomplex structure J . Let γ be the integral curve starting from the chosen point of the gradientvector field generated by the Riemannian metric obtained by ω and J . The curve γ convergesto a certain fixed point. The S1-orbit of γ is a J-holomorphic disk such that its symplectic areais equal to π/2 times its Maslov index. Such a J-holomorphic disk is bounded by an S1-orbitin TGZ and called a gradient holomorphic disk, see [CK19, Section 2] for more details. As thelevel of T ′GZ varies, the symplectic area of this gradient disk changes due to Archimedes, whilethe Maslov index of the disk does not change. It implies that the monotone torus must becontained in the level zero. �

CHEKANOV TORUS AND GELFAND–ZEITLIN TORUS 9

Corollary 13 says that the torus T ′GZ is completely contained in the S1-orbit of the image of

the open disk D(√

2) under the composition map ρ ◦ ∆D in (1) and (3). By intersecting thefundamental domain, we obtain a curve ΓGZ determined by

ΓGZ := Θ(

(ρ ◦∆D)−1(T ′GZ) ∩ F(√

2)).

Since T ′GZ is a smooth torus, ΓGZ is a simple closed curve not passing through the origin. Thenthere are two possibilities :

(1) ΓGZ does not bound the closed region,(2) ΓGZ bounds the closed region.

We claim that (2) holds.

Proposition 14. The curve ΓGZ ⊂ D(√

2)\{0} bounds the closed region.

To see Proposition 14, we collect some facts. Let (X,ω) be a closed monotone symplecticmanifold and L a monotone Lagrangian submanifold. For each homotopy class β ∈ π2(X,L) ofMaslov index two, a generic choice of ω-compatible almost complex structure makes the modulispace of stable maps in the class β from (D, ∂D) to (X,L) transversal. The virtual dimensionof this moduli space is exactly dimension of L. This monotonicity condition ensures that themoduli space is closed. We then count the number of stable map passing through a generic pointof L at the marking point on the boundary of the disk D. Such a number is called a countinginvariant of L in X and is denoted by nβ. The sum of counting invariants is meant to be

∑β nβ

where the summation is taken over all homotopy class β ∈ π2(X,L). By the dimension reason,almost all nβ = 0 and the sum is finite because of Gromov’s compactness theorem.

Lemma 15 ([EP97]). Suppose that φ is a symplectomorphism from (X,ω) to (X ′, ω′). If Land L′ are monotone Lagrangian submanifolds related by φ, then the sum of counting invariantsbounded by L in X is equal to that of counting invariants by L′ in X ′.

The sum of counting invariants of TGZ in O can be computed from the disk potential of TGZ

in [Kim21, Theorem A].

Lemma 16 ([Kim21]). The sum of counting invariants of TGZ in O is five.

Proof of Proposition 14. Suppose that the curve ΓGZ does not bound the closed region. Thenthe torus T ′GZ is Hamiltonian isotopic to the product of equators of S2 × S2. If so, the sum ofcounting invariants of T ′GZ in S2×S2 is four. According to Lemma 15 and Lemma 16, the sumof counting invariants of T ′GZ has to be five. We have derived a contradiction. �

Recall that the chosen Γ for (4) is contained in H(√

2). Let

ΓCS := Θ(Γ).

Let us compare two simple closed curves ΓCS and ΓGZ. A priori, the simple closed curve ΓGZ

might be complicated so that it may not be contained in a branch, while ΓCS is contained inthe branch D(

√2)\{z | Im(z) = 0, z ≥ 0}.

We shall deform one to the other via a Hamiltonian isotopy. For this purpose, we recall onelemma concerning Hamiltonian isotopy class of loops in a two dimensional exact symplecticmanifold. (Here, the dimension condition is necessary to apply the Moser argument).

Lemma 17 (Lemma 2.3 in [LM14]). Let (Σ, ω := dα) be an exact symplectic manifold of realdimension two. Let Γ1 and Γ2 be simple closed curves in Σ. Suppose that

(1) Γ0 and Γ1 are isotopic and(2)

∫∂Γ0

α =∫∂Γ1

α.

10 YOOSIK KIM

Then Γ0 and Γ1 are related by via a compactly supported Hamiltonian isotopy.

