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

. Lagrangian Geometry of the Gauss Images of 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 ( M 2 n , ω ) φ : L − → immersion symplectic mfd. . Definition . . . φ ∗ ω = 0 1 “Lagrangian immersion” ⇐ ⇒ ( ⇔ φ : “isotropic ”) def . . dim L = n . 2 φ − 1 TM /φ ∗ TL T ∗ L linear isom. � ∈ ∈ v �− → α v := ω ( v , · ) 5 / 42

. Hamiltonian Deformations . → ( M 2 n , ω ) immersion with φ 0 = φ φ t : L − V t := ∂φ t ∂ t ∈ C ∞ ( φ − 1 t TM ) def φ t : Lagr. imm. for ∀ t “Lagrangian deformation” ⇐ ⇒ ⇒ α V t ∈ Z 1 ( L ) for ∀ t ⇐ closed def α V t ∈ B 1 ( L ) for ∀ t “Hamiltonian deformation” ⇐ ⇒ exact Hamil. deform. = ⇒ Lagr. deform. The difference between Lagr. deform. and Hamil. deform. is equal to H 1 ( L ; R ) � Z 1 ( L ) / B 1 ( L ) . . 6 / 42

. Characterization of Hamiltonian Deformations in terms of isomonodromy deformations . φ t : L − → M : Lagr. deform. Suppose 1 γ [ ω ] integral ( ∃ γ ) . { φ t } : Hamil. deform. . φ − 1 → ∃ ( E , ∇ ) t E − − − − −   ⇕     φ − 1 t ∇   � � flat A family of flat connections { } φ t φ − 1 L − − − − − → ( M , ω ) t ∇ has same holonomy . homom. ρ : π 1 ( L ) − → U ( 1 ) (“isomonodromy deformation”) . 7 / 42

1. Lagrangian Submanifolds in Complex Hyperquadrics . Complex Hyperquadrics . . Q n ( C ) := { [ z ] ∈ C P n + 1 | z 2 n + 1 = 0 } ⊂ C P n + 1 0 + z 2 1 + · · · + z 2 . Real Grassmann manifolds of oriented 2 -planes . Gr 2 ( R n + 2 ) ⊂ Λ 2 R n + 2 � := { [ W ] | [ W ] is an oriented 2 -dim. vect. subsp. of R n + 2 } . Identification √ Gr 2 ( R n + 2 ) ∋ [ W ] ← � → [ a + − 1b ] ∈ Q n ( C ) where { a , b } : an orth. basis of [ W ] compatible with its ori. M = Q n ( C ) � � Gr 2 ( R n + 2 ) � SO ( n + 2 ) / SO ( 2 ) × SO ( n ) 8 / 42 is a cpt. Herm. symm. sp. of rank 2 with the invariant

Q n ( C ) � � Gr 2 ( R n + 2 ) � SO ( n + 2 ) / SO ( 2 ) × SO ( n ) . Oriented hypersurface in a sphere . N n ֒ → S n + 1 ( 1 ) ⊂ R n + 2 x : the position vector of points of N n n : the unit normal vector field of N n in S n + 1 ( 1 ) . . “Gauss map” . √ G : N n ∋ p �− → [ x ( p ) + − 1 n ( p )] = x ( p ) ∧ n ( p ) ∈ Q n ( C ) is a Lagrangian immersion. . . Proposition . Deformation of N n = Hamiltonian deformation of G . 9 / 42

Remark. ( 2 n + 1 ) -dimensional real Stiefel manifold V 2 ( R n + 2 ) := { ( a , b ) | a , b ∈ R n + 2 orthonormal } � SO ( n + 2 ) / SO ( n ) the standard Einstein-Sasakian manifold over Q n ( C ) . The natural projections ϕ : V 2 ( R n + 2 ) ∋ ( a , b ) �− → a ∈ S n + 1 ( 1 ) , π : V 2 ( R n + 2 ) ∋ ( a , b ) �− → a ∧ b ∈ Q n ( C ) . US n + 1 = L = ˆ ˜ N n V 2 ( R n + 2 ) = P ✲ Leg. S n SO ( 2 ) � S 1 ϕ π � ❄ ❄ ❄ Q n ( C ) ⊃ π (ˆ N n S n + 1 N ) = G ( N n ) = L ✲ ori.hypsurf. Lag. N n of N n ֒ Here the Legendrian life ˜ → S n + 1 ( 1 ) to V 2 ( R n + 2 ) is defined by N n ∋ p �− → ( x ( p ) , n ( p )) ∈ V 2 ( R n + 2 ) . 10 / 42

