Toward classification of biharmonic homogeneous hypersurfaces in - PowerPoint PPT Presentation

Toward classification of biharmonic homogeneous hypersurfaces in compact symmetric spaces Takashi Sakai （酒井 高司） Tokyo Metropolitan University & OCAMI 2019 Workshop on the Isoparametric Theory Beijin Normal University, Beijin June 6th, 2019 Takashi Sakai Biharmonic hypersurfaces in compact symmetric spaces

This talk is based on joint work with Shinji Ohno and Hajime Urakawa. S. Ohno, T. Sakai and H. Urakawa, Biharmonic homogeneous hypersurfaces in compact symmetric spaces , Differential Geom. Appl. 43 (2015), 155–179. S. Ohno, T. Sakai and H. Urakawa, Biharmonic homogeneous submanifolds in compact Lie groups and compact symmetric spaces , Hiroshima Math. J. 49 (2019), no. 1, 47–115. Takashi Sakai Biharmonic hypersurfaces in compact symmetric spaces

Harmonic maps ( M m , g ) , ( N n , h ) : Riemannian manifolds ϕ : M − → N smooth map ∫ E ( ϕ ) = 1 ∥ dϕ ∥ 2 v g energy functional 2 M Definition ϕ : M − → N : harmonic def ⇐ ⇒ ϕ is a critical point of E ⇐ ⇒ τ ( ϕ ) := trace B ϕ = 0 tension field � ∫ � d � � E ( ϕ t ) = − ⟨ τ ( ϕ ) , V ⟩ v g dt � M t =0 where V is the variation vector field of ϕ t with ϕ 0 = ϕ . Takashi Sakai Biharmonic hypersurfaces in compact symmetric spaces

Biharmonic maps ∫ E 2 ( ϕ ) = 1 ∥ τ ( ϕ ) ∥ 2 v g bienergy functional 2 M Definition (Eells-Lemaire) ϕ : M − → N : biharmonic def ⇐ ⇒ ϕ is a critical point of E 2 ⇐ ⇒ τ 2 ( ϕ ) := J ( τ ( ϕ )) = 0 bitension field (Guoying Jiang) ( V ∈ Γ( ϕ − 1 TN )) J ( V ) = ∆ V − R ( V ) Jacobi operator ∆ V = − ∑ m i =1 {∇ e i ∇ e i V − ∇ ∇ ei e i V } rough Laplacian R ( V ) = ∑ m i =1 R h ( V, dϕ ( e i )) dϕ ( e i ) ϕ : harmonic = ⇒ ϕ : biharmonic We call ϕ proper biharmonic if it is biharmonic, but not harmonic. Takashi Sakai Biharmonic hypersurfaces in compact symmetric spaces

B.-Y. Chen conjecture B.-Y. Chen conjecture Every biharmonic submanifold of the Euclidean space R n must be harmonic (minimal). Theorem (Akutagawa-Maeta 2013) Every complete properly immersed biharmonic submanifold of the Euclidean space R n is minimal. Generalized B.-Y. Chen conjecture (Caddeo-Montaldo-Piu) Every biharmonic submanifold of a Riemannian manifold of non-positive curvature must be harmonic (minimal). Takashi Sakai Biharmonic hypersurfaces in compact symmetric spaces

Purpose of this talk Examples of proper biharmonic submanifolds in S n √ A small sphere S n − 1 (1 / 2) ⊂ S n (1) . The Clifford hypersurface √ √ S n − p (1 / 2) × S p − 1 (1 / 2) ⊂ S n (1) ( n − p ̸ = p − 1) . The goal of our project Classify all proper biharmonic homogeneous hypersurfaces in irreducible compact symmetric spaces. Takashi Sakai Biharmonic hypersurfaces in compact symmetric spaces

Biharmonic isometric immersions Theorem (Jiang 2009, Ohno-S-Urakawa 2015) Let ϕ : ( M m , g ) → ( N n , h ) be an isometric immersion. Assume ⊥ τ ( ϕ ) = 0 . Then, ϕ is biharmonic if and only if that ∇ m m ∑ ∑ R h ( ) ( ) τ ( ϕ ) , dϕ ( e i ) dϕ ( e i ) = B ϕ A τ ( ϕ ) e i , e i , i =1 i =1 where A ξ denotes the shape operator of ϕ w.r.t. ξ ∈ T ⊥ M . Corollary Assume that the sectional curvature of ( N n , h ) is non-positive. Let ϕ : ( M m , g ) → ( N n , h ) be an isometric immersion satisfying ⊥ τ ( ϕ ) = 0 . Then, ∇ ϕ : biharmonic ⇐ ⇒ ϕ : harmonic Takashi Sakai Biharmonic hypersurfaces in compact symmetric spaces

