Absolutely area-minimizing cones.
over some minimal submanifolds in S nTakashi Sakai.

Tokyo Metropolitan University
Variational problems for curves and surfaces and related topics
at Nara Women’s University
July 1, 2009

Takashi Sakai

Area-minimizing cones over some minimal submanifolds in S nTable of contents
Methods to show area-minimizing property
Geometric measure theory
Calibration
Area-nonincreasing retraction
Integral geometry
Plan of this talk1

Motivation and history of area-minimizing cones

2

Area-minimizing cones over some minimal orbits ofs-representations

3

Weakly reﬂective submanifolds and austere submanifolds

4

Special Lagrangian cones
Takashi Sakai

Area-minimizing cones over some minimal submanifolds in S nTable of contents
Methods to show area-minimizing property
Geometric measure theory
Calibration
Area-nonincreasing retraction
Integral geometry
Plan of this talk1

Motivation and history of area-minimizing cones

2

Area-minimizing cones over some minimal orbits ofs-representations

3

Weakly reﬂective submanifolds and austere submanifolds

4

Special Lagrangian cones
Takashi Sakai

Area-minimizing cones over some minimal submanifolds in S nIntroduction and Motivation
Plateau problem
For given a Jordan curve Γ in Rn , does there exist an
area-minimizing surface with boundary Γ ?

.

.
(Douglas and Rado).

AnswerYES!

by geometric measure theory
But such a surface may have singularities“integral current”
Theorem (Osserman)
An area-minimizing surface M 2 in R3 with smooth boundary has
no singularities.
Takashi Sakai

Area-minimizing cones over some minimal submanifolds in S nIntroduction and Motivation
Plateau problem
For given a Jordan curve Γ in Rn , does there exist an
area-minimizing surface with boundary Γ ?

.

.
(Douglas and Rado).

AnswerYES!

by geometric measure theory
But such a surface may have singularities“integral current”
Theorem (Osserman)
An area-minimizing surface M 2 in R3 with smooth boundary has
no singularities.
Takashi Sakai

Area-minimizing cones over some minimal submanifolds in S nIntroduction and Motivation
Plateau problem
For given a Jordan curve Γ in Rn , does there exist an
area-minimizing surface with boundary Γ ?

.

.
(Douglas and Rado).

AnswerYES!

by geometric measure theory
But such a surface may have singularities“integral current”
Theorem (Osserman)
An area-minimizing surface M 2 in R3 with smooth boundary has
no singularities.
Takashi Sakai

Area-minimizing cones over some minimal submanifolds in S nIntroduction and Motivation
Theorem (Almgren)
The dimension of the singular set of an area-minimizing.
submanifold M k is equal or less than k − 2.
A minimal surface M 2 in Rn (n ≥ 4) may have 0-dimensional.
singular set, i.e. isolated singularities.
Questions
What kind of singularities can appear on area-minimizing
submanifolds?
Study area-minimizing cones, which are tangent cones atisolated singularities.

Takashi Sakai

Area-minimizing cones over some minimal submanifolds in S nIntroduction and Motivation
Theorem (Almgren)
The dimension of the singular set of an area-minimizing.
submanifold M k is equal or less than k − 2.
A minimal surface M 2 in Rn (n ≥ 4) may have 0-dimensional.
singular set, i.e. isolated singularities.
Questions
What kind of singularities can appear on area-minimizing
submanifolds?
Study area-minimizing cones, which are tangent cones at
isolated singularities.

Takashi Sakai

Area-minimizing cones over some minimal submanifolds in S nArea-minimizing cones
M k−1 ⊂ S n−1 (1) ⊂ Rn
CM= {tx | x ∈ M, t ≥ 0} ⊂ Rn

1
CM= {tx | x ∈ M, 0 ≤ t ≤ 1}

Deﬁnition
1 has minimum volume
CM is said to be area-minimizing if CM
among all submanifolds with boundary M .

Proposition
M ⊂ S n−1 : minimal

Takashi Sakai=⇒CM ⊂ Rn : minimal.

