HAMILTONIAN NON-DISPLACEABILITY OF GAUSS IMAGESOF ISOPARAMETRIC HYPERSURFACESHIROSHI IRIYEH, HUI MA, REIKO MIYAOKA, AND YOSHIHIRO OHNITA

Abstract. In this article we study the Hamiltonian non-displaceability ofGauss images of isoparametric hypersurfaces in the spheres as Lagrangiansubmanifolds embedded in complex hyperquadrics.

1. Introduction
The purpose of this article is to build a bridge between submanifold theory andsymplectic geometry. Here, by submanifolds we mean the family of isoparametric
hypersurfaces, originated from geometric optics in Italy, which is systematicallydeveloped by Élie Cartan. The latter is the Lagrangian intersection theory proposed
by A. Weinstein and V.I. Arnold. Needless to say, A. Floer’s contribution is quite
important [10]. Recently, there has been intensive progress for the Lagrangian Floer
theory of toric ﬁbers. In order to proceed the study, we need, on one hand, a lot of
concrete examples, and on the other hand, the common properties such examples
share.
The isoparametric hypersurfaces are compact oriented hypersurfaces embeddedin the standard sphere with constant principal curvatures. They provide a richclass containing inﬁnitely many both homogeneous and non-homogeneous examples. Their topological and diﬀerential geometric data are well investigated, andmanageable in various calculations. Münzner shows that the number g of distinct
constant principal curvatures κ1 > · · · > κg must be 1, 2, 3, 4 or 6, and the multiplicities mi of κi are related by mi = mi+2 (i mod g) [19]. Without loss of generality,
we may assume m1 ≤ m2 . There are inﬁnitely many non-homogeneous examples
with g = 4, [24], [9]. The theory has already been applied to various mathematicalproblems (e.g., [28], [11], [27]).
Through the Gauss map G, isoparametric hypersurfaces N n in S n+1 yield a niceclass of Lagrangian submanifolds Ln = G(N n ) embedded in the complex hyperquadric Qn (C) (n ≥ 2), which is a rank two Hermitian symmetric space of compact
type. B. Palmer [25] pointed out the minimality of L, and the second and the fourthauthors discussed the Hamiltonian stability and related properties of L [15]–[18].
As a ﬁrst step to obtain the Floer homology of L, which has been achieved in
the cases g = 1, 2 [21], [14], we investigate the Hamiltonian non-displaceability of2010 Mathematics Subject Classiﬁcation. Primary 53C40; Secondary 53C42, 53D12.
Key words and phrases. isoparametric hypersurface, Gauss image, complex hyperquadric, Lagrangian submanifold, Hamiltonian non-displaceablity, Floer homology.
The authors were partly supported by JSPS Grant-in-Aid for Young Scientists (B)No. 24740049, NSFC grant No. 11271213, JSPS Grant-in-Aid for Scientiﬁc Research (B)No. 15H03616, JSPS Grant-in-Aid for Scientiﬁc Research (S) No. 23224002 and (C) No. 15K04851,
respectively.
1

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HIROSHI IRIYEH, HUI MA, REIKO MIYAOKA, AND YOSHIHIRO OHNITA

L. Here, a compact Lagrangian submanifold L embedded in a symplectic manifold (M, ω) is said to be Hamiltonian non-displaceable, if L ∩ φ(L) ̸= ∅ for any φbelonging to the group Ham(M, ω) of all Hamiltonian diﬀeomorphisms of M .
Our main result is as follows:
Theorem 1.1. Let N n be a compact oriented isoparametric hypersurface of thestandard sphere S n+1 (1) in Rn+2 . Then its Gauss image Ln = G(N n ) is Hamiltonian non-displaceable in (Qn (C), ωstd ) except for the following few remaining cases:,(g, n, m1 , m2 ) = (3, 3, 1, 1),
N = ZSO(3)
2 +Z2
SO(2)×SO(k+2)(g, n, m1 , m2 ) = (4, 2k + 2, 1, k), N =(k ≥ 1),Z2 ×SO(k)(g, n, m1 , m2 ) = (6, 6, 1, 1),