Biharmonic hypersurfaces Theorem (Ou 2010, Ohno-S-Urakawa 2015) Let ϕ : ( M n − 1 , g ) → ( N n , h ) be an isometric immersion of 1 codimension 1 . Assume that the mean curvature H := n − 1 ∥ τ ( ϕ ) ∥ is non-zero constant. Then ρ h ( ξ ) = ∥ B ϕ ∥ 2 ξ ⇐ ⇒ ϕ is biharmonic where ρ h is the Ricci transform of ( N, h ) , and ξ is a local unit normal vector field along ϕ . In particular, if ( N, h ) is an Einstein manifold, i.e., ρ h = c Id , then ∥ B ϕ ∥ 2 = c. ⇐ ⇒ ϕ is biharmonic Takashi Sakai Biharmonic hypersurfaces in compact symmetric spaces

Biharmonic isoparametric hypersurfaces in S n M ⊂ S n (1) : isoparametric hypersurface g = 1 , 2 , 3 , 4 or 6 0 < θ < π/g Principal curvatures λ 1 > λ 2 > · · · > λ g are given by ( ) θ + ( i − 1) π λ i = cot ( i = 1 , . . . g ) g m i = m i +2 ( subscription mod g ) Theorem (Ichiyama-Inoguchi-Urakawa 2008, 2010) 1 A small sphere and the Clifford hypersurface are the only proper biharmonic isoparametric hypersurfaces in S n . 2 Classification of all proper biharmonic homogeneous hypersurfaces in C P n and H P n . Takashi Sakai Biharmonic hypersurfaces in compact symmetric spaces

Hermann actions ( G, K 1 ) , ( G, K 2 ) : symmetric pairs of compact type i.e. G : compact connected semisimple Lie group G θ i θ 2 0 ⊂ K i ⊂ G θ i θ i ∈ Aut( G ) ( i = 1 , 2) i = id G s.t. G � ❅ π 1 π 2 ✠ � ❅ ❘ K 2 \ G G/K 1 g = k 1 ⊕ m 1 = k 2 ⊕ m 2 { X ∈ g | dθ 1 ( X ) = − X } ∼ m 1 = = T π 1 ( e ) ( G/K 1 ) { X ∈ g | dθ 2 ( X ) = − X } ∼ m 2 = = T π 2 ( e ) ( K 2 \ G ) Takashi Sakai Biharmonic hypersurfaces in compact symmetric spaces

Hermann actions ( G, K 1 ) , ( G, K 2 ) : symmetric pairs of compact type i.e. G : compact connected semisimple Lie group G θ i θ 2 0 ⊂ K i ⊂ G θ i θ i ∈ Aut( G ) ( i = 1 , 2) i = id G s.t. K 2 × K 1 ↷ G � ❅ π 1 π 2 � ✠ ❘ ❅ K 2 ↷ G/K 1 K 2 \ G ↶ K 1 Hermann actions ❅ � ❅ ❘ � ✠ K 2 \ G/K 1 orbit space g = k 1 ⊕ m 1 = k 2 ⊕ m 2 { X ∈ g | dθ 1 ( X ) = − X } ∼ m 1 = = T π 1 ( e ) ( G/K 1 ) { X ∈ g | dθ 2 ( X ) = − X } ∼ m 2 = = T π 2 ( e ) ( K 2 \ G ) Takashi Sakai Biharmonic hypersurfaces in compact symmetric spaces

Fundamental properties of Hermann actions Definition G ↷ ( M, g ) : hyperpolar ⇐ ⇒ There exists a flat section S , i.e. S is a closed, flat, totally geodesic submanifold of M , and all G -orbits meet S perpendicularly. Proposition Hermann actions K 2 ↷ G/K 1 , K 2 \ G ↶ K 1 , K 2 × K 1 ↷ G are hyperpolar. ∵ ) a ⊂ m 1 ∩ m 2 : maximal abelian subspace A := exp a ⊂ G toral subgroup Then G = K 2 AK 1 A ⊂ G is a section of K 2 × K 1 ↷ G π 1 ( A ) ⊂ G/K 1 is a section of K 2 ↷ G/K 1 Takashi Sakai Biharmonic hypersurfaces in compact symmetric spaces

