.

.

Geometry of Lagrangiansubmanifolds related to

isoparametric hypersurfaces

Yoshihiro OHNITA

Osaka City University Advanced Mathematical Institute (OCAMI)& Department of Mathematics, Osaka City University

2014 ICM Satellite Conference onReal and Complex Submanifolds

(The 18th International Workshop on Differential Geometry)NIMS, Daejion, Korea

August 11, 2014

[Submanifolds in Riemannian manifolds](Higher dimensional generalization of curves and surfaces inEuclidean space) It is one of the most fundamental objects inDifferential Geometry.

(M, g): a Riemannian manifold with a Riem. metric g.Let

φ : N −→ M

be a smooth immersion or embedding.Our main research interests are

...1 Deformations and Moduli Spaces for Submanifolds

...2 Geometric Variational Problems for Submanifolds

...3 Lie Group Theoretic Methods in Finite and InfiniteDimensions

[Lagrangian Submanifolds in Symplectic Manifolds]

φ : L −→ (M2n, ω)symplectic mfd.

immersion

.Definition..

.

“Lagrangian immersion”⇐⇒def

...1 φ∗ω = 0(⇔ φ : “isotropic ”)

...2 dim L = n

φ−1TM/φ∗TL � T∗L linear isom.

∈ ∈

v 7−→ αv := ω(v, · )

φt : L −→ (M2n, ω ) immersion with φ0 = φ

Vt :=∂φt

∂t∈ C∞(φ−1

tTM)

“Lagrangian deformation” ⇐⇒defφt : Lagr. imm. for∀t

⇐⇒ αVt ∈ Z1(L)closed

for ∀t

“Hamiltonian deformation” ⇐⇒defαVt ∈ B1(L)

exactfor ∀t

Hamil. deform. =⇒ Lagr. deform.The difference between Lagr. deform. and Hamil. deform. isequal to H1(L; R) � Z1(L)/B1(L).

.Simplest Example..

.

(M, ω) = (R2, dx ∧ dy): a plane equipped with the std. areaform

In this case,

L ⊂ (M, ω) 1-dim. Lagr. submfd.

is nothing but a plane curve.

L ⊂ (M, ω) 1-dim. compact Lagr. submfd.

is nothing but a compact plane curve.

.Characterization of Hamiltonian Deformations in terms ofisomonodromy deformations..

.

φt : L −→ M : Lagr. deform.

Suppose1γ[ω] integral (∃ γ).

{φt} : Hamil. deform.

⇕

A family of flat connections{φ−1

t∇}

has same holonomyhomom.π1(L) −→ U(1)(“isomonodromy deformation”)

.

.

φ−1t

E −−−−−→ ∃(E,∇)φ−1

t∇

flat

y yL

φt−−−−−→ (M, ω)

[Lagrangian Submanifolds in Kahler Manifolds]

(M, ω, J, g) : Kahler manifoldφ : L −→ M : Lagr. imm.

B : second fundamental form of φ↕

S : a symmetric 3-tensor field of degree 3 on LS(X , Y , Z) := ω(B(X , Y), Z) (∀ X , Y , Z ∈ TL)

H : mean curvature vector field of φ↕αH : “mean curvature form”of φ

.Proposition (Dazard)... dαH = φ∗ρM where ρM : Ricci form of M.

[Homogeneous Lagrangian Submanifolds of Kahler manifolds(in the sense of Riemannian Geometry)]

.Definition..

.

(M, ω, J, g) : Kahler manifold.K ⊂ Aut(M, ω, J, g): connected Lie subgroup,

L = K · x ⊂ M: a Lagrangian orbit“homogeneous Lagrangian submanifold ”

[Hamiltonian minimality and stability (Y. G. Oh (1990)]Suppose L : compact (without boundary).

φ : Hamiltonian minimal (or H-minimal)⇐⇒def

∀φt : L −→ M Hamil. deform. with φ0 = φ

d

dtVol (L , φ∗

tg)∣∣∣∣t=0

= 0

⇐⇒ “Hamiltonian minimal equation (HME)”

δαH = 0

.Proposition..

.L : compact homog. Lagr. submfd. of Kahler mfd. M=⇒ L is Hamiltonian minimal

Assume φ : H-minimal.

