Harmonic maps, Toda frames and extended Dynkin diagrams

Harmonic maps, Toda frames and extended Dynkin diagrams

Emma Carberry University of Sydney

Katharine Turner University of Chicago

3rd of December, 2011

Coxeter automorphism on G ^C /T ^C and conditions for it to preserve a real form

de Sitter spheres S ₁ ²ⁿ and isotropic flag bundles

Toda integrable system and relationship to cyclic primitive maps from a surface into G/T

Solution in terms of ODEs (finite type)

Applications to superconformal tori in S ₁ ²ⁿ

Applications to Willmore tori in S ³ .

Coxeter automorphism on G ^C /T ^C

Let G ^C be a simple complex Lie group and T ^C a Cartan subgroup.

The homogeneous space G ^C /T ^C is naturally a k -symmetric space.

That is, we have an automorphism σ : G ^C → G ^C with σ ^k = 1 and

(G _σ ^C ) _id ⊂ T ^C ⊂ G ^C _σ .

Recall that a non-zero α ∈ (t ^C ) ^∗ is a root with root space G ^α ⊂ g ^C if

[H, R _α ] = α(H)R _α ∀H ∈ t, R _α ∈ G ^α .

Choose a set of simple roots, that is roots {α ₁ , . . . , α _N } such that every root can be written uniquely as

α =

N

X

j=1

m _j α _j ,

where all m _j ∈ Z ⁺ or all m _j ∈ Z ⁻ . The height of α is h(α) = P N

j=1 m _j and the root of minimal height is called the lowest root.

Let η ¹ , . . . , η ^N ∈ t ^C be the dual basis to α ₁ , . . . , α _N and σ : G ^C → G ^C be conjugation by

exp( 2πi k

N

X

j=1

η _j ) (Coxeter automorphism).

Then σ has order k , where k − 1 is the maximal height of a root

of g ^C .

Choose a set of simple roots, that is roots {α ₁ , . . . , α _N } such that every root can be written uniquely as

α =

N

X

j=1

m _j α _j ,

where all m _j ∈ Z ⁺ or all m _j ∈ Z ⁻ . The height of α is h(α) = P N

j=1 m _j and the root of minimal height is called the lowest root.

Let η ¹ , . . . , η ^N ∈ t ^C be the dual basis to α ₁ , . . . , α _N and σ : G ^C → G ^C be conjugation by

exp( 2πi k

N

X

j=1

η _j ) (Coxeter automorphism).

Then σ has order k , where k − 1 is the maximal height of a root

Let G be a real simple Lie group with Cartan subgroup T and assume that the Coxeter automorphism preserves the real form G.

I will describe class of harmonic maps from the surface into G/T which are given simply by solving ordinary differential equations and give a relationship between these maps and the Toda equations.

This will generalise work of Bolton, Pedit and Woodward for the

case when G is compact.

Example: SO(2n, 1)

Let R ^2n,1 denote R ²ⁿ⁺¹ with the Minkowski inner product x ¹ y ¹ + x ² y ² + · · · + x ²ⁿ y ²ⁿ − x ²ⁿ⁺¹ y ²ⁿ⁺¹

Consider the de Sitter group SO(2n, 1) of orientation preserving isometries of R ^2n,1 . Take as Cartan subgroup

T = diag (1, SO(2), . . . , SO(2), SO(1, 1)) .

Define a ˜ _k ∈ t ^∗ , k = 1, . . . , n by a ˜ _k

diag

0,

0 a ₁

−a ₁ 0

, . . .

0 a _n a _n 0

= a _k .

Take as simple roots of so(2n, 1, C ) the roots α 1 = i˜ a ₁ ,

α k = i˜ a _k − i˜ a _k−1 for 1 < k < n and α _n = ˜ a _n − i˜ a _n−1 .

The lowest root is then α ₀ = −˜ a n − i a ˜ n−1 , which is of height

−2n + 1.

