Proceedings of The Fifteenth International
Workshop on Diﬀ. Geom. 15(2011) 1-2

Certain Lagrangian Submanifolds in Hermitian SymmetricSpaces and Hamiltonian Stability Problems
Yoshihiro OHNITA
Department of Mathematics, Osaka City University,& Osaka City University Advanced Mathematical Institute (OCAMI),
3-3-138 Sugimoto, Sumiyoshi-ku, Osaka 558-8585, Japane-mail : ohnita@sci.osaka-cu.ac.jp(2010 Mathematics Subject Classification : Primary 53C40, Secondary 53D12, 53C42)

Abstract. In this article we shall provide an exposition on our recent works andrelated topics in geometry of compact homogeneous Lagrangian submanifolds in Hermitiansymmetric spaces such as complex Euclidean spaces, complex projective spaces, complex
hyperquadrics, and so on.

1

Introduction

Submanifold geometry is one of major subjects in classical and modern diﬀerential geometry. In submanifold geometry it is of especial interest to study various
types of submanifolds in symmetric spaces and related geometric variational problems such as minimal submanifold theory and harmonic map theory.
In this article we shall focus on the Lie theoretic approach to Lagrangian submanifolds in Kähler manifolds and related Hamiltonian variational problem. The
purpose of this article is to provide an exposition on recent results and related research problems in geometry of compact homogeneous Lagrangian submanifolds in
Hermitian symmetric spaces such as complex Euclidean spaces, complex projective
spaces, complex hyperquadrics, and so on.
About 1990’s ([36], [37]) Y.-G. Oh ﬁrst considered the volume minimizing property of Lagrangian submanifolds in a Kähler manifold under Hamiltonian deformations. He introduced and investigated the notions of Hamiltonian minimality and
Key words and phrases: Please write Key words and phrases
Lagrangian submanifold, Hermitian symmetric space, Hamiltonian stability.
* This work was supported by JSPS Grant-in-Aid for Scientific Research (S) No. 23224002

1

Hamiltonian stability for Lagrangian submanifolds.
A Lagrangian submanifold is called homogeneous if it is obtained as an orbit
under the Hamiltonian group action of an analytic subgroup of the automorphismgroup of a Kähler manifold. Since we know that any compact homogeneous Lagrangian submanifold in a Kähler manifold is always Hamiltonian minimal, such Lagrangian submanifolds in Kähler manifolds provide many good examples of Hamiltonian minimal Lagrangian submanifold. Theoretically it is possible to analyze thesecond variations of compact homogeneous Lagrangian submanifolds by Harmonic
Analysis over Compact Homogeneous Spaces in order to determine their Hamiltonian stability. It is a quite natural and interesting problem to classify compact
homogeneous Lagrangian submanifolds in speciﬁc Kähler manifolds and to determine their Hamiltonian stability.
Problem. Classify compact homogeneous Lagrangian submanifolds in speciﬁc
Kähler manifolds such as Hermitian symmetric spaces, generalized ﬂag manifolds
with invariant Kähler metrics, toric manifolds and so on.
Problem. Determine the Hamiltonian stability of compact homogeneous Lagrangian submanifolds in such Kähler manifolds.
Compact homogeneous Lagrangian submanifolds with parallel second fundamental form in complex space forms Cn , CP n , CH n have already been classiﬁed
completely by H. Naitoh and M. Takeuchi ([33], [34], [35]) and from their classiﬁcation they all can be constructed by using the standard embeddings of symmetricR-spaces (of type U (r)). There exists compact homogeneous Lagrangian submanifolds with non-parallel second fundamental form in complex projective spaces, moregenerally in complex space forms. The classiﬁcation problem of compact homogeneous Lagrangian submanifolds in complex projective spaces can be converted toa problem on prehomogeneous vector spaces. L. Bedulli and A. Gori ([8]) classiﬁed compact homogeneous Lagrangian submanifolds obtained as orbits of simple
compact Lie groups, by using the classiﬁcation theory of prehomogeneous vector
spaces due to Mikio Sato and Tatsuo Kimura ([45]). The classiﬁcation of Bedulli
and Gori includes so many new compact homogeneous Lagrangian submanifolds
with non-parallel second fundamental form in complex projective spaces. We shall
explain what are known about the Hamiltonian stability results for those compact
homogeneous Lagrangian submanifolds in complex space forms.
Real forms of compact Hermitian symmetric spaces form a nice class of Lagrangian submanifolds in Hermitian symmetric spaces of rank greater than 1, which
are nothing but totally geodesic Lagrangian submanifolds and are canonically embedded symmetric R-spaces ([48]). The stability of each real form as minimal submanifolds has been determined previously by M. Takeuchi ([48]). By using his
results we can determine the Hamiltonian stability of each real form and we observe three types of examples of Hamiltonian unstable real forms in some Hermitiansymmetric spaces of rank greater than 1. We shall also mention recent progress in2

research by the Japanese diﬀerential geometry group on real forms of compact Hermitian symmetric spaces as Lagrangian submanifolds. From the classiﬁcation resultsof extrincic symmetric submanifolds in Riemannian symmetric spaces (J. Berndt,
J. Eschenburg, H. Naitoh and T. Tsukada [9]) we know that all Lagrangian submanifolds in Hermitian symmetric spaces of rank greater than 1 with (non-totally
geodesic) parallel second fundamental form are explicitly expressed as canoincal
Lagrangian deformation of some real forms.
Therefore it is an interesting and important problem to construct and classify compact homogeneous Lagrangian submanifolds in Hermitian symmetric spaces
which do not necessarily have parallel second fundamental form.
The complex hyperquadric is one of compact rank 2 Hermitian symmetric spaces
and it can be canonically isometric to the real Grassmann manifold of oriented 2dimensional vector subspaces. The Gauss map of each oriented hypersurface inthe unit standard sphere provides a Lagrangian immersion into the complex hyperquadric. We observe that the Gauss map of oriented hypersurface with constantprincipal curvatures in the unit standard sphere, so called isoparametric hypersurface, is a minimal Lagrangian immersion and the image of the Gauss map of compact
isoparametric hypersurface in the unit standard sphere is a compact smooth minimal Lagrangian submanifold embedded in the complex hyperquadric. The structure
and classiﬁcation theory of isoparametric hypersurfaces in the standard sphere are
well-developed since Elie Cartan’s work. The Gauss images of compact isoparametric hypersurfaces in the unit standard sphere form a nice class of compact minimal
Lagrangian submanifolds embedded in the complex hyperquadric. We can observethat the homogeneity of original isoparametric hypersurfaces in the standard sphereis equivalent to the homogeneity of its Gauss image in the complex hyperquadric.
All homogeneous isoparametric hypersurfaces are known to be obtained as principal orbits of the isotropy representation of Riemannian symmetric spaces of rank
2 (W.-Y. Hsiang and H. B. Lawson, Jr. [19], R. Takagi and T. Takahashi [46]).
Non-homogeneous isoparametric hypersurfaces in the standard sphere were discovered ﬁrst by H. Ozeki and M. Takeuchi ([41]) and generalized by D. Ferus, H.
Karcher and H. F. Münzner ([14]). They can be constructed by the representationsof Cliﬀord algebras (isoparametric hypersurfaces of OT-FKM type).
We shall discuss the properties of compact minimal Lagrangian submanifolds
embedded in the complex hyperquadric obtained as the Guass images of compact
isoparametric hypersurfaces in the unit standard sphere. Especially we classiﬁed allcompact homogeneous Lagrangian submanifolds in the complex hyperquadric anddetermined the Hamiltonian stability of the Guass images of compact homogeneous
isoparametric hypersurfaces in the unit standard sphere ([24], [26]).
There are several related questions and further problems to be studied in future.
This article is organized as follows: In Section 2 we shall recall the fundamental deﬁnitions and properties on Lagrangian immersions, Hamiltonian deformations and moment maps. In Section 3 we shall treat Lagrangian submanifolds inKähler manifolds and explain the notions of the mean curvature form, Hamiltonian minimality, Hamiltonian stability, strictly Hamiltonian stability and globally
Hamiltonian stability. In Section 4 we shall discuss fundamental examples of com-3