N = ZSO(4),2 +Z2

where ωstd denotes the standard induced Kähler form of Qn (C) ⊂ CP n+1 .
Here we should mention the preceding results and a remark. We know that any
Gauss image L = G(N ) is monotone. A compact monotone Lagrangian submanifold
L in (M, ω) is said to be wide if the Floer homology HF (L) satisﬁes
HF (L) ∼= H∗ (L; Z2 ) ⊗ Λ,and narrow if HF (L) = 0. See Sections 2 and 3 for details. Notice here if L is wide,
then L is Hamiltonian non-displaceable. Y.G. Oh [21] showed that L is wide forany real form L of a compact irreducible Hermitian symmetric space M (see also[14]). In particular, all real forms of Qn (C), consisting of the Lagrangian sphereS n and the real quadric (S k × S n−k )/Z2 (1 ≤ k ≤ n − 1), are wide. Note that they
coincide with the Gauss image of isoparametric hypersurfaces with g = 1 and 2,
respectively. Moreover, when g = 3 and n = 6, 12 or 24, the Gauss image L is wide,
although L is not a real form (Corollary 4.4). It is an interesting future problem
to decide L is wide or not in the remaining cases.
The paper is organized as follows: In Section 2, we prepare fundamental properties of the Gauss images of isoparametric hypersurfaces in the standard sphere,
especially the formula of their minimal Maslov number (Proposition 2.1). In Section
3, we recall Damian’s lifted Floer homology and its spectral sequence [7], which is
our main tool in the later sections. In Section 4, we examine the topology of Gauss
images when g = 3. As a result we obtain that the Gauss image with g = 3 is a
Z2 -homology sphere, and it is wide if n = 6, 12 or 24. In Section 5 and Section 6,
we prove the Hamiltonian non-displaceability of Gauss images with g = 4 or 6 andm1 ≥ 2, respectively. In the ﬁnal section, we propose several open problems and a
conjecture.
Throughout this article any manifold is smooth and connected.
Acknowledgments. This work was done mostly during the authors’ short visits
to RIMS in Kyoto, Japan in 2014 and 2015, and TSIMF in Sanya, China, in 2014.
The authors would like to express their sincere gratitude to both institutions forthe excellent research circumstance and generous support.
2. Gauss images of isoparametric hypersurfaces
Let N n be an oriented hypersurface embedded in the unit standard sphereS(1) ⊂ Rn+2 . Denote by x the position vector of points of N and n the
n+1

HAMILTONIAN NON-DISPLACEABILITY OF GAUSS IMAGES

3

unit normal vector ﬁeld of N in S n+1 (1). It is a fundamental fact in diﬀerential
geometry that the Gauss map deﬁned by
G : N n ∋ p 7−→ x(p) ∧ n(p) ∼= [x(p) +√f 2 (Rn+2 ) ∼−1n(p)] ∈ Gr
= Qn (C)is always a Lagrangian immersion into the complex hyperquadric Qn (C). Herethe complex hyperquadric Qn (C) is identiﬁed with the real Grassmann manifold
f 2 (Rn+2 ) of oriented 2-dimensional vector subspaces of Rn+2 , which has a symGr
metric space expression SO(n + 2)/(SO(2) × SO(n)).
It follows from [25] that the Gauss map G : N n → Qn (C) from an isoparametric hypersurface N n in S n+1 (1) is a minimal Lagrangian immersion into Qn (C).
Moreover, the “Gauss image”of G is a compact minimal Lagrangian submanifold
Ln = G(N n ) ∼= N n /Zg embedded in Qn (C), where G : N n → G(N n ) = Ln is thecovering map with the Deck transformation group Zg [15], [16], [23].
Let us brieﬂy recall the notion of monotone Lagrangian submanifold. Given a
Lagrangian manifold L in a symplectic manifold (M, ω), two homomorphisms Iµ,L :π2 (M, L) → Z and Iω : π2 (M, L) → R are deﬁned as follows. For a smooth map
u : (D2 , ∂D2 ) → (M, L) in the class A ∈ π2 (M, L), there is a unique trivialization,
up to homotopy, of the pull-back bundle u∗ T M ∼= D2 × Cn as a symplectic bundle.
12nThis gives a map ũ from S = ∂D to Λ(C ), the set of Lagrangian vector subspaces
in Cn . By using the Maslov class∫µ ∈ H 1 (Λ(Cn ), Z), one can deﬁne Iµ,L (A) := µ(ũ).
Next Iω is deﬁned by Iω (A) := D2 u∗ ω. Then a Lagrangian submanifold L in M
is said to be monotone, if there exists a constant λ > 0 such that
Iµ,L = λIω .
Denote by NL ∈ Z+ the positive generator of the image of Iµ,L . We call NL theminimal Maslov number of L.
We obtain the following properties of the Gauss image:
Proposition 2.1 ([23], [17]). The Gauss image Ln = G(N n ) of an isoparametric
hypersurface N n in S n+1 is a compact monotone Lagrangian submanifold embedded
in Qn (C), and its minimal Maslov number NL is given by
NL =

