Lagrangian Geometry of the Gauss Images of Isoparametric - - PowerPoint PPT Presentation

. .

Lagrangian Geometry

• f the Gauss Images
• f Isoparametric Hypersurfaces

Yoshihiro OHNITA

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

Workshop on the Isoparametric Theory, Beijing Normal University, Beijing, P . R. China, June 1-7, 2019 June 2, 2019

1 / 42

This talk is based on Joint works with Hui Ma (Tsinghua University, Beijing, P . R. China), Hiroshi Iriyeh (Ibraki University, Mito, Japan), Reiko Miyaoka (Tohoku Univerisity, Sendai, Japan).

2 / 42

. Plan of this talk . .

• 1. Lagrangian submanifolds in symplectic manifolds
• 2. Lagrangian submanifolds in Einsterin-K¨

ahler manifolds

• 3. Lagrangian submanifolds in complex hyperquadrics
• 4. Gauss images of isoparametric hypersurfaces in spheres
• 5. Hamiltonian non-displaceability of a Lagrangian

submanifold

• 6. Hamiltonian non-displaceability of Gauss images of

isoparametric hypersurfaces

• 7. Open problems and related questions

3 / 42

• 0. Isoparametric Theory and Symplectic Geometry

It is an interesting question to ask what relationship is between “Isoparametric Theory” ⇐ ⇒

??

“Symplectic Geometry ” “Isoparametric Hypersurfaces” ⇐ ⇒

??

“Lagrangian Submanifolds”

4 / 42

• 3. 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

− → αv := ω(v, · )

5 / 42

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

Vt := ∂φt

∂t ∈ C∞(φ−1

t TM)

“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. is equal to H1(L; R) Z1(L)/B1(L).

6 / 42

. Characterization of Hamiltonian Deformations in terms of isomonodromy deformations . . φt : L − → M : Lagr. deform. Suppose 1 γ [ω] integral (∃ γ). {φt} : Hamil. deform. ⇕ A family of flat connections { φ−1

t ∇

} has same holonomy homom. ρ : π1(L) − → U(1) (“isomonodromy deformation”) . . φ−1

t E −

− − − − → ∃(E, ∇)

φ−1

t ∇

flat

   

• 

  

• L

φt

− − − − − → (M, ω)

7 / 42

• 1. Lagrangian Submanifolds

in Complex Hyperquadrics

. 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.

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

is a cpt. Herm. symm. sp. of rank 2 with the invariant

8 / 42

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 − → [x(p) + √ −1n(p)] = x(p) ∧ n(p) ∈ Qn(C) is a Lagrangian immersion. . Proposition . . Deformation of Nn = Hamiltonian deformation of G

9 / 42

• 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 ϕ : V2(Rn+2) ∋ (a, b) − → a ∈ Sn+1(1), π : V2(Rn+2) ∋ (a, b) − → a ∧ b ∈ Qn(C).

❄ USn+1 = V2(Rn+2) = P

ϕ

Sn Sn+1 ✲

˜

L = ˆ Nn

Leg.

❄

• ❄

Nn

• ri.hypsurf.

✲

π

SO(2) S1 Qn(C) ⊃ π(ˆ N) = G(Nn) = L

Lag. Here the Legendrian life ˜

Nn of Nn ֒

→ Sn+1(1) to V2(Rn+2) is defined by Nn ∋ p − → (x(p), n(p)) ∈ V2(Rn+2).

10 / 42

• 2. The Gauss images of isoparametric

hypersurfaces

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

. Suppose . .

Nn ֒

→ Sn+1(1) ⊂ Rn+2 with constant principal curvatures “isoparametric hypersurface” . “Gauss map” . . G : Nn ∋ p − →

• 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.

11 / 42

. Suppose . .

Nn ֒

→ Sn+1(1) ⊂ Rn+2 has g constant principal curvatures. “isoparametric hypersurface” [E. Cartan, M¨ unzner] Nn extends to a compact embedded algebraic hypersurface of Sn+1(1) defined by a real homogeneous polynomial F of degree g, so called Cartan-M¨ unzner polynomial. The isoparametric function

f = F|Sn+1(1) is given by f(x) = F(x) = cos gt(x)

(x ∈ Sn+1(1)), where t is a spherical distance function to a focal manifold.

g must be 1, 2, 3, 4 or 6 (M¨

unzner, 1981-82). For each p ∈ N, a unit normal geodesic at p to N

x(θ) := cos θ x(p) + sin θ n(p)

(0 ≤ θ ≤ 2π) satisfies

x(θ) ∈ N

⇐ ⇒ θ ∈ π

g Z.