Symmetric triads (Ikawa 2011) Hereafter we assume that θ 1 θ 2 = θ 2 θ 1 . g = ( k 1 ∩ k 2 ) + ( m 1 ∩ m 2 ) + ( k 1 ∩ m 2 ) + ( m 1 ∩ k 2 ) ad( a ) -inv. ad( a ) -inv. a ⊂ m 1 ∩ m 2 : maximal abelian subspace For λ ∈ a { } X ∈ m 1 ∩ m 2 | (ad H ) 2 X = −⟨ λ, H ⟩ 2 X ( H ∈ a ) m λ := { } X ∈ m 1 ∩ k 2 | (ad H ) 2 X = −⟨ λ, H ⟩ 2 X ( H ∈ a ) V λ := Σ := { λ ∈ a \ { 0 } | m λ ̸ = { 0 }} W := { λ ∈ a \ { 0 } | V λ ̸ = { 0 }} � Σ := Σ ∪ W ( � Σ , Σ , W ) : symmetric triad with multiplicities Takashi Sakai Biharmonic hypersurfaces in compact symmetric spaces

Orbit space of Hermann actions � { } ∩ � ⟨ λ, H ⟩ ̸∈ π Z , ⟨ α, H ⟩ ̸∈ π � 2 + π Z a reg = H ∈ a λ ∈ Σ ,α ∈ W P : a cell, (a connected component of a reg ) K 2 \ G/K 1 ∼ = P ∼ = a /W (˜ Σ , Σ , W ) For each orbit K 2 π 1 ( x ) , there exists H ∈ P uniquely so that x = exp H . H ∈ P ← → regular orbit H ∈ ∂P ← → singular orbit P is a simplex in a , and the cell decomposition of P gives a stratification of the orbit types. Takashi Sakai Biharmonic hypersurfaces in compact symmetric spaces

Case of type III- A 2 ( G, K 1 , K 2 ) = ( SU (6) , SO (6) , Sp (3)) � Σ = Σ = W = { e i − e j | 1 ≤ i, j ≤ 3 , i ̸ = j } � { } � ⟨ e i − e j , H ⟩ ̸∈ π � 2 Z (1 ≤ i < j ≤ 3) H ∈ a a reg = ⟨ e i − e j , H ⟩∈ π Z ⟨ e i − e j , H ⟩∈ π 2 + π Z Takashi Sakai Biharmonic hypersurfaces in compact symmetric spaces

Case of type III- A 2 ( G, K 1 , K 2 ) = ( SU (6) , SO (6) , Sp (3)) � Σ = Σ = W = { e i − e j | 1 ≤ i, j ≤ 3 , i ̸ = j } � { } � ⟨ e i − e j , H ⟩ ̸∈ π � 2 Z (1 ≤ i < j ≤ 3) H ∈ a a reg = ⟨ e i − e j , H ⟩∈ π Z ⟨ e i − e j , H ⟩∈ π 2 + π Z Takashi Sakai Biharmonic hypersurfaces in compact symmetric spaces

Case of type III- A 2 ( G, K 1 , K 2 ) = ( SU (6) , SO (6) , Sp (3)) � Σ = Σ = W = { e i − e j | 1 ≤ i, j ≤ 3 , i ̸ = j } � { } � ⟨ e i − e j , H ⟩ ̸∈ π � 2 Z (1 ≤ i < j ≤ 3) H ∈ a a reg = ⟨ e i − e j , H ⟩∈ π Z ⟨ e i − e j , H ⟩∈ π 2 + π Z Takashi Sakai Biharmonic hypersurfaces in compact symmetric spaces

Minimal orbits For x = exp H ( H ∈ a ) , the tension field τ H of K 2 π 1 ( x ) is dL − 1 x ( τ H ) ∑ ∑ = − m ( λ ) cot ⟨ λ, H ⟩ λ + n ( α ) tan ⟨ α, H ⟩ α. λ ∈ Σ+ α ∈ W + ⟨ λ,H ⟩̸∈ π Z ⟨ α,H ⟩̸∈ ( π/ 2)+ π Z Theorem (Ikawa 2011, Ohno 2016) 1 There exists a unique minimal orbit K 2 π ( x ) ⊂ G/K 1 in each strata of P ∼ = K 2 \ G/K 1 . 2 K 2 π 1 ( x ) ⊂ G/K 1 minimal ⇐ ⇒ π 2 ( x ) K 1 ⊂ K 2 \ G minimal ⇐ ⇒ K 2 xK 1 ⊂ G minimal Takashi Sakai Biharmonic hypersurfaces in compact symmetric spaces