Area-minimizing cones over some minimal submanifolds in S nArea-minimizing cones
M k−1 ⊂ S n−1 (1) ⊂ Rn
CM= {tx | x ∈ M, t ≥ 0} ⊂ Rn

1CM= {tx | x ∈ M, 0 ≤ t ≤ 1}

Deﬁnition
1 has minimum volume
CM is said to be area-minimizing if CM
among all submanifolds with boundary M .

Proposition
M ⊂ S n−1 : minimal

Takashi Sakai=⇒CM ⊂ Rn : minimal.

Area-minimizing cones over some minimal submanifolds in S nHistory and Examples of area-minimizing cones
1969 (Bombieri-DeGiorgi-Giusti) M = S n × S n ⊂ S 2n+1 (n ≥ 3),then CM ⊂ R2n+2 is area-minimizing.
1972 (Lawson) M = S k × S l ⊂ S k+l+1 (k + l ≥ 7),then CM ⊂ Rk+l+2 is area-minimizing.
1974 (Simoes)
M = S 2 × S 4 ⊂ S 7 , then CM ⊂ R8 is area-minimizing.
M = S 1 × S 5 ⊂ S 7 , then CM ⊂ R8 is NOT area-minimizing.
1991 (Lawlor) provided a criterion for a cone to be area-minimizing,
and classiﬁed all area-minimizing cones over product of spheres.
1988 (Cheng), 1994 (Kerckhove), 2000 (Hirohashi-Kanno-Tasaki),2002 (Kanno)
Cones over some symmetric R-spaces are area-minimizing.
Takashi Sakai

Area-minimizing cones over some minimal submanifolds in S nHistory and Examples of area-minimizing cones
1969 (Bombieri-DeGiorgi-Giusti) M = S n × S n ⊂ S 2n+1 (n ≥ 3),then CM ⊂ R2n+2 is area-minimizing.
1972 (Lawson) M = S k × S l ⊂ S k+l+1 (k + l ≥ 7),then CM ⊂ Rk+l+2 is area-minimizing.
1974 (Simoes)
M = S 2 × S 4 ⊂ S 7 , then CM ⊂ R8 is area-minimizing.
M = S 1 × S 5 ⊂ S 7 , then CM ⊂ R8 is NOT area-minimizing.
1991 (Lawlor) provided a criterion for a cone to be area-minimizing,and classiﬁed all area-minimizing cones over product of spheres.
1988 (Cheng), 1994 (Kerckhove), 2000 (Hirohashi-Kanno-Tasaki),2002 (Kanno)
Cones over some symmetric R-spaces are area-minimizing.
Takashi Sakai

Area-minimizing cones over some minimal submanifolds in S nHistory and Examples of area-minimizing cones
1969 (Bombieri-DeGiorgi-Giusti) M = S n × S n ⊂ S 2n+1 (n ≥ 3),then CM ⊂ R2n+2 is area-minimizing.
1972 (Lawson) M = S k × S l ⊂ S k+l+1 (k + l ≥ 7),then CM ⊂ Rk+l+2 is area-minimizing.
1974 (Simoes)
M = S 2 × S 4 ⊂ S 7 , then CM ⊂ R8 is area-minimizing.
M = S 1 × S 5 ⊂ S 7 , then CM ⊂ R8 is NOT area-minimizing.
1991 (Lawlor) provided a criterion for a cone to be area-minimizing,and classiﬁed all area-minimizing cones over product of spheres.
1988 (Cheng), 1994 (Kerckhove), 2000 (Hirohashi-Kanno-Tasaki),2002 (Kanno)
Cones over some symmetric R-spaces are area-minimizing.
Takashi Sakai

Area-minimizing cones over some minimal submanifolds in S nHistory and Examples of area-minimizing cones
1969 (Bombieri-DeGiorgi-Giusti) M = S n × S n ⊂ S 2n+1 (n ≥ 3),then CM ⊂ R2n+2 is area-minimizing.
1972 (Lawson) M = S k × S l ⊂ S k+l+1 (k + l ≥ 7),then CM ⊂ Rk+l+2 is area-minimizing.
1974 (Simoes)
M = S 2 × S 4 ⊂ S 7 , then CM ⊂ R8 is area-minimizing.
M = S 1 × S 5 ⊂ S 7 , then CM ⊂ R8 is NOT area-minimizing.
1991 (Lawlor) provided a criterion for a cone to be area-minimizing,and classiﬁed all area-minimizing cones over product of spheres.
1988 (Cheng), 1994 (Kerckhove), 2000 (Hirohashi-Kanno-Tasaki),2002 (Kanno)
Cones over some symmetric R-spaces are area-minimizing.
Takashi Sakai