pact homogeneous Lagrangian submanifolds in complex Euclidean spaces, complex
projective spaces, complex space forms and compact Hermitian symmetric spaces,
and their Hamiltonian stability. In Section 5 we shall explain the Gauss map construction of compact minimal Lagrangian submanifolds in complex hyperquadrics
from isoparametric hypersurfaces in the unit standard sphere. In Section 6 weshall describe our classiﬁcation of compact homogeneous Lagrangian submanifolds
in complex hyperquadrics. In Section 7 we shall describe our Hamiltonian stability
results of all compact homogeneous minimal Lagrangian submanifolds embedded incomplex hyperquadrics, which are obtained as the Gauss images of compact homogeneous isoparametric hyersurfaces in the unit standard sphere. In Section 8 weshall mention related questions and further problems to be studied.

2

Lagrangian submanifolds, Hamiltonian deformations and momentmaps.

Let (M, ω) be a 2n-dimensional symplectic manifold with a symplectic form
ω. A Lagrangian immersion φ : L −→ M is deﬁned as a smooth immersion of ann-dimensional smooth manifold L into M satisfying the condition φ∗ ω = 0.
The most elementary example of a symplectic manifold is a plane M = R2
equipped with the standard area from ω = dx ∧ dy. In this case a 1-dimensional
Lagrangian submanifold is a plane curve and a 1-dimensional compact Lagrangiansubmanifold is nothing but a closed plane curve.
The normal bundle φ−1 T M/φ∗ T L of a Lagrangian immersion φ : L −→ M canbe identiﬁed with the cotangent bundle T ∗ L of L :φ−1 T M/φ∗ T L ∋ v 7−→ αv := ω(v, ·) ∈ T ∗ L
A Lagrangian deformation is deﬁned as a smooth one-parameter family of Lagrangian immersions φt : L −→ M with φ = φ0 . Let αVt := ω(Vt , dφ(·)) be 1-forms∞−1t
on L corresponding to its variational vector ﬁeld Vt := ∂φT M ). The∂t ∈ C (φLagrangian deformation is characterized by the closedness of αVt , for each t. Furthermore, if αVt is exact for each t, then {φt } is called a Hamiltonian deformationof φ = φ0 .
Suppose that a connected Lie group K has the Hamiltonian group action on(M, ω) with the moment map µK : M −→ k∗ . If a Lagrangian submanifold L in M
is preserved by the group action of K, thenL ⊂ µ−1
K (α)
where

for ∃ α ∈ z(k∗ ),z(k∗ ) := {α ∈ k∗ | Ad∗ (a)α = α for all a ∈ K}.

Assume that M and K are compact. Then an orbit L = K · x of K is Lagrangian
if and only if
L = K · x = µ−1
for ∃ α ∈ z(k∗ ) ∼= c(k),K (α)
where c(k) denotes the center of k.

4

3

Lagrangian submanifolds in Kähler manifolds and Hamiltonian stability

Assume that (M, ω, J, g) is a Kähler manifold with a complex structure J anda Kähler metric g. Here ω is the Kähler form deﬁned by ω(X, Y ) = g(JX, Y ) for
each X, Y ∈ T M .
Let φ : L → M be a Lagrangian immersion into a Kähler manifold. Let
S(X, Y, Z) := ω(B(X, Y ), Z) = g(JB(X, Y ), Z)(X, Y, Z ∈ T L)

be a symmetric 3-tensor ﬁeld on L deﬁned by the second fundamental form B of L
in M . Letn∑
H = tr B =B(ei , ei )i=1denote the mean curvature vector ﬁeld of φ, where {ei } is an orthonormal basis ofTx L. The corresponding 1-form αH ∈ Ω1 (L) is called the mean curvature form of
φ. It is well-known that the mean curvature form αH satisﬁes the following identitydαH = φ∗ ρM ,where ρM denotes the Ricci form of M . In particular, if M is Einstein-Kähler, thenαH is closed.
In Riemannian geometry a submanifold vanishing the mean curvature vector
ﬁeld, i.e. H = 0, is called a minimal submanifold. A 1-dimensional minimal
submanifold is nothing but a geodesic. It is well-known that there is no compact
minimal submanifold in a Euclidean space, more generally in a simply connectedcomplete Riemannian manifold of nonnegative sectional curvatures (cf. [22, Vol. II,
p.379, Note. 27]).
Let Aut(M, ω, J, g) be the automorphism group of the Kähler structure of(M, ω, J, g). If a Lagrangian submanifold L embedded in M is obtained as an orbitof a connected Lie subgroup K of Aut(M, ω, J, g), then we call L a homogeneous
Lagrangian submanifold of a Kähler manifold M . Let k denote the Lie algebra ofK. Then it is elementary that the mean curvature form αH of a Lagrangian orbit
L = K · p of K in M is given by the formula(3.1)

1
αH (X̃) = − div(J X̃)|L2for each X ∈ k.
Assume that (M, ω, g, J) is an Einstein-Kähler manifold with nonzero Einsteinconstant κ ̸= 0. It is known that the map µ̃ : M → k∗ deﬁned by(3.2)⟨µ̃, X⟩ : M ∋ x 7−→

1(divJX)(x) ∈ R
2κ

(X ∈ k)is a moment map of the group action of K with respect to ω. We call µ̃ the canonicalmoment map for the group action of a Lie subgroup K. Therefore we know that5

Lemma 3.1. the mean curvature form αH of a Lagrangian orbit of K in M isexpressed as
αH (X̃) = −κ ⟨µ̃, X⟩|L

(3.3)

(X ∈ k).

