Some constructions of harmonic maps, 50 years after Eells ... · 5/14/2014 · Integrable systems...

Some constructions of harmonic maps, 50years after Eells–Sampson

Lund University, 14 May 2014

John C. Wood

University of Leeds, UK

Dedicated to the memory of James Eells

[email protected]

http://www1.maths.leeds.ac.uk/Pure/staff/wood/wood.html

Harmonic maps [Eells–Sampson, 1964]

Let ϕ : (M, g)→ (N, h) be a smooth map between compactsmooth Riemannian manifolds. The energy or Dirichlet integralof ϕ is

E (ϕ) =

∫Me(ϕ)ωg =

∫M

12 |dϕ|

2 ωg

where ωg = volume measure and, for any p ∈ M,

|dϕp|2 = Hilbert–Schmidt square norm of dϕp = g ijhαβ ϕαi ϕ

βj .

The map ϕ is called harmonic if the first variation of E forvariations ϕt of the map ϕ vanishes at ϕ, i.e., d

dtE (ϕt)∣∣t=0

= 0.We compute

d

dtE (ϕt)

∣∣t=0

= −∫M

⟨τ(ϕ), v

⟩ωg

where v = ∂ϕt/∂t|t=0 is the variation vector field of (ϕt), andτ(ϕ) = div dϕ is the tension field of ϕ given as follows:

Tension field of a map ϕ [Eells–Sampson]

τ(ϕ) = div dϕ = −d∗∇dϕ = Trace∇dϕ

=m∑i=1

∇dϕ(ei , ei ) =m∑i=1

∇ϕei(dϕ(ei )

)− dϕ(∇M

eiei )

;

τ(ϕ)γ = g ij

(∂2ϕγ

∂x i∂x j− Γk

ij

∂ϕγ

∂xk+ Lγαβ

∂ϕα

∂x i∂ϕβ

∂x j

)= ∆Mϕγ + g(grad ϕα, grad ϕβ) Lγαβ .

∆M = Laplacian on functions f : M → R:

∆Mf = div grad f = div df = −d∗df = Trace∇df

=m∑i=1

ei(ei (f )

)−(∇M

eiei)f

=1√|g |

∂

∂x i

(√|g | g ij ∂f

∂x j

).

Examples of harmonic maps[Eells–Sampson]

ϕ : M → N is harmonic ⇐⇒τ(ϕ) ≡ Trace∇dϕ = 0 (Harmonic equation)

1. ϕ : Rm ⊇ U → Rn is harmonic iff ∆ϕ = 0(∆ = usual Laplacian on Rm).

2. ϕ : (M, g)→ Rn is harmonic iff ∆Mϕ = 0(∆M = Laplace–Beltrami operator on (M, g) ).

3. ϕ : R ⊇ U → N or S1 → N is harmonic iff it defines a geodesicparametrized linearly.

4. ±-Holomorphic maps between Kahler manifolds are harmonic;in fact they give absolute minima of the energy functional.

5. Weakly conformal maps between Riemann surfaces are±-holomorphic, so harmonic with respect to any Hermitian metrics.

The Eells–Sampson Existence Theorem

Theorem (Eells–Sampson, 1964)

Let M and N be compact Riemannian manifolds with the sectionalcurvatures of N all non-positive. Then, in any homotopy class ofmaps from M to N, there is a harmonic map.

This was used by Y.-T. Siu [1980] and many others to prove‘strong rigidity’ of suitably negatively curved compact Kahlermanifolds by showing that the Eells–Sampson harmonic map in thehomotopy class of the homotopy equivalence is holomorphic orantiholomorphic.

A uniqueness statement was given by P. Hartman [1967], seebelow.

Joseph Sampson went on to write several papers on harmonicmaps, for example, extending Siu’s theorem on holomorphicity ofharmonic maps into manifolds of suitably negative curvature[Sampson, 1986], and sharpening Hartman’s uniqueness result to:

Maps into manifolds of positive curvature

Theorem (Sampson, 1978)

Let f : M → N be a harmonic map between compact Riemannianmanifolds with the sectional curvatures of N all non-positive. Iff (M) contains a point at which the sectional curvatures are allstrictly negative, and if f (M) is not contained in a closed geodesic,then f is the only harmonic map in its homotopy class.

(Hartman’s result required real-analyticity, or that the sectionalcurvatures at all points of f (M) must be strictly negative.)

On the other hand, quoting Jim Eells’ obituary [D. Toledo, 2008]:

In spite of the great success of his paper withSampson, Jim’s main interest always was in harmonicmaps to other targets, in particular to positively curvedones, where no general existence theorem is available, ...

We shall now discuss this case!

Harmonic maps into the 2-sphere

Some results of Eells and his students. Recall: ±-holomorphicmaps between Kahler mfds are harmonic. Conversely,

Theorem (Holomorphicity, Eells–JCW, 1976)

Any harmonic map from a compact Riemann surface of genus p toS2 which has degree D with |D| ≥ p is holomorphic orantiholomorphic.

