DIFFERENTIAL GEOMETRY OF
LAGRANGIAN SUBMANIFOLDS AND
HAMILTONIAN VARIATIONAL PROBLEMSHUI MA AND YOSHIHIRO OHNITA

Abstract. In this article we shall provide a survey on our recent works([25],[26]) and their environs on diﬀerential geometry of Lagrangian submanifolds in speciﬁc symplectic Kähler manifolds, such as complex projective spaces, complex space forms, Hermitian symmetric spaces and Kähler
C-spaces. We shall discuss (1) Hamiltonian minimality and Hamiltonian
stability of Lagrangian submanifolds in Hamiltonian volume minimizing
problem, (2) classiﬁcation problem of homogeneous Lagrangian submanifolds from the viewpoint of Lagrangian orbits and moment maps, (3) tightness problem of Lagrangian submanifolds. Moreover we shall give attention
to Lagrangian submanifolds in complex hyperquadrics, which are compact
Hermitian symmetric spaces of rank 2. The relationship between certain
minimal Lagrangian submanifold in complex hyperquadrics and isoparametric hypersurfaces in spheres will be emphasized. Recently we gave acomplete classiﬁcation of compact homogeneous Lagrangian submanifolds
in complex hyperquadrics and we determined the Hamiltonian stabilityof ALL compact minimal Lagrangian submanifolds embedded in complex
hyperquadrics which are obtained as the Gauss images of homogeneous
isoparametric hypersurfaces in spheres.

Introduction
A Lagrangian submanifold L immersed in a symplectic manifold (M 2n , ω) is
an n-dimensional submanifold on which the pull-back of the symplectic form
ω vanishes. The study of Lagrangian submanifolds L in Kähler manifolds(M 2n , ω, J, g) is a fruitful area in diﬀerential geometry of submanifolds. Fromvarious viewpoints of Riemannian geometry and symplectic geometry, there
appear many interesting works on Lagrangian submanifolds in speciﬁc Kähler
manifolds such as complex space forms, Hermitian symmetric spaces, generalized ﬂag manifolds and so on. Throughout this article, we treat compact
immersed or embedded Lagrangian submanifolds without boundary.
In this article we shall explain our recent works on Lagrangian submanifolds
in complex hyperquadrics M 2n = Qn (C) and their environs. The complex
hyperquadric M 2n = Qn (C) is a compact Hermitian symmetric space of rank
2000 Mathematics Subject Classiﬁcation. Primary 53C42, Secondary 53C40, 53D12.
Key words and phrases. Lagrangian submanifolds, Hamiltonian stability, minimal submanifolds, isoparametric hypersurfaces.
The ﬁrst named author is partially supported by NSFC grant No. 10971111 and NKBRPC
No. 2006CB805905. The second named author is partially supported by JSPS Grant-in-Aid
for Scientiﬁc Research (A) No. 19204006 and Osaka City University Priority Research on“Mathematical knots and wide-angle evolution to scientiﬁc objects”.
1

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H. MA AND Y. OHNITA

2. There is a relationship between Lagrangian geometry in the complex hyperquadrics Qn (C) and hypersurface geometry in the unit standard sphereS n+1 (1). Via the “Gauss maps ”oriented hypersurfaces in S n+1 (1) give Lagrangian submanifolds immersed in Qn (C). Especially the Gauss images oforiented hypersurfaces with constant principal curvatures, so called “isoparametric hypersurfaces”, in S n+1 (1) provide a nice class of compact minimal
Lagrangian submanifolds embedded in Qn (C). By using the results of isoparametric hypersurface theory, we shall discuss the properties of such Lagrangiansubmanifolds, a classiﬁcation of compact homogeneous Lagrangian submanifolds and the Hamiltonian stability/instability of the Gauss images of homogeneous isoparametric hypersurfaces, in the complex hyperquadrics.
This article is based on the talk given by the second named author at “AHarmonic Map Fest”in honor of Professor John C. Wood on the occasion ofhis 60th birthday, Cagliari in Italy, September 7-10, 2009. The second named
author would like to thank the organizers of the conference, especially Professor
Stefano Mondaldo and Eric Loubeau, for their excellent organization and kind
hospitality.
1. Lagrangian submanifolds in symplectic manifolds
and Hamiltonian deformations
Let (M, ω) be a 2n-dimensional symplectic manifold with a symplectic form
ω. A Lagrangian immersion φ : L −→ M is a smooth immersion of an ndimensional (maximal dimensional) smooth manifold L into M satisfying thecondition φ∗ ω = 0. For a Lagrangian immersion φ : L −→ M , by the nondegeneracy of ω the natural linear bundle map φ−1 T M/φ∗ T L ∋ v 7→ αv :=ω(v, φ∗ (·)) ∈ T ∗ L becomes a linear bundle isomorphism and thus we havea linear isomorphism C ∞ (φ−1 T M/φ∗ T L) → Ω1 (L). Here Ωi (L) denotes thevector space of smooth i-forms on L.
A Lagrangian deformation is deﬁned as a one-parameter smooth family of
Lagrangian immersions φt : L −→ M with φ = φ0 . Let αVt be the 1-form on L
corresponding to its variational vector ﬁeld Vt := ∂φt /∂t ∈ C ∞ (φ−1t T M ). The
Lagrangian deformation is characterized by the condition that for each t, αVt
is closed, i.e., αVt ∈ Z 1 (L). Furthermore, if αVt is exact, i.e., αVt ∈ B 1 (L), foreach t, then {φt } is called a Hamiltonian deformation of φ = φ0 . Here Z i (L)denotes the vector space of smooth closed i-forms on L and B i (L) denotes thevector space of smooth exact i-forms on L.
There is a characterization of a Hamiltonian deformation in terms of “isomonodromy deformation”as follows (cf. [25], [44]). Suppose that [(1/2π)ω] ∈H 2 (M, R) is an integral cohomology class. Then there is a complex line √bundleL over M with a U (1)-connection ∇ whose curvature coincides with −1ω.
Let φt : L −→ M be a Lagrangian deformation. For each t, we take the pull−1
back complex line bundle φ−1t L over L with the pull-back connection φt ∇through φt and thus we have a family of ﬂat connections {φ−1t ∇}. Then a
Lagrangian deformation {φt } is a Hamiltonian deformation if and only if afamily of ﬂat connections {φ−1t ∇} have the same holonomy homomorphism
π1 (L) −→ U (1).