In particular L = K · p is a minimal Lagrangian orbit if and only if L = K · p ⊂µ̃−1 (0).
The notion of Hamiltonian minimality and Hamiltonian stability was introduced
and discussed ﬁrst by Y. G. Oh ([36]).
For the simplicity, suppose that L is compact without boundary.
Definition. A Lagrangian immersion φ is called Hamiltonian minimal (shortly, Hminimal ) or Hamiltonian stationary if under every Hamiltonian deformation {φt }the ﬁrst variation of the volume vanishes, that isd
Vol(L, φ∗ g)|t=0 = 0.
dt
The H-minimal Lagrangian submanifold equation isδαH = 0,where δ denotes the codiﬀerential operator relative to the induced metric φ∗ g on L.
Definition. An H-minimal Lagrangian immersion φ is called Hamiltonian stable(shortly, H-stable) if under every Hamiltonian deformation {φt } the second variationof the volume is nonnegative, that isd2
Vol(L, φ∗ g)|t=0 ≥ 0.
dt2
The second variational formula was given as follows ([37]):d2
Vol(L, φ∗ g)|t=0
2
dt
∫
( 1
)
=
⟨∆L α, α⟩ − ⟨R̄α, α⟩ + 2⟨α ⊗ α ⊗ αH , Sα⟩ + ⟨αH , α⟩2 dv,
L

where we set α = αV0 ∈ B1 (L). Here⟨R̄α, α⟩ =n∑

RicM (ei , ej )α(ei )α(ej ).

i,j=1

6

Definition. An H-minimal Lagrangian immersion φ is called strictly Hamiltonian
stable (shortly, strictly H-stable) if the following two conditions are satisﬁed:(a) φ is Hamiltonian stable.
(b) The null space of the second variation on Hamiltonian deformations coincides
with the vector subspace spanned by inﬁnitesimal holomorphic isometries ofM (assume that M is simply connected, or more generally H 1 (M, R) = {0}).
Remark 3.2. If L is strictly Hamiltonian stable, then L has local minimum volume
under every Hamiltonian deformation.
Corollary 3.3. Suppose that L is minimal Lagrangian and M is Einstein-Kählerwith Einstein constant κ. Then L is Hamiltonian stable if and only if the ﬁrst
(positive) eigenvalue λ1 of the Laplacian ∆0L of L acting on on functions satisﬁes
λ1 ≥ κ.
Proposition 3.4 (cf. [25]). Any compact homogeneous Lagrangian submanifold ina Kähler manifold is always H-minimal.
Theoretically it is possible to analyze the second variations of compact homogeneous Lagrangian submanifolds by Harmonic Analysis over Compact HomogeneousSpaces in order to determine their (strictly) Hamiltonian stability. It is a quite
natural and interesting problem to classify compact homogeneous Lagrangian submanifolds in speciﬁc Kähler manifolds and to determine their Hamiltonian stability.

4

ExamplesNot so many examples of compact Hamiltonian stable Lagrangian submanifolds
are known.
4.1

Hamiltonian stable Lagrangian submanifolds in Cn and CP n

Example 1. A circle S 1 (r) ⊂ C = R2 with radius r > 0 on the Euclidean plane is
a compact 1-dimensional H-minimal (not minimal!) Lagrangian submanifold in R2
which is globally strictly H-stable, because of the isoperimetric inequality on R2 .
Example 2. A great or small circle S 1 (r) ⊂ S 2 (1) with radius 0 < r ≤ 1 on the 2dimensional standard unit sphere is a compact 1-dimensional H-minimal (minimal7

if and only if r = 1) Lagrangian submanifold in S 2 (1), which is globally strictly
H-stable, because of the isoperimetric inequality on S 2 (1).
Example 3. The product of two circles in Example 1 S 1 (r1 ) × S 1 (r2 ) ⊂ C2 = R2
is also a compact 2-dimensional H-minimal Lagrangian submanifold in R2 , whichis strictly H-stable. Its global H-stability is conjectured by Y. G. Oh and it is still
open. Example 2 is constructed from this product example as follows: For eachr1 > 0, r2 > 0 with (r1 )2 + (r2 )2 = 1,S 1 (r1 ) × S 1 (r2 )π S1?
S 1 (r)⊂⊂S 3 (1) ⊂

C2

π S1
?
CP 1 = S 2 ( 12 )Note that r1 = r2 if and only if S 1 ⊂ CP 1 = S 2 ( 12 ) is a great circle (i.e. a geodesic).
More generally, the product of n+1 circles in Example 1 S 1 (r1 )×· · ·×S 1 (rn+1 ) ⊂Cn+1 = R2(n+1) is also a compact (n + 1)-dimensional H-minimal (never minimal!)
Lagrangian submanifold in Cn+1 = R2(n+1) , which is strictly Hamiltonian stable([37]). The above construction can be generalized as follows:
S 1 (r1 ) × · · · × S 1 (rn+1 ) ⊂ S 2n+1 (1) ⊂ Cn+1
π S1
?
Tn

⊂

π S1
?
CP n

Then T n ⊂ CP n is a compact H-minimal Lagrangian submanifold which is strictly
H-stable ([37]). Note that it is minimal in CP n if and only if r1 = · · · = rn+1 .
Example 4. A great circle S 1 (1) ⊂ S 2 (1) in Example 2 can be considered as a realprojective line RP 1 ⊂ CP 1 . Generally, the real projective subspaces RP n ⊂ CP n ,which exhaust all totally geodesic (by deﬁnition S = 0) Lagrangian submanifolds

8in CP n , are compact strictly Hamiltonian stable and globally Hamiltonian stable
minimal (in this case totally geodesic) Lagrangian submanifold ([36]).

S1 · Sn
π S1
?
RP n

⊂

S 2n+1 (1)

⊂

π S1
?
CP n

⊂

Cn+1

Here the submanifold S 1 · S n is the inverse image of RP n by the Hopf ﬁbration
π : S 2n+1 (1) → CP n and we have
S1 · Sn =∪c Rn+1 ∩ S 2n+1 (1) ∼= (S 1 × S n )/Z2 ,c∈S 1

which is isometric to a real hyperquadricQ1,n+1 (R) ∼= (SO(2) × SO(n + 1))/S ′ (O(1) × O(n)),
where








ε
0







0






 0 A


′




S (O(1) × O(n)) := 
|
ε
=
±1,
A
∈
O(1),
B
∈
O(n)
.






B 0 






0






0 ε
Then S 1 · S n is a compact minimal submanifold in S 2n+1 (1) and a compact Hamiltonian minimal (never minimal) Lagrangian submanifold in Cn+1 which is strictly
Hamiltonian stable ([2], [4], [5]).
Example 5. Let U (p) and SU (p) be the unitary group and special unitary groupof degree p. Let M(p, C) denotes the complex vector space of all complex p × p
matrices. Then U (p) and SU (p) are naturally embedded into the vector spaceM(p, C). Here set n = p2 − 1.