2n
=
g

{

m1 + m2 ,2m1 ,if g is even,
if g is odd.

(2.1)

The Gauss image G(N n ) is orientable if and only if 2n/g is even.
Remark 2.1. It is well-known that 1 ≤ NL ≤ 2n for a closed monotone Lagrangiansubmanifold L in Qn (C). The upper bound is sharp since the Lagrangian sphereS n ⊂ Qn (C) attains NL = 2n. The Gauss images of isoparametric hypersurfaces
with g = 1 and g = 2 are S n and (S k ×S n−k )/Z2 (1 ≤ k ≤ n−1), respectively. They
provide real forms of Qn (C), which are totally geodesic. Notice that the minimal
Maslov number of (S k × S n−k )/Z2 (1 ≤ k ≤ n − 1) is equal to n. We observe from
(2.1) that Gauss images of isoparametric hypersurfaces with g ≥ 3 provide examplesof Lagrangian submanifolds in Qn (C) with relatively large minimal Maslov numbers(for the case of CP n , see [13]). On the other hand, NL = 2 is taken only when
m1 = m2 = 1, and such isoparametric hypersurfaces are homogeneous [6], [26], [8].

4

HIROSHI IRIYEH, HUI MA, REIKO MIYAOKA, AND YOSHIHIRO OHNITAWhen g ≥ 3, all Gauss images with NL = 2 are given respectively [17] by
 SO(3)(g, m1 , m2 ) = (3, 1, 1),
 (Z2 +Z2 )Z3 ,SO(2)×SO(3)L=(Z2 ×1)Z4 , (g, m1 , m2 ) = (4, 1, 1), SO(4)(g, m1 , m2 ) = (6, 1, 1).
(Z2 +Z2 )Z6 ,3. Damian’s lifted Floer homology
In this section we recall necessary facts from the Floer theory for monotone
Lagrangian submanifolds.
3.1. Lagrangian Floer homology. Let (M, ω) be a closed symplectic manifold
and let L ⊂ (M, ω) be a closed connected monotone embedded Lagrangian submanifold with NL ≥ 2. Consider a Hamiltonian isotopy {φt }0≤t≤1 deﬁned bya time-dependent Hamiltonian H : [0, 1] × M → R. The time-one map φ1 iscalled a Hamiltonian diﬀeomorphism of M . Assume that L ⋔ φ1 (L) ̸= ∅. The
Floer complex CF (L) is deﬁned as a free module over Z2 generated by all pointsof the intersection L ∩ φ1 (L). Consider a time-dependent family J = {Jt }0≤t≤1
of almost complex structures on M compatible with ω. Floer’s boundary operator∂J : CF (L) → CF (L) is deﬁned by counting (modulo 2) the isolated J-holomorphic
strips v : R × [0, 1] → M connecting pairs of points of L ∩ φ1 (L) with boundary in
L ∪ φ1 (L). For a generic choice of (H, J), the homology HF (L) := H∗ (CF (L), ∂J )is well-deﬁned and called the Floer homology of L with Z2 -coeﬃcient, which isinvariant under the Hamiltonian isotopies of L (see [10], [20] and [22] for moredetails).
Let us regard the intersection C := L ∩ φ1 (L) as points in L. For any two points
p, q ∈ C, consider an isolated J-holomorphic strip v : R × [0, 1] → M from p to q,
which deﬁnes a path γ in L from p to q by γ(s) := v(s, 0). We denote by Γ all
the paths γ given by this procedure. Notice that to reconstruct the Floer complex(CF (L), ∂J ) it is enough to know the above collection (C, Γ) of points and paths.
3.2. Damian’s spectral sequence. Now we recall the deﬁnition of Damian’s
lifted Floer homology for monotone Lagrangian submanifolds (see [7] for details).
Let (M, ω) be a closed symplectic manifold and L ⊂ M be a closed embedded
monotone Lagrangian submanifold with NL ≥ 3. We start with the data (C, Γ) andﬁx an arbitrary covering π : L̄ → L. For any point p ∈ C, denote by {pi }i∈I theelements of the ﬁber π −1 (p). Consider the set Γ̄ of all the lifts of the paths of Γ tothe covering space L̄. For any points pi , qj (i, j ∈ I), the cardinality of elements inΓ̄ which connect pi with qj is ﬁnite. We
by n(pi , qj ) its parity. Let CF L̄ (L)∪ denote−1
be the free Z2 -module generated by p∈C π (p). One can deﬁne the boundary
operator ∂ L̄ on CF L̄ (L) by the formula∑∂ L̄ (pi ) =n(pi , qj )qj .
π(qj )=q∈C