12 / 42

x(θ) := cos θ x(p) + sin θ n(p)

(0 ≤ θ ≤ 2π)

13 / 42

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

. Suppose . .

Nn ֒

→ Sn+1(1) ⊂ Rn+2 with constant principal curvatures “isoparametric hypersurface” . “Gauss map” . . G : Nn ∋ p − →

• Lagr. imm. x(p) ∧ n(p) ∈ Qn(C)

Nn −

→

Zg

Ln = G(Nn) Nn/Zg ֒

→ Qn(C)

• cpt. embedded minimal Lagr. submfd

. . . Proposition . .

2n g is even (resp. odd) ⇐

⇒ G(N) ⊂ Qn(C) is orientable (resp. non-orientable).

14 / 42

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

. Suppose . .

Nn ֒

→ Sn+1(1) ⊂ Rn+2 with constant principal curvatures “isoparametric hypersurface” . “Gauss map” . . G : Nn ∋ p − →

• Lagr. imm. x(p) ∧ n(p) ∈ Qn(C)

Nn −

→

Zg

Ln = G(Nn) Nn/Zg ֒

→ Qn(C)

• cpt. embedded minimal Lagr. submfd

. Proposition . .

g = 1 or 2 ⇔ G(N) ⊂ Qn(C) is totally geodesic (a real form).

15 / 42

Construction of isoparametric hypersurfaces:

Principal orbits of the isotropy representations of Riemannian symmetric pairs (U, K) of rank 2 = ⇒ All homogeneous isopara. hypersurf. (Hsiang-Lawson,

• R. Takagi-T. Takahashi)

Algebraic construction of Cartan-M¨ unzner polynomials by representations of Clifford algebras in case g = 4 (Ozeki-Takeuchi, Ferus-Karcher-M¨ unzner) OT-FKM type = ⇒ So many non-homogeneous isopara. hypersurf.

Classification of isoparametric hypersurfaces:

g = 1: Nn = Sn, a great or small sphere; g = 2: Nn = Sm1(r1) × Sm2(r2), (n = m1 + m2, 1 ≤ m1 ≤ m2 ≤ n − 1, r2

1 + r2 2 = 1), 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 (Cecil-Chi-Jensen, Immervoll, Chi).

16 / 42

. Two invariants of Lagrangian submanifolds . .

L: Lagr. submfd. of a sympl. mfd. (M, ω)

Define two kinds of group homomorphisms

Iµ,L : π2(M, L) → Z

and

Iω,L : π2(M, L) → R.

17 / 42

. The invariant Iµ,L . . For a smooth map u : (D2, ∂D2) → (M, L) with

A = [u] ∈ π2(M, L), choose a trivialization of the pull-back bdl.

as a symplectic vect bdl. (which is unique up to the homotopy).

u−1TM D2 × Cn.

This gives a smooth map ˜

u : S1 = ∂D2 → Λ(Cn).

Here Λ(Cn): Grassmann mfd. of Lagrangian vect. subsp. of Cn. Using the Moslov class µ ∈ H1(Λ(Cn), Z) Z, we define

Iµ,L(A) := µ(˜ u).

18 / 42

. The invariant Iω,L . . Next, Iω,L : π2(M, L) → R is defined by

Iω,L(A) :=

∫

D2 u∗ω.

Note that

Iµ,L is invariant under symplectic diffeomorphisms Iω,L is invariant under Hamiltonian diffeomorphisms but not

under symplectic diffeomorphisms.

19 / 42

. Monotonicity of Lagrangian submanifolds . . A Lagr. submfd. L is monotone ⇐ ⇒

def

Iµ,L = λ Iω,L

(∃λ > 0). ΣL ∈ Z+: a positive generator of Im(Iµ,L) ⊂ Z as an additive subgroup ΣL: minimal Maslov number of L． . Theorem (K. Cieliebak and E. Goldstein 2004, Hajime Ono 2004) . . (M, ω, J, g): Einstein-K¨ ahler mfd. of Einstein const. κ > 0

L: compact minimal Lagr. submfd. of M

= ⇒

L is monotone.