9

U (p) = S 1 · SU (p) ⊂ S 2p −1 (1) ⊂ M(p, C) = Cp
2

π S1
?
SU (p)/Zp

⊂

2

π S1
?
2
CP p −1

Then the irreducible symmetric space SU (p)/Zp is a compact minimal Lagrangiansubmanifold in CP n which is strictly Hamiltonian stable ([3]). U (p) is a compact
minimal submanifold in S 2n+1 (1) and a compact Hamiltonian minimal (never minimal) Lagrangian submanifold in Cn+1 which is strictly H-stable ([2], [4], [5]).
Example 6. Let S(p, C) denotes the complex vector space of all complex symmetricp × p matrices. The symmetric space U (p)/O(p) is standardly embedded into thevector space S(p, C) as follows:
U (p)/O(p) ∋ a O(p) 7−→ t aa ∈ S(p, C)
Here set n = p(p + 1)/2 − 1.
U (p)/O(p) = S 1 · SU (p)/SO(p) ⊂ S p(p+1)−1 (1) ⊂ S(p, C) = Cp(p+1)/2
π S1?
SU (p)/(SO(p)Zp )

⊂

π S1
?
CP (p−1)(p+2)/2

Then the irreducible symmetric space SU (p)/(SO(p)Zp ) is a compact minimal Lagrangian submanifold in CP n which is strictly Hamiltonian stable ([3]). U (p)/O(p)is a compact minimal submanifold in S 2n+1 (1) and a compact Hamiltonian minimal (never minimal) Lagrangian submanifold in Cn+1 which is strictly Hamiltonian
stable ([2], [4], [5]).
Example 7. Let AS(2p, C) denote the complex vector space of all complex skewsymmetric 2p × 2p matrices. The symmetric space U (2p)/Sp(p) is standardly embedded into the vector space AS(2p, C) as follows:
U (2p)/Sp(p) ∋ a Sp(p) 7−→ t aJp a ∈ AS(2p, C),10
where Jp = 

0

−Ip

Ip

0


 and Ip is the identity matrix of degree p. Here set n =p(2p − 1) − 1.

U (2p)/Sp(p) = S 1 · SU (2p)/Sp(p) ⊂ S p(p+1)−1 (1) ⊂ AS(2p, C) = Cp(2p−1)
π S1?
SU (2p)/(Sp(p)Z2p )

⊂

π S1
?
CP (2p+1)(p−1)

Then the irreducible symmetric space SU (p)/Zp is a compact minimal Lagrangiansubmanifold in CP n which is strictly H-stable ([3]). U (2p)/Sp(p) is a compact minimal submanifold in S 2n+1 (1) and a compact H-minimal (never minimal) Lagrangiansubmanifold in Cn+1 which is strictly H-stable ([2], [4], [5]).
Example 8. Let C be the Cayley algebra over R and CC be the complexiﬁcationof C. Let J = J(3, C) be the real vector space of all 3 × 3 Hermitian matrices over
C and JC be the complexiﬁcation of J. J and JC have the structures of the Jordan
algebras as t he multiplication ◦, the cross product ×, the inner product ( , ), thecubic product ( , , ), the determinant det and the Hermitian inner product on
JC . The simply connected compact Lie groups of exceptional types E6 and F4 aredeﬁned as
E6 := {a ∈ GL(CC ) | det(a(X)) = det(X), ⟨a(X), a(Y )⟩ = ⟨X, Y ⟩},F4 := {a ∈ E6 | ⟨a(X), a(Y )⟩ = ⟨X, Y ⟩} = {a ∈ E6 | a(I3 ) = I3 }.
Refer Ichiro Yokota’s article [52] for an excellent article on exceptional Lie groups.
The symmetric space S 1 · E6 /F4 is standardly embedded into the vector space JC
as follows:
S 1 · E6 /F4 ∋ a F4 7−→ a(I3 ) ∈ JC .
Here set n + 1 = 27.

11

S 1 · E6 /F4
π S1?
E6 /(F4 Z3 )⊂S 53 (1) ⊂ JC = C27

⊂

π S1
?
CP 26

Then the irreducible symmetric space E6 /(F4 Z3 ) is a compact minimal Lagrangiansubmanifold embedded in CP n which is strictly Hamiltonian stable ([3]). S 1 ·E6 /F4 is a compact minimal submanifold embedded in S 2n+1 (1) and a compact
Hamiltonian minimal (never minimal) Lagrangian submanifold embedded in Cn+1
which is strictly Hamiltonian stable ([2], [4], [5]).
The compact symmetric spaces appearing in the above examples
S 1 · S n , U (p), U (p)/O(p), U (2p)/Sp(p), S 1 · E6 /F4and their Lagrangian embedding in complex Euclidean spaces exhaust so-called
irreducible symmetric R-spaces of type U (r) and their standard embeddings (see[48]). A symmetric R-space can be characterized as a compact symmetric spaceobtained as an orbit of the isotropy representation of a Riemannian symmetricspace. All Lagrangian submanifolds in Cn and CP n described in Examples 1-8 haveparallel second fundamental form ∇S = 0. Lagrangian submanifolds in complex
space forms with ∇S = 0 are already completely classiﬁed by Hiroo Naitoh and
Masaru Takeuchi ([33], [34], [35]). According to the classiﬁcation theory, all compact
Lagrangian submanifolds embedded in simply connected complete complex space
forms of constant holomorphic sectional curvature c (complex Euclidean spaces
Cn , complex projective spaces CP n , complex hyperbolic spaces CH n ) are given as
follows (cf. [38]):(1) Suppose that L is a compact Lagrangian submanifold embedded in Cn with
∇S = 0. Then L is congruent to the standard embedding of a symmetricR-space of type U (r), that is, an irreducible symmetric R-space of type U (r)or a Riemannian product of symmetric R-spaces of type U (r).
(2) Suppose that L is a compact Lagrangian submanifold embedded in CP n with
∇S = 0. Then L is locally isometric to a symmetric space M0 × M1 × · · · ×
Mr , where L0 is of Euclidean type, dim M0 ≥ r − 1 and Mi is one of thefollowing irreducible symmetric spaces of compact type: (a) S n , (b) SU (p),(c) SU (p)/SO(p), (d) SU (2p)/Sp(p), (e) E6 /F4 . Let π : S 2n+1 → CP n be
the Hopf ﬁbration. Then π −1 (L) is also a Lagrangian submanifold embedded
in Cn+1 of the case (1).
(3) Suppose that L is a compact Lagrangian submanifold embedded in CH n
with ∇S = 0. Then L is locally isometric to a symmetric space M0 × M1 ×