DIFFERENTIAL GEOMETRY OF LAGRANGIAN SUBMANIFOLDS

3

A fundamental fact on the relationship of Lagrangian orbits and moment
maps of the Hamiltonian group action on a symplectic manifold is as follows :any Lagrangian orbit of Hamiltonian group action G on a symplectic manifold(M, ω) with moment map µ appears as components of the level set µ−1 (α) forsome α ∈ z(g∗ ), where g∗ is the dual space of Lie algebra g of G andz(g∗ ) := {α ∈ g∗ | Ad∗ (a)α = α for all a ∈ G}.
If M and G are compact and connected, then each Lagrangian orbit coincides
with the level set µ−1 (α) for some α ∈ z(g∗ ) ∼= c(g), the center of g.
Here we mention about elementary examples of the moment maps related
to Hermitian symmetric spaces (cf. [42]). Suppose that (U, K) is an Hermitiansymmetric pair of compact type or noncompact type. Let u = k + p be thecanonical decomposition of the Hermitian symmetric Lie algebra of (U, K) andthere is Z ∈ c(k) such that ad(Z)|p corresponds with the standard complex
structure J on p invariant under the linear isotropy action of K. The momentmap µ̃ : p → k∗ ∼= k of the isotropy representation of K on p is given by theformulaµ̃(p) − µ̃(0) = (ad(p))2 (Z)
(1.1)
∗ ∼
for each p ∈ p, where µ̃(0) ∈ c(k). The moment map µ : U/K → k = k of theisotropy action of K on U/K is given by⟨µ(aK), ξ⟩ − ⟨µ(eK), ξ⟩ = ⟨(Ad(a) − 1)Z, ξ⟩(1.2)

for each ξ ∈ k and each aK ∈ U/K, where µ(eK) ∈ c(k). If we deﬁne µ asµ(eK) := Z, then⟨µ(aK), ξ⟩ = ⟨Ad(a)Z, ξ⟩for each ξ ∈ k. Using the projection πk : u → k with respect to u = k + p, we
can express µ asµ(aK) − µ(eK) = πk ◦ (Ad(a) − 1)(Z) = πk (Ad(a)(Z)) − Z
for each aK ∈ U/K.
2. Lagrangian submanifolds in Kähler manifolds
2.1. Hamiltonian minimality and Hamiltonian stability. Let (M, ω, J, g)
be a Kähler manifold with complex structure J and Kähler metric g andφ : L −→ M be a Lagrangian immersion. Let B and A denote the second fundamental form and the shape operator of submanifold L in (M, g). Let
H denote the mean curvature vector ﬁeld of φ and the corresponding 1-form
αH ∈ Ω(L) is called the mean curvature form of φ. Submanifolds with vanishing mean curvature vector ﬁeld H = 0 are called minimal submanifolds inRiemannian geometry. It is known ([13]) that the mean curvature form of a
Lagrangian immersion always satisﬁes the identitydαH = φ∗ ρM ,(2.1)

where ρM denotes the Ricci form of M deﬁned by ρM (X, Y ) = RicM (JX, Y )and RicM denotes the Ricci tensor ﬁeld of (M, ω, J, g). It follows from the Codazzi equation of Riemannian submanifolds. Thus if (M, ω, J, g) is an EinsteinKähler manifold, then αH is closed, i. e. αH ∈ Z 1 (L).

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H. MA AND Y. OHNITA

The notions of Hamiltonian minimality and Hamiltonian stability were introduced and investigated ﬁrst by Y. G. Oh ([35], [37]).
Definition 2.1. A Lagrangian immersion φ : L → M is called Hamiltonian minimal, shortly H-minimal, or Hamiltonian stationary if it has extremal
volume under every Hamiltonian deformation of φ.
The Euler-Lagrange equation of this variational problem isδαH = 0,(2.2)

where δ denotes the co-diﬀerential operator with respect to the induced metricφ∗ g on L.
Definition 2.2. A Hamiltonian minimal Lagrangian immersion φ : L → M
is called Hamiltonian stable, shortly H-stable if the second variation of thevolume is nonnegative under every Hamiltonian deformation of φ.
A Lagrangian immersion φ : L → M is called Hamiltonian volume minimizing, or globally Hamiltonian stable, shortly globally H-stable, if φ has minimum
volume under every Hamiltonian deformation of φ.
The Hamiltonian version of the second variational formula is given as follows([37]) : Suppose that φ is an H-minimal Lagrangian immersion. For each
Hamiltonian deformation φt : L → M with φ0 = φ and Vt = ∂φt /∂t ∈C ∞ (φ−1
t T M ),d2
Vol (L, φ∗t g)|t=0
2
dt
∫
)
( 1
=
⟨∆L α, α⟩ − ⟨R(α), α⟩ − 2⟨α ⊗ α ⊗ αH , S⟩ + ⟨αH , α⟩2 dv,

(2.3)

L

where dv denotes the Riemannian measure of φ∗ g and ∆1L denotes the Hodgede Rham Laplace operator of (L, φ∗ g) acting on the vector space Ω1 (L) of
smooth 1-forms on L and α := αV0 ∈ B 1 (L). Here,⟨R(α), α⟩ :=n∑

RicM (ei , ej )α(ei )α(ej ),i,j=1

where {ei } is a local orthonormal frame on L andS(X, Y, Z) := ω(B(X, Y ), Z) = g(JB(X, Y ), Z)(2.4)

for each X, Y, Z ∈ C ∞ (T L), which is a symmetric 3-tensor ﬁeld on L deﬁned by
the second fundamental form B of submanifold L in M . The Jacobi diﬀerential
operator or second variational operator of a Hamiltonian minimal Lagrangian
immersion φ is the self-adjoint linear diﬀerential operator Jφ : B 1 (L) → B 1 (L)deﬁned by
∫
d2
∗
(2.5)
Vol (L, φt g)|t=0 = ⟨Jφ (α), α⟩dv,
dt2
L
where
(2.6)
Jφ (α) = ∆1L α − R(α) − 2A∗H (α) + ⟨αH , α⟩αH .

DIFFERENTIAL GEOMETRY OF LAGRANGIAN SUBMANIFOLDS

5

Note that A∗H (α) = α ◦ AH . The linearized Hamiltonian minimal Lagrangian
equation is the linear diﬀerential equationJφ (α) = ∆1L α − R(α) − 2A∗H (α) + ⟨αH , α⟩αH = 0(2.7)