20 / 42

Monotone Lagrangian Submanifolds in Einstein-K¨ ahler Manifolds

(M, ω, J, g): simply connected Einstein-K¨ ahler mfd. with κ > 0,

L: compact monotone Lagrangian submfd. of M

. Proposition (Hajime Ono, Japan J. Math. 2004) . .

nLΣL = 2γc1,

where γc1 := min{c1(M)(A) | A ∈ H2(M; Z), c1(M)(A) > 0},

nL := min{k ∈ Z+ | ⊗kE trivial }. E cplx. line bdle.

flat πE πL

U(1)-connection ∇ M Einstein-K¨

ahler mfd.

✲ E|L ❄ ❄ L

Lag.

✲

Here 1

γω = c1(P, ∇) for some nonzero γ > 0.

21 / 42

Minimal Maslov number of the Gauss images of isoparametric hypersurfaces

Nn ⊂ Sn+1(1) ⊂ Rn+2: oriented hypersurface of Sn+1(1)

ˆ

Nn := {(x(p), n(p)) ∈ V2(Rn+2) | p ∈ N}: Legendrian lift of Nn.

unit sphere tangent bundle of Sn+1 = Stiefel mfd. of o.n. 2-frames of Rn+2

P = T1Sn+1 = V2(Rn+2)

SO(n+2) SO(n)

ρ(π1(L)) π G Gauss map

SO(2) U(1) M = Qn(C) = Gr2(Rn+2)

SO(n+2) SO(2)×SO(n)

complex hyperquadric =real ori. 2-plane Grassmann mfd.

✲ N ⊂ Sn+1(1)

• ˆ

L = ˆ N

Leg.

❄ ❄ L = G(N)

Gauss image Lag.

✲

The Gauss Map is defined by G :

N

− → ˆ

N

− → G(N) ⊂ Qn(C) =

Gr2(Rn+2) p

− → (x(p), n(p)) − → [x(p) + √ −1n(p)] = x(p) ∧ n(p)

22 / 42

Minimal Maslov number of the Gauss images of isoparametric hypersurfaces

Nn ⊂ Sn+1(1): isoparametric hypersurf. with g dist. prin. curv.

ˆ

Nn := {(x(p), n(p)) ∈ V2(Rn+2) | p ∈ N}: Legendrian lift of Nn.

simply conn. Einstein Sasakian homog. sp.

P = V2(Rn+2) =

SO(n+2) SO(n)

= T1Sn+1 ρ(π1(L)) Zg π G Gauss map

SO(2)

U(1) Einstein-K¨ ahler HSS

M = Qn(C) =

SO(n+2) SO(2)×SO(n)

complex hyperquadric

✲ N ⊂ Sn+1(1)

• ˆ

L = ˆ N

• min. Leg embed.

❄ ❄ L = G(N)

Gauss image

• min. Lag. embed.

✲

. Theorem (Hui Ma-O. (2010), JDG (2014)) . . ΣL = 2n

g .

23 / 42

[Hamiltonian minimality and stability (Y. G. Oh (1990)] Suppose L: compact (without boundary). φ : “Hamiltonian minimal ” ⇐ ⇒

def ∀φt : L −

→ M Hamil. deform. with φ0 = φ

d dt Vol (L, φ∗

t g)

• t=0 = 0

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

def ∀ {φt} : Hamil. deform. of φ0 = φ

d2 dt2 Vol (L, φ∗

t g)

• t=0 ≥ 0

24 / 42

[Homogeneous Lagrangian Submanifolds of K¨ ahler manifolds (in the sense of Riemannian Geometry)] . Definition . . (M, ω, J, g) : K¨ ahler manifold.

K ⊂ Aut(M, ω, J, g): connected Lie subgroup, L = K · x ⊂ M: a Lagrangian orbit

“homogeneous Lagrangian submanifold ” . Proposition . .

L: compact homog. Lagr. submfd. of K¨

ahler mfd. M = ⇒

Lis Hamiltonian minimal

25 / 42

In my preceding joint works with Hui Ma, we have done .

• Lagr. orbits and moment maps of cpt. isometry subgrps.

. . (1) Classification of all compact homogeneous Lagr. submfds. in Qn(C) ([MO 2009] Math. Z.). . 2nd variations of volume under Hamil. deform. . . (2) Determination of Hamiltonian stability, Hamiltonian rigidity and strict Hamiltonian stability for the Guass images of all homogeneous isoparametric hypersurfaces: . .