Then writing η j for the dual basis of t ^C , conjugation by Q = exp πi

n

n

X

j=1

η j

= diag

1, R π n

, R

2π n

, . . . , R r π n

, −I ₂

is an automorphism of order 2n.

It is not hard to prove directly in this case that the real form

SO(2n, 1) is preserved by the Coxeter automorphism.

Let h·, ·i denote the complex bilinear form

hz , wi = z ¹ w ¹ + z ² w ² + · · · + z ²ⁿ w ²ⁿ − z ²ⁿ⁺¹ w ²ⁿ⁺¹ on C ²ⁿ⁺¹ .

A subspace V ⊂ C ²ⁿ⁺¹ is isotropic if hu, v i = 0 for all u, v ∈ V . Geometrically,

SO(2n, 1)/T = SO(2n, 1)/(1 × SO(2) × · · · × SO(2) × SO(1, 1)) is the full isotropic flag bundle

Fl(S ₁ ²ⁿ ) = {V ₁ ⊂ V ₂ ⊂ · · · ⊂ V _n−1 ⊂ T ^C S ₁ ²ⁿ | V _j is an

isotropic sub-bundle of dimension j}

We now give conditions under which a choice of real form g of a simple complex Lie algebra g ^C , Cartan subalgebra t ^C and simple roots α _j yield a Coxeter automorphism σ = Ad _exp(

k

P

j=1

η

) which

preserves the real Lie algebra g.

The condition for the Coxeter automorphism σ to preserve g is that for the simple roots α 1 , . . . , α N we have

¯

α _j ∈ {−α ₀ , . . . , −α _N },

where α(X) = ¯ α( ¯ X) and α ₀ is the lowest root.

We will now use a Cartan involution to express this reality

condition in terms of the extended Dynkin diagram.

A Cartan involution for g is an involution Θ of g ^C such that hX , Y i _Θ = −hX , Θ(Y )i

is positive definite on g, where h·, ·i denotes the Killing form.

Alternatively, it is an involution for which k ⊕ im is compact, where

k = +1-eigenspace of Θ m = −1-eigenspace of Θ.

We may choose a Cartan involution which preserves the given

Cartan subalgebra t

Proposition

Let g be a real simple Lie algebra, t a Cartan subalgebra and Θ be a Cartan involution preserving t. Choose simple roots α ₁ , . . . , α _N and let σ = Ad _exp(

k

P

j=1

η

) be the corresponding Coxeter automorphism of g ^C . Then the following are equivalent:

1

σ preserves the real form g,

2

σ commutes with Θ,

3

Θ defines an involution of the extended Dynkin diagram for

g ^C consisting of the usual Dynkin diagram augmented with

the lowest root α 0 .

For a Θ-stable Cartan subalgebra t, t is maximally

compact ⇔ Θ defines a permutation of the Dynkin diagram for g ^C and so when t is maximally compact (e.g. g is compact), the real form g is preserved by any Coxeter automorphism defined by simple roots for t.

The more interesting case is when we have an involution of the extended Dynkin diagram which does not restrict to an

involution of the Dynkin diagram (i.e. t is not maximally compact).

Call these non-trivial involutions.

E ₈

α

α

α

α

α

α

α

α

α

D _N . . .

α

α

α

α

α

α

C _N . . .

α

α

α

α

α

B _N . . .

α

α

α

α

α

. . . A _N

α

α

α

α

α

E 7 α

α

α

α

α

α

α

α

E ₆

α

α

α

α

α

α

α

F ₄

α

α

α

α

α

G ₂

α

α

α

There are nontrivial involutions for all root systems except

E ₈ , F ₄ and G ₂ .

Theorem

Every involution of the extended Dynkin diagram for a simple complex Lie algebra g ^C is induced by a Cartan involution of a real form of g ^C .