Damian [7, Proposition 2.6] proved that (CF L̄ (L), ∂ L̄ ) is a complex and its homology HF L̄ (L) := H∗ (CF L̄ (L), ∂ L̄ ) is called the lifted Floer homology of L, which isalso invariant under the Hamiltonian isotopies of L.
The following spectral sequence is an adaptation of Biran’s construction (see [2,Theorem 5.2.A]) to the lifted Floer homology, which is the main technical tool usedin the sequel sections.

HAMILTONIAN NON-DISPLACEABILITY OF GAUSS IMAGES

5

Theorem 3.1 (Damian’s spectral sequence, [7, Theorem 2.9]). Denote by Λ =Z2 [T, T −1 ] the algebra of Laurent polynomials over Z2 and Λi ⊂ Λ the subspace of
homogeneous elements of degree i. There exists a spectral sequence {Erp,q , dr } which
satisﬁes the following properties:
L̄(1) E0p,q = Cp+q−pN⊗ ΛpNL , d0 = [∂0L̄ ] ⊗ 1.
Lp,q(2) E1 = Hp+q−pNL (L̄, Z2 ) ⊗ ΛpNL , d1 = [∂1L̄ ] ⊗ T −NL , where[∂1L̄ ] : Hp+q−pNL (L̄; Z2 ) → Hp+q−1−(p−1)NL (L̄; Z2 )is induced by ∂1L̄ .
(3) For any r ≥ 1, Erp,q has the form Erp,q = Vrp,q ⊗ΛpNL with dr = δr ⊗T −rNL ,where each Vrp,q is a vector space over Z2 and δr : Vrp,q → Vrp−r,q+r−1 is ahomomorphism deﬁned for every p, q and satisﬁes δr ◦ δr = 0. Moreover,
p,q
Vr+1
=

Ker(δr : Vrp,q → Vrp−r,q+r−1 ).
Im(δr : Vrp+r,q−r+1 → Vrp,q )L̄In particular, V0p,q = Cp+q−pN, V1p,q = Hp+q−pNL (L̄; Z2 ), δ1 = [∂1L̄ ].
Lp,q ∼(4) Erp,q collapses at (ν + 1)-step and for any p ∈ Z, ⊕q∈Z E∞= HF L̄ (L),dim L+1
where ν = [ NL ].