In our case, two simple closed curves ΓCS and ΓGZ in D(√

2)\{0} bound regions havingthe same area since the bounded regions can be lifted to a disk bounded by TCS and by T ′GZrespectively. The monotonicity of TCS and T ′GZ ensures the bounded regions are same. ByStokes’ theorem, the second condition holds. For the first condition, we can apply the followingwell-known fact from differential topology.

Lemma 18 (Theorem 3.1 in Chapter 8 of [Hir94]). Let X be a connected n-manifold andf, g : Dk → X embedding of the k-disk, 0 ≤ k ≤ n. If k = n and X is orientable, assume that fand g both preserve orientation. Then f and g are isotopic.

If f(Dk)∪g(Dk) is contained in X−∂X, an isotopy between them can be realized by a smoothisotopy of X having compact support.

We now discuss the relation between TCS and by T ′GZ. A choice of isotopy from ΓCS to ΓGZ in

D(√

2)\{0} gives rise to a Lagrangian isotopy between TCS and by T ′GZ. Proposition 11 further

claim that the Lagrangian isotopy arising from the Hamiltonian isotopy in D(√

2)\{0} can beextended to an ambient Hamiltonian isotopy of CP 1 × CP 1.

Proof of Proposition 11. For simplicity of notation, let us set(X := CP 1 × CP 1, ω := 2(ωFS ⊕ ωFS)

),

while presenting the proof. Suppose that ` : T 2 × [0, 1] → D(√

2) × D(√

2) is a Lagrangianisotopy arising from the Hamiltonian isotopy from Lemma 17. Since ` is a Lagrangian isotopy,the pull-backed form is of the following form

(∆D ◦ `)∗ω = `∗ωstd = αs ∧ dswhere {αs} is a family of one forms.

We shall show that the Lagrangian isotopy ∆D ◦ ` is exact. Then this Lagrangian isotopy canbe extended to an ambient Hamiltonian isotopy of CP 1 × CP 1 as desired, see [Pol01, Section6], [Oh15, Section 3.6] for instance.

It remain to show that for each s, αs is exact. Note that ωstd = dη. The pull-back of theprimitive η is

`∗η = f(x, s)ds+ η′swhere {η′s} is a family of one forms on T 2 × {s}. To show that αs is exact, we claim that theintegration of η over any loop remains constant through the isotopy. Let Γs be an isotopy fromΓCS to ΓGZ from Lemma 17. We then have an isotopy of loops

γs := (ρ ◦∆D)(Θ−1(Γs)

).

Then∫γsη is constant through the isotopy because of the construction of the isotopy. We denote

by γ′0 a Lefschetz thimble bounded by TCS in D(√

2)×D(√

2). Let γ′s be the isotoped circle. In

other words, γ′s :=(`s ◦ `−1

0

)(γ′0) where `s := `(·, s) : T 2 → D(

√2)× D(

√2). Then

∫γ′sη = 0 for

all s. Hence, the claim is verified.The claim yields that the integration of η′s over each loop in T 2 is independent to s. Then

dxf = αs as desired. �

Proof of Theorem 4. It follows from Lemma 9, Lemma 10, and Proposition 11. �

CHEKANOV TORUS AND GELFAND–ZEITLIN TORUS 11

References

[AF08] Peter Albers and Urs Frauenfelder, A nondisplaceable Lagrangian torus in T ∗S2, Comm. Pure Appl.Math. 61 (2008), no. 8, 1046–1051. MR 2417887

[Aur07] Denis Auroux, Mirror symmetry and T -duality in the complement of an anticanonical divisor, J.Gokova Geom. Topol. GGT 1 (2007), 51–91. MR 2386535

[Aur15] , Infinitely many monotone Lagrangian tori in R6, Invent. Math. 201 (2015), no. 3, 909–924.MR 3385637

[BC12] Paul Biran and Octav Cornea, Lagrangian topology and enumerative geometry, Geom. Topol. 16(2012), no. 2, 963–1052. MR 2928987

[Bir01] P. Biran, Lagrangian barriers and symplectic embeddings, Geom. Funct. Anal. 11 (2001), no. 3, 407–464. MR 1844078

[Che96] Yu. V. Chekanov, Lagrangian tori in a symplectic vector space and global symplectomorphisms, Math.Z. 223 (1996), no. 4, 547–559. MR 1421954

[CK19] Yunhyung Cho and Yoosik Kim, Monotone Lagrangians in flag varieties, Int. Math. Res. Not. IMRN(2019).