More precisely, let g ^C be a simple complex Lie algebra with Cartan subalgebra t ^C and choose simple roots α ₁ , . . . , α _N for the root system ∆(g ^C , t ^C ). Given an involution π of the

extended Dynkin diagram for ∆, there exists a real form g of g ^C

and a Cartan involution Θ of g preserving t = g ∩ t ^C such that Θ

induces π and t is a real form of t ^C . The Coxeter automorphism

σ determined by α ₁ , . . . , α _N preserves the real form g.

Primitive Maps and Loop Groups

The Coxeter automorphism σ : g → g of order k induces a Z k -grading

g ^C =

k−1

M

j=0

g ^σ _j , [g ^σ _j , g ^σ _l ] ⊂ g ^σ _j+l ,

where g ^σ _j denotes the e ^j

-eigenspace of σ.

We have the reductive splitting g = t ⊕ p with

p ^C =

k −1

M

j=1

g ^σ _j , t ^C = g ^σ ₀ ,

and if ϕ is a g-valued form we may decompose it as

ϕ = ϕ t + ϕ p .

A smooth map f of a surface into a symmetric space (G/K , σ) is harmonic if and only if for a smooth lift F : U → G of

f : U → G/K , the form ϕ = F ⁻¹ dF has the property that for each λ ∈ S ¹

ϕ _λ = λϕ ⁰ _p + ϕ _k + λ ⁻¹ ϕ ⁰⁰ _p satisfies the Maurer-Cartan equation

dϕ _λ + 1

2 [ϕ _λ ∧ ϕ _λ ] = 0.

Moreover given a family of flat connections as above, we can

recover a harmonic map f : U → G/K on any simply connected

U.

A smooth map f of a surface into a symmetric space (G/K , σ) is harmonic if and only if for a smooth lift F : U → G of

f : U → G/K , the form ϕ = F ⁻¹ dF has the property that for each λ ∈ S ¹

ϕ _λ = λϕ ⁰ _p + ϕ _k + λ ⁻¹ ϕ ⁰⁰ _p satisfies the Maurer-Cartan equation

dϕ _λ + 1

2 [ϕ _λ ∧ ϕ _λ ] = 0.

Moreover given a family of flat connections as above, we can

recover a harmonic map f : U → G/K on any simply connected

U.

When k > 2, the condition that

ϕ _λ = λϕ ⁰ _p + ϕ _k + λ ⁻¹ ϕ ⁰⁰ _p

satisfies the Maurer-Cartan equation dϕ _λ + 1

2 [ϕ λ ∧ ϕ λ ] = 0.

characterises (not merely harmonic but) primitive maps ψ of a surface into the k -symmetric space G/K .

ψ is primitive if for a smooth lift F : U → G of ψ : U → G/K , ϕ ⁰ = F ⁻¹ ∂F takes values in g ^σ ₀ ⊕ g ^σ ₁ . We say that F is a primitive frame.

Primitive maps ψ are in particular harmonic.

For studying maps into G/T it is helpful to consider the twisted loop group

Ω ^σ G = {γ : S ¹ → G : γ(e

λ)} = σ(γ(λ))}

and corresponding twisted loop algebra Ω ^σ g. The (possibly doubly infinite) Laurent expansion

ξ(λ) = X

j

ξ _j λ ^j , ξ _j ∈ g ^σ _j ⊂ g ^C , Φ −j = ¯ Φ _j

allows us to filtrate Ω ^σ g ^C by finite-dimensional subspaces

Ω ^σ _d = {ξ ∈ Ωg | ξ _j = 0 whenever |j| > d}.

Suppose ξ : R ² → Ω ^σ _d satisfies the Lax equation

∂ξ

∂z = [ξ, λξ _d + 1 2 ξ d−1 ].

Then ϕ λ (z ) =

λξ d (z ) + 1

2 ξ d−1 (z )

dz +

λ ⁻¹ ξ −d (z) + 1

2 ξ d−1 (z) d ¯ z satisfies the Maurer-Cartan equation and so defines a primitive map f : R ² → G/T .

