On Lagrangian submanifolds in complex hyperquadrics and Hamiltonian - PowerPoint PPT Presentation

On Lagrangian submanifolds in Qn ( C ) On Lagrangian submanifolds in complex hyperquadrics and Hamiltonian volume variational problem Hui Ma (Joint work with Yoshihiro Ohnita) Department of Mathematical Sciences Tsinghua University, Beijing, 100084, China The 10th Pacific Rim Geometry Conference Osaka-Fukuoka, 2011

On Lagrangian submanifolds in Qn ( C ) Contents 1 Backgrounds 2 Lagrangian submanifolds in complex hyperquadrics and hypersurfaces in spheres 3 Hamiltonian stability of the Gauss images of isoparametric hypersurfaces 4 Classification of Homogeneous Lagrangian submanifolds in complex hyperquadrics 5 Further questions

On Lagrangian submanifolds in Qn ( C ) Backgrounds Hamiltonian minimality and Hamiltonian stability (Y.-G. Oh (1990)) ( M, ω, J, g ) : K¨ ahler manifold, ϕ : L − → M Lagr. imm. H : mean curvature vector field of ϕ � α H := ω (H , · ) : “mean curvature form”of ϕ dα H = ϕ ∗ ρ M where ρ M : Ricci form of M . (Dazord) If M is Einstein-K¨ ahler, then dα H = 0. Suppose L : compact without boundary ϕ : “Hamiltonian minimal” (or “ H -minimal ”) ∀ ϕ t : L − ⇐ ⇒ → M Hamil. deform. with ϕ 0 = ϕ def d dt Vol ( L, ϕ ∗ t g ) | t =0 = 0 ⇐ ⇒ δα H = 0 minimal = ⇒ H-minimal

On Lagrangian submanifolds in Qn ( C ) Backgrounds Assume ϕ : H -minimal. ∀ { ϕ t } : Hamil. deform. of ϕ 0 = ϕ ϕ : “Hamiltonian stable ” ⇐ ⇒ d 2 def dt 2 Vol ( L, ϕ ∗ t g ) | t =0 ≥ 0 The Second Variational Formula d 2 dt 2 Vol ( L, ϕ ∗ t g ) | t =0 = � � L α, α � − � R ( α ) , α � − 2 � α ⊗ α ⊗ α H , S � + � α H , α � 2 � �△ 1 dv L where ∈ B 1 ( L ) α := α ∂ϕt � � ∂t � t =0 � n Ric M ( e i , e j ) α ( e i ) α ( e j ) � R ( α ) , α � := { e i } : o.n.b. of T p L i,j =1 S ( X, Y, Z ) := ω ( h ( X, Y ) , Z ) sym. 3-tensor field on L

On Lagrangian submanifolds in Qn ( C ) Backgrounds Corollary M : Einstein-K¨ ahler manifold with Einstein constant κ . → M : compact minimal Lagr. submfd. (i.e. α H ≡ 0) L ֒ Then L is Hamiltonian stable ⇐ ⇒ λ 1 ≥ κ . Here λ 1 : the first (positive) eigenvalue of the Laplacian of L on C ∞ ( L ). (B. Y. Chen - P. F. Leung - T. Nagano , Y. G. Oh)

On Lagrangian submanifolds in Qn ( C ) Backgrounds Fact (H. Ono, Amarzaya-Ohnita) Assume M : compact homogeneous Einstein - K¨ ahler mfd. with κ > 0 . → M : compact minimal Lagr. submfd. L ֒ Then λ 1 ≤ κ. λ 1 = κ ⇐ ⇒ L is Hamiltonian stable.

On Lagrangian submanifolds in Qn ( C ) Backgrounds Trivial Hamiltonian deformations X : holomorphic Killing vector field of M = ⇒ α X = ω ( X, · ) is closed α X = ω ( X, · ) is exact if H 1 ( M, R ) = { 0 } . = ⇒ If M is simply connected, more generally H 1 ( M, R ) = { 0 } , each holomorphic Killing vector field of M generates a volume-preserving Hamiltonian deformation of ϕ . Def. Such a Hamiltonian deformation of ϕ is called trivial.

