INTEGRAL GEOMETRY AND HAMILTONIAN VOLUMEMINIMIZING PROPERTYTAKASHI SAKAI

1. Introduction
This paper is a survey of our recent works [9], [10], [11] and [20] on integral
geometry and its application for some geometric variational problems.
Let G be a Lie group and K a closed subgroup of G. We assume that G has aleft invariant Riemannian metric that is also right invariant under elements of K,
then G/K is a homogeneous space with an invariant Riemannian metric. Consider
now two compact submanifolds M and N of G/K, one fixed and the other moving
under the action of g ∈ G. Then it becomes a measurable function on G if weassign an integral invariant I(M ∩ gN ) of submanifold M ∩ gN for each g ∈ G. It
is called the kinematic formula that represents the equality between the integralZ(1.1)
I(M ∩ gN )dµG (g)G

and some geometric invariants of M and N , where dµG is the invariant measureof G. This integral has been studied by many geometers from various viewpoints.
In particular, it is called Poincaré formula if we take vol(M ∩ gN ) as an integral
invariant in (1.1). For example, in the case where M and N are submanifolds of areal space form, this integral is equal to a constant times the product of the volumesof M and N . This was studied by Poincaré, Blaschke, Santaló and others (see [22]for reference). Santaló [21] showed that if M and N are complex submanifolds ofa complex projective space, then the Poincaré formula is expressed as a constanttimes the product of the volumes of two submanifolds.
Furthermore, Chern [4] and Federer [5] obtained a remarkable result stated asfollows: Let G(Rn ) be the group of orientation preserving isometries of Euclidiann-space Rn . Assume that 0 ≤ 2l ≤ p + q − n. Then for any compact submanifolds
M and N of dimensions p and q in Rn , we have the equality
(1.2)

Z

G(Rn )µ2l (M ∩ gN )dµG(Rn ) (g) =lXa(p, q, n, k, l)µ2k (M )µ2(l−k) (N )k=0where each constant a(p, q, n, k, l) depends only on the indicated parameters. Hereµ2i is the integral invariant which appears in coefficients of the Weyl tube formulaand the generalized Gauss-Bonnet formula (see [6]). We shall explain this invariant

2000 Mathematics Subject Classification. Primary 53C40 ; Secondary 53C65.
Key words and phrases. Poincaré formula, kinematic formula, Lagrangian submanifold, Hamiltonian stability, Hamiltonian volume minimizing property.
12TAKASHI SAKAI

in Section 3. C. S. Chen [3] showed that for any closed surfaces M and N in R3
µZ
¶
Z
2
(1.3)
κ ds dµG(R3 ) (g)G(R3 )M ∩gN
ZZ= π 3 vol(M ) (khN k2 + 2(H N )2 )dµN + π 3 vol(N )(khM k2 + 2(H M )2 )dµM
NM

where κ and ds are the curvature and the arc-element of the intersection curve
M ∩ gN . We denote by hX , H X and dµX the second fundamental form, the mean
curvature and the invariant volume element of a submanifold X, respectively. When
G is the unitary group U (n + 1) acting on the complex projective space CP n , M
and N are complex submanifolds letting I(M ∩ gN ) be the integral of Chern classleads to the kinematic formula of Shifrin [23].
Afterward Howard [8] defined integral invariants induced from an invariant homogeneous polynomial of the second fundamental form of M ∩ gN , and he achieved
more general kinematic formula, where G is unimodular and acts transitively on
the sets of tangent spaces to each of M and N . He put the kinematic formulas
listed above into a uniform shape. However, we must have a nontrivial calculationin order to get concrete forms. For a polynomial of degree 0, his formulation gives
exactly the Poincaré formula. In Section 1, using his formulation we attempt to givesome sufficient conditions that the Poincaré formulas for complex submanifolds in
Hermitian homogeneous spaces can be expressed as a constant times of volumes of
two submanifolds. On the other hand, we can not consider invariant homogeneouspolynomials of odd degree in the case of real space forms. So our interest goes to theconcrete form of the kinematic formuls for integral invariants defined by invariant
homogeneous polynomial of degree 2. In Section 2, we give explicit expressions of
the kinematic formulas of this case, completely.
The kinematic formulas are interesting in their own forms, and furthermore they
have been applied to several geometric problems (see [12] and [22] for references).
For example, Zhou [26] was obtained an extention of Chen’s formula, and applied
it to describe a sufficient condition for Hadwiger’s containment problem in higher
dimensions. The Poincaré formulas have been applied to show some volume minimizing properties of certain submanifolds (see [13], [14] and [24]). In the present
paper, we attempt to apply the Poincaré formula to show Hamiltonian volume
minimizing properties of minimal Lagrangian submanifolds in Kähler manifolds.
The equator on S 2 has the least length among all its images under area bisecting deformations. This is a well-known theorem by Poincaré and a special case ofisoperimetric inequality for closed curves on S 2 . This theorem stands in the intersection of symplectic geometry and Riemannian geometry. If we interpret S 2 as a
Kähler manifold and the equator as a minimal Lagrangian submanifold, then areabisecting deformations of the equator are nothing but Hamiltonian deformations.
Considering S 2 as CP 1 and the equator as a real form RP 1 ⊂ CP 1 , it is natural togeneralize Poincaré’s theorem to the case RP n ⊂ CP n . In 1990, Y.-G. Oh [16] and
B. Kleiner actually showed that the standard real form RP n ⊂ CP n is Hamiltonian
volume minimizing, applying Poincaré formula and Arnold-Givental inequality. In
Section 4, we explain an unified method of proving the Hamiltonian volume minimizing properties of real forms in Hermitian symmetric spaces of compact type andpose a conjecture in terms of integral geometry. Finally we show the conjecturein the case S 1 (1) × S 1 (1) ⊂ S 2 (1) × S 2 (1) and establish its Hamiltonian volume
minimizing property and the uniqueness modulo isometric group actions.