Back to the Gauss image, since ν = [ dimNL+1] = [ (n+1)g2n ], one can easily get
L
Corollary 3.2. For a Gauss image Ln = G(N n ) ⊂ Qn (C), g ≥ 3 and any p, q ∈ Z,
we havep,q(1) E2p,q = E∞
if and only if g = 3 and (m1 , m2 ) = (2, 2), (4, 4), (8, 8).
p,qp,qif and only if g = 3, (m1 , m2 ) = (1, 1) or g = 4.
(2) E3 = E∞p,q(3) E4p,q = E∞
if and only if g = 6, (m1 , m2 ) = (1, 1) or (2, 2).
4. Topology of Gauss images of isoparametric hypersurfaces with
g=3
An isoparametric hypersurface with g = 3, the so called Cartan hypersurface,is given by a tube around the standard embedding of the projective plane FP 2 in
S 3d+1 , where F is the division algebra R, C, H or O, and d = 1, 2, 4 or 8 [6]. The
multiplicity satisﬁes m = m1 = m2 = d, and n = 3m. In this section, we ﬁrst
observe the topology of Gauss images of isoparametric hypersurfaces with g = 3,
then apply Biran and Cornea’s general fact to obtain their wideness except for thecase n = 3.
Lemma 4.1. The Gauss image L3 of g = 3 is a Z2 -homology sphere.
Proof. Recall that all homogeneous isoparametric hypersurfaces N n ⊂ S n+1 (1) canbe obtained as principal orbits of compact Riemannian symmetric pairs (U, K) ofrank 2, and that isoparametric hypersurfaces with g = 3 must be homogenous.
For the case when (g, m, n) = (3, 1, 3), in [15, p.780] we have a homogenous spaceSO(3)expression N = K/K0 ∼= SU (2)/K̃[a] ,= SU (2)/K̃0 , and L = K/K[a] ∼= Z2 ⊕Z2 ∼where K̃[a] is generated by K̃0 and
(
)
(
)
√
√
√
√
±

1+ −1
2√
−1+ −1
2

1+ −1
2
√
1− −1
2

=: ±A0 ,±−1+ −12√−1+ −12

1+ −1
2√−1− −1
2=: ±A20 ,6

HIROSHI IRIYEH, HUI MA, REIKO MIYAOKA, AND YOSHIHIRO OHNITA(
with A30 = −

1
0

0
1

)
. By a direct computation, we obtain [K̃[a] , K̃[a] ] = K̃0 .

Hence
H1 (L3 ; Z) ∼= Z3 .
= π1 (L3 )/[π1 (L3 ), π1 (L3 )] ∼= K̃[a] /[K̃[a] , K̃[a] ] ∼= K̃[a] /K̃0 ∼Since L3 is orientable, by the Poincaré duality and universal coeﬃcient theorem,
we obtain
if k = 0, 3,
 Z
Z3 if k = 1,
Hk (L3 ; Z) ∼=0
if k = 2.
Therefore, by the universal coeﬃcients theorem, L3 is a Z2 -homology sphere.