Corollary (Non-existence, Eells–JCW, 1976)

There is no harmonic degree one map from a 2-torus to a 2-sphere,no matter which metrics are chosen in the domain and target,

In contrast, we have

Theorem (Existence, L. Lemaire, 1978a)

For every pair of integers p ≥ 1, D with |D| < p there is acompact Riemann surface M of genus p, a metric on the 2-sphereS2, and a harmonic map of degree D from M into S2 which is notholomorphic or antiholomorphic.

Higher dimensions

If the codomain contains no 2-sphere, there are always harmonicmaps:

Theorem (Lemaire, 1978b; Sacks–Uhlenbeck, 1977)

If M is a compact surface and N a compact Riemannian manifoldwith π2(N) = 0, then every homotopy class of maps has aharmonic map.

If the domain is of higher dimension, we can find harmonic maps ifwe don’t mind changing the metric (the Rendering Problem):

Theorem (Eells–M.J. Ferreira, 1991)

If M and N are compact Riemannian manifolds, with dimM ≥ 3,in any homotopy class there is a map which is harmonic withrespect to a conformally equivalent metric on the domain.

We shall mainly discuss maps from surfaces.

Harmonic maps from surfaces

• From the Composition Law [Eells–Sampson], harmonic mapsfrom surfaces are conformally invariant, i.e., the compositionϕ ψ of a harmonic map ϕ : M2 → N from a surface with aweakly conformal map ψ : M ′2 → M2 of surfaces is harmonic.

• [Eells–Sampson] A volume-minimizing conformal map from asurface is energy-minimizing. In fact, a weakly conformal map isharmonic if and only if it is a minimal branched immersion in thesense of [Gulliver–Osserman–Royden, 1973].

• [JCW and others] Any harmonic map from the 2-sphere isweakly conformal and so is a minimal branched immersion.

Questions 1) How can we construct harmonic maps/minimalbranched immersions to higher dimensional manifolds?

2) When are harmonic maps holomorphic in some sense, say atwistorial sense?

Integrable systems methods

We concentrate on constructing and understanding harmonic mapsfrom surfaces to Lie groups and symmetric spaces.

• K. Uhlenbeck [1989] showed how to write the harmonicequation as a ‘zero-curvature equation’ by introducing a ‘spectralparameter’. She found a transform called ‘adding a uniton’ whichgave new harmonic maps from old ones; doing this transform afinite number of times gave all harmonic maps from S2 to U(n),and thus, in theory, to any compact Lie group or (Riemannian)symmetric space.

• We can make the procedure for U(n) completely algebraicwithout the need to integrate or solve any equations[Ferreira–Simoes–JCW, 2010].• Further work: [Svensson–JCW, 2012], using the Grassmannianmodel of loop groups [G. Segal, 1989] to understand this for otherclassical groups and symmetric spaces, and for G2 [N. Correia–R.Pacheco, 2012] but questions remain.

Harmonic maps from surfaces to CPn

By Eells–Sampson, holomorphic maps M2 → CPn are harmonic,as are antiholomorphic maps.

When n = 1 and M2 = S2, this accounts for all harmonic maps.

When n > 1, we can find harmonic maps which are not±-holomorphic, as follows:

A harmonic map ϕ = [Φ] : M2 → CPn which is not ±-holomorphichas two Gauss transforms, a ∂′- and a ∂′′-transform:

G ′(ϕ) =[π⊥ϕ

∂Φ

∂z

], G ′′(ϕ) =

[π⊥ϕ

∂Φ

∂z

].

These can be extended over points where the derivatives are zero,and are both harmonic. [If ϕ is holomorphic (resp.antiholomorphic), only G ′(ϕ) (resp. G ′′(ϕ)) is defined.]

[J. Eells–JCW, 1983] All harmonic maps from S2 → CPn areobtained from holomorphic maps by applying the ∂′-Gausstransform up to n times. Skip to Twistor spaces

Harmonic maps into Lie groups

For any smooth map, ϕ : M → N, we have (a) d∇dϕ = 0. Theharmonic map equation reads (b) d∗∇dϕ = 0. For a mapϕ : M → G to a Lie group, set A = Aϕ = 1

2ϕ−1dϕ, a g-valued

1-form. Then the first two equations become

(a) dA + [A,A] = 0 , (b) d∗A = 0 .

The first equation is an integrability condition: given a g-valued1-form A, there are local solutions ϕ : M → G to 1

2ϕ−1dϕ = A if

and only if (a) holds.

For a map from a Riemann surface, write Aϕ = Aϕz dz + Aϕz dz ,then (a) reads

∂zAz − ∂zAz + 2[Az ,Az ] = 0 . (1)

Further, the harmonic equation (b) for ϕ can be written as eitherof the equivalent equations:

(b1) ∂zAz + [Az ,Az ] = 0 , (b2) ∂zAz + [Az ,Az ] = 0 . (2)

Introduction of spectral parameter

For each λ ∈ S1, set

Aλ = 12 (1− λ−1)Azdz + 1

2 (1− λ)Azdz ; (3)

FACT: dAλ + [Aλ,Aλ] = 0 iff (1) and (2) all hold, i.e, iffA = 1

2ϕ−1dϕ for some harmonic map ϕ : M2 → U(n).