for α ∈ B 1 (L).
For an H-minimal Lagrangian immersion φ : L → M , denote by E0 (φ) :=Ker(Jφ ) the null space of the second variation on B 1 (L), or equivalently, thevector space of all solutions to the linearized Hamiltonian minimal Lagrangian
equation (2.7), and then the dimension n(φ) := dim(E0 (φ)) of E0 (φ) is called
the nullity of φ.
Suppose that X is a holomorphic Killing vector ﬁeld deﬁned on M . Thenthe corresponding 1-form αX := ω(X, ·) on M is closed. If H 1 (M, R) = {0},then αX = ω(X, ·) is exact, i.e. X is a Hamiltonian vector ﬁeld on M .
Suppose that M is simply connected, more generally H 1 (M, R) = {0}. Theneach holomorphic Killing vector ﬁeld X of M generates a volume-preserving
Hamiltonian deformation of φ and thus{φ∗ αX | X is a holomorphic Killing vector ﬁeld on M } ⊂ E0 (φ) ⊂ B 1 (L).
Set nhk (φ) := dim{φ∗ αX | X is a holomorphic Killing vector ﬁeld on M }, whichis called the holomorphic Killing nullity of φ. Such a Hamiltonian deformationof φ is called trivial.
Definition 2.3. Assume that φ is an H-minimal Lagrangian immersion. Then
we call φ strictly Hamiltonian stable if the following two conditions are satisﬁed:(i) φ is Hamiltonian stable.
(ii) The null space of the second variation on Hamiltonian deformations
coincides with the vector subspace consisting of inﬁnitesimal deformations induced by trivial Hamiltonian deformations of φ. Namely,
nhk (φ) = n(φ).
Note that if L is strictly Hamiltonian stable, then L has an isolated local
minimum volume, up to the congruence, under every Hamiltonian deformation.
An H-minimal Lagrangian immersion φ is called Hamiltonian rigid if nhk (φ) =n(φ) (cf. Yng-Ing Lee [23]).
Definition 2.4. Assume that (M, ω, J, g) is a Kähler manifold and G is ananalytic subgroup of its automorphism group Aut(M, ω, J, g). We call a Lagrangian orbit L = G · x ⊂ M of G a homogeneous Lagrangian submanifold of
M.
Then the following is an easy but useful observation.
Proposition 2.1. Any compact homogeneous Lagrangian submanifold in aKähler manifold is always Hamiltonian minimal.
Proof. Since αH is an invariant 1-form on L, δαH is a constant function on L.
Hence by the divergence theorem we obtain δαH = 0.
SetG̃ := {a ∈ Aut(M, ω, J, g) | a(L) = L}.

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H. MA AND Y. OHNITA

Then G ⊂ G̃ and G̃ is the maximal subgroup of Aut(M, ω, J, g) preserving L.
Moreover we have nhk (φ) = dim(Aut(M, ω, J, g)) − dim(G̃).
2.2. First eigenvalue of minimal Lagrangian submanifolds in EinsteinKähler manifolds. In the case of minimal Lagrangian submanifolds in EinsteinKähler manifolds, by the second variational formula (2.3), the Hamiltonian
stability condition is simpliﬁed as follows :Corollary 2.1 (B. Y. Chen - P. F. Leung - T. Nagano [11], Y. G. Oh [35]).
Assume M is an Einstein-Kähler manifold with Einstein constant κ and φ :L → M is a minimal Lagrangian immersion of a compact smooth manifold L
into M (i.e. αH ≡ 0). Then L is Hamiltonian stable if and only if
λ1 ≥ κ ,where λ1 denotes the ﬁrst (positive) eigenvalue of the Laplacian of L acting on
C ∞ (L).
On the other hand, there is an upper bound of the ﬁrst eigenvalue λ1 in thefollowing homogeneous Einstein-Kähler manifold case:
Theorem 2.1 ([45], [46], [5]). Assume that M is a compact homogeneous
Einstein-Kähler manifold with Einstein constant κ > 0. Let L ,→ M be acompact minimal Lagrangian submanifold immersed in M . Thenλ1 ≤ κ .
It is a natural question what compact minimal Lagrangian submanifolds
attain its equality. Combining it with Corollary 2.1, in this case we get a variational characterization of such minimal Lagrangian submanifolds as follows :L is Hamiltonian stable if and only if λ1 = κ.
2.3. Examples of Hamiltonian stable Lagrangian submanifolds.
Question. What compact Lagrangian submanifolds in a Kähler manifold is a
Hamiltonian stable H-minimal Lagrangian submanifold ?
Example 2.1.
(1) Circles on a plane S 1 ⊂ R2 ∼= C, great circles and11
2 ∼small circles on a sphere S ⊂ S = CP , closed circles on a hyperbolic
plane S 1 ⊂ H 2 ∼= CH 1 are elementary examples of compact strictly
Hamiltonian stable H-minimal Lagrangian submanifolds.
(2) The real projective space embedded in the complex projective space
RP n ⊂ CP n as a totally geodesic Lagrangian submanifold is strictly
Hamiltonian stable ([35]). In fact, it is Hamiltonian volume minimizing(Kleiner and Oh, cf. [35], [36]). See also [2].
(3) A product of n+1 circles S 1 (r0 )×· · ·×S 1 (rn ) ⊂ Cn+1 and the quotient
space T n ⊂ CP n by the standard S 1 -action are strictly Hamiltonian
stable ([37], [48]). Note that T n ⊂ CP n is minimal if and only if
r0 = · · · = rn . F. Urbano ([60]) showed that any Hamiltonian stable
minimal Lagrangian torus in CP 2 is congruent to T 2 ⊂ CP 2 with
r0 = r1 = r2 (see also [10]).

DIFFERENTIAL GEOMETRY OF LAGRANGIAN SUBMANIFOLDS

7(4) Compact irreducible minimal Lagrangian submanifolds
SU (p)/SO(p) · Zp , SU (p)/Zp , SU (2p)/Sp(p) · Z2p and E6 /F4 · Z3
standardly embedded in CP n are strictly Hamiltonian stable ([5], cf. [2]).
Remark that they are not totally geodesic but they all satisfy ∇S = 0.
(5) Let (V3 , ρ3 ) be the irreducible unitary representation of SU (2) of degree
3, where the representation space V3 consists of all complex homogeneous polynomials with respect to z0 , z1 of degree 3. The minimal
Lagrangian orbit ρ3 (SU (2))[z03 + z13 ] ⊂ CP 3 is a 3-dimensional compact embedded strictly Hamiltonian stable Lagrangian submanifoldwith ∇S ̸= 0 (L. Bedulli and A. Gori [7], [43]). It gives a negative
answer to Problem 4.2 in [5, p.606].
(6) M. Takeuchi ([56]) classiﬁed all compact totally geodesic Lagrangiansubmanifolds in compact irreducible Hermitian symmetric spaces. He
proved that they all are real forms of Hermitian symmetric spaces,
i.e. the ﬁxed point subset of anti-holomorphic isometries, and are given
as symmetric R-spaces L canonically embedded in compact Hermitiansymmetric spaces M . If a symmetric R-space L is of Hermitian type,then L is canonically embedded in a compact Hermitian symmetricspace M = L × L as a diagonal subset and thus L is homologically
volume-minimizing in M = L × L, in particular L is Hamiltonian stable and moreover L is strictly Hamiltonian stable, because of theoremsof Lichnerowicz and Matsushima on the ﬁrst eigenvalue of compact
Einstein-Kähler manifolds with positive Einstein constant. In the casewhen a symmetric R-space L is not of Hermitian type, the Hamiltonian
stability of L are given as in Table 1 ([56], [35], [5], [25, p. 775]). Thena compact totally geodesic Lagrangian submanifold L embedded in acompact irreducible Hermitian symmetric space M is NOT Hamiltonian stable if and only if (Qp,q (R) = (S p−1 × S q−1 )/Z2 , Qp+q−2 (C)) (q − p ≥ 3),(U (2p)/Sp(p), SO(4p)/U (2p)) (p ≥ 3),(L, M ) =
 (T · E /F , E /T · E ).
64
7
6Here Gp,q (F) denotes the Grassmann manifold of all p-dimensional subspacesof Fp+q for each F = R, C, H, and P2 (K) the Cayley projective plane. Qp,q (R)and Qn (C) denote the real and complex hyperquadric of dimension n. Hereeach M is equipped with the standard Kähler metric of Einstein constant 1/2and λ1 denotes the ﬁrst eigenvalue of the Laplacian of L on smooth functions.
The totally geodesic Lagrangian torus S 1 ×S 1 ∼= Q2,2 (R) ⊂ Q2 (C) ∼= S 2 ×S 2
2
2
in S × S is Hamiltonian stable, and more strongly it is Hamiltonian volume
minimizing (H. Iriyeh, H. Ono and T. Sakai [21]).
At present non-trivial known examples of Hamiltonian volume minimizing
Lagrangian submanifolds in Kähler manifolds are only RP n ⊂ CP n and totally
geodesic Lagrangian torus S 1 × S 1 ⊂ S 2 × S 2 . It is a natural problem toinvestigate whether a given Hamiltonian stable Lagrangian submanifold in aKähler manifold is Hamiltonian volume minimizing or not.