□

Lemma 4.2. The Gauss image Ln = G(N n ) of a Cartan isoparametric hypersurface N n ⊂ S n+1 (1) (i.e., g = 3 and n = 3, 6, 12, 24) is a Z2 -homology sphere.
Proof. The Z2 -homology of N n is known [19] as
for k = 0, n, Z2 ,
Z2 ⊕ Z2 , for k = m, 2m,
Hk (N ; Z2 ) ∼=0,otherwise,
where n = 3m and m ∈ {1, 2, 4, 8}. Since G : N n → Ln is a covering map and Z3
acts on N n as a Deck transformation, Theorem 2.4 in [5, p.120] implies
Hk (Ln ; Z2 ) = Hk (N/Z3 ; Z2 ) ∼= Hk (N ; Z2 )Z3 = 0,for k ̸= 0, m, 2m, 3m(= n).
On the other hand, when m ∈ {2, 4, 8}, n is even and χ(L) = χ(N n )/3 = 6/3 = 2
holds. Putting l := dimZ2 Hm (L; Z2 ), we obtain 2 = χ(L) = 1 + l + l + 1 = 2 + 2l,
which yields l = 0. Hence L is a Z2 -homology sphere.
Combing with Lemma 4.1, we complete the proof.
□
Let us give several applications of Lemma 4.2. Firstly we recall Biran andCornea’s result [3] stated in [4, Proposition 1.6 (3)]:
Proposition 4.3. Let L be a closed monotone Lagrangian submanifold embedded
in a closed symplectic manifold (M, ω). If Hi (L; Z2 ) = 0 for any i ∈ Z withi ≡ −1 mod NL , then HF (L) ∼= H∗ (L; Z2 ) ⊗ Λ.
By Proposition 2.1, Lemma 4.2 and Proposition 4.3, we obtain
Corollary 4.4. Let Ln = G(N n ) ⊂ Qn (C) be the Gauss image of an isoparametric
hypersurface N with g = 3 and n = 6, 12, 24. Then L is wide. Hence, #(L∩φ(L)) ≥2 for any φ ∈ Ham(Qn (C), ωstd ) such that L intersects with φ(L) transversally.
Secondly, recall Biran and Khanevsky’s result:
Proposition 4.5 ([4, Corollary 1.5]). Let Ln be a closed monotone Lagrangiansubmanifold embedded in Qn (C) with NL ≥ 2 and n ≥ 2. If HF (L) ̸= 0, then
L ∩ S n ̸= ∅, where S n denotes the totally geodesic Lagrangian sphere of Qn (C).
Combining Corollary 4.4 with Proposition 4.5, we obtain
Corollary 4.6. Let Ln = G(N n ) ⊂ Qn (C) be as in Corollary 4.4. Then φ(L) ∩S n ̸= ∅ for any φ ∈ Ham(Qn (C), ωstd ).

HAMILTONIAN NON-DISPLACEABILITY OF GAUSS IMAGES7

For any n-dimensional submanifold R in Qn (C), it holds∫
Vol(SO(n + 2))#(ϕ(R) ∩ S n )dµSO(n+2) (ϕ) ≤ 2
Vol(R),
Vol(S n )SO(n+2)where ϕ ∈ SO(n + 2) ⊂ Ham(Qn (C), ωstd ) ((6.7) in [14]). Finally, we give an estimate of the volume of Gauss images deformed by Hamiltonian isotopies of Qn (C).
Namely, applying above to R = φ(L), we obtain
Corollary 4.7. For a Lagrangian submanifold L in Corollary 4.4,
1
Vol(S n ) ≤ Vol(φ(L))
2holds for any φ ∈ Ham(Qn (C), ωstd ).
Remark 4.1.
(1) From Gorodski and Podesta’s classiﬁcation [12] of all compact tight Z2 -homology Lagrangian spheres in simply connected compact
homogeneous Kähler manifolds, we know that any compact tight Z2 -homology
Lagrangian sphere in Qn (C) is the standard real form S n up to congruence.
(2) Biran and Cornea [3] proved that any even dimensional closed Lagrangiansubmanifold L embedded in Qn (C) with H1 (L; Z) = 0 must be a Z2 homology sphere. Notice that the Gauss images L with g = 3 are Z2 homology spheres with H1 (L; Z) ∼= Z3 .
5. Hamiltonian Non-displaceability of Gauss images of isoparametric
hypersurfaces with g = 4
Now we consider the Gauss images L of isoparametric hypersurfaces with g = 4.
nSince NL = 2n4 = 2 , NL ≥ 3 leads to n ≥ 6. Thus except for the only one caseSO(2)×SO(3)n = 4 when L = (Z2 ×1)Z4 , the argument in Subsection 3.2 can work and thelifted Floer homology HF L̄ (L) is well-deﬁned for any covering L̄ of L.
Proposition 5.1. Let L = G(N ) ⊂ Qn (C) be the Gauss image of an isoparametric
hypersurface N with g = 4 and m1 ≥ 2. Then HF L̄ (L) ̸= 0 for L̄ = N .
Proof. Consider the Z4 -covering L̄ = N n over Ln . We argue by contradiction.
Assume that HF L̄ (L) = 0. Since ν = [ dimNL+1] = 2, Damian’s spectral sequenceLp,q{Er , dr } collapses at 3-step. Then
0 = E30,q = V30,q =Ker(δ2 : V20,q → V2−2,q+1 )Im(δ2 : V22,q−1 → V20,q )and thus we have the following exact sequence
V22,q−1 → V20,q → V2−2,q+1
for any q ∈ Z. Since
V22,q−1=