In that case we can find, at least locally, an S1-family of smoothmaps Φ = Φλ : M2 → U(n) with

12 Φ−1

λ dΦλ = Aλ (λ ∈ S1) , (4)

equivalently,

∂zΦλ = (1− λ−1)ΦλAϕz and ∂zΦλ = (1− λ)ΦλA

ϕz . (5)

Such a Φλ : M2 → G is called an extended solution (associated toϕ). Note that Φ1 is constant, take it equal to I and thenΦ−1 = Aϕ for some constant A ∈ G .

Harmonic maps from unitons

Any smooth map ϕ defines a connection Aϕ = 12ϕ−1dϕ, and thus

a covariant derivative Dϕ = d + Aϕ, on the trivial bundleCn = M2 × Cn ; writing Aϕ = Aϕz dz + Aϕz dz , since Aϕz represents∂ϕ/∂z , harmonicity of ϕ is equivalent to holomorphicity of Aϕz .

A uniton for ϕ is a subbundle α of the trivial bundle which isholomorphic, i.e, closed under Dϕ

z , and closed under Aϕz .

Then [K. Uhlenbeck, 1989]:

(i) If ϕ : M2 → U(n) is harmonic and α is a uniton for ϕ,then ϕ = ϕ(πα − π⊥α ) is harmonic.

(ii) Any harmonic map ϕ : S2 → U(n) can written as a finiteproduct of unitons:

ϕ = const. (πα1 − π⊥α1) · · · (παr − π⊥αr

) . (6)

Unitons and extended solutions

(i) of previous page is proved by showing the stronger result forextended solutions:

(i) Let Φ : M2 → U(n) be an extended solution. Then α is auniton for ϕ if and only if Φ = Φ(πα− π⊥α ) is an extended solution.

(ii) has counterpart as well:

(ii) Any extended solution Φ : S2 → U(n) can written as afinite product of unitons:

Φ = (πα1 + λπ⊥α1) · · · (παr + λπ⊥αr

) (7)

Building all harmonic maps

To build all harmonic maps we add unitons α1, α2, . . . starting withthe constant map, giving a sequence of harmonic mapsϕ0 = const., ϕ1 = ϕ0(πα1 − π⊥α1

),ϕ2 = ϕ1(πα2 − π⊥α2

) = ϕ0(πα1 − π⊥α1)(πα2 − π⊥α2

), . . ..To do this, we must know the possible unitons at each stage.For this, we must find a meromorphic basis for the trivial bundleCn with respect to Dϕi

z for each i ; in general, this involves solving∂-problems.

[M.J. Ferreira, B. Simoes, JCW, 2010]Every harmonic map can be built by adding unitons which aregiven in terms of projections along previous unitons ofmeromorphic functions, without solving ∂-problems.

This gives a completely explicit algebraic parametrization of allharmonic maps S2 → U(n) by meromorphic functions.

Skip to Twistor spaces Skip to harmonic maps into G2/SO(4)

The general construction

TheoremAll non-constant harmonic maps from S2 → U(n) are given as a

product (6) in the following way for some r ∈ 0, 1, . . . , n − 1 :

Choose an r × n array of meromorphic functions Hi ,j : S2 → Cn,(i = 0, . . . , r − 1, j = 1, . . . , n); these could be zero.For each i = 0, . . . , r − 1 set

αi+1 = spani∑

s=k

C isH

(k)s−k,j | j = 1, . . . , n, k = 0, . . . , i,

where C is =

∑1≤i1<···<is≤i π

⊥αis· · ·π⊥αi1

.

Then ϕ : M2 → U(n), ϕ = ϕ0(πα1 − πα⊥1 ) · · · (παr − πα⊥r ) is

harmonic, and all harmonic maps M2 → U(n) of finite unitonnumber arise in this way.

Here the superscript (k) denotes the k’th derivative.

The first few formulae

(i) α1 is the subbundle α1 = α(0)1 = spanH0,j .

(ii) α2 is the sum of subbundles α2 = α(0)2 + α

(1)2 where

α(0)2 = spanH0,j + π⊥α1

H1,j ,

α(1)2 = spanπ⊥α1

H(1)0,j .

(iii) α3 is the sum α3 = α(0)3 + α

(1)3 + α

(2)3 where

α(0)3 = spanH0,j + (π⊥α1

+ π⊥α2)H1,j + π⊥α2

π⊥α1H2,j ,

α(1)3 = span(π⊥α1

+ π⊥α2)H

(1)0,j + π⊥α2

π⊥α1H

(1)1,j ,

α(2)3 = spanπ⊥α2

π⊥α1H

(2)0,j .

Skip to Grassmannian model

Examples

A smooth (resp. holomorphic) map α : M2 → Gk(Cn) is the samething as a smooth (resp. holomorphic) subbundle α of Cn.The Grassmannians Gk(Cn) are included in U(n) via the Cartanembbedding B 7→ πB − π⊥B . This is a totally geodesic isometrywhich identifies the disjoint union G∗(Cn) of these Grassmannianswith the subset B ∈ U(n) : B = B∗ = B ∈ U(n) : B2 = I.