[CS10] Yuri Chekanov and Felix Schlenk, Notes on monotone Lagrangian twist tori, Electron. Res. Announc.Math. Sci. 17 (2010), 104–121. MR 2735030

[EP97] Yakov Eliashberg and Leonid Polterovich, The problem of Lagrangian knots in four-manifolds, Geo-metric topology (Athens, GA, 1993), AMS/IP Stud. Adv. Math., vol. 2, Amer. Math. Soc., Provi-dence, RI, 1997, pp. 313–327. MR 1470735

[EP09] Michael Entov and Leonid Polterovich, Rigid subsets of symplectic manifolds, Compos. Math. 145(2009), no. 3, 773–826. MR 2507748

[FOOO12] Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, and Kaoru Ono, Toric degeneration and nondisplaceableLagrangian tori in S2 × S2, Int. Math. Res. Not. IMRN (2012), no. 13, 2942–2993. MR 2946229

[Gad13] Agnes Gadbled, On exotic monotone Lagrangian tori in CP2 and S2 × S2, J. Symplectic Geom. 11(2013), no. 3, 343–361. MR 3100797

[GS83a] Victor Guillemin and Shlomo Sternberg, The Gel’fand-Cetlin system and quantization of the complexflag manifolds, J. Funct. Anal. 52 (1983), no. 1, 106–128. MR 705993

[GS83b] , On collective complete integrability according to the method of Thimm, Ergodic Theory Dy-nam. Systems 3 (1983), no. 2, 219–230. MR 742224

[Hir94] Morris W. Hirsch, Differential topology, Graduate Texts in Mathematics, vol. 33, Springer-Verlag,New York, 1994, Corrected reprint of the 1976 original. MR 1336822

[Kim21] Yoosik Kim, Disk potential functions for quadrics, preprint (2021), arXiv:2107.05839.[LM14] Yankı Lekili and Maksim Maydanskiy, The symplectic topology of some rational homology balls, Com-

ment. Math. Helv. 89 (2014), no. 3, 571–596. MR 3260842[MS17] Dusa McDuff and Dietmar Salamon, Introduction to symplectic topology, third ed., Oxford Graduate

Texts in Mathematics, Oxford University Press, Oxford, 2017. MR 3674984[NNU10] Takeo Nishinou, Yuichi Nohara, and Kazushi Ueda, Potential functions via toric degenerations,

preprint (2010), arXiv:0812.0066.[Oh93a] Yong-Geun Oh, Floer cohomology of Lagrangian intersections and pseudo-holomorphic disks. I,

Comm. Pure Appl. Math. 46 (1993), no. 7, 949–993. MR 1223659[Oh93b] , Floer cohomology of Lagrangian intersections and pseudo-holomorphic disks. II. (CPn,RPn),

Comm. Pure Appl. Math. 46 (1993), no. 7, 995–1012. MR 1223660[Oh15] , Symplectic topology and Floer homology. Vol. 1, New Mathematical Monographs, vol. 28,

Cambridge University Press, Cambridge, 2015, Symplectic geometry and pseudoholomorphic curves.MR 3443239

[OU16] Joel Oakley and Michael Usher, On certain Lagrangian submanifolds of S2 × S2 and CPn, Algebr.Geom. Topol. 16 (2016), no. 1, 149–209. MR 3470699

[Pol01] Leonid Polterovich, The geometry of the group of symplectic diffeomorphisms, Lectures in MathematicsETH Zurich, Birkhauser Verlag, Basel, 2001. MR 1826128

[Via16] Renato Ferreira de Velloso Vianna, Infinitely many exotic monotone Lagrangian tori in CP2, J. Topol.9 (2016), no. 2, 535–551. MR 3509972

[Via17] Renato Vianna, Infinitely many monotone Lagrangian tori in del Pezzo surfaces, Selecta Math. (N.S.)23 (2017), no. 3, 1955–1996. MR 3663599

12 YOOSIK KIM

Department of Mathematics, Pusan National UniversityEmail address: yoosik@pusan.ac.kr