φ : “Hamiltonian stable ”⇐⇒def

∀ {φt} : Hamil. deform. of φ0 = φ

d2

dt2Vol (L , φ∗

tg)∣∣∣∣t=0≥ 0

.The Second Variational Formula..

.

d2

dt2Vol (L , φ∗

tg)∣∣∣∣t=0

=∫L

(⟨△1

Lα, α⟩ − ⟨R(α), α⟩ − 2⟨α ⊗ α ⊗ αH , S⟩+ ⟨αH , α⟩2

)dv

where

α := α ∂φt∂t

∣∣∣∣t=0

∈ B1(L)

⟨R(α), α⟩ :=n∑

i,j=1

RicM(ei , ej)α(ei)α(ej) {ei} : o.n.b. of TpL

S(X , Y , Z) := ω(B(X , Y), Z) sym. 3-tensor field on L

X : holomorphic Killing vector field of M=⇒ αX = ω(X , ·) is closed=⇒ αX = ω(X , ·) is exact if H1(M,R) = {0}.

If M is simply connected, more generally H1(M,R) = {0},each holomorphic Killing vector field of M generates avolume-preserving Hamiltonian deformation of φ.

Def. Such a Hamiltonian deformation of φ is called trivial.

Assume φ : H-minimal.φ : “strictly Hamiltonian stable ”⇐⇒def

(1) φ is Hamiltonian stable(2) The null space of the second variation on Hamiltoniandeformations coincides with the vector subspace induced bytrivial Hamiltonian deformations of φ. That is, n(φ) = nhk (φ).Here n(φ) := dim[ the null space ] andnhk (φ) := dim{φ∗αX | Xa holomorphic Killing vector field of M}.Remark.φ : “Hamiltonian rigid ” (Yng-Ing Lee),⇐⇒def

φ satisfies the second condition (2).

There are nice results on deformation and existence ofHamiltonian minimal Lagrangian submanifolds by Joyce-Y. I.Lee-Schoen, Bettiol-Piccione-Siciliano (equivariant case)related to this condition.

Assume M : Einstein-Kahler manifold of Einstein constant κ.L ↪→ M cpt. minimal Lagr. submfd. (i.e. αH ≡ 0)

Then

L is Hamiltonian stable ⇐⇒ λ1 ≥ κ

Hereλ1 : the first (positive) eigenvalue of the Laplacian of Lon C∞(L).

(B. Y. Chen - T. Nagano - P. F. Leung, Y. G. Oh)

.[Fact] (A. Ros, F. Urbano, Hajime Ono, Amarzaya - O.)..

.

Assume M : cpt. homog. Einstein - Kahler mfd. with κ > 0.L ↪→ M cpt. minimal Lagr. submfd.

Thenλ1 ≤ κ

λ1 = κ ⇐⇒ L is Hamil. stable.

.Question...

.

What Lagrangian submanifolds are H-minimal? Moreover,What H-minimal Lagrangian submanifolds are H-stable?

.Proposition..

.L : compact homog. Lagr. submfd. of Kahler mfd. M=⇒ L is H-minimal

Theoretically it is possible to analyze the second variations ofcompact homogeneous Lagrangian submanifolds by HarmonicAnalysis over Compact Homogeneous Spaces in order todetermine their Hamiltonian stability.

A new and interesting construction of compact H-minimalLagrangian submanifolds in Cn, CPn and toric Kahler manifoldsis studied by the method of toric topologyin A. E. Mironov and T. Panov.

[Classification theorems of compact H-stable minimalLagrangian submanifolds in specific Kahler manifolds].Proposition..

.

Any compact Hamiltonian stable minimal Lagrangiansubmanifold L immersed in CPn must be π1(L) , {1}, morestrongly H1(L;Z) , {1}..Theorem (F. Urbano)..

.Any Hamiltonian stable minimal Lagrangian torus L immersedin CP2 must be the Clliford minimal torus.

.Theorem (Castro-Urbano)..

.

Any Hamiltonian stable minimal Lagrangian torus L immersedin S2 × S2 = Q2(C) must be a totally geodesic Lagrangiantorus S1 × S1 of S2 × S2.