A harmonic map into G∗(Cn) is thus the same as a harmonic mapinto U(n) with image in this subset.

1. If the data (Hi ,j) has only one non-zero entry H0,1, then

α1 = spanH0,1 and α2 = spanH0,1+ spanπ⊥α1(H

(1)0,1 ). The

formula gives ϕ2 = the ∂′-Gauss transform G ′(α1) of α1.

2. More generally, if each column of (Hi ,j) has only one non-zeroentry, then the unitons are nested: αi ⊂ αi+1, and ϕ is a map intoa Grassmannian which is invariant under a natural S1-action ofC.-L. Terng. Skip to Grassmannian model

Extended solutions

Any harmonic map ϕ gives rise to a loop of smooth maps Φλ

called an extended solution associated to ϕ as follows.

Set Aϕ = 12ϕ−1dϕ, write Aϕ = Aϕz dz + Aϕz dz and set

Aϕλ = 12 (1− λ−1)Aϕz dz + 1

2 (1− λ)Aϕz dz .

Then ϕ harmonic implies dAϕλ + [Aϕλ ,Aϕλ ] = 0, the integrability

condition for solving12 Φ−1

λ dΦλ = Aϕλ .

We let Φλ be any smooth solution — these exist locally, at least.Note that Φ1 is constant, we take it to be I , so Φ : M2 → ΩU(n),and Φ−1 equals ϕ up to left multiplication by an element of U(n).ϕ finite uniton number iff has an associated extended solutionwhich is algebraic, i.e., a Laurent polynomial in λ.An extended solution associated to product (6) is:

Φ = (πα1 + λπ⊥α1) · · · (παr + λπ⊥αr

) .

The Grassmannian model

ΩalgU(n) = r∑

k=−sλkTk ∈ ΩU(n) | r , s ∈ N

H = L2(S1,Cn) ⊃ H+ = ∑k≥0

akλk ∈ H | ak ∈ Cn

We have a map ΩalgU(n) 3 Φ 7→ ΦH+ ⊂ H, and this gives abijection from ΩalgU(n) to

W ⊂ H| λrH+ ⊂W ⊂ λsH+ for some r , s ∈ N, λW ⊂W .

algebraic extended solutions Φ : M2 → ΩalgU(n)

⇐⇒λrH+ ⊂W ⊂ λsH+, λW ⊂W , ∂W ⊂W , λ∂W ⊂W

We then call W a complex extended solution.

Construction of general complex extendedsolution

By multiplying by λs , can take s = 0. Note that, for r > 0,

H+/λrH+ = Cn ⊕ λCn ⊕ · · · ⊕ λr−1Cn.

Theorem (Guest 2002)

Let r > 0, and suppose that X is a holomorphic subbundle ofH+/λ

rH+. Then

W = X + λX(1) + · · ·+ λr−1X(r−1) + λrH+

is a complex extended solution. Any complex extended solution Wwith λrH+ ⊂W ⊂ H+ is given in this way for some holomorphicsubbundle X .

Here

X(i) = span∂kσ | σ a local section of X , k ≤ i.

Filtrations

Let W = ΦH+ be a complex extended solution withλrH+ ⊂W ⊂ H+.

DefinitionWe call a filtration

W = Wr ⊂Wr−1 ⊂ · · · ⊂W1 ⊂W0 = H+

of complex extended solutions satisfying

λWi−1 ⊂Wi ⊂Wi−1 (i = 1, . . . , r)

a λ-filtration of W .

Uniton factorization of Φ ⇐⇒ λ-filtration of W .

Filtrations – 2

Proposition (Svensson–JCW, 2012)

Let W = ΦH+ be a complex extended solution withλrH+ ⊂W ⊂ H+ and consider any λ-filtration:

W = Wr ⊂Wr−1 ⊂ · · · ⊂W1 ⊂W0 = H+

of W . Define a sequence Φi : M2 → ΩU(n) inductively by

Φ0 = I , and Φi = (πα1 + λπ⊥α1) · · · (παi + λπ⊥αi

),

where αi = P0Φ−1i−1Wi . Then Wi = ΦiH+.

Let S ik be the coefficient of λ−k in

Φi−1 = (πi + λ−1π⊥i ) · · · (π1 + λ−1π⊥1 ). Then

αi =∑i−1

s=0(S i−1s )Ps(Wi )

Skip to Harmonic maps into the orthogonal group

Two extreme λ-filtrations

Let W be a complex extended solution with λrH+ ⊂W ⊂ H+.Define, for i = 0, . . . , r ,

• W Ui = (λi−rW ) ∩H+ (the Uhlenbeck filtration);

• W Si = W + λiH+ (the Segal filtration).

Writing Wi = ΦiH+, Φi =∑i

k=0 λkT i

k , then

• αi = Im(T i0)∗ = ImS i

0 for the Uhlenbeck unitons;

• αi = kerT ii for the Segal unitons.

Proposition

(see [Svensson–JCW, 2012]) For any λ-filtrationW = Wr ⊂Wr−1 ⊂ · · · ⊂W1 ⊂W0 = H+ of W , we have

W Si ⊂Wi ⊂W U

i (i = 0, . . . , r).