INTEGRAL GEOMETRY AND HAMILTONIAN VOLUME MINIMIZING PROPERTY

3

2. Poincaré formula of complex submanifolds
In this section, we shall explain about the generalized Poincaré formula in Riemannian homogeneous spaces. And we make assertions for Poincaré formula forcomplex submanifolds in Hermitian manifolds.
We begin with a definition of the angle between subspaces. Let E be a finite
dimensional real vector space with an inner product h·, ·i. For two vector subspaces
V and W of dimensions p and q in E with p + q ≤ n, we take orthonormal bases
v1 , . . . , vp and w1 , . . . , wq of V and W respectively. Then we define σ(V, W ), theangele between V and W , byσ(V, W ) = kv1 ∧ · · · ∧ vp ∧ w1 ∧ · · · ∧ wq k,
where
kx1 ∧ · · · ∧ xk k = det(hxi , xj i).
This definition is independent of the choice of orthonormal bases.
Let G/K be a Riemannian homogeneous space. We denote by o the origin of
G/K. For x and y in G/K and vector subspaces V in Tx (G/K) and W in Ty (G/K)
we define σK (V, W ), the angle between V and W , byZ
−1
−1
σK (V, W ) =σ((gx )−1
∗ V, k∗ (gy )∗ W )dµK (k),K

where gx and gy are elements of G with gx o = x and gy o = y. We note that thisdefinition is independent of the choice of gx and gy .
With these notations, the generalized Poincaré formula for homogeneous spaces
can be stated as follows:
Theorem 2.1 (Howard [8]). Suppose that G is unimodular and let M and N be
submanifolds of G/K with dim M + dim N ≥ dim(G/K). Then we have
Z
Z
(2.1)
vol(M ∩ gN )dµG (g) =σK (Tx⊥ M, Ty⊥ N )dµM ×N (x, y).
M ×N

G

Let Vo be a p-dimensional subspace of To (G/K). Then a p-dimensional submanifold M of G/K is said to be of type Vo if for each x in M there exists gx in G with(gx )−1∗ (Tx M ) = Vo . Equality (2.1) implies that if M is a submanifold of G/K oftype Vo and N of type Wo for some subspaces Vo and Wo of To (G/K), then σK isa constant function on M × N andZ(2.2)
vol(M ∩ gN )dµG (g) = σK (Vo⊥ , Wo⊥ )vol(M )vol(N ).
G

It is clear that if G/K is a real space form then all (dim Vo )-dimensional submanifolds are of type Vo . So (2.2) yields the Poincaré formula in real space forms. In thecase where G/K is a complex projective space CP n , any p-dimensional complex
submanifolds is a type Vo for any p-dimensional complex subspace Vo in To (G/K).
Thus for any complex submanifolds M and N of complex dimensions p and q with
p + q ≥ n, we haveZ
(2.3)
vol(M ∩ gN )dµU (n+1) (g)U (n+1)=

vol(CP p+q−n )vol(U (n + 1))
vol(M )vol(N ).
vol(CP p )vol(CP q )This was first obtained by Santaló [21]. For further details, see [8] as a reference.