1

g = 1, 2, 3 ([MO 2009] Math. Z.).

. .

2

g = 4, (U, K) is of classical type ([MO 2014] J. Differential

Geom.). . .

3

g = 6 and g = 4, (U, K) is of exceptional type G2, GI, EIII

([MO 2015] Tohoku Math. J.).

• Problem. Study Hamiltonian stability of the Gauss images of

non-homogeneous isopara. hypersurf. of OT-FKM type.

26 / 42

. Hamiltonian stability of Gauss images of homogeneous isoparametric hypersurfaces . .

Nn ⊂ Sn+1(1)：homogeneous isoparametric hypersurface

with g distinct principal curvatures and their multiplicities (m1, m2)

Ln = G(Nn) ⊂ Qn(C)：the Gauss image of Nn

. .

1

In case g = 1, Ln = Sn ⊂ Qn(C)(totally geodesic Lagrangian sphere) is Hamiltonian stable. . .

2

In case g = 2, Ln = (Sm1 × Sm2)/Z2 ⊂ Qn(C) (real quadrics) is Hamiltonian stable if and only if m2 − m1 ≥ 3. . .

3

In case g = 3, Ln ⊂ Qn(C) is Hamiltonian stable. . .

4

In case g = 4 except for (m1, m2) = (9, 6), Ln ⊂ Qn(C) is Hamiltonian stable if and only if m2 − m1 ≥ 3. In case g = 4 with (m1, m2) = (9, 6), Ln ⊂ Qn(C) is Hamiltonian stable. . .

5

In case g = 6, Ln ⊂ Qn(C) is Hamiltonian stable.

27 / 42

• 4. Hamiltonian non-displaceability of a Lagrangian

submanifold

. Hamiltonian diffeomorphisms of a symplectic manifold . . (M, ω) : sympl. mfd． ϕ: Hamiltonian diffeomorphism of (M, ω) ⇐ ⇒

def

∃ Ht (t ∈ [0, 1]): time-dependent Hamiltonians ∃ ϕt : M → M (t ∈ [0, 1]): an isotopy of M with ϕ = ϕ1 satisfying the Hamiltonian equation

dϕt(x) dt

= (XHt)ϕt(x) and ϕ0(x) = x (x ∈ M), where XHt: a Hamiltonian vector field corresponding to a Hamiltonian Ht with respect to ω. {ϕt}t∈[0,1] : Hamiltonian isotopy of (M, ω).

28 / 42

. Hamiltonian diffeomorphism group of a symplectic manifold . . Haml(M, ω) := {ϕ | Hamiltonian diffeomorphisms of (M, ω)}. Then Haml(M, ω) is a subgroup of Symp0(M, ω) Haml(M, ω): Hamiltonian diffeomorphism group of (M, ω).

29 / 42

30 / 42

31 / 42

32 / 42

33 / 42

5, Hamiltonian non-displaceability of Lagrangian Submanifolds

. .

L: Lagrangian submanifold in M L is Hamiltonian non-displaceable

⇐ ⇒

def

For ∀ϕ ∈ Haml(M, ω),

L ∩ ϕ(L) ∅ . L is Hamiltonian displaceable

⇐ ⇒

def

∃ϕ ∈ Haml(M, ω) such that

L ∩ ϕ(L) = ∅ .

34 / 42

. Question . . What Lagrangian submanifolds are Hamiltonian non-displaceable? In the 2-dimensional standard sphere S2(1), a great circle S1(1) ⊂ S2(1) is Hamiltonian non-displaceable, a small circle S1(r) (0 < r < 1) ⊂ S2(1) is Hamiltonian displaceable.

35 / 42

• 6. Hamiltonian non-displaceability of Gauss

images of isoparametric hypersurfaces

. Theorem (IMMO (H. Iriyeh, Hui Ma, R. Miyaoka and O.), Bull. LMS 2016) . .