Not so many examples of compact H-stable Lagrangiansubmanifolds are known..Example 0...

.

A circleS1(r) ⊂ C = R2

with radius r > 0 on the Euclidean plane

...1 cpt. 1-dim. H-minimal (not minimal!) Lagr. submfd. in R2

...2 globally strictly H-stable(by the isoperimetric inequality on R2)

[Examples of compact Hamiltonian stable H-minimalLagrangian submanifolds in Cn+1]

...1 Circles S1(r) in C. S , 0, αH , 0,∇S = 0

...2 Q2,n+1(R) ⊂ Cn+1, U(p)/O(p) ⊂ Cp(p+1)/2, U(p) ⊂ Cp2,

U(2p)/Sp(p) ⊂ Cp(2p−1), T1 ·E6/F4 ⊂ C27 (Amarzaya - O.)S , 0, αH , 0,∇S = 0

...3 Their Riemannian products L = L0 × · · · × Lk .(Amarzaya - O.)

S , 0, αH , 0,∇S = 0

They all are strictly Hamiltonian stable.

Letπ : Cn+1 ⊃ S2n+1 −→ CPn

denote the Hopf fibration.

[Examples of compact Hamiltonian stable H-minimalLagrangian submanifolds in CPn]

...1 Real projective subspaces RPn ⊂ CPn S = 0

...2 SU(p)/SO(p) · Zp ⊂ CP(p−1)(p+2)/2, SU(p)/Zp ⊂ CPp2−1,SU(2p)/Sp(p) · Z2p ⊂ CP(p−1)(2p+1)

E6/F4 · Z3 ⊂ CP26

(Amarzaya - O., Tohoku Math. J. 2003)S , 0,∇S = 0, αH = 0,

...3 L = π(L), L is an example of the previous page(Amarzaya - O.) .S , 0,∇S = 0, generically αH , 0

...4 ρ3(SU(2))[z30+ z3

1] ⊂ CP3 River Chiang Lagrangian

∇S , 0, αH = 0(L. Bedulli - A. Gori, Transf. Groups 2007. O., OsakaJ. Math. 2007)

...5 (SU(3) × SU(3))/T2 · Z4 ⊂ CP5 ∇S , 0, αH = 0(Petrecca-Podesta, Tohoku Math. J. 2012)

They all are also strictly Hamiltonian stable.

[Real forms of cpt. Herm. symm. sp.(classified by Professor Masaru Takeuchi)]M: cpt. irred. Herm. sym. sp. of rank ≥ 2

(M. Takeuchi, Y. G. Oh, Amarzaya-O.)L real form of M ⇐⇒ L tot. geod. Lagr. submfd. of M

(L ,M)tot. geod.

Lagr. submfd.

=

(Qp+1,q+1(R) = (Sp × Sq)/Z2,

Qp+q(C)) (q − p ≥ 3)(U(2p)/Sp(p), SO(4p)/U(p)) (p ≥ 3),(T · E6/F4, E7/T · E6).

⇐⇒ L is NOT Hamil. stable.

[Concrete Examples of compact Hamiltonian volumeminimizing Lagrangian submanifolds]

...1 Circles S1(r) in C.

...2 Great or small circles S1(r) in S2 = CP1.

...3 Real projective subspaces RPn ⊂ CPn (Kleiner-Y. G. Oh).

...4 Totally geodesic Lagrangian torus S2 × S2 inS2 × S2 = Q2(C)(H. Iriyeh-Hajime Ono-Takashi Sakai).

...5 Totally geodesic Lagrangian sphere Sn ⊂ Qn(C)(H. Iriyeh-Takashi Sakai-H. Tasaki).Remark: It is a calibrated submanifold if n is even, but it isnot so if n is odd.

Y. G. Oh’s Conjecture (Math. Z. 1993).Are the Clifford tori Tn+1 = S1(r0) × · · · S1(rn) ⊂ Cn+1 andTn = π(Tn+1) ⊂ Cn+1 Hamiltonian volume minimizing or not?

.Complex Hyperquadrics..

.Qn(C) := {[z] ∈ CPn+1 | z2

0+ z2

1+ · · ·+ z2

n+1= 0} ⊂ CPn+1

.Real Grassmann manifolds of oriented 2-planes..