4

TAKASHI SAKAI

By the representation theory of compact Lie groups, we can show that σK isconstant for complex submanifolds in an almost Hermitian homogeneous space
under some assumptions. Hence Poincaré formula can be expressed as a constanttimes of volumes of two submanifolds. More precisely,
Proposition 2.2 ([11], [20]). Let G be a unimodular Lie group and G/K an almost Hermitian homogeneous space of complex dimension n. Assume that K acts
irreducibly on the exterior algebras ∧p (To (G/K))(1,0) and ∧q (To (G/K))(1,0) with
p + q ≤ n. Then there exists a positive constant C such that for any almost complex submanifolds M and N of G/K of complex dimensions (n − p) and (n − q)respectively
Z
G

vol(M ∩ gN )dµG (g) = Cvol(M )vol(N )holds.

Theorem 2.3 ([20]). Let G/K be an irreducible Hermitian symmetric space ofcomplex dimension n. Assume that K acts irreducibly on the exterior algebra∧p (To (G/K))(1,0) . Then for any complex submanifolds M and N of G/K of complex dimensions (n − p) and (n − q) respectively with p + q ≤ n, we haveZ(n − p)!(n − q)!vol(K)vol(M ∩ gN )dµG (g) =vol(M )vol(N ).
n!(n − p − q)!
G

Remark 2.4. If G/K = CP n and G = U (n + 1), then K-action on To (G/K) is orbit
equivalent to the canonical action of a unitary group U (n). So K acts irreduciblyon ∧p (To (G/K))(1,0) for any p. Thus Theorem 2.3 is actually an extension of theformula (2.3) due to Santaló.

In Table 1, we give p when K acts irreducibly on ∧p (To (G/K))(1,0) for irreducible
Hermitian symmetric spaces. Although we show the case of compact type, it is clearthat their non-compact duals also give the same results of Table 1.

A III
D III
BD I
C I
E III
E V II

compact type
SU (l)/S(U (m) × U (l − m))any p (if m = 1)
p=1 (if m ≥ 2)
SO(2l)/U (l)p = 1, 2SO(2l)/SO(2) × SO(2l − 2)
p 6= l − 1SO(2l + 1)/SO(2) × SO(2l − 1) any p
Sp(l)/U (l)p = 1, 2(e6(−78) , so(10) + R)
p = 1, 2, 3(e7(−133) , e6 + R)
p = 1, 2, 3, 4
Table 1

3. Kinematic formulas in real space forms
Here we shall review some definitions and fundamental properties with respect
to the kinematic formula in Reimannian homogeneous spaces obtained by Howard[8]. And we determine the concrete forms of kinematic formulas of degree 2 in real
space forms, completely.

INTEGRAL GEOMETRY AND HAMILTONIAN VOLUME MINIMIZING PROPERTY

5Let Vo be a linear subspace of To (G/K). We define a vector space II(Vo ) to be©
ªII(Vo ) = h | h : Vo × Vo → Vo⊥ ; symmetric bilinear ,where Vo⊥ is the normal space of Vo in To (G/K). An element h ∈ II(Vo ) can bethought of as the second fundamental form of a submanifold of G/K which passthrough o and have Vo as the tangent space at o. Let K(Vo ) be the stabilizer of Vo
in K, that is, K(Vo ) = {k ∈ K | k∗ Vo = Vo }. This group K(Vo ) acts on II(Vo ) in
the following manner:(3.1)(kh)(u, v) = k∗ h(k∗−1 u, k∗−1 v)(u, v ∈ Vo )for k ∈ K(Vo ) and h ∈ II(Vo ). Here we may consider a polynomial P on thevector space II(Vo ) which is invariant under K(Vo ), that is, P(kh) = P(h) for allk ∈ K(Vo ) and h ∈ II(Vo ). In addition, let M be a submanifold of G/K of type
Vo . Then for each x ∈ M there exists gx ∈ G such that (gx )−1∗ (Tx M ) = Vo . Thus
gx−1 M is a submanifold of G/K which pass through o with To (gx−1 M ) = Vo . For
the second fundamental form hM
x of M at x ∈ M , we defineg −1 MxP(hMx ) = P(ho).