The Uhlenbeck unitons αi = Im(T i0)∗

Let W be a complex extended solution with λrH+ ⊂W ⊂ H+;W = X + λX(1) + · · ·+ λr−1X(r−1) + λrH+.

Fix a meromorphic spanning set Lj for X , withLj = λo(j)

∑rk=0 λ

k Ljk where Lj0 6= 0. Then αi is spanned by

i−1∑s=0

S i−1s (Ljs)(k) : k + o(j) ≤ r − i).

For example, if X is spanned by L = L0 + λL1 + λ2L2 + λ3L3, then

α1 = spanL(k)0 : k ≤ 3 ,

α2 = spanπ1L(k)0 + π⊥1 L

(k)1 : k ≤ 2 ,

α3 = spanS20 L

(k)0 + S2

1 L(k)1 + S2

2 L(k)2 : k ≤ 1 ,

α4 = spanS30 L0 + S3

1 L1 + S32 L2 + S3

3 L2 .

These are equivalent to formulae of B. Dai and C.-L. Terng (2007).

The Segal unitons αi = kerT ii


Fix a meromorphic spanning set for Lj for X , withLj =

∑r−1k=0 λ

kLk,j . Then αi =∑i−1

s=0 Si−1s PsW .

Explicitly:

α1 = spanL0,j

α2 = spanπ1L0,j + π⊥1 L1,j , π⊥1 L

(1)0,j

α3 = spanπ2π1L0,j + (π2π⊥1 + π⊥2 π1)L1,j + π⊥2 π

⊥1 L2,j ,

(π⊥1 + π⊥2 )L(1)0,j + π⊥2 π

⊥1 L

(1)1,j , π

⊥2 π⊥1 L

(2)0,j

etc. These are equivalent to the ‘Lisbon’ formulae(Ferreira–Simoes–JCW).

Mixture of Segal and Uhlenbeck steps


Fix a meromorphic spanning set for Lj for X , withLj =

∑r−1k=0 λ

kLk,j = λo(j)∑r

k=0 λk Ljk .

Let Wi be obtained from W by u Uhlenbeck steps and r − i − uSegal steps, in any order. Then

αi = span i−1∑s=0

S i−1s (Ljs−k+u)(k) : 0 ≤ o(j) + k − u ≤ i − 1

+ span

i−1∑s=0

S i−1s (Ljs)(k) : o(j) + k − u < 0

. (8)

Skip to Final slide

Grassmannian solutions

Recall the totally geodesic Cartan embedding

G∗(Cn) = ∪kGk(Cn) → U(n), V 7→ πV − πV⊥ .

This identifies G∗(Cn) with√I = g ∈ U(n) | g2 = I.

Harmonic maps ϕ : M2 → G∗(Cn)

⇐⇒

Harmonic maps ϕ : M2 →√I ⊂ U(n).

W−λ = Wλ ⇔ Φλ = Φ−λΦ−1−1 =⇒ Φ−1 : M2 →

√I

Conversely, if ϕ : M2 →√I has an extended solution, then it also

has an extended solution Φ : M2 → ΩU(n) with Φλ = Φ−λΦ−1−1,

i.e.,with W−λ = Wλ. Skip to Real Harmonic Maps

S1-invariant solutions – 1

Consider the action of S1 on ΩU(n) given by

(γ · η)(λ) = η(λγ)η(γ)−1 (γ ∈ S1, η ∈ ΩU(n)).

An extended solution Φ is S1-invariant if ΦλΦγ = Φλγ for allλ, γ ∈ S1.

Proposition

Let Φ : M2 → ΩU(n) be an S1-invariant polynomial extendedsolution of degree r . Let α1, . . . , αr and β1, . . . , βr be thecorresponding Segal and Uhlenbeck unitons, respectively. Then

• α1 ⊂ α2 ⊂ · · · ⊂ αr ;

• αi are holomorphic subbundles with respect to the standardholomorphic structure on Cn;

• ∂αi ⊂ αi+1;

• βi = αr+1−i .

S1-invariant solutions – 2

The corresponding harmonic map ϕ = Φ−1 is then given by

ϕ = α1 + α⊥2 ∩ α3 + · · ·+ α⊥r : M2 → G∗(Cn).

Conversely, given any sequence of holomorphic subbundlesα1 ⊂ · · · ⊂ αr of Cn satisfying ∂αi ⊂ αi+1, then

Φ = (πα1 + λπ⊥α1) · · · (παr + λπ⊥αr

)

is an S1-invariant extended solution.

Example

Proposition

Suppose that ϕ : M2 → Gk(Cn) has a polynomial extendedsolution Φ : M2 → ΩU(n). Then it has a polynomial extendedsolution Φ of degree r ≤ 2k satisfying

Φλ = Φ−λΦ−1−1.

In particular, if ϕ : M2 → CPn, then Φ is of degree r ≤ 2 =⇒ Φ isS1-invariant.

Hence Φ = (πα1 + λπ⊥α1)(πα2 + λπ⊥α2

) where α1 ⊂ α2 ⊂ Cn areholomorphic subbundles with ∂α1 ⊂ α2, and

ϕ = α⊥1 ∩ α2.