.

Gr2(Rn+2) ⊂ Λ2Rn+2

:={ [W] | [W] is an oriented 2-dim. vect. subsp. of Rn+2 }

Identification

Gr2(Rn+2) ∋ [W] ←→ [a +

√−1b] ∈ Qn(C)

where {a, b}: an orth. basis of [W] compatible with its ori.

Qn(C) � Gr2(Rn+2) � SO(n + 2)/SO(2) × SO(n)

is a cpt. Herm. symm. sp. of rank 2.

.Example 3...

.

Consider an algebraic equation

F(x0, · · · , xn+1) := −(x0)2−· · ·−(xp)

2+(xp+1)2+· · ·+(xn+1)

2 = 0

where p, q are non-negative integers with p + q = n. Then

Nn := Sn+1(1) ∩ F−1(0) � Sp × Sq Clifford hypersurface.

Moreover we observe that each point of Nn defines a point ofQn(C) and the map G : Nn → Qn(C) is a 2 : 1 immersion ifp, q ≥ 1 and an embedding if p = 0 or q = 0.

...1 The image G(Nn) is a cpt. totally geodesic (S = 0) Lagr.submfd. in Qn(C).

...2 G(Nn) is Hamiltonian stable if and only if |p − q| < 3.

[Generalization of Example]Consider a homogeneous polynomial F : Rn+2 −→ R of degreeg (Cartan-Munzner polynomial) satisfying the PDE∆ F = c rg−2,

∥dF∥2 = g2r2g−2,

where c := g2 (m2 − m1)/2 and r = ∥x∥2 (x ∈ Rn+2)s. t.

Nn = Sn+1(1) ∩ F−1(s) (∃ s ∈ (−1, 1)).

is a oriented smooth hypersurface embedded in Sn+1(1) withconstant principal curvatures.Such a family of hypersurfaces is called an isoparametric family(E. Cartan)..Theorem (Munzner)...g must be 1, 2, 3, 4 or 6.

.Gauss map construction of Lagrangian Submanifolds inQn(C)... Qn(C) � Gr2(Rn+2) � SO(n + 2)/SO(2) × SO(n)

.Oriented hypersurface in a sphere..

.

Nn ↪→ Sn+1(1) ⊂ Rn+2

x : the position vector of points of Nn

n : the unit normal vector field of Nn in Sn+1(1)

.“Gauss map”..

.

G : Nn ∋ p 7−→ [x(p) +√−1n(p)] = x(p) ∧ n(p) ∈ Qn(C)

is a Lagrangian immersion.

.Lagrangian Submanifolds in Complex Hyperquadrics... Qn(C) � Gr2(Rn+2) � SO(n + 2)/SO(2) × SO(n)

.Oriented hypersurface in a sphere..

.

Nn ↪→ Sn+1(1) ⊂ Rn+2

x : the position vector of points of Nn

n : the unit normal vector field of Nn in Sn+1(1)

.“Gauss map”..

.

G : Nn ∋ p 7−→ [x(p) +√−1n(p)] = x(p) ∧ n(p) ∈ Qn(C)

is a Lagrangian immersion.

.Proposition...Deformation of Nn = Hamiltonian deformation of G

.Lagrangian Submanifolds in Complex Hyperquadrics... Qn(C) � Gr2(Rn+2) � SO(n + 2)/SO(2) × SO(n)

.Oriented hypersurface in a sphere..

.

Nn ↪→ Sn+1(1) ⊂ Rn+2 with constant principal curvatures(“isoparametric hypersurface”)

.“Gauss map”..

.

G : Nn ∋ p 7−→Larg. imm.

[x(p) ∧ n(p)] ∈ Qn(C)

Nn −→ G(Nn) � Nn/Zg ↪→ Qn(C)cpt. embedded minimal Lagr. submfd

.

.

Here g := #{distinct principal curvatures of Nn},

m1 ≤ m2 : multiplicities of the principal curvatures.

.Lagrangian Submanifolds in Complex Hyperquadrics... Qn(C) � Gr2(Rn+2) � SO(n + 2)/SO(2) × SO(n)

.Oriented hypersurface in a sphere..