Maps f obtained in this simple way are said to be of finite type.

Suppose ξ : R ² → Ω ^σ _d satisfies the Lax equation

∂ξ

∂z = [ξ, λξ _d + 1 2 ξ d−1 ].

Then ϕ λ (z ) =

λξ d (z ) + 1

2 ξ d−1 (z )

dz +

λ ⁻¹ ξ −d (z) + 1

2 ξ d−1 (z) d ¯ z satisfies the Maurer-Cartan equation and so defines a primitive map f : R ² → G/T .

Maps f obtained in this simple way are said to be of finite type.

The equation 1

2 (X (ξ) − iY (ξ)) =

λξ _d + 1 2 ξ _d−1 defines vector fields X , Y on Ω ^σ _d .

Assume the vector fields X , Y are complete (e.g. G is compact). The vector fields X , Y commute and so define an action

(x, y) · ξ(λ) = X ^x ◦ Y ^y (ξ(λ))

of R ² on Ω _d . Define ξ(z, λ) := (x , y ) · ξ ⁰ (λ) for any initial ξ ⁰ (λ) ∈ Ω _d , where z = x + iy. Then

ϕ _λ (z ) =

λξ _d (z ) + 1

2 ξ d−1 (z ) dz +

λ ⁻¹ ξ −d (z) + 1

2 ξ d−1 (z)

d ¯ z

satisfies the Maurer-Cartan equation and so defines a primitive

map f : ² → G/T .

For the Coxeter automorphism on G/T , g ^σ ₀ = t and g ^σ ₁ is the sum of the simple and lowest root spaces.

We say that a primitive map ψ / frame F is in addition cyclic if the image of F ⁻¹ ∂F contains a cyclic element.

An element of g ^σ ₀

⊕ g ^σ ₁

is cyclic if its projection to each of the root spaces G ^α

, . . . , G ^α

, G ^α

is non-zero.

I will now describe the relationship between cyclic primitive

maps into G/T and the Toda equations.

For the Coxeter automorphism on G/T , g ^σ ₀ = t and g ^σ ₁ is the sum of the simple and lowest root spaces.

We say that a primitive map ψ / frame F is in addition cyclic if the image of F ⁻¹ ∂F contains a cyclic element.

An element of g ^σ ₀

⊕ g ^σ ₁

is cyclic if its projection to each of the root spaces G ^α

, . . . , G ^α

, G ^α

is non-zero.

I will now describe the relationship between cyclic primitive

maps into G/T and the Toda equations.

Toda equation

The classical 1-dimensional affine Toda integrable system describes the motion of finitely many particles of equal mass arranged in a circle, joined by “exponential springs".

0

1 n

2 n − 1

. . . . . .

m d ² x _j

dt ² = e ^(x

^−x

⁾ − e ^(x

^−x

⁾ .

We may generalise this to any simple Lie algebra as 2 d ² Ω

dt ² =

n

X

j=0

m _j e ^2α

^(Ω) [R _α

, R −α

]

or for a 2-dimensional domain 2Ω z¯ z =

N

X

j=0

m _j e ^2α

^(Ω) [R _α

, R −α

] (1)

where Ω : C → it is a smooth map, m _j ∈ R ⁺ satisfies m _π(j) = m _j

and R _α

are root vectors satisfying R _α

= R _−α

.

To recover the classical Toda equation:

1

Take the standard simple roots for su(n + 1).

2

Set m ₀ = 1 and let

α ₀ = −

N

X

j=1

m _j α _j

be the expression for the lowest root α 0 .

3

Choose root vectors R _α

so that [R _α

, R −α

] is the dual of α _j with respect to the Killing form.

Notice that the extended Dynkin diagram for su(n + 1) looks like

. . .

