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Toward classification of biharmonic homogeneous hypersurfaces in compact symmetric spaces Takashi Sakai Tokyo Metropolitan University & OCAMI 2019 Workshop on the Isoparametric Theory Beijin Normal University, Beijin

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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

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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

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Harmonic maps

(Mm, g), (Nn, h) : Riemannian manifolds ϕ : M − → N smooth map E(ϕ) = 1 2 ∫

M

∥dϕ∥2vg energy functional Definition ϕ : M − → N : harmonic

def

⇐ ⇒ ϕ is a critical point of E ⇐ ⇒ τ(ϕ) := trace Bϕ = 0 tension field d dt

  • t=0

E(ϕt) = − ∫

M

⟨τ(ϕ), V ⟩vg where V is the variation vector field of ϕt with ϕ0 = ϕ.

Takashi Sakai Biharmonic hypersurfaces in compact symmetric spaces

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Biharmonic maps

E2(ϕ) = 1 2 ∫

M

∥τ(ϕ)∥2vg bienergy functional Definition (Eells-Lemaire) ϕ : M − → N : biharmonic

def

⇐ ⇒ ϕ is a critical point of E2 ⇐ ⇒ τ2(ϕ) := J(τ(ϕ)) = 0 bitension field (Guoying Jiang) J(V ) = ∆V − R(V ) (V ∈ Γ(ϕ−1TN)) Jacobi operator ∆V = − ∑m

i=1{∇ei∇eiV − ∇∇eieiV }

rough Laplacian R(V ) = ∑m

i=1 Rh(V, dϕ(ei))dϕ(ei)

ϕ : harmonic = ⇒ ϕ : biharmonic We call ϕ proper biharmonic if it is biharmonic, but not harmonic.

Takashi Sakai Biharmonic hypersurfaces in compact symmetric spaces

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B.-Y. Chen conjecture

B.-Y. Chen conjecture Every biharmonic submanifold of the Euclidean space Rn must be harmonic (minimal). Theorem (Akutagawa-Maeta 2013) Every complete properly immersed biharmonic submanifold of the Euclidean space Rn 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

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Purpose of this talk

Examples of proper biharmonic submanifolds in Sn A small sphere Sn−1(1/ √ 2) ⊂ Sn(1). The Clifford hypersurface Sn−p(1/ √ 2) × Sp−1(1/ √ 2) ⊂ Sn(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

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Biharmonic isometric immersions

Theorem (Jiang 2009, Ohno-S-Urakawa 2015) Let ϕ : (Mm, g) → (Nn, h) be an isometric immersion. Assume that ∇

⊥τ(ϕ) = 0. Then, ϕ is biharmonic if and only if m

∑

i=1

Rh( τ(ϕ), dϕ(ei) ) dϕ(ei) =

m

∑

i=1

Bϕ ( Aτ(ϕ)ei, ei ) , where Aξ denotes the shape operator of ϕ w.r.t. ξ ∈ T ⊥M. Corollary Assume that the sectional curvature of (Nn, h) is non-positive. Let ϕ : (Mm, g) → (Nn, h) be an isometric immersion satisfying ∇

⊥τ(ϕ) = 0. Then,

ϕ : biharmonic ⇐ ⇒ ϕ : harmonic

Takashi Sakai Biharmonic hypersurfaces in compact symmetric spaces

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Biharmonic hypersurfaces

Theorem (Ou 2010, Ohno-S-Urakawa 2015) Let ϕ : (Mn−1, g) → (Nn, h) be an isometric immersion of codimension 1. Assume that the mean curvature H :=

1 n−1∥τ(ϕ)∥

is non-zero constant. Then ϕ is biharmonic ⇐ ⇒ ρh(ξ) = ∥Bϕ∥2 ξ where ρh 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., ρh = c Id, then ϕ is biharmonic ⇐ ⇒ ∥Bϕ∥2 = c.

Takashi Sakai Biharmonic hypersurfaces in compact symmetric spaces

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Biharmonic isoparametric hypersurfaces in Sn

M ⊂ Sn(1) : isoparametric hypersurface g = 1, 2, 3, 4 or 6 0 < θ < π/g Principal curvatures λ1 > λ2 > · · · > λg are given by λi = cot ( θ + (i − 1)π g ) (i = 1, . . . g) mi = mi+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 Sn.

2 Classification of all proper biharmonic homogeneous

hypersurfaces in CP n and HP n.