It is easy to check that this definition is independent of the choice of gx ∈ G by theinvariance of P under K(Vo ). Now we can define the integral invariant of M withrespect to a polinomial P byZ
(3.2)
I P (M ) =P(hMx )dµM .
MWhen P is an invariant homogeneous polynomial of degree l, we call I P (M ) an
integral invariant of degree l. We remark that I P is invariant under G. Many of
the invariants that are usually dealt with are of the form I P . For example, if P ≡ 1then I P (M ) = vol(M ). An integration of square of the mean curvature on M iscalled the Willmore functional.
We also define a vector space EII(To (G/K)) as follows:
EII(To (G/K)) = {h | h : To (G/K) × To (G/K) → To (G/K); symmetric bilinear} .
Since K acts on EII(To (G/K)) as in (3.1), we can also define integral invariantsfrom polynomials on EII(To (G/K)) that are invariant under K in the same mannerwith (3.2).
With these preliminaries, the kinematic formula in Riemannian homogeneous
spaces can now be stated as follows:
Theorem 3.1. (Howard [8] paragraph 4.10) Let G/K be a Riemannian homogeneous space and assume that G is unimodular. Let Vo and Wo be linear subspaces of
To (G/K) with dim(Vo ) + dim(Wo ) ≥ dim(G/K) and P a homogeneous polynomialof degree l on EII(To (G/K)) which is invariant under K such thatZ
(3.3)
σ(Vo⊥ , k∗ Wo⊥ )1−l dµK (k) < ∞.
K

Then there exists a finite set of pairs {Qα , Rα }α∈A such that(1) each Qα is a homogeneous polynomial on II(Vo ) invariant under K(Vo ),(2) each Rα is a homogeneous polynomial on II(Wo ) invariant under K(Wo ),(3) deg Qα + deg Rα = l for each α,6TAKASHI SAKAI(4) for all compact submanifolds (possibly with boundaries) M of type Vo and Nof type Wo in G/K the kinematic formulaZ
X
I P (M ∩ gN )dµG (g) =(3.4)
I Qα (M )I Rα (N )Gαholds.

Remark 3.2. The inequality (3.3) is the convergence condition for the integral. If
G/K is a real space form and G is the group either of all isometries or of allorientation preserving isometries of G/K, then the condition (3.3) can be replacedby the manageable inequality l ≤ dim(M ) + dim(N ) − dim(G/K) + 1.
In his proof of Theorem 3.1 the integration on G was reduced to that on K. From
this fact we arrive at the “transfer principle”, a method of transferring kinematicformulas from one homegeneous space to any other homogeneous space with thesame isotropy subgroup. Refer to [8] for detailed discussions of these facts.
Let us consider the case where G is the group of all isometries of a real spaceform. Then we can identify K with O(To (G/K)), the orthogonal group of To (G/K),
via the isomorphism of K to O(To (G/K)) with a 7→ a∗ . It is clear that if Vo is ap-dimensional subspace of To (G/K) then K(Vo ) = O(Vo ) × O(Vo⊥ ). If deg P = 0
then (3.4) is entirely Poincaré’s formula. Here, we note that there are no homogeneous polynomials of odd degree on II(Vo ) invariant under K(Vo ). The following
polynomials W2l are well-known invariant homogeneous polynomials on II(Vo ) ofdegree 2l.

hki11i1
hki11i2...
hki11i2l
 hk 1
hki21i2...
hki21i2l 


i2 i1
X


.
.
.
.

,
.
.
.
.
W2l (h) =
det 
.
.
.
.

 kl

l
l
. . . hki2l−1
 hi2l−1 i1 hki2l−1
i2
i2l hki2ll i1
hki2ll i2...
hki2ll i2lwhere the summation is over 1 ≤ i1 , i2 , . . . , i2l ≤ p and p + 1 ≤ k1 , k2 , . . . , kl ≤ n.
k
We note that the (a, b)-component of above matrix is hia[(a+1)/2], where [x] meansib
the greatest integer [x] not greater than x. It is remark that these polynomials arecharacterized as the invariant polynomials which vanish identically on the second
fundamental forms of some generalized cylinders. For each submanifold M of G/K,
we introduce the integral invariants µ2l (M ) defined byµ2l (M ) = I W2l (M ).
Indeed, W2l can be also expressed as a homogeneous polynomial of the componentsof the curvature tensor of M from Gauss formula, so µ2l (M ) is an intrinsic invariantof M . For these integral invariants µ2l , the Cheren-Federer kinematic formula(1.2) holds. In fact, this formula holds in any real space forms by the transfer
principle. The value of the constants a(p, q, n, k, l) were first computed by Chern [4].
Afterward Nijenhuis [15] gave a simpler expression.
The space of homogeneous polynomials of degree 2 invariant under K(Vo ) is
spanned by the two polynomialsÃ!2X ¡ ¢2X X
kk
Q1 (h) =hij ,Q2 (h) =
hii ,
i,j,k