All such αi are given by α1 = ft−1, α2 = ft for some holomorphicf : S2 → CPn and t ∈ 0, 1, . . . , n, whence ϕ = G (t)(f ),recovering [Eells–JCW,1983].

Harmonic maps ϕ : M2 → SO(n) – 1

A complex extended solution W with λrH+ ⊂W ⊂ H+ is said tobe real if

W⊥

= λ1−rW .

Equivalently, a polynomial extended solution Φ : M2 → ΩU(n) ofdegree r is said to be real if Φ = λ−rΦ.

Then W = ΦH+ with Φ−1dΦ taking values in so(n). If r = 2s,then

λ−sΦ : M2 → ΩSO(n) ⊂ ΩU(n).

LemmaSuppose that W is real. Let

Wr−1 = W + λr−1H+, Wr−2 = (λ−1Wr−1) ∩H+.

Then Wr−2 is also real.

Harmonic maps ϕ : M2 → SO(n) – 2


Any extended solution Φ : M2 → ΩSO(n) which is a finite Laurentpolynomial in λ, has a factorization

Φ = λ−sη1 · · · ηr where ηi = παi + λπ⊥αi

where r = 2s and

1 each subfactor η1 · · · ηk : M2 → ΩU(n) is an extendedsolution;

2 each quadratic subfactor λ−1η2j−1η2j takes values in ΩSO(n).

Taking W with W−λ = Wλ gives all harmonic maps of finiteuniton number into a real Grassmannian.

Real extended solutions with r odd – 1

Let O(2m)/U(m) = space of orthogonal complex structures∼= V ∈ Gm(C2m) : V⊥ = V . Cartan embeddingGm(C2m)→ U(2m) embeds this in U(2m).

For a map J : M2 → O(2m)/U(m) this gives

ϕ : M2 → O(2m), ϕ = i(πV − π⊥V ) = J

Example

W is real with r = 1 if and only if W = V + λH+ with Vholomorphic and maximally isotropic. This corresponds to aholomorphic map J : M2 → O(2m)/U(m). This has extendedsolution

πV + λπ⊥V : M2 → ΩU(2m) , giving W = V + λH+ .

Real extended solutions with r odd – 2


Assume that Φ : M2 → ΩU(n) is a real extended solution of odddegree r . Then n = 2m and Φ has a factorization

Φ = η0η1 · · · ηr−1, η0 = πV0 + λπV⊥0, ηi = παi + λπ⊥αi

,

and

1 each subfactor (πV0 + λπV⊥0)η1 · · · ηk : M2 → ΩU(2m) is an

extended solution;

2 each quadratic subfactor λ−1η2j−1η2j takes values inΩSO(2m).

3 V0 : M2 → O(2m)/U(m) is holomorphic.

If additionally, W−λ = Wλ, this gives all harmonic maps of finiteisotropy order into O(2m)/U(m).

Example

Take r = 3, n = 6 and X = spanH0 + λH1 + λ2H2, λH3 + λ2H4.Then

W = spanH0 + λH1 + λ2H2, λH3 + λ2H4+ λ(H0 + λH1)(1)

+ λ2(H0)(2) + λ2(H3)(1) + λ3H+ .

W is real if and only if spanH0,H(1)0 ,H3 is a maximal isotropic

subbundle of C6, and

〈H0,H2〉C + 〈H1,H1〉C = 〈H0,H1〉C = 〈H1,H3〉C + 〈H0,H4〉C = 0.

Thus we obtain a harmonic map into O(6); generically this hasuniton number 3.If H1 = H4 = 0, then Wλ = W−λ so that we get a harmonic mapinto O(6)/U(3), generically of uniton number 3.

Explicit formulae and Sp(n)

Explicit formulae provided by mixed Segal–Uhlenbeck formulaeMixed formulae

Similarly, for Sp(n), replacing ‘real’ by ‘symplectic’ and complexconjugation by J, we obtain:

1 an explicit version of R. Pacheco’s factorization (2006);

2 all harmonic maps of finite uniton number into Sp(n),quaternionic Grassmannians and the space of quaternioniccomplex structures Sp(n)/U(n).

In summary, we have found explicit formulae for all harmonic mapsof finite uniton number into classical compact Lie groups and theirinner symmetric spaces.

http://www.maths.leeds.ac.uk/pure/staff/wood/wood.html

The genesis of twistor lifts 1: Gauss maps

Let M2 be a Riemann surface and ϕ : M2 → Rn an immersion.The Gauss map γ : M2 → G2(Rn) is defined by γ(p) = Im(dϕp).

Theorem [S.S. Chern, 1965] Let ϕ : M2 → Rn be conformal.Then its Gauss map is antiholomorphic iff ϕ is harmonic, i.e.,minimal.Proof. G2(Rn) ∼= Qn−2 =

[ζ0, . . . , ζn−1] ∈ CPn−1 :

∑ζ 2i = 0

.

In a local complex coordinate z = x + iy on M2,

γ(z) = span∂ϕ∂x,∂ϕ

∂y

=[∂ϕ∂x

+ i∂ϕ

∂y

]=[∂ϕ∂z

]∈ Qn−2.