12

· · · × Mr , where L0 is of Euclidean type, dim M0 ≥ r and Mi is one ofirreducible symmetric spaces of compact type (a)∼(e) in the case (2). Let
Cn+1= C × Cn be an (n + 1)-dimensional complex vector1∑n space equippedwith an indeﬁnite Hermitian form F (z, w) := −z0 w̄0 + i=1 zi w̄i for eachz = (z0 , z1 , · · · , zn ), w = (w0 , w1 , · · · , wn ) ∈ Cn+1
. Set H 2n+1 (4/c) := {z ∈
1
n+1
2n+1
n
C1 | F (z, z) = 4/c} and π : H(4/c) → HP be the natural Riemannian
submersion with ﬁber S 1 . Then π −1 (L) = S 1 × M̃ . Here S 1 = {z ∈ C |F (z, z) = −r02 } for some r0 > 0, r1 > 0 with −20 + r12 = 4/c and L̃ is a
Lagrangian submanifold embedded in Cn of the case (1).
Theorem 4.1 ([50], [38]). Let M = M̃ (c) be a complex space form and L be
a compact Lagrangian submanifold immersed in M . If L is H-minimal and has
nonnegative sectional curvatures, then L has parallel second fundamental form, i.e.
∇S = 0. The converse also holds.
This statement does not hold in general for compact Lagrangian submanifolds
of Hermitian symmetric spaces of rank greater than 1 (See Remark 7.1).
Theorem 4.2 ([51], [10]). If L is a H-stable minimal Lagrangian torus immersed
in CP 2 , then L is a Cliﬀord torus in CP 2 (n = 2, r1 = r2 = r3 in Example 3), in
particular ∇S = 0.
Theorem 4.3 ([4],[5]). If L is a compact Lagrangian submanifold embedded ina simply connected complete complex space form (that is, Cn , CP n , CH n ) with
∇S = 0, then L is H-stable.
Example 9. Let V3 be the vector space of complex homogeneous polynomials withtwo variables z0 , z1 of degree 3 and ρ3 be the irreducible unitary representation of
SU (2) on V3 . Then
ρ3 (SU (2))[z03 + z13 ] ⊂ CP 3
is a 3-dimensional compact embedded strictly H-stable minimal Lagrangian submanifold with ∇S ̸= 0 (Bedulli and Gori [7], independently [39]).
Moreover Bedulli and Gori [8] provided a nice classiﬁcation of compact homogeneous Lagrangian submanifolds in CP n obtained as Lagrangian orbits of compact
simple K ⊂ SU (n + 1) by applying the classiﬁcation theory of prehomogeneous
vector spaces (Mikio Sato and Tatsuo Kimura [45]). Such Lagrangian orbits were13

classiﬁed as
16 types of examples = [5 types of examples with∇S = 0]+[11 types of examples with∇S ̸= 0]
A 5-dimensional minimal Lagrangian orbit (SU (2) × SU (2))/T 1 · Z4 ⊂ CP 5 ofnon-simple SU (2) × SU (2) with non-parallel second fundamental form is H-stable(Petrecca and Podesta [43]).
Recently a new and interesting construction of compact non-homogeneous Hminimal Lagrangian submanifolds embedded in Cn and CP n is studied by themethod of toric topology in A. E. Mironov and T. Panov ([30]).
At present I do not know any counter example to the following questions:
Problem. Is it true that a compact Hamiltonian minimal Lagrangian submanifold
embedded in a complex projective space is Hamiltonian stable? Or is there a compact
Hamiltonian unstable Hamiltonian minimal Lagrangian submanifold embedded in acomplex projective space?
Problem. Is it true that a compact Hamiltonian minimal Lagrangian submanifold
embedded in a complex Euclidean space is Hamiltonian stable? Or is there a compact
Hamiltonian unstable Hamiltonian minimal Lagrangian submanifold embedded in acomplex Euclidean space?
The following problem is also interesting but still open:
Problem. Classify all compact homogeneous Lagrangian submanifolds in complex
hyperbolic space form CH n .
4.2

Hamiltonian stability of real forms of Hermitian symmetric spaces

The Hamiltonian stability of each compact Lagrangian submanifold embedded
as a real form L in compact irreducible Hermitian symmetric spaces M of higher
rank are known as below ([48]). Here each M is equipped with the standard Kähler
metric of Einstein constant 12 and λ1 denotes the ﬁrst eigenvalue of the Laplacianof L on functions.
Concerned with recent other related works on real forms of compact Hermitiansymmetric spaces as Lagrangian submanifolds, Hiroshi Iriyeh, Takashi Sakai, Hiroyuki Tasaki and Makiko Tanaka make progress on the Tightness and Lagrangian14intersection Floer homology for real forms for compact Hermitian symmetric spaces([49], [21]).
From the classiﬁcation results of extrincic symmetric submanifolds in Riemannian symmetric spaces (J. Berndt, J. Eschenburg, H. Naitoh and T. Tsukada [9])we know that all Lagrangian submanifolds in Hermitian symmetric spaces of rank
greater than 1 with (non-totally geodesic) parallel second fundamental form are explicitly expressed as a Lagrangian deformation of a real form which is a canonicallyembedded symmetric R-spaces of type U (r).

M

L

Einstein

λ1

H-stablestableGp,q (C), p ≤ q

Gp,q (R)

Yes

1
2

Yes

No

Yes

Yes

G2p,2q (C), p ≤ q

Gp,q (H)Yes

1
2

Gm,m (C)

U (m)

No

1
2

Yes

No

SO(2m)/U (m)SO(m), m ≥ 5

Yes

1
2

Yes

No

SO(4m)/U (2m), m ≥ 3

U (2m)/Sp(m)

No

m
4m−2

No

No

Sp(2m)/U (2m)Sp(m), m ≥ 2

Yes

1
2

Yes

Yes

Sp(m)/U (m)U (m)/O(m)

No

Yes

No

Qp+q−2 (C), q − p ≥ 3

Qp,q (R), p ≥ 2

No

1
2
p
p+q−2

No

No

Qp+q−2 (C), 0 ≤ q − p < 3

Qp,q (R), p ≥ 2

No

1
2

Yes

No

Qq−1 (C), q ≥ 3

Q1,q (R)

Yes

1
2

Yes

Yes

Yes

Yes

E6 /T Spin(10)P2 (K)Yes

1
2

E6 /T Spin(10)G2,2 (H)/Z2

Yes

1
2

Yes

No

E7 /T E6

SU (8)/Sp(4)Z2

Yes

1
2

Yes

No

E7 /T · E6

T · E6 /F4

No

1
6

No

No

Podestá [44] classiﬁed compact homogeneous Lagrangian submanifolds L of positive Euler characteristic in complex Grassmann manifolds Grk (C) and he showedthat they are totally geodesic Lagrangian submanifolds L = Grp (Rn ) ⊂ Grp (Cn )with p(n − p) even, L = Grp (Hm ) ⊂ Gr2p (C2m ), L = Gr2 (R8 ) ∼= SO(7)/U (3) · Z2 ⊂Gr2 (C8 ) and L = Gr2 (R7 ) ∼= G2 /U (2) · Z2 ⊂ Gr2 (C7 ).