k

i

INTEGRAL GEOMETRY AND HAMILTONIAN VOLUME MINIMIZING PROPERTY

7

where 1 ≤ i, j ≤ p, p + 1 ≤ k ≤ n. And if 2 ≤ p ≤ n − 1 these two polynomials are
independent. Geometrically, Q1 (h) is the square of the norm of the second fundamental form, and Q2 (h) is p2 times the square of the mean curvature. However, it
is convenient for us to take the basis
W2 = Q2 − Q1 ,Up = pQ1 − Q2 .

In the case of the integral invariant of degree 2, the Chern-Federer formula (1.2)
can be stated as follows:
Proposition 3.3. Assume that 2 ≤ p + q − n. Then there exist constants a(p, q, n)
so that for any compact submanifolds M p and N q of a real space from G/K thekinematic formulaZ
I W2 (M ∩ gN )dµG (g) = a(p, q, n)I W2 (M )vol(N ) + a(q, p, n)vol(M )I W2 (N )Gholds.

On the other hand, Up is characterized as the invariant polynomial which vanish
identically on the second fundamental forms of p-dimensional sphere in Rn . Hence
we easily have the following:
Proposition 3.4. (Howard [8] paragraph 8.5) Assume that 2 ≤ p + q − n. Then
there exist constants b(p, q, n) so that for any compact submanifolds M p and N q ofa real space from G/K the kinematic formulaZ
I Up+q−n (M ∩ gN )dµG (g) = b(p, q, n)I Up (M )vol(N ) + b(q, p, n)vol(M )I Uq (N )Gholds.

The space of invariant homogeneous polynomials of degree 2 on II(Vo ) is spannedby W2 and Up . Hence if we can give explicitly the constants a(p, q, n) and b(p, q, n)appearing in Propositions 3.3 and 3.4, then we can obtain the concrete form of
kinematic formulas of this case completely.
Theorem 3.5 ([10]). In Propositions 3.3 and 3.4
a(p, q, n) =b(p, q, n) =2(p + q − n − 1) vol(SO(n + 1))vol(S p+q−n ),p−1
vol(S p )vol(S q )2(p + q − n + 2)(p + q − n − 1) vol(SO(n + 1))vol(S p+q−n ).
(p + 2)(p − 1)vol(S p )vol(S q )In the previous Theorem we investigated the kinematic formulas for the integral
invariants of degree 2 under the condition 2 ≤ p + q − n. However, it remains
the case of p + q − n = 1, that is, the intersection becomes a curve. In this casethe polynomial Q1 coincides with Q2 and these polynomials are the square of thecurvature of the curve.
Let M p and N n−p+1 be compact submanifolds of a real space form G/K ofdimension n. Then the intersection M ∩ gN becomes a curve for almost all g ∈ G.
Here we take the total square curvature of M ∩ gN as an integral invariant of thecurve, and denoteZ
2

I κ (M ∩ gN ) =From Theorem 3.1, we have the following:κ2 ds.

M ∩gN

8TAKASHI SAKAI

Proposition 3.6. There exist constants c(p, n) and d(p, n) such thatZ
2
I κ (M ∩ gN )dµG (g)G¡¢= c(p, n)I W2 (M ) + d(p, n)I Up (M ) vol(N )¡¢+vol(M ) c(n − p + 1, n)I W2 (N ) + d(n − p + 1, n)I Un−p+1 (N ) .
We can give the explicit value of c(p, n) and d(p, n).