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H. MA AND Y. OHNITA

Table 1. Symmetric R-space L of non-Hermitian type canonically embedded in compact Hermitian symmetric spaces
M
L
EinsteinGp,q (C), p ≤ q
Gp,q (R)Yes
G2p,2q (C), p ≤ q
Gp,q (H)Yes
Gm,m (C)U (m)No
SO(2m)/U (m)SO(m), m ≥ 5
Yes
SO(4m)/U (2m), m ≥ 3
U (2m)/Sp(m)No
Sp(2m)/U (2m)Sp(m), m ≥ 2
YesSp(m)/U (m)U (m)/O(m)No
Qp+q−2 (C), q − p ≥ 3
Qp,q (R), p ≥ 2
No
Qp+q−2 (C), 0 ≤ q − p < 3 Qp,q (R), p ≥ 2
No
Qq−1 (C), q ≥ 3
Q1,q (R)Yes
E6 /T · Spin(10)P2 (K)Yes
E6 /T · Spin(10)G2,2 (H)/Z2
Yes
E7 /T · E6
SU (8)/Sp(4)Z2
Yes
E7 /T · E6
T · E6 /F4
No

λ1
1
2
1
2
1
2
1
2
m
4m−2
1
2
1
2
p
p+q−2
1
2
1
2
1
2
1
2
1
2
1
6

H-stable stable
Yes
No
Yes
Yes
Yes
No
Yes
No
No
No
Yes
Yes
Yes
No
No
No
Yes
No
Yes
Yes
Yes
Yes
Yes
No
Yes
No
No
No

f(c) be a simply connected complete complex space form with constantLet Mf(c) is a complex Euclidean spaceholomorphic sectional curvature c, that is, Mnn
C , a complex projective space CP or a complex hyperbolic space form CH n .
Lagrangian submanifolds with ∇S = 0 in complex space forms were completelyclassiﬁed by Hiroo Naitoh and Masaru Takeuchi [31], [32], [33], [34].
f(c) be a compact Lagrangian submanTheorem 2.2 ([4],[3],[6]). Let Ln ,→ Mnifold with ∇S = 0 embedded in C or CP n . Then L is strictly Hamiltonian
stable.
It gives a positive answer to Problem 4.1 in [5, p.606].
Problem. Let L ,→ CP n be a compact minimal Lagrangian submanifold
embedded in a complex projective space. Is it true that λ1 = κ, or equivalently,
L is Hamiltonian stable? (At present we do not know any counter example
yet.)2.4. Classification of homogeneous Lagrangian submanifolds in complex projective spaces. By using the classiﬁcation theory of prehomogeneous vector spaces due to M. Sato and T. Kimura [52], L. Bedulli and A. Gori[8] provided a classiﬁcation of compact homogeneous Lagrangian submanifolds
in CP n which are obtained as Lagrangian orbits of compact connected simple
Lie subgroups of SU (n + 1). Such Lagrangian submanifolds are classiﬁed into
16 types of examples, which consist of 5 types of examples with ∇S = 0 ( (2),(4) in Example 2.1) and 11 types of examples with ∇S ̸= 0 (including (5) in
Example 2.1 as the simplest non-trivial one).
The classiﬁcation of compact homogeneous Lagrangian submanifolds in complex hyperquadrics Qn (C) are described in Subsection 3.5.

DIFFERENTIAL GEOMETRY OF LAGRANGIAN SUBMANIFOLDS

92.5. Tightness of Lagrangian submanifolds. Suppose that M 2n = G/K
is an Hermitian symmetric space of compact type and Ln is a compact Lagrangian submanifold embedded in M = G/K. Let Symp(M ) and Ham(M )denote the group of all symplectic diﬀeomorphisms of M and the group of all
Hamiltonian diﬀeomorphisms of M .
Let L be a real form of an Hermitian symmetric space of compact type. Assume that the minimal Maslov numbger ΣL ≥ 2. Then for each ϕ ∈ Hamil(M )with transversal intersection L ∩ aL the Arnold-Givental inequality♯(L ∩ ϕL) ≥ SB(L, Z2 )holds (Y.-G. Oh [39], [40], [41]).
Definition 2.5. We call L globally tight (resp. locally tight) if for each a ∈ G(resp. a in a neighborhood of the identity of G) the intersection L ∩ aL is not
empty and the equality♯(L ∩ aL) = SB(L, Z2 )holds provided that the intersection L ∩ aL is transverse. Heren∑
SB(L, Z2 ) :=rank(Hi (L, Z2 ))i=0

is the sum of Betti numbers of L over Z2 .
Problem. Classify compact globally tight or locally tight Lagrangian submanifolds in compact Hermitian symmetric spaces, more generally simply connected compact homogeneous Kähler manifolds, that is, Kähler C-spaces orgeneralized ﬂag manifolds with invariant Kähler metrics.
Theorem 2.3 ([36]). Let L be a compact locally tight Lagrangian submanifold embedded in CP n . Then L is a totally geodesic Lagrangian submanifold
RP n (n ≥ 2) of CP n , or L is a great or small circle of S 2 ∼= CP 1 (n = 1).
Theorem 2.4 ([22]). Let L be a compact locally tight Lagrangian submanifold
embedded in S 2 × S 2 ∼= Q2 (C). Then L is a totally geodesic Lagrangian submanifold S 2 embedded in S 2 × S 2 as a diagonal subset, or L = S 1 (a) × S 1 (b) ⊂S 2 × S 2.
Due to the results of Masaru Takeuchi and Shoshichi Kobayashi on thestandard embeddings of R-spaces, we know
Theorem 2.5 ([57]). Any real form of an Hermitian symmetric space of compact type is locally tight.
In his theory on integral geometry of homogeneous spaces, R. Howard gives
Theorem 2.6 ([18]). A totally geodesic Lagrangian submanifold RP n of CP n
is globally tight.
The global tightness was applied to prove the Hamiltonian volume minimizing property for RP n ⊂ CP n (Kleiner-Oh, cf. [36]).
More generally, recently Makiko Sumi Tanaka and H. Tasaki proved

