Geometry of Lagrangian submanifolds and isoparametric ...ohnita/2011/BeamerWuhan1107.pdf · Einstein-Kahler manifolds¨...2 Lagrangian submanifolds in complex hyperquadrics and isoparametric

.

.

Geometry of Lagrangiansubmanifolds and

isoparametric hypersurfaces

Yoshihiro OHNITA

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

The 4th International Workshop on Nonlinear MathematicalPhysics & The 10th National Conference on Integrable

Systems,July 25-29, 2011,

Wuhan Institute of Physics and Mathematics, China

Plan of Talk

...1 Lagrangian submanifolds in symplectic manifolds andEinstein-Kahler manifolds

...2 Lagrangian submanifolds in complex hyperquadrics andisoparametric hypersurfaces in spheres

...3 Related further problems

We assume that any manifold in my talk is smooth andconnected.

[Lagrangian Submanifolds]

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” ⇐⇒def

φt : Lagr. imm. for∀t

⇐⇒ αVt ∈ Z1(L)closed

for ∀t

“Hamiltonian deformation” ⇐⇒def

αVt ∈ B1(L)exact

for ∀t

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

.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−1t∇

flat

y yL

φt−−−−−→ (M, ω)

[Moment Map and Lagrangian Submanifolds]

SupposeK : Lie groupK has Hamiltonian group action on (M, ω) with “moment map”

µK : M −→ k∗.

.Proposition..

.

Suppose that M and K are compact. ThenL = K · x is a Lagrangian orbit of K through x in M⇐⇒

L = K · x = µ−1K

(α) for ∃ α ∈ z(k∗) � c(k).

Herez(k∗) := {α ∈ k∗ | Ad∗(a)α = α for ∀ a ∈ K }.

.Characterization of homogeneous Lagrangiansubmanifolds in K ahler manifolds...(M, ω, J, g) : compact Kahler manifold with dimC H1,1(M) = 1.

K ⊂ Aut(M, ω, J, g)) : cpt. Lie subgroup..L = K · [v] ⊂ M Lagr. submfd.

⇕.

.

complexified orbit (Zariski open)

KC · [v] ⊂ M is Stein

⇓M \ K C · [v] is a complex analytic variety of codimension 1.

[Hamiltonian minimality and stability (Y. G. Oh (1990)](M, ω, J, g): Kahler manifold.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

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 byholomorphic Killing vector fields of M. That is, n(φ) = nhk (φ).

.Proposition..

.

L is strictly Hamiltonian stable =⇒ L has local minimumvolume under every Hamiltonian deformation.

.Complex Hyperquadrics..

.Qn(C) := {[z] ∈ CPn+1 | z2

0+ · · ·+ z2

n+1= 0}

.Real Grassmann manifolds of oriented 2-planes..

.

Gr2(Rn+2)

:={ [W] | [W] is an oriented 2-dim. vect. subsp. of Rn+2 }⊂ Λ2Rn+2

Identification

Gr2(Rn+2) ∋ [W] ←→ [a +

√−1b] ∈ Qn(C)

where {a, b}: an orthonormal basis of W compatible with itsorientation.

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

is a compact Hermitian symmetric spaces of rank 2.

.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

Remark. (2n + 1)-dimensional real Stiefel manifold

V2(Rn+2) := {(a, b) | a, b ∈ Rn+2 orthonormal } � SO(n+2)/SO(n)

the standard Einstein-Sasakian manifold over Qn(C).The natural projections

p1 : V2(Rn+2) ∋ (a, b) 7−→ a ∈ Sn+1(1),

p2 : V2(Rn+2) ∋ (a, b) 7−→ a ∧ b ∈ Qn(C).

?

UTSn+1 = V2(Rn+2)

p1 Sn

Sn+1

-Nnψ

Legend.

?�

?Nn

ori.hypsurf.-

p2 S1

Qn(C) ⊃ p1(ψ(L)) = G(Nn)

Lagr.