Takashi Sakai Biharmonic hypersurfaces in compact symmetric spaces

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Hermann actions

(G, K1), (G, K2) : symmetric pairs of compact type i.e. G : compact connected semisimple Lie group θi ∈ Aut(G) (i = 1, 2) θ2

i = idG

s.t. Gθi

0 ⊂ Ki ⊂ Gθi

G G/K1 π1 π2 K2\G

  • ✠

❅ ❅ ❘

g = k1 ⊕ m1 = k2 ⊕ m2 m1 = {X ∈ g | dθ1(X) = −X} ∼ = Tπ1(e)(G/K1) m2 = {X ∈ g | dθ2(X) = −X} ∼ = Tπ2(e)(K2\G)

Takashi Sakai Biharmonic hypersurfaces in compact symmetric spaces

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Hermann actions

(G, K1), (G, K2) : symmetric pairs of compact type i.e. G : compact connected semisimple Lie group θi ∈ Aut(G) (i = 1, 2) θ2

i = idG

s.t. Gθi

0 ⊂ Ki ⊂ Gθi

K2 × K1 ↷ G K2 ↷ G/K1 π1 π2 K2\G ↶ K1 K2\G/K1

  • rbit space
  • ✠

❅ ❅ ❘ ❅ ❅ ❘

  • ✠

Hermann actions g = k1 ⊕ m1 = k2 ⊕ m2 m1 = {X ∈ g | dθ1(X) = −X} ∼ = Tπ1(e)(G/K1) m2 = {X ∈ g | dθ2(X) = −X} ∼ = Tπ2(e)(K2\G)

Takashi Sakai Biharmonic hypersurfaces in compact symmetric spaces

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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 K2 ↷ G/K1, K2\G ↶ K1, K2 × K1 ↷ G are hyperpolar. ∵) a ⊂ m1 ∩ m2 : maximal abelian subspace A := exp a ⊂ G toral subgroup Then G = K2AK1 A ⊂ G is a section of K2 × K1 ↷ G π1(A) ⊂ G/K1 is a section of K2 ↷ G/K1

Takashi Sakai Biharmonic hypersurfaces in compact symmetric spaces

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Symmetric triads (Ikawa 2011)

Hereafter we assume that θ1θ2 = θ2θ1. g = (k1 ∩ k2) + (m1 ∩ m2) + (k1 ∩ m2) + (m1 ∩ k2) ad(a)-inv. ad(a)-inv. a ⊂ m1 ∩ m2 : maximal abelian subspace For λ ∈ a mλ := { X ∈ m1 ∩ m2 | (adH)2X = −⟨λ, H⟩2X (H ∈ a) } Vλ := { X ∈ m1 ∩ k2 | (adH)2X = −⟨λ, H⟩2X (H ∈ a) } Σ := {λ ∈ a \ {0} | mλ ̸= {0}} W := {λ ∈ a \ {0} | Vλ ̸= {0}}

  • Σ := Σ ∪ W

( Σ, Σ, W) : symmetric triad with multiplicities

Takashi Sakai Biharmonic hypersurfaces in compact symmetric spaces

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Orbit space of Hermann actions

areg = ∩

λ∈Σ,α∈W

{ H ∈ a

  • ⟨λ, H⟩ ̸∈ πZ, ⟨α, H⟩ ̸∈ π

2 + πZ } P : a cell, (a connected component of areg) K2\G/K1 ∼ = P ∼ = a/W(˜ Σ, Σ, W) For each orbit K2π1(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

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Case of type III-A2

(G, K1, K2) = (SU(6), SO(6), Sp(3))

  • Σ = Σ = W = {ei − ej | 1 ≤ i, j ≤ 3, i ̸= j}

areg = { H ∈ a

  • ⟨ei − ej, H⟩ ̸∈ π

2 Z (1 ≤ i < j ≤ 3) } ⟨ei − ej, H⟩∈ πZ ⟨ei − ej, H⟩∈ π 2 + πZ

Takashi Sakai Biharmonic hypersurfaces in compact symmetric spaces

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Case of type III-A2

(G, K1, K2) = (SU(6), SO(6), Sp(3))

  • Σ = Σ = W = {ei − ej | 1 ≤ i, j ≤ 3, i ̸= j}

areg = { H ∈ a

  • ⟨ei − ej, H⟩ ̸∈ π

2 Z (1 ≤ i < j ≤ 3) } ⟨ei − ej, H⟩∈ πZ ⟨ei − ej, H⟩∈ π 2 + πZ

Takashi Sakai Biharmonic hypersurfaces in compact symmetric spaces

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Case of type III-A2

(G, K1, K2) = (SU(6), SO(6), Sp(3))

  • Σ = Σ = W = {ei − ej | 1 ≤ i, j ≤ 3, i ̸= j}

areg = { H ∈ a

  • ⟨ei − ej, H⟩ ̸∈ π

2 Z (1 ≤ i < j ≤ 3) } ⟨ei − ej, H⟩∈ πZ ⟨ei − ej, H⟩∈ π 2 + πZ

Takashi Sakai Biharmonic hypersurfaces in compact symmetric spaces

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Minimal orbits

For x = exp H (H ∈ a), the tension field τH of K2π1(x) is dL−1

x (τH)

= − ∑

λ∈Σ+ ⟨λ,H⟩̸∈πZ

m(λ) cot⟨λ, H⟩λ + ∑

α∈W + ⟨α,H⟩̸∈(π/2)+πZ

n(α) tan⟨α, H⟩α. Theorem (Ikawa 2011, Ohno 2016)

1 There exists a unique minimal orbit K2π(x) ⊂ G/K1 in each

strata of P ∼ = K2\G/K1.