Area-minimizing cones over some minimal submanifolds in S nHistory and Examples of area-minimizing cones
1969 (Bombieri-DeGiorgi-Giusti) M = S n × S n ⊂ S 2n+1 (n ≥ 3),then CM ⊂ R2n+2 is area-minimizing.
1972 (Lawson) M = S k × S l ⊂ S k+l+1 (k + l ≥ 7),then CM ⊂ Rk+l+2 is area-minimizing.
1974 (Simoes)
M = S 2 × S 4 ⊂ S 7 , then CM ⊂ R8 is area-minimizing.
M = S 1 × S 5 ⊂ S 7 , then CM ⊂ R8 is NOT area-minimizing.
1991 (Lawlor) provided a criterion for a cone to be area-minimizing,and classiﬁed all area-minimizing cones over product of spheres.
1988 (Cheng), 1994 (Kerckhove), 2000 (Hirohashi-Kanno-Tasaki),2002 (Kanno)
Cones over some symmetric R-spaces are area-minimizing.
Takashi Sakai

Area-minimizing cones over some minimal submanifolds in S nHistory and Examples of area-minimizing cones
1969 (Bombieri-DeGiorgi-Giusti) M = S n × S n ⊂ S 2n+1 (n ≥ 3),then CM ⊂ R2n+2 is area-minimizing.
1972 (Lawson) M = S k × S l ⊂ S k+l+1 (k + l ≥ 7),then CM ⊂ Rk+l+2 is area-minimizing.
1974 (Simoes)
M = S 2 × S 4 ⊂ S 7 , then CM ⊂ R8 is area-minimizing.
M = S 1 × S 5 ⊂ S 7 , then CM ⊂ R8 is NOT area-minimizing.
1991 (Lawlor) provided a criterion for a cone to be area-minimizing,
and classiﬁed all area-minimizing cones over product of spheres.
1988 (Cheng), 1994 (Kerckhove), 2000 (Hirohashi-Kanno-Tasaki),2002 (Kanno)
Cones over some symmetric R-spaces are area-minimizing.
Takashi Sakai

Area-minimizing cones over some minimal submanifolds in S nArea-nonincreasing retraction
Φ : Rn −→ CM

diﬀerentiable retractionis area-nonincreasing if
J(dΦ)x = sup{∥(dΦ)x (u1 ) ∧ · · · ∧ (dΦ)x (uk )∥} ≤ 1for all x ∈ Rn , where u1 , . . . uk are orthonormal vectors of Tx (Rn )Proposition
If there exists an area-nonincreasing retraction Φ : Rn −→ CM ,then CM is area-minimizing.
Proof .

1Vol(CM) ≤ Vol(Φ(N )) ≤ Vol(N ),1 ⊂ Φ(N ).
for any submanifold N with boundary M , since CM
Takashi Sakai

Area-minimizing cones over some minimal submanifolds in S nArea-nonincreasing retraction
Φ : Rn −→ CM

diﬀerentiable retractionis area-nonincreasing if
J(dΦ)x = sup{∥(dΦ)x (u1 ) ∧ · · · ∧ (dΦ)x (uk )∥} ≤ 1for all x ∈ Rn , where u1 , . . . uk are orthonormal vectors of Tx (Rn )Proposition
If there exists an area-nonincreasing retraction Φ : Rn −→ CM ,then CM is area-minimizing.
Proof .