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H. MA AND Y. OHNITA

Theorem 2.7 ([58], [59]). Any real form of an Hermitian symmetric space ofcompact type is globally tight.
Concerned with the tightness for real forms of Kähler C-spaces, very recently
H. Iriyeh, T. Sakai, H. Tasaki showed
Theorem 2.8 (H. Iriyeh, T. Sakai and H. Tasaki, 2010). A real form Fk1 ,··· ,kr (R)of Fk1 ,··· ,kr (C) is globally tight.
3. Lagrangian Submanifolds in Complex Hyperquadrics
3.1. Complex hyperquadrics and real Grassmann manifolds of oriented 2-planes. The complex hyperquadricf 2 (Rn+2 ) ∼Qn (C) ∼= Gr= SO(n + 2)/(SO(2) × SO(n))
is a compact Hermitian symmetric space of rank 2, where2
Qn (C) := {[z] ∈ CP n+1 | z02 + z12 + · · · + zn+1
= 0},

f 2 (Rn+2 ) := {W | oriented 2-dimensional vector subspace of Rn+2 }.
Gr
f 2 (Rn+2 ) is given by
The identiﬁcation between Qn (C) and Gr
CPn+1⊃ Qn (C) ∋ [a +√f 2 (Rn+2 ) ⊂−1b] ←→ W = a ∧ b ∈ Gr2∧

Rn+2 .

Here {a, b} is an orthonormal basis of W compatible with its orientation. In
case n = 2, then Q2 (C) ∼= S 2 × S 2 . If n ≥ 3, then Qn (C) is irreducible.
Note that the Einstein constant κ of the standard Kähler metric on Qn (C) ∼
=
n+2
n+2
f
Gr2 (R ) induced from the standard inner product of R
is equal to n.
3.2. Lagrangian submanifolds in complex hyperquadrics and hypersurfaces in spheres. Let N n ,→ S n+1 (1) ⊂ Rn+2 be an oriented hypersurface
immersed or embedded in the (n + 1)-dimensional unit standard sphere. Let
x and n denote the position vector of points of N n and the unit normal vector
ﬁeld of N n in S n+1 (1), respectively. It is a fundamental fact in symplectic
geometry that the Gauss map deﬁned by√
G : N n ∋ p 7−→ [x(p) + −1n(p)] = x(p) ∧ n(p) ∈ Qn (C)is always a Lagrangian immersion.
Proposition 3.1 ([25]). Any deformation of an oriented hypersurface N n in
S n+1 (1) gives a Hamiltonian deformation of G in Qn (C). Conversely, anysmall Hamiltonian deformation of G in Qn (C) is obtained from a deformationof an oriented hypersurface N n in S n+1 (1).
The (2n + 1)-dimensional real Stiefel manifold
V2 ({Rn+2 ) := {(a, b) | a, b ∈ Rn+2 orthonormal} ∼= SO(n + 2)/SO(n)of oriented 2-frames in Rn+2 has the standard η-Einstein Sasakian manifold
structure over Qn (C). Deﬁne the natural projections p1 , p2 as
p1 : V2 (Rn+2 ) ∋ (a, b) 7−→ a ∈ S n+1 (1),p2 : V2 (Rn+2 ) ∋ (a, b) 7−→ a ∧ b ∈ Qn (C).

DIFFERENTIAL GEOMETRY OF LAGRANGIAN SUBMANIFOLDS

11

ψÑ n
V2 (Rn+2 ) = V2 (Rn+2 )Legend.
∼p2 S 1
p1 S n=?

?

?

- S n+1 (1)Nnori.hypsurf.

Qn (C) ⊃ p2 (ψ(N n )) = G(N n )Lagr.

Here the Legendrian lift Ñ n of N n ,→ S n+1 (1) to V2 (Rn+2 ) is deﬁned by N n ∋p 7−→ (x(p), n(p)) ∈ V2 (Rn+2 ).
More generally, assume that N m is an oriented m-dimensional submanifold immersed in S n+1 (1). It is a classical fact that the conormal bundle
νN∗ is a Lagrangian submanifold in the cotangent vector bundle T ∗ S n+1 (1)of the unit standard sphere S n+1 (1). Notice that the unit cotangent bundle U (T ∗ S n+1 (1)) is diﬀeomorphic to V2 (Rn+2 ). Furthermore, the unit cotangent bundle U (T ∗ S n+1 (1)) is a circle bundle over the space Geod+ (S n+1 (1)) oforiented geodesics of S n+1 , which is isomorphic to the real Grassmann manf 2 (Rn+2 ) of oriented 2-planes in Rn+2 and thus the complex hyperifold Gr
quadric Qn (C). Hence U (T ∗ S n+1 (1)) carries the canonical contact structure
and then the unit conormal bundle U (νN∗ ) of N is a Legendrian submanifoldof U (T ∗ S n+1 (1)). Then the projection of U (νN∗ ) gives a Lagrangian immersion
in Qn (C). We have the following diagram :νN∗-

T ∗ S n+1 (1)Lag.
?

U(νN∗ )?

p2 (U(νN∗ ))

-

Leg.
-

Lag.

?

U (T ∗ S n+1 (1))∼=V2 (Rn+2 )p2 S 1

p1 S n?

?

Qn (C)

S n+1 (1) ⊃ N mimm. submfd.

std
3.3. The mean curvature form formula. Let gQ
be the standard Kählern (C)metric of Qn (C) induced from the standard inner product of Rn+2 . Note thatstd
the Einstein constant of gQ
is equal to n. Let κi (i = 1, · · · , n) denote then (C)nprincipal curvatures of N ⊂ S n+1 (1). Choose an orthonormal frame {ei } on
N n ⊂ S n+1 (1) such that the second fundamental form h of N n in S n+1 (1) withrespect to n is diagonalized as h(ei , ej ) = κi δij and let {θi } be its dual coframe.
std
Then the induced metric G ∗ gQ
on N n by the Gauss map G is given as
n (C)
std
=
G ∗ gQn (C)n∑(1 + κ2i )θi ⊗ θi .

i=1

Let H denote the mean curvature vector ﬁeld of G. Then the mean curvatureform of the Gauss map G is expressed in terms of the principal curvatures asfollows:

12

H. MA AND Y. OHNITA

Lemma 3.1 (Palmer [51]).
(

(

))
n
∏
√
αH = d Im log (1 + −1κi ).
i=1In case n = 2, if N 2 ⊂ S 3 (1) is a minimal surface, then the Gauss map
f 2 (R4 ) ∼G : N 2 −→ Gr= Q2 (C) ∼= S 2 × S 2 is a minimal Lagrangian immersion.
See also Castro-Urbano [9]. More generally, if N n ⊂ S n+1 (1) is an oriented
austere hypersurface in S n+1 (1), introduced by Harvey-Lawson ([17]), then theGauss map G : N n −→ Qn (C) is a minimal Lagrangian immersion.
And if N n ⊂ S n+1 (1) is an oriented hypersurface in S n+1 (1) with constantprincipal curvatures, then the Gauss map G : N n −→ Qn (C) is a minimal
Lagrangian immersion.
Note that more minimal Lagrangian submanifolds of complex hyperquadrics
can be obtained from Gauss maps of certain oriented hypersurfaces in spheres
through Palmer’s formula and there are compact oriented rotational nonminimal hypersurfaces embedded in spheres with n ≥ 3 whose Gauss maps
are minimal Lagrangian immersions. ([24]).
3.4. The Gauss maps of isoparametric hypersurfaces in S n+1 (1). Assume that N n ,→ S n+1 (1) ⊂ Rn+2 is a compact oriented hypersurface embedded in the standard sphere with constant principal curvatures, so called“isoparametric hypersurface”. Let g denote the number of distinct principal
curvatures of N n in S n+1 (1) and m1 , m2 , · · · , mg denote the multiplicities ofthe principal curvatures k1 < · · · < kg . Then the image of the Gauss map
G : N n −→ Qn (C) is a compact minimal Lagrangian submanifold embedded
in Qn (C) and the Gauss map gives a covering map
N n −→ Ln = G(N n ) ∼= N n /Zg ,→ Qn (C)Zg

with Deck transformation group Zg .
By the famous theorems of Münzner ([29], [30]), we know that mi (i =1, · · · , g) satisfy mi = mi+2 for each i, i.e., m1 = m3 = · · · , m2 = m4 = · · · ,and g must be 1, 2, 3, 4 or 6. By using the results of [38], [47], we can show
Proposition 3.2 ([26]). L = G(N n ) is a compact monotone and cyclic Lagrangian submanifold embedded in Qn (C) and its minimal Maslov number ΣLis given by2n
ΣL == m1 + m2 .
g
All isoparametric hypersurfaces in spheres are classiﬁed into homogeneous
one and non-homogeneous ones. A hypersurface N n in S n+1 (1) is homogeneous
if it is obtained as an orbit of a compact connected subgroup G of SO(n +2). Obviously a homogeneous hypersurface in S n+1 (1) is an isoparametric
hypersurface.
Due to W. Y. Hsiang and H. B. Lawson, Jr. ([19]), see also Ryoichi Takagi
and Tsunero Takahashi ([53]), any homogeneous isoparametric hypersurface
in a sphere can be obtained as a principal orbit of the isotropy representationof a Riemannian symmetric pair (U, K) of rank 2 (see Table 2).

DIFFERENTIAL GEOMETRY OF LAGRANGIAN SUBMANIFOLDS

13

Table 2. Homogeneous isoparametric hypersurfaces in spheres
g
1

T ype
S1×
BDII
2 BDII×
BDII
3

AI2

(U, K)(S 1 × SO(n + 2), SO(n + 1))
n ≥ 1, [R ⊕ A1 ](SO(p + 2) × SO(n + 2 − p),SO(p + 1) × SO(n + 1 − p))
1 ≤ p ≤ n − 1, [A1 ⊕ A1 ]
(SU (3), SO(3)) [A2 ]dimNnm1 , m2
n

K/K0
Sn

n

p, n − p

S p × S n−p

3

1, 1

SO(3)
Z2 +Z2
SU (3)T2
Sp(3)
Sp(1)3F4Spin(8)SO(5)T2S(U (2)×U (m))
S(U (1)×U (1)×U (m−2))

3
3

a2
AII2

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

6
12

2, 2
4, 4

3

EIV

(E6 , F4 ) [A2 ]

24

8, 8

4
4

b2
AIII2

4

BDI2

4

CII2

4

DIII2

4

EIII

(E6 , U (1) · Spin(10)) [BC2 ]

30

6, 9

6
6

g2
G

(G2 × G2 , G2 ) [G2 ](G2 , SO(4)) [G2 ]

12
6

2, 2
1, 1

(SO(5) × SO(5), SO(5)) [B2 ]82, 2(SU (m + 2), S(U (2) × U (m)))
4m − 22,m ≥ 2, [BC2 ](m ≥ 3), [B2 ](m = 2)2m − 3(SO(m + 2), SO(2) × SO(m))
2m − 2
1,m ≥ 3, [B2 ]m−2(Sp(m + 2), Sp(2) × Sp(m))
8m − 24,m ≥ 2, [BC2 ](m ≥ 3), [B2 ](m = 2)4m − 5(SO(10), U (5)) [BC2 ]18
4, 5

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

Cartan showed that any isoparametric hypersurface in a sphere with g ≤3 distinct principal curvatures must be homogeneous. The classiﬁcation of
isoparametric hypersurfaces in spheres with four or six distinct principal curvatures are still open. In the case g = 6, all of the principal curvatures musthave the same multiplicity m1 = m2 ([30]), and m1 = m2 must be 1 or 2([1]). In either case, the known examples belong to the family of homogeneous hypersurfaces. In the case m = 1, the isoparametric hypersurface must
be homogeneous due to Dormeister-Neher [14] (See also Miyaoka [27]), but
for m1 = m2 = 2 the classiﬁcation is still open so far. In the case g = 4,
the Cliﬀord algebra construction of non-homogeneous isoparametric hypersurfaces in the sphere were discovered ﬁrst by Hideki Ozeki and Masaru Takeuchi([49], [50]) and generalized by D. Ferus, H. Karcher and H. F. Münzner [15](so called “isoparametric hypersurfaces of OT-FKM type”(cf. [28]). Recently
T. Cecil, Q.-S. Chi and G. Jensen ([12]) and S. Immervoll ([20]) showed thatisoparametric hypersurfaces in the sphere with g = 4 except for the cases of(m1 , m2 ) = (3, 4), (4, 5), (6, 9), (7, 8) are either homogeneous or of OT-FKM
type.
Proposition 3.3 ([25]). N n is homogeneous if and only if G(N n ) is homogeneous.