5

Lagrangian submanifolds in complex hyperquadrics and hypersurface
geometry in S n+1 (1)Next we shall discuss Lagrangian submanifolds in complex hyperquadricsf 2 (Rn+2 ) ∼Gr= Qn (C) ∼= SO(n + 2)/SO(2) × SO(n), which is a compact irreducible15

f 2 (Rn+2 ) denotes the real Grassmann
Hermitian symmetric space of rank 2. Here Gr
n+2
manifold of oriented 2-planes in R
and Qn (C) a complex hypersurface of CP n+1
2
2
2
deﬁned by z0 + z1 + · · · + zn+1 = 0.
Let N n ⊂ S n+1 (1) ⊂ Rn+2 be an oriented hypersurface immersed in the unit
sphere. Now we denote by x its position vector of points p of N n and by n the unit
normal vector ﬁeld to N n in S n+1 (1). Then the Gauss map is deﬁned as
G : N n ∋ p 7−→ x(p) ∧ n(p) ∼= [x(p) +√f 2 (Rn+2 ) ∼−1n(p)] ∈ Gr
= Qn (C).

Here [x(p) ∧ n(p)] denotes an oriented 2-plane in Rn+2 spanned by two vectors x(p)and n(p). Then G is always a Lagrangian immersion.
The mean curvature form formula was shown by [42] as follows:
( n
))
)
( (
n
∑
∏
√
αH = −d,arc cotκi = d Im log (1 + −1κi )
i=1

i=1

where κi (i = 1, · · · , n) denotes the principal curvatures of N n ⊂ S n+1 (1).
Now suppose that N n ⊂ S n+1 (1) is a compact oriented hypersurface with constant principal curvatures, so called isoparametric hypersurface. Then by the above
mean curvature form formula the Gauss map G : N n → Qn (C) is a minimal Lagrangian immersion. By Münzner’s famous result ([31], [32]), the number g ofdistinct principal curvatures must be g = 1, 2, 3, 4, 6. Then the image of the Gauss
map G : N n −→ Qn (C) is a compact embedded minimal Lagrangian submanifold
L = G(N n )(∼= N n /Zg ) ⊂ Qn (C).
See [40] for the details. By Münzner’s results ([31]) we know that 2n/g = m1 + m2
if g is even and 2n/g = 2m1 if g is odd.
Note that g = 1 or 2 if and only if G(N n ) is a totally geodesic Lagrangiansubmanifold in Qn (C).
Theorem 5.1 ([25], [40]). L = G(N n ) is a monotone and cyclic Lagrangian submaifold in Qn (C) with minimal Maslov number 2n/g. Moreover, L = G(N n ) isorientable (resp. non-orientable) if and only if 2n/g is an even (resp. odd) integer.
Recently H. Li, H. Ma and G. Wei constructed a class of non-isoparametricand non-minimal compact oriented rotational hypersurfaces whose Gauss maps areminimal Lagrangian immersions into complex hyperquadrics for n ≥ 3 ([23])Classification Theory of isoparametric hypersurfaces of the standard
sphere: All isoparametric hypersurfaces in S n+1 (1) are classiﬁed into
• Homogeneous ones (Hsiang-Lawson [19], R. Takagi-T. Takahashi [46]) can beobtained as principal orbits of the isotropy representations of Riemanniansymmetric pairs (U, K) of rank 2.

16

– g = 1 : N n = S n , a great or small sphere.
– g = 2, N n = S m1 × S m2 , (n = m1 + m2 , 1 ≤ m1 ≤ m2 ), the Cliﬀord
hypersurfaces.
Sp(3)F4– g = 3, N n is homogeneous, N n = ZSO(3), SUT (3), Sp(1)
2
3 , Spin(8)2 +Z2(E. Cartan).

– g = 6: Only homogeneous examples are known now.
∗ g = 6, m1 = m2 = 1: homogeneous (Dorfmeister-Neher,
R. Miyaoka [27]).
∗ g = 6, m1 = m2 = 2: homogeneous (R. Miyaoka [28]).
• Non-homogenous isoparametric hypersurfaces in the standard sphere werediscovered ﬁrst by H. Ozeki and M. Takeuchi ([41]) and generalized by D.
Ferus, H. Karcher and H. F. Münzner ([14]). They can be constructed by the
representations of Cliﬀord algebras (isoparametric hypersurfaces of OT-FKMtype).
– g = 4: except for (m1 , m2 ) = (7, 8), either homogeneous or OT-FKM
type (Cecil-Chi-Jensen [11], Immervoll [20], Chi [12, 13]).

6

Classification of compact homogeneous Lagrangian submanifolds incomplex hyperquadrics

In [24] we classiﬁed compact homogeneous Lagrangian submanifolds in complex
hyperquadrics. Here we brieﬂy explain our classiﬁcation theory.
First we observed that N n ⊂ S n+1 (1) is homogeneous (i.e. an orbit of a compact
Lie subgroup K ⊂ SO(n + 2)) if and only if G(N n ) is homogeneous in Qn (C) ([24,p.759, Proposition 3.1]). By Hsiang-Lawson [19] and Takagi-Takahashi [46], weknow that all homogeneous isoparametric hypersurfaces N n ⊂ S n+1 (1) can beobtained as principal orbits of isotropy representation of Riemannian symmetricpairs (U, K) of rank 2. We should notice that the group action by K of (U, K) is
the maximal group action of cohomogeneity 1 on the standard sphere ([19]).
Let g = k+p be the canonical decomposition as a symmetric Lie algebra and a be
a maximal Abelian subspace of p. For each regular element H of a∩S n+1 (1), we havea homogeneous isoparametric hypersurface in the unit sphere N n := (AdK)H ⊂f 2 (p) ∼S n+1 (1) ⊂ Rn+2 ∼= p. Its Gauss image is G(N n ) = (AdK)[a] ⊂ Gr
= Qn (C).
Then the canonical moment map µ̃ of the action of K on Qn (C) induced by theadjoint action of K on p is given as follows :√f 2 (p) ∋ [a + −1b] = [W ] 7−→ −[a, b] ∈ k ∼µ̃ : Qn (C) ∼
= k∗
= Gr
where {a, b} is an orthonormal basis of W ⊂ p compatible with its orientation.
Hence we obtain
G(N n ) = µ̃−1 (0) .