Here the Legendrian life Nn of Nn ↪→ Sn+1(1) to V2(Rn+2) isdefined by Nn ∋ p 7−→ (x(p), n(p)) ∈ V2(Rn+2).

(Conormal bundle construction)More generally for a given Nm ⊂ Sn+1(1) submanifold,

?

T∗Sn+1(1) ⊂ Qn+1(C)

?

UT∗Sn+1(1) � V2(Rn+2)

p1 S1

Qn(C)

-

-

?

ν∗N

U(ν∗N)

Lag.

Leg.

? ?p1(U(ν∗

N))

Lag.-

p2 Sn

Sn+1(1) ⊃ Nm

imm. submfd.

In the case m = n and Nn oriented, it coincides with our Gaussmap construction.

The Mean Curvature Form Formula (B. Palmer, 1997).Lemma..

.

αH =d

Im

logn∏

i=1

(1 +√−1κi)

,

= − d

n∑i=1

arccot κi

where H : mean curvature vector field of G,κi (i = 1, · · · , n) : principal curvatures of Nn ⊂ Sn+1(1).

N2 ⊂ S3 min. surf. ⇒ G : N2 → Q2(C) � S2×S2 min. Lagr. imm.

Nn ⊂ Sn+1 austere min. hypersurf. ⇒ G : Nn → Qn(C) min. Lagr. imm.



.

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.



.


.“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.

[Isoparametric Hypersurfaces in Spheres]

E. Cartan, Munzner

Nn: oriented hypersurface embedded in Sn+1(1) ⊂ Rn+2 with gdistinct principal curvatures:

k1 > k2 > · · · > kg

and corresponding multiplicities

m1,m2, · · · ,mg .

.Theorem (Munzner)..

.

Set kα = cot θα (α = 1, · · · , g) with 0 < θ1 < · · · < θg < π.Then

θα = θ1 + (α − 1)π

g(α = 1, · · · , g), (1)

mα = mα+2 indices modulo g. (2)

∃ F : Rn+2 −→ [−1, 1] ⊂ R (Cartan-Munzner polynomial)∆ F = c rg−2,

∥gradF∥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)).

The isoparametric family is given in this way..Theorem (Munzner)..

.

(1) g must be 1, 2, 3, 4 or 6.

(2) If g = 6, then m1 = m2.

.Theorem (Abresch)...If g = 6, then m1 = m2 = 1 or 2.

All isoparametric hypersurfaces in Sn+1(1) are classified into

Homogeneous ones (Hsiang-Lawson,R. Takagi-T. Takahashi) can be obtained as principal orbitsof the isotropy representations of Riemannian symmetricpairs (U,K) of rank 2.

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

SO(3)

Z2+Z2, SU(3)

T2 , Sp(3)

Sp(1)3 , F4

Spin(8);

g = 6: Only homog. examples are known now.g = 6,m1 = m2 = 1: homog. (Dorfmeister-Neher, R.Miyaoka)g = 6,m1 = m2 = 2: homog. (R. Miyaoka)

Non-homogenous ones exist (H.Ozeki- M.Takeuchi) andare almost classified (Ferus-Karcher-Munzner,Cecil-Chi-Jensen, Immervoll).

g = 4: except for (m1,m2) = (4, 5), (6, 9), (7, 8), eitherhomog. or OT-FKM type.

.Isoparametric hypersurfaces and integrable systems..

.

Ferapontov (1995):

Nn = Sn+1(1) ∩ F−1(0) ⊂ Sn+1(1) ⊂ Rn+2 :

cpt. isopara. hypersurf. with g distinct principal curvatures.The Hamiltonian system for u = u(t , x) ∈ Rn+2

∂

∂tu(t , x) =

1g

∂

∂x(gradF)(u(t , x))

can be restricted to a hydrodynamic type system inNn = Sn+1(1) ∩ F−1(0)

∂

∂tu(t , x) =

1g

A(∂

∂xu(t , x)

)where A denotes the shape operator (Weingarten map) of Nn

in Sn+1 wrt. unit normal n.In the case Nn is homogeneous, it is an integrable system.