2. The Gauss images of isoparametric hypersurfaces Q n ( C ) � � Gr 2 ( R n + 2 ) � SO ( n + 2 ) / SO ( 2 ) × SO ( n ) . Suppose . N n ֒ → S n + 1 ( 1 ) ⊂ R n + 2 with constant principal curvatures “isoparametric hypersurface” . . “Gauss map” . G : N n ∋ p Lagr. imm. x ( p ) ∧ n ( p ) ∈ Q n ( C ) �− → L n = G ( N n ) � N n / Z g ֒ N n − → Q n ( C ) → Z g cpt. embedded minimal Lagr. submfd . . Here g := # { distinct principal curvatures of N n } , m 1 , m 2 : multiplicities of the principal curvatures. . 11 / 42

. Suppose . N n ֒ → S n + 1 ( 1 ) ⊂ R n + 2 has g constant principal curvatures. “isoparametric hypersurface” . unzner] N n extends to a compact embedded [E. Cartan, M¨ algebraic hypersurface of S n + 1 ( 1 ) defined by a real homogeneous polynomial F of degree g , so called Cartan-M¨ unzner polynomial . The isoparametric function f = F | S n + 1 ( 1 ) is given by ( x ∈ S n + 1 ( 1 )) , f ( x ) = F ( x ) = cos gt ( x ) 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 ≤ 2 π ) satisfies θ ∈ π x ( θ ) ∈ N ⇐ ⇒ g Z . 12 / 42

x ( θ ) := cos θ x ( p ) + sin θ n ( p ) ( 0 ≤ θ ≤ 2 π ) 13 / 42

Q n ( C ) � � Gr 2 ( R n + 2 ) � SO ( n + 2 ) / SO ( 2 ) × SO ( n ) . Suppose . N n ֒ → S n + 1 ( 1 ) ⊂ R n + 2 with constant principal curvatures “isoparametric hypersurface” . . “Gauss map” . G : N n ∋ p Lagr. imm. x ( p ) ∧ n ( p ) ∈ Q n ( C ) �− → N n − L n = G ( N n ) � N n / Z g ֒ → → Q n ( C ) Z g cpt. embedded minimal Lagr. submfd . . . Proposition . 2 n g is even (resp. odd) ⇐ ⇒ G ( N ) ⊂ Q n ( C ) is orientable (resp. non-orientable). . . 14 / 42

Q n ( C ) � � Gr 2 ( R n + 2 ) � SO ( n + 2 ) / SO ( 2 ) × SO ( n ) . Suppose . → S n + 1 ( 1 ) ⊂ R n + 2 with constant principal curvatures N n ֒ “isoparametric hypersurface” . . “Gauss map” . G : N n ∋ p Lagr. imm. x ( p ) ∧ n ( p ) ∈ Q n ( C ) �− → N n − L n = G ( N n ) � N n / Z g ֒ → Q n ( C ) → Z g cpt. embedded minimal Lagr. submfd . . Proposition . g = 1 or 2 ⇔ G ( N ) ⊂ Q n ( 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 : N n = S n , a great or small sphere; g = 2 : N n = S m 1 ( r 1 ) × S m 2 ( r 2 ) , ( n = m 1 + m 2 , 1 ≤ m 1 ≤ m 2 ≤ n − 1 , r 2 1 + r 2 2 = 1 ), Clifford hypersurfaces; g = 3 : N n is homog., N n = SO ( 3 ) SU ( 3 ) Sp ( 3 ) F 4 Z 2 + Z 2 , T 2 , Sp ( 1 ) 3 , Spin ( 8 ) (E. Cartan); g = 6 : N n is homog. m 1 = m 2 = 1 : homog. (Dorfmeister-Neher, R. Miyaoka) m 1 = m 2 = 2 : homog. (R. Miyaoka) g = 4 : N n 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 I µ, L : π 2 ( M , L ) → Z and I ω, L : π 2 ( M , L ) → R . . 17 / 42

. The invariant I µ, L . For a smooth map u : ( D 2 , ∂ D 2 ) → ( M , L ) with A = [ u ] ∈ π 2 ( M , L ) , choose a trivialization of the pull-back bdl. as a symplectic vect bdl. (which is unique up to the homotopy). u − 1 TM � D 2 × C n . This gives a smooth map u : S 1 = ∂ D 2 → Λ( C n ) . ˜ Here Λ( C n ) : Grassmann mfd. of Lagrangian vect. subsp. of C n . Using the Moslov class µ ∈ H 1 (Λ( C n ) , Z ) � Z , we define I µ, L ( A ) := µ (˜ u ) . . 18 / 42

. The invariant I ω, L . Next, I ω, L : π 2 ( M , L ) → R is defined by ∫ D 2 u ∗ ω. I ω, L ( A ) := Note that I µ, L is invariant under symplectic diffeomorphisms I ω, L is invariant under Hamiltonian diffeomorphisms but not under symplectic diffeomorphisms. . 19 / 42

. Monotonicity of Lagrangian submanifolds . A Lagr. submfd. L is monotone ⇐ ⇒ def I µ, L = λ I ω, L ( ∃ λ > 0 ) . Σ L ∈ Z + : a positive generator of Im ( I µ, L ) ⊂ Z as an additive subgroup Σ L : 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