.

Nn ↪→ Sn+1(1) ⊂ Rn+2 with constant principal curvatures(“isoparametric hypersurface”)

.“Gauss map”..

.

G : Nn ∋ p 7−→Lagr. imm.

x(p) ∧ n(p) ∈ Qn(C)

Nn −→Zg

Ln = G(Nn) � Nn/Zg ↪→ Qn(C)

cpt. embedded minimal Lagr. submfd

.

.

Here g := #{distinct principal curvatures of Nn},

m1,m2 : multiplicities of the principal curvatures.

.Lagrangian Submanifolds in Complex Hyperquadrics..

.

Joint works with Hui Ma (Tsinghua University, Bejing)

L ⊂ Qn(C)

...1 Construction of a nice class of certain compact minimalLagrangian submanifolds L in Qn(C) by the Gauss mapsof isoparametric hypersurfaces in the unit spheres.

...2 Classification of all compact homogeneous Lagrangiansubmanifolds L in Qn(C).

...3 Determination of the (strictly) Hamiltonian stability for allcompact homogeneous minimal Lagrangian submanifoldsL in Qn(C).

Construction of isoparametric hypersurfaces:Principal orbits of the isotropy representations ofRiemannian symmetric pairs (U,K) of rank 2 =⇒All homogeneous isopara. hypersurf. (Hsiang-Lawson,R. Takagi-T. Takahashi)Algebraic construction of Cartan-Munzner polynomials byrepresentations of Clifford algebras in case g = 4(Ozeki-Takeuchi, Ferus-Karcher-Munzner) =⇒So many non-homogeneous isopara. hypersurf.

Classification of isoparametric hypersurfaces:g = 1 : Nn = Sn, a great or small sphere;g = 2 : Nn = Sm1 × Sm2 , (n = m1 + m2, 1 ≤ m1 ≤ m2),Clifford hypersurfaces;g = 3: Nn is homog., Nn =

SO(3)Z2+Z2

, SU(3)

T2 , Sp(3)

Sp(1)3 , F4

Spin(8)

(E. Cartan);g = 6: Nn is homog.

m1 = m2 = 1: homog. (Dorfmeister-Neher, R. Miyaoka)m1 = m2 = 2: homog. (R. Miyaoka)

g = 4: Nn is either homog. or OT-FKM type except for(m1,m2) = (7, 8) (Cecil-Chi-Jensen, Immervoll, Chi).

.Lagrangian Submanifolds in Complex Hyperquadrics... Qn(C) � Gr2(Rn+2) � SO(n + 2)/SO(2) × SO(n)

.Oriented hypersurface in a sphere..

.

Nn ↪→ Sn+1(1) ⊂ Rn+2 with constant principal curvatures(“isoparametric hypersurface”)

.“Gauss map”..

.

G : Nn ∋ p 7−→Larg. imm.

x(p) ∧ n(p) ∈ Qn(C)

Nn −→Zg

Ln = G(Nn) � Nn/Zg ↪→ Qn(C)

cpt. embedded minimal Lagr. submfd

.Problems on Ln = G(Nn)..

.

We shall discuss the following problems on compact minimalLagrangian submanifolds in Qn(C) obtained as the Gaussimages of isoparametric hypersurfaces in spheres.

...1 Properties of the Gauss images Ln = G(Nn) in Qn(C) ascompact embedded Lagrangian submanifolds.

...2 Classification of compact homogeneous Lagrangiansubmanifolds in Qn(C).

...3 Strictly Hamiltonian stability of the Gauss imagesLn = G(Nn) in Qn(C) as compact minimal Lagrangiansubmanifolds.

From isopara. hypersurf. theory, we know

2ng

=

m1 + m2 if g is even,

2m1 if g is odd.

.Theorem (H. Ma-O.)..

.

...1 If 2ng is even, then Ln = G(N) � Nn/Zg is orientable.

...2 If 2ng is odd, then Ln = G(N) � Nn/Zg is non-orientable.

.Theorem (H. Ma-O.)..

.

Ln = G(Nn) is a monotone and cyclic Lagrangian submanifoldwhose minimal Maslov number is equal to

ΣL =2ng

=

m1 + m2 if g is even,

2m1 if g is odd.