.


.“Gauss map”..

.


x(p) ∧ n(p) ∈ Qn(C)

Nn −→Zg


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..

.

...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.



.


.“Gauss map”..

.


x(p) ∧ n(p) ∈ Qn(C)

Nn −→Zg


cpt. embedded minimal Lagr. submfd.

.Proposition...N

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

[Classification Problem of Homog. Lagr. Submfds.].Classification of Homogeneous Lagr. submfds. in CPn

(Bedulli and Gori, math.DG/0604169)..

.

16 examples of minimal Legr. orbits in CPn

= [5 examples with ∇S = 0] +[11 examples with ∇S , 0]

K ⊂ SU(n + 1) : cpt. simple subgroup..L = K · [v] ⊂ CPn Lagr. submfd.

⇕.

.


KC · [v] ⊂ CPn is Stein

⇑Classification Theory of “Prehomogeneous vectorspaces”(Mikio Sato and Tatsuo Kimura)


(Bedulli and Gori, math.DG/0604169)..

.




⇕.

.


KC · [v] ⊂ CPn is Stein



(Bedulli and Gori, Comm. Anal. Geom., 2008,)..

.




⇕.

.


K C · [v] ⊂ CPn is Stein


.Classification of Homogeneous Lagrangian submanifoldsin Qn(C) (Hui Ma and O., Math. Z. 2009)..

.

SupposeG ⊂ SO(n + 2) : cpt. subgroup ,

L = G · [W] ⊂ Qn(C) Lagr. submfd.

⇓.

.

There exists

Nn ⊂ Sn+1(1) ⊂ Rn+2 : cpt. homog. isopara. hypersurf.

such that(a) L = G(N) and L is a cpt. minimal Lagr. submfd., or(b) L belongs to a certain Lagrangian deformation of G(N)consisting of compact homgeneous Lagrangian submanifolds.

.Classification of Homogeneous Lagrangian submanifoldsin Qn(C) (Hui Ma and O., Math. Z. 2009)..

.

SupposeG ⊂ SO(n + 2) : cpt. subgroup ,

L = G · [W] ⊂ Qn(C) Lagr. submfd.

⇓.

.

There exists

Nn ⊂ Sn+1(1) ⊂ Rn+2 : cpt. homog. isopara. hypersurf.

such that(a) L = G(N) and L is a cpt. minimal Lagr. submfd., or(b) L belongs to a certain Lagrangian deformation of G(N)consisting of compact homgeneous Lagrangian submanifolds.

.W.Y.Hsiang-H.B.Lawson’s theorem (1971)..

.

There is a compact Riemannian symmetric pair (U,K) of rank 2such that

N = Ad(K)v ⊂ Sn+1(1) ⊂ Rn+2 = p,

where u = k+ p is the canonical decomposition of (U,K).

.Moment map..

.

The moment map of the induced action of K onQn(C) = Gr2(p) is given by

µ : Qn(C) = Gr2(p) ∋ [W] 7−→ [a, b] ∈ k � k∗

where {a, b} : orthonormal basis of W compatible with theorientation of [W].

The case (b) happens only when (U,K) is one of...1 (S1 × 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).

In the first two cases, it is elementary and well-known todescribe all Lagrangian orbits of the natural actions ofK = SO(2) on Q1(C) � S2 and K = SO(2) × SO(2) onQ2(C) � S2 × S2. Also in the last two cases there existone-parameter families of Lagrangian K -orbits in Qn(C) andeach family contains Lagrangian submanifolds which can NOTbe obtained as the Gauss image of any homogeneousisoparametric hypersurface in a sphere. The fourth one is anew family of Lagrangian orbits.

.If (U,K) is (S1 × SO(3), SO(2)),...then L is a small or great circle in Q1(C) � S2.

.If (U,K) is (SO(3) × SO(3), SO(2) × SO(2)),..

