The 9th International Workshop on Differential Geometry
Kyungpook National University
Saturday 13 November, 2004

Integral geometry and Hamiltonian
volume minimizing property, II
Application of integral geometry
to variational problems

Depertment of Mathematics,
Tokyo Metropolitan University
Takashi Sakai
tsakai@comp.metro-u.ac.jp1

Hamiltonian stability of Lagrangian submanifolds
S 1 ⊂ S 2 : equator
S 1 is an “unstable”
minimal submanifold
But!
L : area-bisecting curve⇒ vol(L) ≥ vol(S 1)(1) #(S 1 ∩ L) ≥ 2
Z(2)

SO(3)

#(c ∩ gc0)dµ(g) = 4vol(c)vol(c0)4vol(S 1)vol(L) =≥ZZSO(3)SO(3)#(S 1 ∩ gL)dµ(g)#(S 1 ∩ gS 1)dµ(g)= 4vol(S 1)22

(M 2m, ω) : symplectic manifold
Lm ⊂ M : Lagrangian submanifold(i.e. ω|T L ≡ 0)
F : M × [0, 1] → R, C ∞Ã Xt ∈ X(M ) : Hamiltonian vector fieldi(Xt)ω = dFt

Ã φt ∈ Diff(M ) s.t.
dφt= Xt ◦ φ t ,dt

φ0 = idM

Definitionφ ∈ Diff(M ) : Hamiltonian isotopy∃ F : M × [0, 1] → R

s.t. φ = φ1Ham(M, ω) := {Hamiltonain isotopy of M }3

(M, J, ω) : Kähler manifold
L ⊂ M : Lagrangian submanifold
Definition
L : Hamiltonian minimal¯

def d ¯⇐⇒ ¯¯vol(φtL) = 0
dt t=0∀ {φ }
t −²<t<² ⊂ Ham(M, ω)

L : Hamiltonian stable¯

d2 ¯¯vol(φtL) ≥ 0⇐⇒ 2 ¯dt t=0def∀ {φ }
t −²<t<² ⊂ Ham(M, ω)

L : Hamiltonian volume minimizingdef⇐⇒ vol(φL) ≥ vol(L)∀ φ ∈ Ham(M, ω)

4

Theorem (Oh)(M, J, ω) : Kähler-Einstein manifoldRicci form ρ = cωL ⊂ M : minimal Lagrangian submanifold
Then, L is Hamiltonian stable ⇔ λ1(L) ≥ cλ1(L) : first eigenvalue of the Laplacian on L
Examples of Hamiltonian stable minimal
Lagrangian submanifold(1) RP n, Clifford torus ⊂ CP n

(Oh)(2) G/K : cpt Hermitian symmetric space
L ⊂ G/K : real form, Einstein(Oh)

H-stabilities of non-Einstein cases were
determined by Amarzaya-Ohnita(3) L ⊂ CP n : cpt minimal Lagrangian submanifold with parallel 2nd fundamental form
(Amarzaya-Ohnita)

5Conjecture (Oh)M : Kähler-Einstein manifold
τ : anti-holomorphic involutive isometry
L := Fix(τ )Assume that L is also Einstein whose Ricci curvature is positive⇒ vol(φL) ≥ vol(L)(∀φ ∈ Ham(M ))Theorem (Kleiner-Oh)
L := RP n ⊂ CP n
⇒ vol(φL) ≥ vol(L)(∀φ ∈ Ham(CP n))

Theorem (Goldstein) arXiv:math.DG0406334(1) RP 2m ⊂ CP n minimizes volume among theisotropic submanifold in its Z2 homology
class.
(2) RP 2m−1 ⊂ CP n is Hamiltonian volume minimizing.

6Theorem (Iriyeh-Ono-S.)
L := S 1(1) × S 1(1) ⊂ S 2(1) × S 2(1)⇒ vol(φL) ≥ vol(L)(∀φ ∈ Ham(S 2 × S 2))

Moreover, vol(φL) = vol(L)⇒∃ g ∈ Isom(S 2(1) × S 2(1)) s.t. φL = gL

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Application of integral geometryN ⊂ G/K : submanifold of type VoIf M ⊂ G/K : submanifold which satisfies(1) dim M + dim N = dim(G/K)(2) for almost all x ∈ M
σK (Tx⊥M, Vo⊥)= max{σK (V, Vo⊥) | V ∈ Gp(To(G/K))} =: A(3) for almost all g ∈ G#(M ∩ gN ) = HIN(M, N )then, for ∀M 0 ⊂ G/K Z2 homologous to M ,Z

1vol(M ) =σK (Tx⊥M, Vo⊥)dµ(x)A M
Z1#(M ∩ gN )dµ(g)=Avol(N ) GZ
1≤#(M 0 ∩ gN )dµ(g)Avol(N ) GZ
1=σK (Tx⊥M 0, Vo⊥)dµ(x)A M0≤ vol(M 0)8Lê Hông Vân
Gr (Fr+s) ⊂ Gr (Fr+n) minimizes volume in its

Z2 homology class.