.Lagrangian Submanifolds in Complex Hyperquadrics... Qn(C) � Gr2(Rn+2) � SO(n + 2)/SO(2) × SO(n)

.Oriented hypersurface in a sphere..

.

Nn ↪→ Sn+1(1) ⊂ Rn+2 with constant principal curvatures(“isoparametric hypersurface”)

.“Gauss map”..

.

G : Nn ∋ p 7−→Larg. imm.

x(p) ∧ n(p) ∈ Qn(C)

Nn −→Zg

Ln = G(Nn) � Nn/Zg ↪→ Qn(C)

cpt. embedded minimal Lagr. submfd.

.Proposition...N

n is homogeneous ⇔ Ln = G(Nn) is homogeneous

[Hamiltonian Stability Problem]Nn ↪→ Sn+1(1): cpt. embedded isopara. hypersurf..H-stability of the Gauss map. (Palmer)..

.

Its Gauss map G : N → Qn(C) is H-stable⇐⇒ Nn = Sn ⊂ Sn+1 (g = 1).

.Question...Hamiltonian stability of its Gauss image G(Nn) ⊂ Qn(C)?

.Main result..

.

We have determined the Hamiltonian stability of Gauss imagesof ALL homogeneous isoparametric hypersurfaces (byHarmonic Analysis on cpt. homog. sp. G(Nn) � K/K[a] case bycase).

g = 1 : L is strictly Hamil. stableg = 2 : L is not Hamil. stable ⇐⇒ m2 − m1 ≥ 3

: L is Hamil. stable but not strictly Hamil. stable⇐⇒ m2 − m1 = 2

: L is strictly Hamil. stable ⇐⇒ m2 − m1 < 2

=⇒ L = Qp,q(R) tot. geod.g = 3 : L is strictly Hamil. stable =⇒ homog.(E. Cartan)

(H. Ma - O., Math. Z. 2008. arXiv:0705.0694[math.DG])

g = 4 :{Homog. case ?Non-homog. cace ??

(Ozeki-Takeuchi, Ferus-Karcher-Munzner,Cecil-Chi-Jensen, Immervoll)

.Theorem (Hui Ma-O.)..

.

g = 6 : L = SO(4)/(Z2 + Z2) · Z6 (m1 = m2 = 1)L = G2/T2 · Z6 (m1 = m2 = 2) homog.=⇒ L is strictly Hamil. stable.

.Theorem (Hui Ma and O.)..

.

g = 4, homogeneous :(1) L = SO(5)/T2 · Z4 (m1 = m2 = 2) is Hamil. stable.(2) L = U(5)/(SU(2) × SU(2) × U(1)) · Z4

(m1 = 4,m2 = 5) is Hamil. stable.(3) L = (SO(2) × SO(m))/(Z2 × SO(m − 2)) · Z4

(m1 = 1,m2 = m − 2,m ≥ 3)m2 − m1 ≥ 3 ⇐⇒ L is NOT Hamil. stable.m2 − m1 = 2 =⇒ L is Hamil. stable but not strictly Hamil. stable.m2 − m1 = 1 or 0 =⇒ L is strictly Hamil. stable.

(4) L = S(U(2) × U(m))/S(U(1) × U(1) × U(m − 2))) · Z4

(m1 = 2,m2 = 2m − 3,m ≥ 2)m2 − m1 ≥ 3 ⇐⇒ L is NOT Hamil. stable.m2 − m1 = 1 or − 1 =⇒ L is strictly Hamil. stable.

(5) L = Sp(2) × Sp(m)/(Sp(1) × Sp(1) × Sp(m − 2))) · Z4

(m1 = 4,m2 = 4m − 5,m ≥ 2)m2 − m1 ≥ 3 ⇐⇒ L is NOT Hamil. stable.m2 − m1 = −1 =⇒ L is strictly Hamil. stable.

.Theorem (Hui Ma-O.)..

.

g = 4, homogeneous :(6) L = U(1) · Spin(10)/(S1 · Spin(6)) · Z4

(m1 = 6,m2 = 9, thus m2 − m1 = 3!)=⇒ L is strictly Hamil. stable !