On Lagrangian submanifolds in Qn ( C ) Backgrounds Strictly Hamiltonian stability Assume ϕ : L → ( M, ω, J, g ) : H -minimal. ϕ : “strictly Hamiltonian stable ” ⇐ ⇒ def (1) ϕ is Hamiltonian stable (2) The null space of the second variation on Hamiltonian deformations coincides with the vector subspace induced by trivial Hamiltonian deformations of ϕ , i.e., n ( ϕ ) = n hk ( ϕ ). Here, n ( ϕ ) := dim[ the null space ] and n hk ( ϕ ) := dim { ϕ ∗ α X | X is a holomorphic Killing vector field of M } . If L is strictly Hamiltonian stable, then L has local minimum volume under each Hamiltonian deformation.

On Lagrangian submanifolds in Qn ( C ) Backgrounds Elementary examples Circles on a plane S 1 ⊂ R 2 ∼ = C , great circles and small circles on a sphere S 1 ⊂ S 2 ∼ = C P 1 , are compact Hamiltonian stable H-minimal Lagrangian submanifolds.

On Lagrangian submanifolds in Qn ( C ) Backgrounds (Oh) The real projective space totally geodesic embedded in the complex projective space R P n ⊂ C P n is strictly Hamiltonian stable. It is Hamiltonian volume minimizing (Kleiner-Oh).

On Lagrangian submanifolds in Qn ( C ) Backgrounds (Oh) The ( n + 1)-torus T n +1 r 0 , ··· ,r n = S 1 ( r 0 ) × · · · × S 1 ( r n ) ⊂ C n +1 is strictly Hamiltonian stable H-minimal Lagrangian submanifold in C n +1 . r 0 , ··· ,r n is not minimal in C n +1 ( ∄ closed minimal submanifolds in T n +1 C n +1 ). ⇒ It is not stable under arbitrary deformation of T n +1 r 0 , ··· ,r n . It is H-minimal in C n +1 . It is strictly Hamiltonian stable. Is it Hamiltonian volume minimizing? (Oh’s conjecture, still open)

On Lagrangian submanifolds in Qn ( C ) Backgrounds (Oh, H. Ono) The quotient space by S 1 -action r 0 , ··· ,r n /S 1 ⊂ C P n T n +1 is strictly Hamiltonian stable H-minimal Lagrangian submanifold in C P n . 1 If r 0 = · · · = r n = √ n +1 , then it is minimal (“Clifford torus ”), otherwise, not minimal but H-minimal. It is strictly Hamiltonian stable for any ( r 0 , · · · , r n ) Is the Clifford torus Hamiltonian volume minimizing? (Oh’s conjecture, still open)

On Lagrangian submanifolds in Qn ( C ) Backgrounds (Amarzaya-Ohnita) Compact irreducible minimal Lagrangian submanifolds ( p − 1)( p +2) SU ( p ) /SO ( p ) · Z p ⊂ C P 2 SU ( p ) / Z p ⊂ C P p 2 − 1 SU (2 p ) /Sp ( p ) · Z 2 p ⊂ C P ( p − 1)(2 p +1) E 6 /F 4 · Z 3 ⊂ C P 26 embedded in complex projective spaces are strictly Hamiltonian stable. They are not totally geodesic but their second fundamental forms are parallel.

On Lagrangian submanifolds in Qn ( C ) Backgrounds (R. Chiang,Bedulli-Gori, Ohnita) The minimal Lagrangian orbit ρ 3 ( SU (2))[ z 3 0 + z 3 1 ] ⊂ C P 3 is a compact embedded Hamiltonian stable submanifold with non-parallel second fundamental form.