(F = R, C, H)

Liu
The Pontryagin cycle Pi ⊂ SO(n) minimizes
volume in its Z2 homology classes.
(i = 1, 2, . . . , n − 1)
TasakiG/K : compact simply connected irreducible
symmetric spaceκ : maximal sectional curvature
The Helgason sphere S p(κ) ⊂ G/K minimizes
volume in the non-null homotopic class9Theorem (Goldstein) arXiv:math.DG0406334(1) RP 2m ⊂ CP n minimizes volume among theisotropic submanifold in its Z2 homology
class.
(2) RP 2m−1 ⊂ CP n is Hamiltonian volume minimizing.
Proof.
(1) even dimensional case#(RP 2m ∩ CP n−m) = 1(a) odd dimensional case
S 2m+1⊃ ψt(S 2m−1)

2:1CP nSt⊂S 2n+32:1?

?

-⊃ φt(RP 2m−1)?

Lt?

⊂CP n+1(St =)¯2m−1¯ x ∈ ψt(S),n+1(cos θx, sin θ) ∈ C⊕ C ¯¯0≤θ≤π10

Hamiltonian volume minimality
Problem
Does a real form of a Hermitian symmetricspace of compact type which is Hamiltonian
stable have Hamiltonian volume minimizing
property?
Arnold-Givental inequality
G/K : cpt Hermitian symmetric space
L ⊂ G/K : real form
minimal Maslov number ≥ 2⇒ #(L ∩ φL) ≥dimXL

rankHi(L, Z2)

i=0(∀φ ∈ Ham(G/K), L t φL)

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If, for all Lagrangian submanifold N of G/K,Z
G#(L ∩ gN )dµG(g) ≤ Cvol(L)vol(N )vol(G) X
where C =rankHi(L, Z2),vol(L)2 i
then we have
ZCvol(L)vol(φL) ≥≥#(L ∩ g ◦ φ(L))dµ(g)

ZG X
G i

rankHi(L, Z2)dµ(g)= vol(G)X

rankHi(L, Z2).

i

Hence

vol(φL) ≥ vol(L).

12Formulation due to Howard
ZGZ#(M ∩gN )dµ(g) =M ×N

σK (Tx⊥M, Ty⊥N )dµ(x, y)

For V p, W q ⊂ To(G/K)
Z

σK (V, W ) =K

σ(V, k∗−1W )dµ(k)σK is a function on Gp(To(G/K))×Gq (To(G/K))
invariant under K

Ã σK is a function on the orbit space ofK y Gp(To(G/K))
Our program
Express σK concretely using the isotropy invariants, and obtain an estimation by the inequality.
ExampleG/K = CP n
U (n) y SO(2n)/SO(p)×SO(2n−p) = G̃p(To(G/K))
Hermann action

Ã multiple Kähler angle
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The case of S 2 × S 2
G = SO(3) × SO(3),K = SO(2) × SO(2)To(G/K) = To1 (S 2) ⊕ To2 (S 2)
K y G̃2(To(G/K)) = SO(4)/SO(2) × SO(2)=: G0/K 0
Ad(K) = K 0
Decompose g0 = k0 ⊕ m0, and take a maximalabelian subspace a0 in m0Ã!
)! ¯¯O X
θ1 0¯X=, θ1, θ2 ∈ Ra0 =¯¯
0 θ2−t X O
then the positive roots ∆ w.r.t. a0 is(Ã∆ = {θ1 + θ2, θ1 − θ2}14

LemmaN, L ⊂ S 2 × S 2 : Lagrange surfaces
in particular, L : product of curves in S 2⇒ σK (Tx⊥N, Ty⊥L)= 4 length(Ellip(sin2 θx, cos2 θx))where 2θx − π/2 is the Kähler angle of Tx⊥Nw.r.t. ω0 ⊕ (−ω0)
proof.
σK (Tx⊥N, Ty⊥L)
Z==

ZSO(2)×SO(2)SO(2)×SO(2)P :2^

ku1 ∧ u2 ∧ k−1(v1 ∧ v2)kdµ(k)|hk1(u01 ∧ u02), k2(v1 ∧ v2)i|dk1dk2

To(G/K) → W ⊃ k2(v1 ∧ v2)

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lemma4π ≤ σK (Tx⊥N, Ty⊥L) ≤ 16
In the 2nd inequality,“=” ⇐⇒ N is also a product of curves in S 2
In this casevol(G) X(8π 2)2
rankHi(L, Z2) =· 4 = 16
222
vol(L) i(4π )Open problems(1) Problem(2) Is a product of Hamiltonian volume minimizing Lagrangian submanifolds again Hamiltonian volume minimizing?
(3) Classify all Hamiltonian volume minimizing
Lagrangian submanifolds in Hermitian symmetirc spaces.
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