Since the harmonicity equation (??) reads (∂/∂z)(∂ϕ/∂z) = 0,ϕ harmonic implies γ antiholomorphic.

Conversely, γ antiholomorphic implies that∂

∂z

(∂ϕ∂z

)= λ

∂ϕ

∂z. But

since LHS is a multiple of the mean curvature, it is normal, soλ = 0, i.e., ϕ is harmonic. This extends to weakly conformal maps.

The genesis of twistor lifts 2: Gauss lifts

Let F be the trivial bundle π : Rn × G2(Rn)→ Rn, then we canthink of the Gauss map as a map ψ : M2 → F called the Gausslift of ϕ which covers ϕ, i.e., ϕ = π ψ. Then Chern’s resultreads: a weakly conformal map ϕ is harmonic iff its Gauss lift ψ isvertically antiholomorphic, i.e., the vertical part of its differentialintertwines the complex structure JM on M2 with −JV where JV isthe complex structure on G2(Rn).

Replacing Rn by an arbitrary Riemannian manifold and F by theGrassmanian bundle G2(TN), this result still holds.

Now suppose that n = 4 and N is oriented. Each pointw ∈ G2(TN) defines an almost complex structure Jw on Tπ(w)N,

namely, rotation by +π/2 on w and w⊥; this lifts to an almostcomplex structure JH on Hw . Then ϕ is conformal iff it ishorizontally holomorphic, i.e., the horizontal part of itsdifferential intertwines the complex structure JM with JH.

Twistor lifts given by Gauss lifts

To combine the statements for vertical and horizontal parts, setJ1 = (JV , JH) and J2 = (−JV , JH). We have

Proposition Let ϕ : M2 → N4 be an immersion. Then ϕ isharmonic iff its Gauss lift ψ : M2 → (G2(TN), J2) is holomorphic.

Now, let π : Z+ → N4 (resp. π : Z− → N4) be the bundle ofpositive (resp. negative) orthogonal complex (i.e., almostHermitian) structures on N4, Then G2(TN) ∼= Z+ × Z− and theGauss lift gives lifts ψ± into Z+ and Z−. We obtain

Theorem (J. Eells and S. Salamon, 1983, 1985)

There is a bijection between nonconstant weakly conformalharmonic maps (or what is the same thing, minimal branchedimmersions) ϕ : M → N and nonvertical J2-holomorphic mapsψ± : M → Z±. The bijection is given by ϕ = π ψ±; conversely,ψ± is obtained from the Gauss lift ψ of ϕ as above.

Twistor spaces - 1

Definition (F.E. Burstall and J.H. Rawnsley, 1990)

A twistor fibration of N (for harmonic maps) is• an almost complex manifold (Z , J) (called the twistor space), and• a fibration π : Z → N such that

for every holomorphic map ψ : M2 → (Z , J) from a Riemannsurface, the composition ϕ = π ψ : M2 → N is harmonic.

ϕ is called the twistor projection of ψ,ψ is called a twistor lift of ϕ.

Problems

• Find twistor fibrations for given Riemannian manifolds N, e.g.,symmetric spaces.• Find twistor lifts of harmonic maps M2 → N from surfaces intothose symmetric spaces.

Twistor spaces - 2

First problem solved by Burstall and Rawnsley [1990]:Twistor fibrations of compact inner symmetric spaces N are givenby canonical fibrations of generalized flag manifolds.

• For N = a complex Grassmannian Gk(Cn), the twistor spaces Zare geometric flag manifolds:

Z = Fd0,...,dt = U(n)/U(d0)× · · · ×U(dt)

= orthogonal decompositions ψ = (ψ0, . . . , ψt) of Cn.

• The twistor projection π : Fd0,...,dt → Gk(Cn) is given bycombining the even-numbered legs of ψ:

πe : Fd0,...,dt → Gk(Cn), ψ = (ψ0, . . . , ψt) 7→[t/2]∑j=0

ψ2j .

Z has a canonical Kahler structure J1 from embedding in aproduct of complex Grassmannians. However, we give Z thealmost complex structure J2 obtained from J1 by reversingorientation on the vertical spaces.

Twistor spaces - 3

Second problem: contributions of Svensson–JCWGiven a harmonic map ϕ : M2 → Gk(Cn) from a surface tocomplex projective space, Burstall [1986] showed that ϕ (or itsorthogonal complement) has a twistor lift if and only if it isnilconformal (see below). We extend this result to(i) [2014] all other classical simply connected compact innerRiemannian symmetric spaces:• the oriented real GrassmanniansGk(Rn) = SO(n)/

(SO(n − k)× SO(k)

)with k(n − k) even,

• the quaternionic GrassmanniansGk(Hm) = Sp(m)

/(Sp(k)× Sp(m − k)

),

• the space SO(2m)/U(m) of +ve orthogonal complex strs on

R2m and the space Sp(m)/U(m) of Lagrangian subspaces of C2m;

(ii) [arXiv] the exceptional symmetric space G2/SO(4).