Theorem 3.7 ([10]). In Proposition 3.6,c(p, n) =d(p, n) =4π
vol(SO(n + 1)),
p − 1 vol(S p )vol(S n−p+1 )12π
vol(SO(n + 1)).
(p + 2)(p − 1) vol(S p )vol(S n−p+1 )4. Hamiltonian volume minimizing properties of minimal Lagrangiansubmanifolds
In the previous sections, we studied the formuation of kinematic formulas. Here
we attempt to apply the Poincaré formula to show Hamiltonian volume minimizing
properties of minimal Lagrangian submanifolds in Kähler manifolds.
Let (M, ω) be a 2n-dimensional closed symplectic manifold with symplectic form
ω and L be an n-dimensional closed submanifold of M . Then L is said to be
Lagrangian if ω|T L ≡ 0. Hamiltonian isotopies of (M, ω) are defined as follows. If
a smooth function F : M × [0, 1] → R is given, then we can uniquely define thevector field Xt on M for each t ∈ [0, 1] such that
ω(Xt , ·) = dF (·, t).
Therefore, we have the flow {φt }t∈[0,1] of diffeomorphisms on M defined by thedifferential equationdφt (x) = Xt (φt (x))
dtwith initial condition φ0 = idM . The time 1-map φ1 of this flow is called a Hamiltonian diffeomorphism. The set of all Hamiltonian diffeomorphisms is denoted byHam(M, ω). We can check that Ham(M, ω) is a subgroup of the identity component
Diff0 (M ) of the diffeomorphism group of M . By definition, a Hamiltonian diffeomorphism φ of M preserves the symplectic structure (i.e., φ∗ ω = ω). Therefore, if
L is a Lagrangian submanifold of M , then φ(L) is also Lagrangian.
Hereafter we restrict our attention to Kähler manifolds to introduce the volume
functional. Let (M, ω, J) be a closed connected Kähler manifold, and L its Lagrangian submanifold. We set Ham(L) = {φ(L) | φ ∈ Ham(M, ω)}. If the volume
functional on Ham(L) is stationary at L, then L is called Hamiltonian minimal.
A Hamiltonian minimal Lagrangian submanifold is said to be Hamiltonian stable
if any second variation of the volume functional at L on Ham(L) is non-negative.
Moreover, if L has the least volume in Ham(L), then L is said to be Hamiltonian
volume minimizing. These notions are first introduced by Y. G. Oh. In his paper[16], he studied the second variational formula for minimal Lagrangian submanifolds
in Kähler manifolds. And he showed a following local criterion.

INTEGRAL GEOMETRY AND HAMILTONIAN VOLUME MINIMIZING PROPERTY

9

Proposition 4.1 (Oh). Let (M, ω, J) is a Kähler-Einstein manifold with Einsteinconstant c and L be its minimal Lagrangian submanifold. Then L is Hamiltonian
stable if and only if λ1 ≥ c, where λ1 denotes the first eigenvalue of Laplace-Beltrami
operator −∆ acting on C ∞ (M ).
This says that local Hamiltonian stabilities of minimal Lagrangian submanifolds
of Kähler-Einstein manifolds are only depend on their properties in Riemanniangeoemtry, namely first eigenvalue of Laplacian. From this proposition, we can have
many Hamiltonian stable minimal Lagrangian submanifolds (See [1] and [16] forreferences).
On the other hand, we have few examples of Hamiltonian volume minimizing
Lagrangian submanifolds. Trivial examples are special Lagrangian submanifolds in
Ricci-flat Kähler manifolds. In fact, they are calibrated submanifolds and haveminimun volume in their homology classes. But, in general, it is difficult to check
whether a minimal Lagrangian submanifold L is Hamiltonian volume minimizing
or not, when L is not a calibrated submanifold. At the end of his paper, Y.G.Oh
posed a following conjecture:
Conjecture 4.2 (Oh). Let M be a Kähler-Einstein manifold with an involutive
anti-holomorphic isometry τ . Suppose that the fixed point set of τ
L := Fix τ
is also a compact Einstein manifold with positive Ricci curvature. Then L is Hamiltonian volume minimizing.
For this conjecture, Kleiner and Oh gave only one non-trivial example RP n ⊂CP n . Recently, Goldstein [7] obtained stronger statement for the volume minimality of real projective spaces in CP n .
Now we set up our problem for Hamiltonian volume minimizing properties of
Lagrangian submanifolds as following.
Problem 4.3. Does a real form of a Hermitian symmetric space of compact typewhich is Hamiltonian stable have Hamiltonian volume minimizing property or not?
Hermitian symmetric spaces of compact type are important examples of KählerEinstein manifolds with positive Ricci curvature. A real form is a totally geodesic Lagrangian submanifold defined by the fixed point set of an involutive antiholomorphic isometry in a Hermitian symmetric space. Hamiltonian stabilities ofreal forms were already determined completely by Amarzaya and Ohnita [1]. So weare interested in their Hamiltonian volume minimizing properties. The reason why
we set the problem like this is that we want to apply the following Arnold-Givental
inequality in the Lagrangian intersection theory. That is a global investigation in
symplectic geometry.
Theorem 4.4 (Oh [17], [18] and [19]). Let (M, ω) be a compact symplectic manifold such that there exists an integrable almost complex structure J for which thetriple (M, ω, J) becomes a compact Hermitian symmetric space. Let L = Fix τ be
the fixed point set of an anti-holomorphic involutive isometry τ on M . Assume
that the minimal Maslov number of L is greater than or equal to 2. Then for any
Hamiltonian diffeomorphism φ of M such that L and φ(L) intersect transversely,10