14

H. MA AND Y. OHNITA

In [25] we classiﬁed all compact homogeneous Lagrangian submanifolds incomplex hyperquadrics Qn (C) by using the theory of homogeneous isoparametric hypersurfaces. We shall mention it in the next subsection.
Consider
2
∧
n
n+2
f
G : N ∋ p 7−→ x(p) ∧ n(p) ∈ Gr2 (R ) ⊂Rn+2 .
∧
Here 2 Rn+2 ∼= o(n + 2) can be identiﬁed with the Lie algebra of all (holomorf 2 (Rn+2 ). Let k̃ be the Lie subalgebraphic) Killing vector ﬁelds on S n+1 (1) or Grof o(n + 2) consisting of all Killing vector ﬁelds tangent to N n or G(N n ) andK̃ be an analytic subgroup of SO(n + 2) generated by k̃. Take the orthogonal
direct sum
2∧
Rn+2 = k̃ + V,
where V is a vector subspace of o(n + 2). The linear map
V ∋ X 7−→ αX |G (N n ) ∈ E0 (G) ⊂ B 1 (G(N n ))is injective and nhk (G) = dim V. Then G(N n ) ⊂ V and thus G(N n ) ⊂f 2 (Rn+2 ) ∩ V. Indeed, for each X ∈ k̃ and each p ∈ N n ,Gr⟨X, x(p) ∧ n(p)⟩ = ⟨Xx(p), n(p)⟩ − ⟨x(p), Xn(p)⟩= 2⟨Xx(p), n(p)⟩ = 0.
Note that G(N n ) is a compact minimal submanifold embedded in the unit
hypersphere of V and by the theorem of Tsunero Takahashi [54] each coordinate
function of V restricted to G(N n ) is an eigenfunction of the Laplace operator
with eigenvalue n. If n is just the ﬁrst (positive) eigenvalue of G(N n ), thenG(N n ) ⊂ Qn (C) is Hamiltonian stable. Moreover if the dimension of the vector
space V is equal to the multiplicity of the ﬁrst eigenvalue n, then G(N n ) ⊂Qn (C) is strictly Hamiltonian stable.
Proposition 3.4. If N n is homogeneous, thenf 2 (Rn+2 ) ∩ V.
G(N n ) = Gr
Proof. Assume that Rn+2 = p and N n = Adp (K)ξ (ξ ∈ a regular ) for af 2 (Rn+2 )∩compact Riemannian symmetric pair (U, K) of rank 2. If X ∧Y ∈ Gr
V, then for any T ∈ k
0 =⟨adp (T ), X ∧ Y ⟩=⟨[T, X], Y ⟩ − ⟨[T, Y ], X⟩=2⟨[T, X], Y ⟩=2⟨T, [X, Y ]⟩and thus we have [X, Y ] = 0. Hence we obtain X ∧ Y ∈ Adp (K)[a] = G(N n ).
∧f 2 (Rn+2 ) → 2 Rn+2 in the following way:Deﬁne µ : Gr
f 2 (Rn+2 ) ∋ [W ] 7−→ a ∧ b ∈µ : Gr2∧

Rn+2 ∼= o(n + 2) = k̃ + V.

DIFFERENTIAL GEOMETRY OF LAGRANGIAN SUBMANIFOLDS

15

f 2 (Rn+2 ) is given by µ := π ◦ µ :The moment map of the action K̃ on Gr
k̃
k̃
n+2
n
f
Gr2 (R ) → k̃. For any p ∈ N ,f 2 (Rn+2 ) ∩ V = µ−1 (0).
K̃(x(p) ∧ n(p)) ⊂ G(N n ) ⊂ Grk̃

It is obvious that N is homogeneous if and only if K̃(x(p) ∧ n(p)) = G(N n ).
nf 2 (Rn+2 ) ∩ V. Then K̃(x(p) ∧Proposition 3.5. Assume that G(N n ) = Grnnn(p)) = G(N ), that is, N is homogeneous.
Proof. Suppose that K̃(x(p) ∧ n(p)) ̸= G(N n ). K̃x(p) ̸= N n . Then there is a
f 2 (Rn+2 ) ∩ V =unit vector c ∈ Tp N n such that c ⊥ Tp K̃x(p). Since x(p)∧c ∈ Grnn
G(N ), there is q ∈ N such that x(p) ∧ c = x(q) ∧ n(q). By the isoparametricproperty of N n we have c = ±n(p), a contradiction.

Corollary 3.1. N n is not homogeneous if and only if
f 2 (Rn+2 ) ∩ V = µ−1 (0).
K̃(x(p) ∧ n(p)) $ G(N n ) $ Grk̃

3.5. Classification of compact homogeneous Lagrangian submanifolds
in complex hyperquadrics. Suppose that G ⊂ SO(n + 2) is a compact
connected Lie subgroup and L = G · [W ] ⊂ Qn (C) is a Lagrangian orbit of G
through a point [W ] ∈ Qn (C), where W is an oriented 2-dimensional vector
subspace of Rn+2 . Denote a unit circle of W by
S 1 (W ) := {v ∈ W | ∥v∥ = 1}.
Then we can show that there is a ﬁnite subset w1 , · · · , wd of S 1 (W ) suchthat for each w ∈ S 1 (W ) \ {w1 , · · · , wd } the orbit G · w of G through w on
S n+1 (1) ⊂ Rn+2 is a compact homogeneous hypersurface in S n+1 (1) ([25]). Set
N n := G · w.
By the theorem of W. H. Hsiang-H. B. Lawson, Jr., there is a compact
Riemannian symmetric pair (U, K) of rank 2 such thatN n = Adp (K)v ⊂ S n+1 (1) ⊂ Rn+2 = p

for some v ∈ S n+1 (1),where u = k + p is the canonical decomposition of the symmetric pair (U, K).
Here we may assume that Adp (K) ⊂ SO(n+2) is a maximal compact subgroupof SO(n + 2) containing G which is orbit-equivalent to the action of G on
S n+1 (1). Then we obtain
Theorem 3.1 (Ma-Ohnita [25]). There exists a compact homogeneous isoparametric hypersurface N n ⊂ S n+1 (1) ⊂ Rn+2 such that(i) L = G(N ) and L is a compact minimal Lagrangian submanifold, or(ii) L is contained in a Lagrangian deformation of G(N ) consisting of compact homogeneous Lagrangian submanifolds.
We shall explain about the second case (ii) a little more in detail. The
second case (ii) happens only when (U, K) is one of(1) (S 1 × SO(3), SO(2)),(2) (SO(3) × SO(3), SO(2) × SO(2)),(3) (SO(3) × SO(n + 1), SO(2) × SO(n)) (n ≥ 3),(4) (SO(m + 2), SO(2) × SO(m)) (n = 2m − 2, m ≥ 3).