α

α

α

α

α

Given a cyclic element W = P N

j=0 r _j R _α

of g ^σ ₁ , we say that a lift F : C → G of ψ : C → G/T is a Toda frame with respect to W if there exists a smooth map Ω : C → it such that

F ⁻¹ F z = Ω z + Ad _{exp Ω} W . We call Ω an affine Toda field with respect to W . Lemma

The affine Toda field equation (1) is the integrability condition for the existence of a Toda frame with respect to W .

Here W = P N

j=0 r _j R α

is a cyclic element of g ^σ ₁ such that

m _π(j) = m _j and R _α

= R _−α

and we take m _j = r _j r _j for

j = 0, . . . , N.

Toda frame and cyclic primitive

Theorem

A map ψ : C → G/T possesses a Toda frame if and only if it has a cyclic primitive frame F for which c ₀ Q N

j=1 c _j ^m

is constant, where

F ⁻¹ F _z | _g

1

=

N

X

j=0

c _j R _α

.

The Toda frame is then cyclic primitive with respect to any W = P N

j=0 r _j R _α

for which

r ₀

N

Y

j=1

r _j ^m

= c ₀

N

Y

j=1

c _j ^m

.

Finite-type

Theorem

Let G be a simple real Lie group, T a Cartan subgroup and

assume that the Coxeter automorphism preserves G. Suppose

ψ : C /Λ → G/T has a Toda frame F : C /Λ → G. Then ψ is of

finite type.

Harmonic maps into S ₁ ²ⁿ

The isotropy order of a harmonic map f of a surface into S ²ⁿ ₁ is the maximal integer r ≥ 0 such that the derivatives

∂ z F , ∂ _z ² F , . . . , ∂ _z ^r f span an isotropic subspace at each point.

If f has the maximal isotropy order r = n we say it is isotropic.

Isotropic surfaces in S ₁ ²ⁿ include all harmonic maps of S ² , and can be expressed holomorphically in terms of a

Weierstrass-type representation (Bryant 84, Ejiri 88)

Harmonic maps f : M ² → S ₁ ²ⁿ with the penultimate isotropy

order r = n − 1 are said to be superconformal.

Applying Gram-Schmidt, we define the harmonic sequence {f ₀ , f ₁ , . . . , f r } of a non-constant harmonic map f : M ² → S ₁ ²ⁿ by

f ₀ = f , f _j+1 = ∂ z f _j − h∂ _z f _j , f _j i

kf _j k ² f _j wherever kf _j k ² 6= 0 and extend by continuity wherever f _j = 0. Then

∂ ¯ z f _j+1 = − kf _j+1 k ²

kf _j k ² f _j for 0 ≤ j ≤ r hf _j , f _k i = 0 unless j = k

and the zeros of the f _j are isolated whenever f _j does not vanish

identically (Hulett 05).

Applying Gram-Schmidt, we define the harmonic sequence {f ₀ , f ₁ , . . . , f r } of a non-constant harmonic map f : M ² → S ₁ ²ⁿ by

f ₀ = f , f _j+1 = ∂ z f _j − h∂ _z f _j , f _j i

kf _j k ² f _j wherever kf _j k ² 6= 0 and extend by continuity wherever f _j = 0. Then

∂ ¯ z f _j+1 = − kf _j+1 k ²

kf _j k ² f _j for 0 ≤ j ≤ r hf _j , f _k i = 0 unless j = k

and the zeros of the f _j are isolated whenever f _j does not vanish

identically (Hulett 05).

Theorem

A harmonic map f : C → S ₁ ²ⁿ has a cyclic primitive lift ψ : C → Fl(S ₁ ²ⁿ ) if and only if it is superconformal and the entries {f ₁ , . . . , f n−1 } of its harmonic sequence are defined everywhere.

We have for each 1 ≤ j ≤ r

f _j = 2 ^j−1 c ₁ . . . c _j F (e _2j + ie _2j+1 ) for each 1 ≤ j ≤ n − 1 where the c _j are root vector coefficients with respect to particular choices of the root vectors appearing in g ^σ ₁

. Corollary

Let f : C/Λ → S ²ⁿ ₁ be a superconformal harmonic map with

globally defined harmonic sequence {f ₁ , . . . , f n }. Then f has a

lift ψ : C /Λ → SO(2n, 1)/T of finite type.