TAKASHI SAKAI

the inequality(4.1)](L ∩ φ(L)) ≥holds.

dim
XL

rankHi (L, Z2 )i=0

Now we state our conjecture.
Conjecture 4.5 (IOS). Let (G/K, ω, J) be a Hermitian symmetric space of compact type and L = Fix τ its real form. If L is Hamiltonian stable, then we haveZ
Cvol(L)vol(N ) ≥](L ∩ gN )dµG (g)
G

where

P

rankH∗ (L; Z2 ))vol(G)
vol(L)2
for any Lagrangian submanifold N .
C=(The assumption that L is Hamiltonian stable is, of course, a necessary condition
for L to be Hamiltonian volume minimizing.
Proposition 4.6. Under the same assumption as Conjecture 4.5, if Conjecture 4.5
is true, then the totally geodesic Lagrangian submanifold L = Fix τ whose minimal
Maslov number is greater than or equal to 2 in (G/K, ω, J) is Hamiltonian volume
minimizing.
Proof. By Theorem 4.4 and Conjecture 4.5, we haveZ
Cvol(L)vol(φ(L)) ≥](L ∩ g ◦ φ(L))dµG (g)
G

≥

Z

dim
XL

rankHi (L, Z2 )dµG (g)G i=0= vol(G)dim
XL

rankHi (L, Z2 )i=02

= Cvol(L) .
Hence,

vol(φ(L)) ≥ vol(L).
Remark 4.7. The method we use here, a combination of Lagrangian intersection
theorems in symplectic geometry and Poincaré formulas in integral geometry, wasfirst pointed out by Kleiner and Oh.
Now we shall discuss the above conjecture in the case S 1 (1) × S 1 (1) ⊂ S 2 (1) ×S (1).
Let G/K be a Riemannian homogeneous space. The linear isotropy representation induces an action of K on the Grassmannian manifold Gp (To (G/K)) consistingof all p-dimensional subspaces in the tangent space To (G/K) at o in a natural way.
Although σK (Tx⊥ L, Ty⊥ N ) is defined as an integral on K, we can consider that it isdefined as an integral on an orbit of K-action on the Grassmannian manifold. So
2