Nn ⊂ Sn+1(1): isoparametric hypersurface

= ⇒ The Gauss image

Ln = G(Nn) ⊂ Qn(C)

is Hamiltonian non-displaceable except for the remaining three cases: (g, n, m1, m2) = (3, 3, 1, 1),

N =

SO(3) Z2+Z2 ,

(g, n, m1, m2) = (4, 2k + 2, 1, k),

N =

SO(2)×SO(k+2) Z2×SO(k)

(k ≥ 1), (g, n, m1, m2) = (6, 6, 1, 1),

N =

SO(4) Z2+Z2 ,

36 / 42

• 7. Open Problems and Related Questions

. Problems and Questions . . Determine whether the lifted Floer homology HF ¯

L(L) is

nonzero or not in the case when (g, n, m1, m2) = (4, 2k + 2, 1, k) (k ≥ 2) (then ΣL = k + 1 ≥ 3). Determine whether the Floer homology HF(L) is nonzero

• r not in the case when

(g, n, m1, m2) = (3, 3, 1, 1), (4, 4, 1, 1) or (6, 6, 1, 1) (then ΣL = 2). When is the Floer homology HF(L) isomorphic to H∗(L; Z2)?

37 / 42

.

• 7. Problems and Questions

. . More generally we should pose the following problem as

• ur goal:

Determine explicitly the Floer homology HF(L) of the Gauss images of isoparametric hypersurfaces in the case when (g, m) = (3, 1), g = 4 or g = 6. Since the Floer homology is based on the Morse homology, it is quite natural to study the following problems: Determine explicitly the homology HF∗(L; Z2) of the Gauss images of isoparametric hypersurfaces in the case when

g = 4 or g = 6.

Construct concretely the Morse homology of the Gauss images of isoparametric hypersurfaces in the case when

g = 3, 4 or g = 6.

38 / 42

. References (MO, IMMO) . . [MO 2009] H. Ma and Y. Ohnita, On Lagrangian submanifolds in complex hyperquadrics and isoparametric hypersurfaces in spheres, Math. Z. 261 (2009), 749–785. [MO 2011] H. Ma and Y. Ohnita, Differential Geometry of Lagrangian Submanifolds and Hamiltonian Variational Problems, Contemp. Math. 542, AMS, 2011, pp. 115-134. [MO 2014] H. Ma and Y. Ohnita, Hamiltonian stability of the Gauss images of homogeneous isoparametric hypersurfaces. I,

• J. Differential Geom. 97 (2014), 275-348.

[MO 2015] H. Ma and Y. Ohnita, Hamiltonian stability of the Gauss images of homogeneous isoparametric hypersurfaces. II, Tohoku Math. J. 67, No.2 (2015), 195-246 . [IMMO 2016] H. Iriyeh, H. Ma, R. Miyaoka and Y. Ohnita: Hamiltonian non-displaceability of Gauss images of isoparametric hypersurfaces, Bull. London Math. Soc. (2016) 48 (5): 802-812.

39 / 42

Thank you very much for your attention !

40 / 42

References

. References (Gromov, Floer) . . [Gromov1985] M. Gromov, Pseudo holomorphic curves in symplectic manifolds, Invent. Math. 82 (1985), 307–347. [FloerJDG1988] A. Floer, Morse theory for Lagrangian

• intersections. J. Differential Geom. 28 (1988), 513–547.

[FloerCPAM1988a] A. Floer, A relative Morse index for the symplectic action. Comm. Pure Appl. Math. 41 (1988), 393–407. [FloerCPAM1988b] A. Floer, The unregularized gradient flow of the symplectic action. Comm. Pure Appl. Math. 41 (1988), 774–813. [FloerJDG1989] A. Floer, Witten’s complex and infinite dimensional Morse theory. J. Differential Geom. 30 (1989), 207–221.

41 / 42

. References (Y. G. Oh, Biran, Damian) . . [YGOh93I] Y.-G. Oh, Floer cohomology of Lagrangian intersections and pseudo-holomorphic disks, I, Comm. Pure

• Appl. Math. 46 (1993), 949–994.

[YGOh95] Y.-G. Oh, Addendum to “Floer cohomology of Lagrangian intersections and pseudo-holomorphic disks, I”,

• Comm. Pure Appl. Math. 48 (1995), 1299–1302.

[YGOhIMRN96] Y.-G. Oh, Floer cohomology, spectral sequences, and the Maslov class of Lagrangian embeddings,

• Int. Math. Res. Not. 7 (1996), 305–346.

[Biran06] P . Biran, Lagrangian non-intersections, Geom. Funct.

• Anal. 16 (2006), 279–326.

[Da 2012] M. Damian, Floer homology on the universal cover, Audin ’ s conjecture and other constraints on Lagrangian submanifolds, Comment. Math. Helv. 87 (2012), 433–462.

42 / 42