.Theorem (Hui Ma-O.)..

.

Suppose that (U,K) is not of type EIII, that is,(U,K) , (E6,U(1) · Spin(10)). Then L = G(N) is NOTHamiltonian 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 Hamiltonianstable.

In my joint works with Hui Ma, we have done

(1) Classification of all compact homogeneous Lagrangiansubmanifolds in complex hyperquadrics (Ma-Ohnita, Math. Z.2009).

(2) Determination of Hamiltonian stability, Hamiltonian rigidityand strict Hamiltonian stability for the Guass images of allhomogeneous isoparametric hypersurfaces:

...1 g = 1, 2, 3 (Ma-Ohnita, Mathematische Zeitschrift 2009).

...2 g = 4, (U,K) is of classical type(Ma-Ohnita Part I, Jourmal of Differential Geometry 97(2014), 275-348).

...3 g = 6 and g = 4, (U,K) is of exceptional type G2,GI,EIII(Ma-Ohnita Part II, to appear in Tohoku MathematicalJournal).

.Intersection theory for Gauss images of isoparametrichypersurfaces (Joint work with Hui Ma and Reiko Miyaoka)..

.

Nn0, Nn

1: two cpt. isopara. hypersurfaces embedded in Sn+1(1)

with g0 and g1 distinct const. prin. curv. , respectively.L0 = G0(Nn

0), L1 = G1(Nn

1): their Gauss images.

f0, f1 : Sn+1(1) → R: corresponding isopara. fcts. which extendto the Cartan-Munzner poly. F0, F1 : Rn+2 → R.

.Lemma..

.

x ∈ Nn0

(resp. Nn1) is a critical point of the function f1|Nn

0= F1|Nn

0

(resp. f0|Nn1= F0|Nn

1) if and only if G0(x) ∈ Ln

0∩ Ln

1.

.Lemma..

.

The set of all critical points of the function f1|Nn0= F1|Nn

0(resp.

f0|Nn1= F0|Nn

1) is invariant under the group action of Zg0 (resp.

Zg1).

.

.

Note that Ln0∩ Ln

1is invariant under the Deck transformation

group Z2 of Gr2(Rn+2) → Gr2(Rn+2)..Lemma..

.

The function f1|Nn0= F1|Nn

0(resp. f0|Nn

1= F0|Nn

1) is a Morse

function on Nn0

(resp. Nn1

) if and only if Ln0

and Ln1

intersecttransversally each other.

Very roughly, we immediately have.Theorem..

.

♯(L0 ∩ L1) is even and

♯(L0 ∩ L1) ≥ 2

if it is finite.

.Theorem..

.

Assume that g1 = 1. Suppose that L0 and L1 intersecttransversally each other. Then

♯(L0 ∩ L1) = 2 (= SB(L1,Z2)).

Remark.By a different method (totally geod. submfd. theory incpt. symm. sp.), H. Tasaki and M. S. Tanaka showed that ifg0 = g1 = 2, then ♯(L0 ∩ L1) = 2 Min{m0

1,m0

2,m1

1,m1

2}+ 2 for

transverse L0 ∩ L1.

Further research is in progress.

Problem.Determine the intersection numbers ♯(L0 ∩ L1) for the Gaussimages of isoparametric hypersurfaces.

Problem.Compute the Lagrangian intersection Floer cohomologyI∗(L0, L1 : Qn(C)) for the Gauss images of isoparametrichypersurfaces.

.Theorem (H. Iriyeh-H. Ma-R. Miyaoka-O.)..

.

Suppose that L = L0 = L1. Assume that g = g0 = g1 = 3 andm = m1 = m2 = 2, 4, or 8, Then

HF(L ,Λ) , {0},

where Λ = Z2[T , T−1]. Hence L is non-displaceable in Qn(C),that is, L ∩ ϕ(L) , ∅ for any Hamiltonian diffeomorphism ofQn(C).

Remark.The case of g0, g1 = 1 or 2 was already treated by HiroshiIriyeh, Takashi Sakai, Hiroyuki Tasaki (J. Math. Soc. Japan, 65(2013)), because L0, L1 are real forms of Qn(C).

Further research is in progress.

Thank you very much for your attention !