INTEGRAL GEOMETRY AND HAMILTONIAN VOLUME MINIMIZING PROPERTY

11σK (·, ·) can be regarded as a function defined on the product of the orbit spacesof such K-actions. In the case where G/K is a real space form, σK (Tx⊥ L, Ty⊥ N ) isconstant since K acts transitively on the Grassmannian manifold. This implies that
the Poincaré formula is expressed as a constant times of the product of the volumesof N and L. In general, such K-actions are not transitive. However, if we candefine an invariant for orbits of this action, which is called an isotropy invariant,then using this we can express the Poincaré formula explicitly.
Next, we define isotropy invariants for surfaces in S 2 × S 2 and give an explicit
Poincaré formula for its Lagrangian surfaces.
Let G be the identity component of the isometry group of S 2 × S 2 , that is,
G = SO(3) × SO(3). Then the isotropy group K at o = (p1 , p2 ) in S 2 × S 2 isisomorphic to SO(2) × SO(2), and S 2 × S 2 can be identified with a coset space
G/K. Assume thet G is equipped with an invariant metric normalized so that G/K
becomes isometric to the product of unit spheres. We decompose the tangent space
To (G/K) as
To (G/K) = Tp1 (S 2 ) ⊕ Tp2 (S 2 ).
We consider the oriented 2-plane Grassmannian manifold G̃2 (To (G/K)). Takean origin Vo := Tp1 (S 2 ) and express G̃2 (To (G/K)) as a coset spaceG̃2 (To (G/K)) = SO(4)/(SO(2) × SO(2)) =: G0 /K 0 .
Now we study the K-action on G̃2 (To (G/K)) and define isotropy invariants. In
this case the actions of K and K 0 on G̃2 (To (G/K)) are equivalent by Ad : K →K 0 . Therefore it suffices to consider the orbit space of the isotropy action ofG̃2 (To (G/K)). It is well known that the orbit space of the isotropy action of asymmetric space of compact type can be identified with a fundamental cell of a
maximal torus. Hence we can define the isotropy invariant by a coordinate of a
maximal torus. We denote by g0 and k0 the Lie algebra of G0 and K 0 , respectively.
Then we have a canonical direct sum decomposition g0 = k0 ⊕ m0 , where
½µ
¶ ¯
¾
¯
O
X
¯
m0 =
X
∈
M
(R)
.
2
¯
−t X O
We take a maximal abelian subspace a0 of m0 as follows:
½µ
¶ ¯
µ
¶
¾
¯
O
X
θ1 0
0
¯
a =
, θ 1 , θ2 ∈ R .
¯X=−t X O
0 θ2Then the set of positive restricted roots of (g0 , k0 ) with respect to a0 is∆ = {θ1 + θ2 , θ1 − θ2 }.

So we have a fundamental cell C of a0 :
½
µ
¶ ¯
µ
¯
O
X
θ1
¯
C= Y =X=¯−t X O
0

0
θ2

¶

,

0 ≤ θ1 + θ2 ≤ π
0 ≤ θ1 − θ2 ≤ π¾.

Thus the isotropy invariants in this case are given by θ1 +θ2 and θ1 −θ2 . S 2 ×S 2 has
a Kähler form ω0 ⊕ ω0 , where ω0 denotes the standard Kähler form of S 2 ∼= CP 1 .
It is easy to see that the geometric meaning of θ1 − θ2 is the Kähler angle of 2dimensional subspace ExpY of To (G/K). On the other hand, S 2 × S 2 has anotherKähler structure ω0 ⊕ (−ω0 ). We can also check that θ1 + θ2 is the Kähler angle of
ExpY with respect to ω0 ⊕ (−ω0 ).
Using these isotropy invariants we obtain the following formula from Theorem
2.1.

12

TAKASHI SAKAI

Theorem 4.8 (IOS [9]). Let N and L be Lagrangian surfaces in S 2 × S 2 . We
assume that L is a product of curves in S 2 . Then we haveZ
Z](L ∩ gN )dµG (g) = 4vol(L)length(Ellip(sin2 θx , cos2 θx ))dµN (x),G

N

where 2θx − π/2 is the Kähler angle of Tx⊥ N with respect to ω0 ⊕ (−ω0 ) andEllip(α, β) denotes an ellipse defined by
y2x2+ 2 = 1.
2αβTheorem 4.8 yields the following immediately.

Corollary 4.9. Let N and L be surfaces of S 2 × S 2 . Suppose that N is Lagrangianand L is a product of curves in S 2 . Then the following inequality holds:
Z(4.2)](L ∩ gN )dµG (g) ≤ 16vol(N )vol(L).
SO(3)×SO(3)Moreover, the equality holds if and only if the Lagrangian surface N is also a productof curves in S 2 .
Let L := S 1 (1) × S 1 (1), then the mimimal Maslov number of L is 2 andP24 · (8π 2 · 8π 2 )i=0 rankHi (L, Z2 )vol(SO(3) × SO(3))== 16.
vol(L)2(4π 2 )2
Thus we have the conclusion by Proposition 4.6:
Theorem 4.10 (IOS [9]). Let L := S 1 (1) × S 1 (1) be a totally geodesic Lagrangian
torus in (S 2 (1) × S 2 (1), ω0 ⊕ ω0 ). Then for any Hamiltonian diffeomorphism φ ∈Ham(S 2 × S 2 ), we havevol(φ(L)) ≥ vol(L).

Moreover, if vol(φ(L)) = vol(L) holds, then there exists an isometry g of S 2 (1) ×
S 2 (1) such that φ(L) = gL.
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Depertment of Mathematics, Tokyo Metropolitan University, 1-1 Minami-osawa, Hachiojishi, Tokyo, 192-0397 Japan
E-mail address: tsakai@comp.metro-u.ac.jp