Ker([∂1L̄ ] : Hq+1−2NL (L̄; Z2 ) → Hq−NL (L̄; Z2 )),Im([∂1L̄ ] : Hq+2−3NL (L̄; Z2 ) → Hq+1−2NL (L̄; Z2 ))V2−2,q+1=

Ker([∂1L̄ ] : Hq−1+2NL (L̄; Z2 ) → Hq−2+3NL (L̄; Z2 )),Im([∂1L̄ ] : Hq+NL (L̄; Z2 ) → Hq−1+2NL (L̄; Z2 ))

(5.1)

8

HIROSHI IRIYEH, HUI MA, REIKO MIYAOKA, AND YOSHIHIRO OHNITA

we see that V22,q−1 = V2−2,q+1 = 0 when 2 ≤ q ≤ n − 2. Then it follows from (5.1)
that
Ker([∂1L̄ ] : Hq (L̄; Z2 ) → Hq−1+NL (L̄; Z2 ))0 = V20,q =Im([∂1L̄ ] : Hq+1−NL (L̄; Z2 ) → Hq (L̄; Z2 ))holds for 2 ≤ q ≤ n − 2.
Putting q = NL = m1 + m2 , we have the exact sequence
H1 (L̄; Z2 ) → Hm1 +m2 (L̄; Z2 ) → H2(m1 +m2 )−1 (L̄; Z2 ).

(5.2)

Recall that by Münzner’s result [19] the Z2 -homology of N is given byfor k = 0, m1 , m2 , 2m1 + m2 , m1 + 2m2 , n,
 Z2 ,Z⊕Z,
for k = m1 + m2 ,Hk (N ; Z2 ) ∼
=
2
2

0,
otherwise.
n

By the assumption that m1 ≥ 2, (5.2) leads to the exact sequence 0 → Z2 ⊕Z2 →0, which is a contradiction.
□
Corollary 5.2. Under the same assumption as in Proposition 5.1, the Gauss image
Ln ⊂ (Qn (C), ωstd ) is Hamiltonian non-displaceable.
Remark 5.1. Note that any non-homogeneous isoparametric hypersurface in thestandard sphere satisﬁes the assumption of Proposition 5.1. By R. Takagi’s result[26] we know that an isoparametric hypersurface N n with g = 4 and m1 = 1 must
be N n = SO(2)×SO(k+2)(n = 2k + 2, m2 = k), in particular homogeneous.
Z2 ×SO(k)6. Hamiltonian Non-displaceability of the Gauss image of an
isoparametric hypersurface with g = 6
When g = 6, NL ≥ 3 implies m1 = m2 = 2 [1]. Thus we focus on the Gauss
image L12 = G(N 12 ) of the isoparametric hypersurface N 12 ⊂ S 13 (1) with g = 6
and m1 = m2 = 2. Compared with the case when g = 4, this case is more
complicated because {Erp,q , dr } collapses at 4-step.
Proposition 6.1. Let L12 = G(N 12 ) ⊂ Q12 (C) be the Gauss image of an isoparametric hypersurface with g = 6 and m1 = m2 = 2. Then HF L̄ (L) ̸= 0 for L̄ = N .
In particular, L is Hamiltonian non-displaceable in Q12 (C).
Proof. Take the Z6 -covering L̄ = N 12 over L12 . Since ν = [ 12+1
4 ] = 3, Damian’s
spectral sequence {Erp,q , dr } collapses at 4-step. Assume that HF L̄ (L) = 0. Then
we haveKer(δ3 : V30,q → V3−3,q+2 )0 = E40,q = V40,q =Im(δ3 : V33,q−2 → V30,q )and the following exact sequence
V33,q−2 → V30,q → V3−3,q+2
for any q ∈ Z.
Notice that
V33,q−2=

Ker(δ2 : V23,q−2 → V21,q−1 ),Im(δ2 : V25,q−3 → V23,q−2 )V23,q−2=

Ker([∂1L̄ ] : Hq−11 (L̄; Z2 ) → Hq−8 (L̄; Z2 )).
Im([∂1L̄ ] : Hq−14 (L̄; Z2 ) → Hq−11 (L̄; Z2 ))(6.1)