Theorem

A harmonic map f : C → S ₁ ²ⁿ has a cyclic primitive lift ψ : C → Fl(S ₁ ²ⁿ ) if and only if it is superconformal and the entries {f ₁ , . . . , f n−1 } of its harmonic sequence are defined everywhere.

We have for each 1 ≤ j ≤ r

f _j = 2 ^j−1 c ₁ . . . c _j F (e _2j + ie _2j+1 ) for each 1 ≤ j ≤ n − 1 where the c _j are root vector coefficients with respect to particular choices of the root vectors appearing in g ^σ ₁

. Corollary

Let f : C/Λ → S ²ⁿ ₁ be a superconformal harmonic map with

globally defined harmonic sequence {f ₁ , . . . , f n }. Then f has a

lift ψ : C /Λ → SO(2n, 1)/T of finite type.

An immersed surface φ : M ² → R ³ is Willmore if it is critical for the Willmore functional

W = Z

M

H ² dA,

where H denotes the mean curvature of φ and dA the area form.

Due to Gauss-Bonnet, it is equivalent to seek critical surfaces

for Z

M

(H ² − K ) dA, = Z

M

(k ₂ − k ₁ ) ² dA

where K is the Gauss curvature and k ₁ , k ₂ are the principal curvatures.

This latter functional is clearly conformally invariant and so we

instead consider immersions into S ³ .

An immersed surface φ : M ² → R ³ is Willmore if it is critical for the Willmore functional

W = Z

M

H ² dA,

where H denotes the mean curvature of φ and dA the area form.

Due to Gauss-Bonnet, it is equivalent to seek critical surfaces

for Z

M

(H ² − K ) dA, = Z

M

(k ₂ − k ₁ ) ² dA

where K is the Gauss curvature and k ₁ , k ₂ are the principal curvatures.

This latter functional is clearly conformally invariant and so we

instead consider immersions into S ³ .

It is not hard to show that W(M ² ) ≥ 4π, with equality if and only if M ² is a (round) sphere.

The Willmore conjecture proposes that W ( C /Λ) ≥ 2π ² for any immersed torus with equality if and only if the torus is

conformally equivalent to

It is not hard to show that W(M ² ) ≥ 4π, with equality if and only if M ² is a (round) sphere.

The Willmore conjecture proposes that W ( C /Λ) ≥ 2π ² for any immersed torus with equality if and only if the torus is

conformally equivalent to

The conformal Gauss map of an immersion φ : M ² → S ³

associates to each point on the surface M ² its central sphere,

that is the oriented 2-sphere in S ³ with the same normal vector

and mean curvature.

A 2-sphere in S ³ is the intersection of S ³ and a hyperplane in R ⁴ ;

S ³ ∩ {x ₁ , x ₂ , x ₃ , x ₄ : a ₁ x ₁ + a ₂ x ₂ + a ₃ x ₃ + a ₄ x ₄ − b = 0}.

For this hyperplane to intersect with S ³ at more than one point requires a ² ₁ + a ² ₂ + a ² ₃ + a ² ₄ − b ² > 0 and hence we can scale (a ₁ , a ₂ , a ₃ , a ₄ , b) so that a ² ₁ + a ² ₂ + a ² ₃ + a ² ₄ − b ² = 1.

De Sitter space S ₁ ²ⁿ is the unit sphere in R ²ⁿ⁺¹ with respect to the Minkowski metric

x ¹ y ¹ + x ² y ² + · · · + x ²ⁿ y ²ⁿ − x ²ⁿ⁺¹ y ²ⁿ⁺¹ .

Thus each 2-sphere in S ³ can be identified with two antipodal points ±(a ₁ , a ₂ , a ₃ , a ₄ , b) ∈ S ⁴ ₁ .