17

g

Type

(U, K)

dim N nm1 , m2

N n = K/K0

1

S1×

(S 1 × SO(n + 2), SO(n + 1))

n

n

Sn

BDII

n ≥ 1, [R ⊕ A1 ]BDII×(SO(p + 2) × SO(n + 2 − p),np, n − p

S p × S n−p

BDIISO(p + 1) × SO(n + 1 − p))
SO(3)Z2 +Z2
SU (3)T2
Sp(3)
Sp(1)32

1 ≤ p ≤ n − 1, [A1 ⊕ A1 ]3

AI2

(SU (3), SO(3)) [A2 ]

3

1, 1

3

a2

(SU (3) × SU (3), SU (3)) [A2 ]

6

2, 2

3

AII2

(SU (6), Sp(3)) [A2 ]

12

4, 4

3

EIV

(E6 , F4 ) [A2 ]

24

8, 8

4

b2

(SO(5) × SO(5), SO(5)) [B2 ]

8

2, 2

4

AIII2

(SU (m + 2), S(U (2) × U (m)))

4m − 22,m ≥ 2, [BC2 ](m ≥ 3), [B2 ](m = 2)
4

BDI2

(SO(m + 2), SO(2) × SO(m))

2m − 32m − 2m ≥ 3, [B2 ]
4

CII2

F4
Spin(8)SO(5)T2S(U (2)×U (m))
S(U (1)×U (1)×U (m−2))

1,

SO(2)×SO(m)Z2 ×SO(m−2)

m−2(Sp(m + 2), Sp(2) × Sp(m))

8m − 2m ≥ 2, [BC2 ](m ≥ 3), [B2 ](m = 2)

4,

Sp(2)×Sp(m)Sp(1)×Sp(1)×Sp(m−2)

4m − 5

4

DIII2(SO(10), U (5)) [BC2 ]

18

4, 5

4

EIII

(E6 , U (1) · Spin(10)) [BC2 ]

30

6, 9

6

g2

(G2 × G2 , G2 ) [G2 ]

12

2, 2

6

G

(G2 , SO(4)) [G2 ]

6

1, 1

U (5)
SU (2)×SU (2)×U (1)U (1)·Spin(10)S 1 ·Spin(6)
G2
T2
SO(4)
Z2 +Z2

Suppose that L is a compact homogeneous Lagrangian submanifold in Qn (C),which is obtained as L = K ′ · [V0 ] for a compact connected Lie subgroup G ofSO(n + 2). Then one can show that there is v ∈ S n+1 (1) such that N n = K ′ ·v ⊂ S n+1 (1) is a homogeneous isoparametric hypersurface in S n+1 (1). By HsiangLawsonś theorem [19] there is a compact Riemannian symmetric pair (U, K) of rank
2 with connected compact K and the symmetric Lie algebra u = k + p such thatp = Rn+2 , K ′ ⊂ Adp (K) and N n = Adp (K)v. By using the moment map argument
and some results from the complete classiﬁcation of cohomogeneity 1 compact group
actions on spheres due to Tohl Asoh [6], we showed that L = K ′ · [V0 ] = K · [V0 ].
Notice that K · [V0 ] = µ̃−1 (η) for some η ∈ c(k), Hence by Lemma 3.1 we obtain
Theorem 6.1 ([24]). Any compact homogeneous minimal Lagrangian submanifold

18

Ln in Qn (C) is the Gauss image G(N n ) of a compact homogeneous isoparametric
hypersurface N n in S n+1 (1).
Moreover, by the Lie algebraic and the moment map arguments, we showed
Lemma 6.2 (([24])).

(1) If (U, K) is (i) (S 1 × SO(3), SO(2)), (ii) (SO(3) ×SO(3), SO(2) × SO(2)), (iii) (SO(3) × SO(n + 1), SO(2) × SO(n)) or (iv)
(SO(m + 2), SO(2) × SO(m)) (n = 2m − 2), then L = µ−1 (ξ) ⊂ Qn (C) forsome ξ ∈ c(k) ∩ Im(µ) ̸= {0}. In this case we have non-trivial families of
Lagrangian orbits in Qn (C).
(2) Otherwise, c(k) ∩ Im(µ) = {0} and L = G(N n ) = µ−1 (0) ⊂ Qn (C), which isa minimal Lagrangian submanifold in Qn (C).
The non-trivial families of Lagrangian orbits in Qn (C) in each case of (1) are
explicitly described as follows:(i) If (U, K) is (S 1 × SO(3), SO(2)), then L is a small or great circle in Q1 (C) ∼=S2.
(ii) If (U, K) is (SO(3) × SO(3), SO(2) × SO(2)), then L is a product of small or
great circles of S 2 in Q2 (C) ∼= S2 × S2.
(iii) If (U, K) is (SO(3) × SO(n + 1), SO(2) × SO(n)) (n ≥ 3), then√
L = K · [Wλ ] ⊂ Qn (C) for some λ ∈ S 1 \ {± −1},
where K · [Wλ ] (λ ∈ S 1 ) is the S 1 -family of Lagrangian or isotropic K-orbits
satisfying(a) K · [W1 ] = K · [W−1 ] = G(N n ) is a totally geodesic Lagrangian submanifold in Qn (C).
√(b) For each λ ∈ S 1 \ {± −1},
K · [Wλ ] ∼= (S 1 × S n−1 )/Z2 ∼= Q2,n (R)is an H-minimal Lagrangian submanifold in Qn (C) with ∇S = 0 andthus ∇αH = 0.
(c) K · [W±√−1 ] are isotropic
dim K · [W±√−1 ] = 0 (points !).

submanifoldsinQn (C)

with(iv) If (U, K) is (SO(m + 2), SO(2) × SO(m)) (n = 2m − 2), then√
L = K · [Wλ ] ⊂ Qn (C) for some λ ∈ S 1 \ {± −1},
where K · [Wλ ] (λ ∈ S 1 ) is the S 1 -family of Lagrangian or isotropic orbits
satisfying

19

(a) K · [W1 ] = K · [W−1 ] = G(N n ) is a minimal (NOT totally geodesic)
Lagrangian submanifold in Qn (C).
√(b) For each λ ∈ S 1 \ {± −1},
K · [Wλ ] ∼= (SO(2) × SO(m))/(Z2 × Z4 × SO(m − 2))is an H-minimal Lagrangian submanifold in Qn (C) with ∇S ̸= 0 and∇αH = 0.
(c) K · [W±√−1 ] ∼= SO(m)/S(O(1) × O(m − 1)) ∼= RP m−1 are isotropic√
submanifolds in Qn (C) with dim K · [W± −1 ] = m − 1.

7

Hamiltonian stability of the Gauss images of homogeneous isoparametic hypersurfaces.