16

H. MA AND Y. OHNITAIn the ﬁrst two cases (1) and (2), it is elementary and well-known to describe
all Lagrangian orbits of the natural actions of K = SO(2) on Q1 ({C) ∼= S 2 andK = SO(2) × SO(2) on Q2 ({C) ∼= S 2 × S 2 . Also in the last two cases (3) and(4), there exist one-parameter families of Lagrangian K-orbits in Qn (C) andeach family contains Lagrangian submanifolds which can NOT be obtained as
the Gauss image of any homogeneous isoparametric hypersurface in a sphere.
The fourth one is a new family of Lagrangian orbits :(1) If (U, K) is (S 1 × SO(3), SO(2)), then L is a small or great circle in
Q1 (C) ∼= S 2.
(2) If (U, K) is (SO(3) × SO(3), SO(2) × SO(2)), then L is a product ofsmall or great circles of S 2 in Q2 (C) ∼= S 2 × S 2.
(3) 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 Lagrangiansubmanifold 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
and thus ∇αH = 0.
(c) K · [W±√−1 ] are isotropic submanifolds in Qn (C) withdim K · [W±√−1 ] = 0 (points !).
(4) 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(a) K · [W1 ] = K · [W−1 ] = G(N n ) is a minimal (NOT totally geodesic)
Lagrangian submanifold√ in Qn (C).
1(b) For each λ ∈ S \ {± −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.
3.6. Hamiltonian stability of the Gauss images of homogeneous isoparametric hypersurfaces in S n+1 (1). Suppose that N n is a compact isoparametric hypersurface embedded in S n+1 (1). Palmer ([51]) 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 G(N n ) =N n /Zg embedded in Qn (C) as a compact minimal Lagrangian submanifold.

DIFFERENTIAL GEOMETRY OF LAGRANGIAN SUBMANIFOLDS

17

g = 1 : N n = S n is a great or small sphere and G(N n ) ∼= S n is strictly
Hamiltonian stable. More strongly, it is stable as a minimal submanifold andhomologically volume-minimizing because it is a calibrated submanifold.
g = 2 : N n = S m1 × S m2 (n = m1 + m2 , 1 ≤ m1 ≤ m2 ) are the socalled Cliﬀord hypersurfaces and G(N n ) = Qm1 +1,m2 +1 (R) ⊂ Qn (C). Thenm2 −m1 ≥ 3 if and only if G(N n ) ⊂ Qn (C) is NOT Hamiltonian stable. In casem2 −m1 ≥ 3, the spherical harmonics of degree 2 on the sphere S m1 ⊂ Rm1 +1 ofsmaller dimension give volume-decreasing Hamiltonian deformations of G(N n ).
If m2 − m1 = 2, then it is Hamiltonian stable but not strictly Hamiltonian
stable. If m2 − m1 = 0 or 1, then it is strictly Hamiltonian stable.
g = 3 : All isoparametric hypersurfaces in the sphere with g = 3 wereclassiﬁed by E. Cartan and they all are homogeneous, so called “Cartan hypersurfaces”.
Theorem 3.2 (Ma-Ohnita [25]). If g = 3, then L = G(N n ) ⊂ Qn (C) is strictly
Hamiltonian stable.
Remark. In case g = 3, each induced metric from Qn (C) is a normal homogeneous metric. It never holds in cases g = 4, 6.
g = 6 : Only homogeneous examples are known now(Dorfmeister-Neher [14],Reiko Miyaoka [27]).
Theorem 3.3 (Ma-Ohnita [26]). If g = 6 and N n is homogeneous, then L =G(N n ) ⊂ Qn (C) is strictly Hamiltonian stable.
g = 4 : In the case g = 4 and N n is homogeneous, we obtain the Hamiltonian
stability of Gauss images for ALL homogeneous isoparametric hypersurfaces
as follows.
Theorem 3.4 (Ma-Ohnita [26]). If g = 4 and N n is homogeneous, then(1)
L = G(N n ) = SO(5)/T 2 · Z4
is strictly Hamiltonian stable.
(2)
L = G(N n ) = U (5)/(SU (2) × SU (2) × U (1)) · Z4
is strictly Hamiltonian stable.
(3)
L = G(N n ) = (SO(2) × SO(m))/(Z2 × SO(m − 2)) · Z4 (m ≥ 3)
is NOT Hamiltonian stable if and only if m ≥ 6, i.e. m2 − m1 =(m − 2) − 1 ≥ 3. If m2 − m1 = (m − 2) − 1 = 2, i.e. m = 5,then it is Hamiltonian stable but not strictly Hamiltonian stable. If
m2 − m1 = (m − 2) − 1 = 0 or 1, i.e. m = 3 or 4, then it is strictly
Hamiltonian stable.
(4)
L = G(N n ) = S(U (2) × U (m))/S(U (1) × U (1) × U (m − 2)) · Z4 (m ≥ 2)

18

H. MA AND Y. OHNITA

is NOT Hamiltonian stable if and only if m ≥ 4, i.e. m2 − m1 =(2m − 3) − 2 ≥ 3. If m2 − m1 = −1 or 1, i.e. m = 2 or 3, then it isstrictly Hamiltonian stable.
(5)
L = G(N n ) = (Sp(2) × Sp(m))/(Sp(1) × Sp(1) × Sp(m − 2)) · Z4 (m ≥ 2)
is NOT Hamiltonian stable if and only if m ≥ 3, i.e. m2 − m1 =(4m − 5) − 4 ≥ 3. If m2 − m1 = 1, i.e. m = 2, then it is strictly
Hamiltonian stable.
(6)
L = G(N n ) = (U (1) · Spin(10))/(S 1 · Spin(6)) · Z4
is strictly Hamiltonian stable.
The last case is obtained as the Gauss image of a principal isotropy orbit of a
Riemannian symmetric pair (U, K) = (E6 , U (1) · Spin(10)) of exceptional type
EIII. For all homogeneous isoparametric hypersurfaces in spheres except forthis case, we can observe that G(N n ) is not Hamiltonian stable if and only if
m2 − m1 ≥ 3. However in the last case (m1 , m2 ) = (6, 9) and thus m2 − m1 = 3
but it is Hamiltonian stable.
In a summary, we obtain the Hamiltonian stability of the Gauss images of
ALL homogeneous isoparametric hypersurfaces in spheres as follows :Theorem 3.5 ([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) = (E6 , U (1)·Spin(10)), then (m1 , m2 ) = (6, 9) but L = G(N ) is strictly Hamiltonian stable.
Problem. Investigate the Hamiltonian stability and other properties of theGauss images of compact non-homogeneous isoparametric hypersurfaces in thesphere with g = 4 as compact minimal Lagrangian submanifolds embedded incomplex hyperquadrics.
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Ann. 264 (1983), 283–302.
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Lagrangian submanifolds in Hermitian symmetric spaces, RIMS Kokyuroku
1236, Geometry of Submanifolds and Related Topics, Nov 2001, RIMS, Kyoto University, Kyoto Japan, 31–48. (revised version: http://math01.sci.osakacu.ac.jp/ ohnita/paper/AmarRIMS01c.pdf)[3] A. Amarzaya and Y. Ohnita, Hamiltonian stability of certain symmetric R-spaces embedded in complex Euclidean spaces, preprint, Tokyo Metropolitan University, 2002.
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Department of Mathematical Sciences, Tsinghua University, Beijing 100084,
P.R. CHINA
E-mail address: hma@math.tsinghua.edu.cn
Osaka City University Advanced Mathematical Institute & Department ofMathematics, Osaka City University, 558-8585, JAPANE-mail address: ohnita@sci.osaka-cu.ac.jp