1Vol(CM) ≤ Vol(Φ(N )) ≤ Vol(N ),1 ⊂ Φ(N ).
for any submanifold N with boundary M , since CMTakashi Sakai

Area-minimizing cones over some minimal submanifolds in S nOrbits of s-representations(G, K) : compact symmetric pair
g=k+m
Ad
Kym∼
= To (G/K) : s-representation

For H ∈ S ⊂ m,Ad(K)H ⊂ S : R-spacea ⊂ m : maximal abelian subspace
R : restricted root system of (g, k)
F : fundamental system of R
C = {H ∈ a | ⟨α, H⟩ > 0 (∀α ∈ F )} : Weyl chamber
Ad(K)C¯ = mC¯ ∩ S : orbit space of K y S
Takashi Sakai

Area-minimizing cones over some minimal submanifolds in S nOrbits of s-representations
For ∆ ⊂ F{C∆ =H ∈ C¯C¯ ∩ S =∪ (⟨α, H⟩ > 0 (α ∈ ∆),⟨β, H⟩ = 0 (β ∈ F − ∆)C∆ ∩ S)}

disjoint union∆⊂FTheorem (Hirohashi-Song-Takagi-Tasaki)
For a subset ∆ ⊂ F there exists unique H ∈ C ∆ ∩ S such that
Ad(K)H is a minimal submanifold in the hypersphere S.
Corollary
When ∆ = {αi } ⊂ F , the orbit Ad(K)H is isolated, so is aminimal submanifold in S.
Takashi Sakai

Area-minimizing cones over some minimal submanifolds in S nOrbits of s-representations
For ∆ ⊂ F{C∆ =H ∈ C¯C¯ ∩ S =∪ (⟨α, H⟩ > 0 (α ∈ ∆),⟨β, H⟩ = 0 (β ∈ F − ∆)C∆ ∩ S)}

disjoint union∆⊂FTheorem (Hirohashi-Song-Takagi-Tasaki)
For a subset ∆ ⊂ F there exists unique H ∈ C ∆ ∩ S such thatAd(K)H is a minimal submanifold in the hypersphere S.
CorollaryWhen ∆ = {αi } ⊂ F , the orbit Ad(K)H is isolated, so is aminimal submanifold in S.
Takashi Sakai

Area-minimizing cones over some minimal submanifolds in S nConstruction of a retraction
Ad(K)H ⊂ S¯ ∥H∥ = 1)
(H ∈ C,Then ∃∆ ⊂ Fs.t. H ∈ C ∆Proposition
For a smooth function f : C¯ → R≥0 which satisfies.

1f (tH) = t(t ≥ 0),

2f |C ∆′ ≡ 0for∀∆′ ⊂ F,

define a mapping ϕ : C¯ → {tH | t ≥ 0} ; x 7→ f (x)H.
Then ϕ extends to a retraction Φ : m → CAd(K)H as
Φ(X) = Ad(k)ϕ(H).

∆′ ̸⊃ ∆,(∀X ∈ m),¯
where X = Ad(k)H (k ∈ K, H ∈ C).
Moreover, Φ is area-nonincreasing if J(dΦx ) ≤ 1 for all x ∈ C.
Takashi Sakai

Area-minimizing cones over some minimal submanifolds in S nCase of type B2
1
H = √ e12f (x) =Takashi Sakai√⟨e1 − e2 , x⟩⟨e1 − e2 , x⟩

Area-minimizing cones over some minimal submanifolds in S nCase of type B2
1
H = e1 √2f (x) =Takashi Sakai√⟨e1 − e2 , x⟩⟨e1 − e2 , x⟩

Area-minimizing cones over some minimal submanifolds in S nCase of type B2
1H = √ (e1 + e2 )2Takashi Sakaif (x) =√2⟨e1 , x⟩⟨e2 , x⟩

Area-minimizing cones over some minimal submanifolds in S nCalibrated geometryf, g) : n-dim. Riemannian manifold(MDeﬁnitionf
α : calibration on M
def
f which satisfies⇐⇒ α is a closed k-form on Mf
for any k-dim. subspace V ⊂ Tx M
α|V ≤ volVf⇐⇒ for any orthonormal system {e1 , . . . , ek } of Tx M
α(e1 , . . . , ek ) ≤ 1.

Takashi Sakai

Area-minimizing cones over some minimal submanifolds in S nCalibrated geometry
Deﬁnitionf : calibrated submanifold w.r.t. a calibration αM ⊂M
def⇐⇒ for all x ∈ M
α|Tx M = volTx M.