.

then L is a product of small or great circles of S2 inQ2(C) � S2 × S2.

.If (U,K) is (SO(3) × SO(n + 1), SO(2) × SO(n)) (n ≥ 2) ,..

.

then

L = K · [Wλ] ⊂ Qn(C) for some λ ∈ S1 \ {±√−1},

where K · [Wλ] (λ ∈ S1) is the S1-family of Lagr. or isotropicK -orbits satisfying

...1 K · [W1] = K · [W−1] = G(Nn) is a tot. geod. Lagr. submfd.in Qn(C).

...2 For each λ ∈ S1 \ {±√−1},

K · [Wλ] � (S1 × Sn−1)/Z2 � Q2,n(R)

is a Lagr. orbit in Qn(C) with ∇S = 0....3 K · [W±√−1] are isotropic orbits in Qn(C) with

dim K · [W±√−1] = 0.

.If (U,K) is (SO(m + 2), SO(2) × SO(m)) (n = 2m − 2),..

.

then

L = K · [Wλ] ⊂ Qn(C) for some λ ∈ S1 \ {±√−1},

where K · [Wλ] (λ ∈ S1) is the S1-family of Lagr. or isotropicorbits satisfying

...1 K · [W1] = K · [W−1] = G(Nn) is a minimal (NOT tot.geod.) Lagr. submfd. in Qn(C).

...2 For each λ ∈ S1 \ {±√−1},

K · [Wλ] � (SO(2) × SO(m))/(Z2 × Z4 × SO(m − 2))

is a Lagr. orbit in Qn(C) with ∇S , 0....3 K · [W±√−1] � SO(m)/S(O(1) × O(m − 1)) � RPm−1 are

isotropic orbits in Qn(C) with dim K · [W±√−1] = m − 1.

[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.


.

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 !


.

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.

.Cohomogeneity 1 special Lagrangian submanifolds intangent bundle over the standard sphere..

.

The cones of Legendrian lifts of isoparametric hypersurfaces tothe unit tangent bundle T1Sn+1 � V2(Rn+2) of the standardsphere provide fundamental examples of special Lagrangiancones in the (non-flat) Ricci-flat Kahler cone.

In the cases of g = 1, 2, Kaname Hashimoto (OCU,D3)-Takashi Sakai (TMU) classify all cohomogeneity 1 specialLagrangian submanifolds in the tangent bundle TSn+1 withrespect to the Stenzel metric deformed from such specialLagrangian cones.

K. Hashimoto and T. Sakai,Cohomegeneity one special Lagrangian submanifolds in thecotangent bundle of the sphere. a preprint (2010), OCAMIPreprint Ser. no.10-19.

It is an interesting problem to study the cases of g = 3, 4, 6. 　

.Extension to Semi-Riemannian case..

.

More recently, Harunobu Sakurai (OCU, D1) studies anextension of Lagrangian propery of the Gauss map and themean curvature form formula to oriented hypersurfaces insemi-Riemannain space forms.

From J. Hahn’s work (Math. Z. 1984, J. Math. Soc. Japan1988), we have many interesting examples of minimalLagrangian submanifolds in semi-Riemannian real Grassmannmaifolds of oriented 2-planes .

.Further questions..

.

...1 Investigate the Hamiltonian stability and other properties ofthe Gauss images of compact non-homogenousisoparametric hypersurfaces, particular OT-FKM type,embedded in spheres with g = 4.

...2 Investigate the relation between our Gauss imageconstruction and Karigiannis-Min-Oo’s results.

...3 Are there similar constructions of Lagrangian subamnifoldsin compact Hermitian symmetric spaces other than CPn,Qn(C) ?

Many Thanks !

Documents

Geometry of Lagrangian submanifolds and isoparametric ...ohnita/2011/BeamerWuhan1107.pdf · Einstein-Kahler manifolds¨...2 Lagrangian submanifolds in complex hyperquadrics and isoparametric