HAMILTONIAN NON-DISPLACEABILITY OF GAUSS IMAGES

9Consequently, V33,q−2 = 0 when q ≤ 10. Similarly, fromV3−3,q+2=

Ker(δ2 : V2−3,q+2 → V2−5,q+3 ),Im(δ2 : V2−1,q+1 → V2−3,q+2 )V2−3,q+2=

Ker([∂1L̄ ] : Hq+11 (L̄; Z2 ) → Hq+14 (L̄; Z2 )),Im([∂1L̄ ] : Hq+8 (L̄; Z2 ) → Hq+11 (L̄; Z2 ))we see that V3−3,q+2 = 0 if q ≥ 2. Thus (6.1) leads to V30,q = 0 for 2 ≤ q ≤ 10. DuetoKer(δ2 : V20,q → V2−2,q+1 ),0 = V30,q =Im(δ2 : V22,q−1 → V20,q )we get the exact sequence V22,q−1 → V20,q → V2−2,q+1 for any 2 ≤ q ≤ 10. Thanks
toV22,q−1=

Ker([∂1L̄ ] : Hq−7 (L̄; Z2 ) → Hq−4 (L̄; Z2 )),Im([∂1L̄ ] : Hq−10 (L̄; Z2 ) → Hq−7 (L̄; Z2 ))V2−2,q+1=

Ker([∂1L̄ ] : Hq+7 (L̄; Z2 ) → Hq+10 (L̄; Z2 )),Im([∂1L̄ ] : Hq+4 (L̄; Z2 ) → Hq+7 (L̄; Z2 ))we derive V20,6 = 0. Now because
0=

V20,6 =

Ker([∂1L̄ ] : H6 (L̄; Z2 ) → H9 (L̄; Z2 )),Im([∂1L̄ ] : H3 (L̄; Z2 ) → H6 (L̄; Z2 ))we get the exact sequence H3 (L̄; Z2 ) → H6 (L̄; Z2 ) → H9 (L̄; Z2 ), which contradicts
the facts in [19] that H3 (L̄; Z2 ) ∼□= H9 (L̄; Z2 ) = 0 and H6 (L̄; Z2 ) ∼= Z2 ⊕ Z2 .
Therefore we complete the proof of Theorem 1.1.
7. Open problems and a conjecture
The following are open problems for further research:(1) We cannot apply the same method above to determine the Hamiltonian
non-displaceability of Gauss images of the following isoparametric hypersurfaces:(i) g = 3, (m1 , m2 ) = (1, 1), n = 3;(ii) g = 4, (m1 , m2 ) = (1, k), n = 2(1 + k);(iii) g = 6, (m1 , m2 ) = (1, 1), n = 6.
Consider how we can do in these cases.
(2) Determine the Floer homology HF (L) of Gauss images for (g, n) = (3, 3)
and g = 4, 6.
(3) Describe H∗ (L; Z2 ) of the Gauss images L = N/Zg for g = 4, 6 explicitly.
According to the present work and a private communication with Hajime Ono, heand the authors give
Conjecture. Any compact connected minimal Lagrangian submanifold in an irreducible Hermitian symmetric space of compact type is Hamiltonian non-displaceable.

10

HIROSHI IRIYEH, HUI MA, REIKO MIYAOKA, AND YOSHIHIRO OHNITA

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Mathematics and Infomatics, College of Science, Ibaraki University, Mito, 310-8512,
Japan.
E-mail address: hiroshi.irie.math@vc.ibaraki.ac.jp
Department of Mathematical Sciences, Tsinghua University, Beijing 100084, P.R.
China
E-mail address: hma@math.tsinghua.edu.cn
Mathematical Institute, Tohoku University, Sendai 980-8578, Japan
E-mail address: r-miyaok@m.tohoku.ac.jp
Osaka City University Advanced Mathematical Institute & Department of Mathematics, Osaka City University, Osaka, 558-8585, Japan
E-mail address: ohnita@sci.osaka-cu.ac.jp