2 K2π1(x) ⊂ G/K1 minimal ⇐

⇒ π2(x)K1 ⊂ K2\G minimal ⇐ ⇒ K2xK1 ⊂ G minimal

Takashi Sakai Biharmonic hypersurfaces in compact symmetric spaces

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Biharmonic orbits of Hermann actions

Proposition (Ikawa-S-Tasaki 2001) Every orbits of Hermann actions satisfy ∇

⊥τ(ϕ) = 0.

Theorem (Ohno-S-Urakawa 2019) For x = exp H (H ∈ a), the orbit K2π1(x) ⊂ G/K1 is biharmonic if and only if ∑

λ∈Σ+ ⟨λ,H⟩̸∈πZ

m(λ)⟨dL−1

x (τH), λ⟩

( 1 − (cot⟨λ, H⟩)2) λ + ∑

α∈W + ⟨α,H⟩̸∈(π/2)+πZ

n(α)⟨dL−1

x (τH), α⟩

( 1 − (tan⟨α, H⟩)2) α = 0. Corollary K2π1(x) ⊂ G/K1 biharmonic ⇐ ⇒ π2(x)K1 ⊂ K2\G biharmonic

Takashi Sakai Biharmonic hypersurfaces in compact symmetric spaces

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Biharmonic orbits of Hermann actions (case of dim a = 1)

When dim a = 1, a regular orbit K2π(x) is a homogeneous hypersurface in G/K1. We determined all proper biharmonic regular orbits of K2 ↷ G/K1, and obtained: Theorem (Ohno-S-Urakawa 2015) When G is simple, and dim a = 1, all the regular orbits K2π1(x) ⊂ G/K1 are classified into the following three classes:

1 There exists a unique regular orbit which is proper biharmonic.

(3 cases)

2 There exist exactly two regular orbits which are proper

  • biharmonic. (7 cases)

3 All the biharmonic regular orbits must be harmonic. (8 cases) Takashi Sakai Biharmonic hypersurfaces in compact symmetric spaces

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Biharmonic orbits of K2 × K1 ↷ G

Theorem (Ohno-S-Urakawa 2019) For x = exp H (H ∈ a), the orbit (K2 × K1) · x is biharmonic in G if and only if ∑

λ∈Σ+\ΣH

m(λ)⟨dL−1

x (τH)x, λ⟩

(3 2 − (cot⟨λ, H⟩)2 ) λ + ∑

α∈W +\WH

n(α)⟨dL−1

x (τH)x, α⟩

(3 2 − (tan⟨α, H⟩)2 ) α + ∑

µ∈Σ+

H

m(µ)⟨dL−1

x (τH)x, µ⟩µ +

∑

β∈W +

H

n(β)⟨dL−1

x (τH)x, β⟩β = 0.

Takashi Sakai Biharmonic hypersurfaces in compact symmetric spaces

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Biharmonic orbits of K2 × K1 ↷ G (case of dim a = 1)

Theorem (Ohno-S-Urakawa 2019) When G is simple, and dim a = 1, all the regular orbits K2xK1 ⊂ G are classified into the following three classes:

1 There exist exactly two regular orbits which are proper

  • biharmonic. (15 cases)

2 All the biharmonic regular orbits must be harmonic. (4 cases) Takashi Sakai Biharmonic hypersurfaces in compact symmetric spaces

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Examples of proper biharmonic homogeneous hypersurfaces

(G, K1, K2) = ( SO(n + 1), SO(n), SO(p) × SO(n − p + 1) ) K2 = SO(p) × SO(n − p + 1) ↷ G/K1 ∼ = Sn K1 = SO(n) ↷ K2\G ∼ =

  • Gp(Rn+1)

K2π1(x) ∼ = Sp−1 × Sn−p ⊂ Sn ∼ = G/K1 ↕ Clifford hypersurface π2(x)K1 ∼ =

  • Fp,n−p,1(R) ⊂
  • Gp(Rn+1) ∼

= K2\G ∼ = SO(n)/(SO(p − 1) × SO(n − p) × SO(1)) universal covering of a real flag manifold

Takashi Sakai Biharmonic hypersurfaces in compact symmetric spaces

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Homogeneous hypersurfaces in compact symmetric spaces (Kollross)

G : connected simple compact Lie group U ⊂ G × G : closed connected subgroup which acts with cohomogeneity one on G Then U-action is orbit equivalent to one of the following:

1 Hermann action of cohomogeneity one, i.e. dim a = 1

(10 commutative cases, 2 non-commutative cases)

2 σ-action of cohomogeneity one,

(2 cases) the adjoint action of SU(2), or the action of {(g, σ(g)) | g ∈ G} on SU(3), where σ ∈ Out(SU(3)).