Suppose that N n is a compact isoparametric hypersurface embedded in S n+1 (1).
Palmer ([42]) showed that its Gauss map G : N n −→ Qn (C) is Hamiltonian stable
if and only if N n = S n ⊂ S n+1 (1) (g = 1).
Problem. Investigate the Hamiltonian stability of its Gauss image L = G(N n ) ∼=N n /Zg embedded in Qn (C) as a compact minimal Lagrangian submanifold.
In the case g = 1, N n = S n is a great or small sphere and G(N n ) ∼= S n isstrictly Hamiltonian stable. More strongly, it is stable as a minimal submanifold([48]). In the case when n is even, it is real homologically volume-minimizing
because it is a calibrated submanifold by an invariant n-form (Gluck, Morgan and
Ziller [15]). It is impossible to be calibrated in the case when n is odd, because
H n (Qn (C); R) = {0}. In the case when n is even, H n (Qn (C), R) ∼= R. In thecase when n is odd, πn (Qn (C)) ∼= {0} for any F . The= Z2 , and Hn (Qn (C), F ) ∼recent result of [21] implies that the totally geodesic Lagrangian submanifold S n ⊂f 2 (Rn+2 ) is Hamiltonian volume minimizing for general n.
Qn (C) ∼= Gr
In the case g = 2, L is not Hamiltonian stable if and only if m2 − m1 ≥ 3, L is
Hamiltonian stable but not strictly Hamiltonian stable if and only if m2 − m1 = 2,
L is strictly Hamiltonian stable if and only if m2 − m1 < 2.
Notice that g = 1 or g = 2 if and only if the Gauss image L = G(N n ) is a totally
geodesic Lagrangian submanifold, that is, a real form Qm1 +1,m2 +1 (R) of Qn (C).
In the case g = 3, L = G(N n ) is strictly Hamiltonian stable (H. Ma - O.[24]).
Remark 7.1. In the case when g = 3, the invariant metric on G(N n ) induced by Gfrom Qn (C) is a normal homogeneous metric ([24]) and hence G(N n ) is a compact
minimal Lagrangian submanifold embedded in Qn (C) with nonnegative sectional
curvatures but ∇S ̸= 0 (compare with Theorem 4.1).

20

Theorem 7.2 (Hui Ma-O.[26]). Suppose that g = 6 and thus L = SO(4)/(Z2 +
Z2 ) · Z6 (m1 = m2 = 1) or L = G2 /T 2 · Z6 (m1 = m2 = 2) is homogeneous. Then
L is strictly Hamiltonian stable.
In the case g = 4, There are homogeneous case and non-homogeneous cace as
we mentioned (Ozeki-Takeuchi, Ferus-Karcher-Münzner, Cecil-Chi-Jensen, Immervoll).
Theorem 7.3 (Hui Ma and O. [26]). Suppose that g = 4 and L = G(N n ) is
homogeneous. Then
(1) L = SO(5)/T 2 · Z4 (m1 = m2 = 2) is strictly Hamiltonian stable.
(2) L = U (5)/(SU (2) × SU (2) × U (1)) · Z4 (m1 = 4, m2 = 5) is strictly Hamiltonian stable.
(3) L = (SO(2) × SO(m))/(Z2 × SO(m − 2)) · Z4 (m1 = 1, m2 = m − 2, m ≥ 3) If
m2 − m1 ≥ 3, then L is NOT Hamiltonian stable. If m2 − m1 = 2, then L is
Hamiltonian stable but not strictly Hamiltonian stable. If m2 − m1 = 1 or 0,then L is strictly Hamiltonian stable.
(4) L = S(U (2) × U (m))/S(U (1) × U (1) × U (m − 2))) · Z4 (m1 = 2, m2 = 2m −3, m ≥ 2) If m2 − m1 ≥ 3, then L is NOT Hamiltonian stable. If m2 − m1 =1 or − 1, then L is strictly Hamiltonian stable.
(5) L = Sp(2) × Sp(m)/(Sp(1) × Sp(1) × Sp(m − 2))) · Z4 (m1 = 4, m2 = 4m −5, m ≥ 2). If m2 − m1 ≥ 3, then L is NOT Hamiltonian stable. If m2 − m1 =−1, then L is strictly Hamiltonian stable.
(6) Suppose that g = 4 is homogeneous and
L = U (1) · Spin(10)/(S 1 · Spin(6)) · Z4 (m1 = 6, m2 = 9, thus m2 − m1 = 3!)
Then L is strictly Hamiltonian stable!
Theorem 7.4 (Hui Ma-O.[26]). Suppose that (U, K) is not of type EIII, that is,(U, K) ̸= (E6 , U (1) · Spin(10)). Then L = G(N ) is NOT Hamiltonian stable if
and only if |m2 − m1 | ≥ 3. Moreover if (U, K) is of type EIII, that is, (U, K) =21

(E6 , U (1) · Spin(10)), then (m1 , m2 ) = (6, 9) but L = G(N ) is strictly Hamiltonian
stable.

8

Related Questions and Further Problems.

8.1

Questions

1. Investigate the Hamiltonian stability and other properties of the Gauss images of compact non-homogenous isoparametric hypersurfaces, particular OTFKM type, embedded in spheres with g = 4.

2. Investigate the relation between our Gauss image construction and KarigiannisMin-Oo’s results.

3. Are there similar constructions of Lagrangian submanifolds in compact Hermitian symmetric spaces other than CP n , Qn (C)? What about in the casesof Gr2 (C), Grr (C)?
8.2

Cohomogeneity 1 special Lagrangian submanifolds in tangent bun-dle over the standard sphere
Let T 1 S n+1 ∼= V2 (Rn+2 ) denote the unit tangent bundle of the standard sphere
n+1
S. The cones over Legendrian lifts of isoparametric hypersurfaces with g distinct principal curvatures to T 1 S n+1 ∼= V2 (Rn+2 ) provide fundamental examples of
special Lagrangian cones in the (non-ﬂat) Ricci-ﬂat Kähler cone. In the cases ofg = 1, 2, Kaname Hashimoto (OCU, D3) and Takashi Sakai (TMU) ([18]) classiﬁed
all cohomogeneity 1 special Lagrangian submanifolds in the tangent bundle T S n+1
with respect to the Stenzel metric deformed from such special Lagrangian cones.
It is an interesting problem to investigate such cohomogeneity 1 special Lagrangian submanifolds in the cases of g = 3, 4, 6.
8.3

Extension to the semi-Riemannian case

It is another interesting problem to study an extension of our Gauss map construction to the semi-Riemannian case.
More recently, my new Ph. D. student, Harunobu Sakurai (OCU, D1), studies
an extension of Lagrangian propery of the Gauss map and the mean curvatureform formula to oriented semi-Riemannian hypersurfaces in semi-Riemannain space
forms. From J. Hahn’s work ([16], [17]), we have many interesting examples ofminimal Lagrangian submanifolds in semi-Riemannian real Grassmann manifolds
of oriented 2-planes .

22

Acknowledgement. This article is based on my lectures at the 14th International
Workshop on Diﬀerential Geometry and Related Fields and the 4th KNUGRGOCAMI Joint Diﬀerential Geometry Workshop at Kyungpook National University
in Korea, on November 2-5, 2011. The author would like to thank Professor YoungJin Suh and Dr. Hyunjin Lee for their kind hospitality and and excellent organization. The author also would like to thank Doctor Shintaro Kuroki (OCAMI),
Professors Takashi Sakai (Tokyo Metropolitan University), Hiroshi Iriyeh (Tokyo
Denki University) and Makiko Sum Tanaka (Tokyo University of Science) for useful
discussion and valuable suggestion.

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23

[10] S. Chang, On Hamiltonian stable minimal Lagrangian surfaces in CP 2 , The
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