Choosing an orientation for the 2-sphere gives a well-defined

element of S ₁ ⁴ .

A 2-sphere in S ³ is the intersection of S ³ and a hyperplane in R ⁴ ;

S ³ ∩ {x ₁ , x ₂ , x ₃ , x ₄ : a ₁ x ₁ + a ₂ x ₂ + a ₃ x ₃ + a ₄ x ₄ − b = 0}.

For this hyperplane to intersect with S ³ at more than one point requires a ² ₁ + a ² ₂ + a ² ₃ + a ² ₄ − b ² > 0 and hence we can scale (a ₁ , a ₂ , a ₃ , a ₄ , b) so that a ² ₁ + a ² ₂ + a ² ₃ + a ² ₄ − b ² = 1.

De Sitter space S ₁ ²ⁿ is the unit sphere in R ²ⁿ⁺¹ with respect to the Minkowski metric

x ¹ y ¹ + x ² y ² + · · · + x ²ⁿ y ²ⁿ − x ²ⁿ⁺¹ y ²ⁿ⁺¹ .

Thus each 2-sphere in S ³ can be identified with two antipodal points ±(a ₁ , a ₂ , a ₃ , a ₄ , b) ∈ S ⁴ ₁ .

Choosing an orientation for the 2-sphere gives a well-defined

Hence we see that the space of oriented 2-spheres in S ³ is naturally identified with S ₁ ⁴ .

The conformal Gauss map f : M ² → S ₁ ⁴ is given explicitly by f (z) = H (z) · Φ(z) + N(z)

where Φ(z) = (φ(z), 1), N = (n, 0).

The conformal Gauss map f is weakly conformal and an immersion away from the umbilic points of φ.

The area form on M ² induced by f is given by (H ² − K )dA

Thus φ : M ² → S ³ is a Willmore immersion without umbilic

points if and only if f : M ² → S ₁ ⁴ is a minimal immersion, or

equivalently is conformal and harmonic.

A minimal immersion f : M ² → S ₁ ⁴ can only have isotropy order r = 1 (superconformal) or r = 2 (isotropic).

Recall that the second fundamental form of f is

II(X , Y ) = (∇ _X Y ) ^⊥ , where ⊥ denotes projection to the orthogonal complement of TM ² in TS ₁ ⁴ .

The curvature ellipse of f at p ∈ M ² is the image of the unit circle in T _p M ² under the second fundamental form.

It is a circle precisely when hf _zz (p), f zz (p)i = 0. This quantity is holomorphic, hence constant when M ² is compact.

The curvature ellipse of f is thus a circle precisely when f is

isotropic. All isotropic f have been constructed by Bryant using

holomorphic data.

A minimal immersion f : M ² → S ₁ ⁴ can only have isotropy order r = 1 (superconformal) or r = 2 (isotropic).

Recall that the second fundamental form of f is

II(X , Y ) = (∇ _X Y ) ^⊥ , where ⊥ denotes projection to the orthogonal complement of TM ² in TS ₁ ⁴ .

The curvature ellipse of f at p ∈ M ² is the image of the unit circle in T _p M ² under the second fundamental form.

It is a circle precisely when hf _zz (p), f zz (p)i = 0. This quantity is holomorphic, hence constant when M ² is compact.

The curvature ellipse of f is thus a circle precisely when f is

isotropic. All isotropic f have been constructed by Bryant using

holomorphic data.

We have seen that the first ellipse of curvature being a non-circular ellipse corresponds to f being superconformal.

For superconformal f : M ² → S ⁴ ₁ the cyclic primitive frame F constructed previously consists of

F = (f , f x , f y , v, w)

where the last two columns of F are determined by the principal directions of the curvature ellipse.

Corollary

A Willmore immersion φ : T ² → S ³ without umbilic points may be constructed either

1

from holomorphic Weierstrass data

2

by integrating a pair of commuting vector fields on a

finite-dimensional space