3 The ρ(H)-action on Sn, CP n, HP n, where ρ is the isotropy

representation of a rank two symmetric space G/H.

4 Exceptional cohomogeneity one action (7 cases) Takashi Sakai Biharmonic hypersurfaces in compact symmetric spaces

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Related works

Inoguchi and Sasahara studied biharmonic homogeneous hypersurfaces in compact symmetric spaces independently. Enoyoshi showed that there are exactly two proper biharmonic

  • rbits of G2-action on

Gr3(ImO), which is an exceptional cohomogeneity one action.

Takashi Sakai Biharmonic hypersurfaces in compact symmetric spaces

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Further problems

Complete classification of biharmonic homogeneous hypersurfaces in irreducible compact symmetric spaces. List of principal curvatures of homogeneous hypersurfaces.

Hermann actions of dim a = 1 with θ1θ2 ̸= θ2θ1. Exceptional cohomogeneity one actions

Biharmonic orbits of Hermann actions when dim a ≥ 2. Inhomogeneous proper biharmonic hypersurfaces in compact symmetric spaces. Balmu¸ s-Montaldo-Oniciuc Conjecture Any biharmonic submanifold in Sn has constant mean curvature. Problem Can we find inhomogeneous proper biharmonic isoparametric hypersurfaces in compact symmetric spaces?

Takashi Sakai Biharmonic hypersurfaces in compact symmetric spaces

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Thank you very much for your attention

Takashi Sakai Biharmonic hypersurfaces in compact symmetric spaces

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  • K. Akutagawa and Sh. Maeta, Properly immersed biharmonic

submanifolds in the Euclidean spaces, Geometriae Dedicata, 164 (2013), 351–355.

  • R. Caddeo, S. Montaldo and P. Piu, On biharmonic maps,
  • Contemp. Math., 288 (2001), 286–290.

B.-Y. Chen, Some open problems and conjectures on submanifolds of finite type, Soochow J. Math., 17 (1991), 169–188.

  • J. Eells and L. Lemaire, Selected Topics in Harmonic Maps,

CBMS, Regional Conference Series in Math., Amer. Math. Soc., 50, 1983.

  • K. Enoyoshi, Principal curvatures of homogeneous

hypersurfaces in a Grassmann manifold Gr3(ImO) by the G2-action, to appear in Tokyo J. Math.

Takashi Sakai Biharmonic hypersurfaces in compact symmetric spaces

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  • T. Ichiyama, J. Inoguchi, H. Urakawa, Biharmonic maps and

bi-Yang-Mills fields, Note di Mat., 28, (2009), 233–275.

  • T. Ichiyama, J. Inoguchi, H. Urakawa, Classifications and

isolation phenomena of biharmonic maps and bi-Yang-Mills fields, Note di Mat., 30, (2010), 15–48.

  • O. Ikawa, The geometry of symmetric triad and orbit spaces of

Hermann actions, J. Math. Soc. Japan 63 (2011), 79–136.

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  • J. Inoguchi and T. Sasahara, Biharmonic hypersurfaces in

Riemannian symmetric spaces I, Hiroshima Math. J. 46 (2016), no. 1, 97–121.

Takashi Sakai Biharmonic hypersurfaces in compact symmetric spaces

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  • J. Inoguchi and T. Sasahara, Biharmonic hypersurfaces in

Riemannian symmetric spaces II, Hiroshima Math. J. G.Y. Jiang, 2-harmonic maps and their first and second variational formula, Chinese Ann. Math., 7A (1986), 388–402; Note di Mat., 28 (2009), 209–232.

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571–612.

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to be weakly reflective, Tokyo J. Math. 39 (2016), no. 2, 537–564.

  • S. Ohno, T. Sakai and H. Urakawa, Biharmonic homogeneous

hypersurfaces in compact symmetric spaces, Differential Geom.

  • Appl. 43 (2015), 155–179.

Takashi Sakai Biharmonic hypersurfaces in compact symmetric spaces

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  • 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. Y.-L. Ou, Biharmonic hypersurfaces in Riemannian manifolds, Pacific J. Math. 248 (2010), no. 1, 217–232. Y.-L. Ou and L. Tang, On the generalized Chen’s conjecture

  • n biharmonic submanifolds, Michigan Math. J., 61 (2012),

531–542.

Takashi Sakai Biharmonic hypersurfaces in compact symmetric spaces