ExampleA complex k-dimensional complex submanifold in a Kähler.
f, J, ω) is calibrated by ωk .
manifold (Mk!
Theorem (Harvey-Lawson)
A calibrated submanifold is volume minimizing in its homologyclass.
.
Takashi Sakai

Area-minimizing cones over some minimal submanifolds in S n

Special Lagrangian submanifolds
f, J, ω, Ω) : Calabi-Yau manifold(MDeﬁnitionf : special Lagrangian submanifold of phase θL⊂Mdef⇐⇒ L is calibrated by Re(eiθ Ω) for some θ ∈ R.
1f
 dim L = 2 dim M⇐⇒
ω|L ≡ 0Im(eiθ Ω|L ) ≡ 0.

ExampleAn n-dim. real subspace V ⊂ Cn is calibrated by Re(Ω0 )⇐⇒ V = gU for some g ∈ SU (n)where U = {(x1 , . . . , xn ) ∈ Cn | xi ∈ R} ⊂ Cn
Takashi Sakai

Area-minimizing cones over some minimal submanifolds in S n.

Definition of austere submanifold

Deﬁnition (Harvey-Lawson)f : austere submanifold
M ⊂Mdef⇐⇒ for all ξ ∈ Tx⊥ M , the set of eigenvalues with theirmultiplicities of the shape operator Aξ of M is invariant
under the multiplication by −1.
.
An austere submanifolds is a minimal submanifold.
A minimal surface is an austere submanifold.

.
Takashi Sakai

Area-minimizing cones over some minimal submanifolds in S n

Special Lagrangian normal bundles
M ⊂ S n : submanifold
Φ : N 1 M × S 1 −→ S 2n+1 ⊂ R2n+2Legendrian(x, ξ, e ) 7−→ (cos θ x, sin θ ξ)iθProposition (Harvey-Lawson)
M ⊂ S n : austere⇐⇒Φ : minimale modifying Φ and showed that
Borrelli-Gorodski deﬁned a map Φ.
e : Legendrian immersionAξ does not have 0-eigenvalue =⇒ ΦeΦ(M) ⊂ S 2n+1 is a minimal Legendrian submanifold .
⇐⇒ CΦ(M⊂ Cn+1 is a special Lagrangian submanifolde)
Takashi Sakai

Area-minimizing cones over some minimal submanifolds in S nDefinition of a weakly reflective submanifoldDeﬁnition (Ikawa-Tasaki-S.)
f : weakly reflective submanifold (WRS)M ⊂M
def⇐⇒ for each x ∈ M and each ξ ∈ Tx⊥ M ,f which satisﬁes
there exists an isometry σξ of Mσξ (x) = x,(dσξ )x ξ = −ξ,σξ (M ) = M.
.

We call σξ a reflection of M with respect to ξ.

.
Takashi Sakai

Area-minimizing cones over some minimal submanifolds in S nWeakly reflective submanifolds
Example√.
S n (1) × S n (1) ⊂ S 2n+1 ( 2) is a weakly reﬂective submanifold.
.
.

Propositionreflective ⊂ WRS ⊂ austere ⊂ minimal

Proposition (Podestà, Ikawa-Tasaki-S.)
Any singular orbit of a cohomogeneity one action on a Riemannian.
manifold is a weakly reflective submanifold.

Takashi Sakai.

Area-minimizing cones over some minimal submanifolds in S nAustere orbits of s-representations (Ikawa-Tasaki-S.)
An orbit Ad(K)H of an irreducible s-representation which is anaustere submanifold in the hypersphere S is one of the following
list:.

.

.

.

1An orbit through a restricted root

2

R = A2 ; H = 2e1 − e2 − e3 , e1 + e2 − 2e3

3

R = A3 ; H = e1 + e2 − e3 − e44

R = Dn ; H = e15

R = D4 ; H = e1 + e2 + e3 ± e46

R = B2 with constant multiplicities;

7α2
R = G2 ; H = α1 + √32H = e1 + e1√+e2Moreover, in the cases (1)〜(5), these austere orbits are weakly
reﬂective submanifolds in S.
Takashi Sakai

Area-minimizing cones over some minimal submanifolds in S nCase of type A2

Takashi Sakai

Area-minimizing cones over some minimal submanifolds in S nCase of type B2

Takashi Sakai

Area-minimizing cones over some minimal submanifolds in S nCase of type G2Takashi Sakai

Area-minimizing cones over some minimal submanifolds in S n