On Lagrangian submanifolds in Qn ( C ) Backgrounds (M. Takeuchi, Oh, Amarzaya-Ohnita) M : cpt. irred. Herm. sym. sp. L : cpt. totally geodesic Lagr. submfd embedded in M .  ( Q p,q ( R ) = ( S p − 1 × S q − 1 ) / Z 2 ,    Q p + q − 2 ( C ))( p ≥ 2 , q − p ≥ 3) ( L, M ) =  ( U (2 p ) /Sp ( p ) , SO (4 p ) /U (2 p ))( p ≥ 3) ,  tot. geod.  Lagr. submfd. ( T · E 6 /F 4 , E 7 /T · E 6 ) . ⇐ ⇒ L is NOT Hamiltonian stable. Takeuchi: All cpt. totally geodesic Lagr. submfds in cpt. irred. Herm. sym. sp. are real forms, i.e., the fixed point subset of involutive anti-holomorphic isometries. Let ( M, ω, J, g ) be an Einstein-K¨ ahler manifold with an involutive anti-holomorphic isometry τ and L := Fix( τ ), Einstein, positive Ricci curvature. Is L Hamiltonian volume minimizing? (Oh’s conjecture, still open)

On Lagrangian submanifolds in Qn ( C ) Backgrounds (Iriyeh-H. Ono-Sakai) Lagr . S 1 (1) × S 1 (1) totally geodesic S 2 (1) × S 2 (1) − − − − − − − − − → is Hamiltonian volume minimizing.

On Lagrangian submanifolds in Qn ( C ) Lagrangian submanifolds in complex hyperquadrics and hypersurfaces in spheres Complex Hyperquadrics Q n ( C ) ∼ = � Gr 2 ( R n +2 ) ∼ = SO ( n + 2) /SO (2) × SO ( n ) a compact Hermitian symmetric space of rank 2 Q n ( C ) := { [ z ] ∈ C P n +1 | z 2 0 + z 2 1 + · · · + z 2 n +1 = 0 } Gr 2 ( R n +2 ) := { W | oriented 2-dimensional vector subspace of R n +2 } � √ → a ∧ b ∈ � Gr 2 ( R n +2 ) Q n ( C ) ∋ [ a + − 1 b ] ← Here { a , b } is an orthonormal basis of W compatible with its orientation. ( Q n ( C ) ∼ = � Gr 2 ( R n +2 ) , g std Q n ( C ) ) is Einstein-K¨ ahler with Einstein constant κ = n . Q 1 ( C ) ∼ = S 2 = S 2 × S 2 Q 2 ( C ) ∼ n ≥ 3, Q n ( C ) is irreducible.

� � � � � � � � On Lagrangian submanifolds in Qn ( C ) Lagrangian submanifolds in complex hyperquadrics and hypersurfaces in spheres Conormal bundle construction Given an oriented submanifold N m ⊂ S n +1 (1) p 1 : V 2 ( R n +2 ) ∋ ( a , b ) �→ a ∈ S n +1 (1) p 2 : V 2 ( R n +2 ) ∋ ( a , b ) �→ a ∧ b ∈ Q n ( C ) Lag . ν ∗ T ∗ S n +1 (1) N Leg . ∼ U ν ∗ U( T ∗ S n +1 (1)) = V 2 ( R n +2 ) N p 2 S 1 p 1 S n Lag . imm . � Q n ( C ) p 2 ( U ( ν ∗ S n +1 (1) N m N )) imm . N n ⊂ S n +1 hypersurface ⇒ This construction is nothing but the following Gauss map.

� � � � � � � � On Lagrangian submanifolds in Qn ( C ) Lagrangian submanifolds in complex hyperquadrics and hypersurfaces in spheres Conormal bundle construction Given an oriented submanifold N m ⊂ S n +1 (1) p 1 : V 2 ( R n +2 ) ∋ ( a , b ) �→ a ∈ S n +1 (1) p 2 : V 2 ( R n +2 ) ∋ ( a , b ) �→ a ∧ b ∈ Q n ( C ) Lag . ν ∗ T ∗ S n +1 (1) N Leg . ∼ U ν ∗ U( T ∗ S n +1 (1)) = V 2 ( R n +2 ) N p 2 S 1 p 1 S n Lag . imm . � Q n ( C ) p 2 ( U ( ν ∗ S n +1 (1) N m N )) imm . N n ⊂ S n +1 hypersurface ⇒ This construction is nothing but the following Gauss map.