Twistor spaces - 4

Second problem: further contributions of [Svensson–JCW,2014]

• We extend Burstall’s result as follows: given a nilconformalharmonic map ϕ from any of the above symmetric spaces and auniton α which is adapted to ϕ, we construct a twistor liftψ : M2 → Fd0,...,dt of ±ϕ associated to the uniton.

• For harmonic maps from a surface to the above symmetricspaces of finite uniton number, we give explicit formulae for themap and its twistor lift in terms of holomorphic data.

Harmonic maps into G2/SO(4)

Think of R7 as the space of imaginary octonions. Recall that R7

has a vector product × given by v ×w = the imaginary part of theoctonion product v · w (v ,w ∈ R7).G2/SO(4) = 3-dimensional subspaces V of R7 which areassociative, i.e., closed under the vector product. Note that anysuch V has a natural orientation coming from a basisv ,w , v × w of V . Thus G2/SO(4) is a submanifold of G3(R7).G2/SO(4) has three twistor spaces, cf. [Salamon, 1989]:

G2/U(1)×U(1)

vv ((G2/U(2)+

((

G2/U(2)−

vvG2/SO(4)

with every fibre isomorphic to CP1.

The twistor spaces of G2/SO(4)

G2/U(2)+ is the quaternionic twistor space with fibre atV ∈ G2/SO(4) consisting of all positive almost Hermitianstructures on V⊥, equivalently, isotropic 2-dimensional subspacesof V⊥ ⊗ C which are coassociative, i.e., enjoy the propertyv × w = 0 for all v ,w ∈ V⊥ ⊗ C.

On the other hand, G2/U(2)− is the twistor space with fibre atV ∈ G2/SO(4) consisting of all negative almost Hermitianstructures on V⊥, equivalently isotropic 2-dimensional subspacesof V⊥ ⊗ C which are not coassociative.

For G2/U(1)×U(1), see the next slide.

Unified description of the twistor spacesSet

F = FRd−s ,...,d0

= ψ = (ψ−s , . . . , ψs) ∈ Fd−s ,...,ds : ψ−i = ψi ∀i.

The twistor projections π : F → Gk(Rn) to real Grassmannians aregiven by the sum of the even legs: π(ψ) =

∑j ψ2j .

We say that (ψi ) is a G2-flag if (G2): ψi × ψj ⊆ ψi+j ∀i , j . Then

G2/U(2)+ = (ψ−1, ψ0, ψ1) ∈ FR2,3 : G2 holds

G2/U(2)− = (ψ−2, ψ−1, ψ0, ψ1, ψ2) ∈ FR1,2,1 : G2 holds

G2/U(1)×U(1) = (ψ−3, ψ−2, ψ−1, ψ0, ψ1, ψ2, ψ3) ∈ FR1,1,1,1 :G2 holds

Since (G2) implies that π(ψ) =∑

j ψ2j is associative, theprojection π restricts to projections from these twistor spaces toG2/SO(4), cf. [Kobak, 1993].

Lifts to the twistor spaces of G2/SO(4)

Theorem (Svensson–JCW, arXiv)

Let ϕ : M2 → G2/SO(4) be harmonic. Then ϕ has a J2-holomorpiclift to one of the above three twistor spaces iff it is nilconformal.

Idea of proof. A harmonic map ϕ : M2 → G3(R7) has a lift toone of the above three flag manifolds F if and only if it isnilconformal. If now, ϕ has image in G2/SO(4), the 3-planes ϕ(p)are associative ∀p ∈ M2; this implies that the legs satisfy (G2).

For example, ϕ has a lift to FR2,3 if and only if it is strongly

conformal. We know that v × w ∈ ϕ for all v ,w ∈ ϕ.Differentiating this twice gives A′ϕv × A′ϕw ∈ ϕ, showing that theGauss bundle G ′(ϕ) is closed under the vector product. By strongconformality, it is isotropic. Either it is of rank 2 and socoassociative as required, or it’s of rank 1 and can be extended toa unique coassociative subbundle of ϕ⊥. Either way, we get ourtwistor lift to G2/U(2)+ .

Almost complex maps into S6

The vector product on R7 defines an almost complex structure Jon S6 by Ju(v) = u × v . A map M2 → S6 is called almostcomplex if it intertwines JM and J. Such maps are harmonic.The next result is a slight reformulation of [Bolton, Vranken andWoodward, 1994] and [Bolton, Pedit and Woodward, 1995].

TheoremLet F : M2 → S6 be a non-constant almost complex map. Theneither (i) F is a weakly conformal map into a totally geodesicS2 = Π3 ∩ S6 where Π3 is associative 3-dimensional subspace ofR7, or(ii) there is a G2-flag

ψ = (ψ−3, ψ−2, ψ−1, ψ0, ψ1, ψ2, ψ3) : M2 → G2/U(1)×U(1)

with ψ0 = f = spanF , and we have a diagram showing the onlypossible second fundamental forms A′ψi ,ψj

:

Maps into G2/SO(4) from a.c. maps – 1

ψ−3// ψ−2

// ψ−1// ψ0

// ψ1// ψ2

//jj ψ3tt

Almost complex maps into S6 can be extended to harmonic mapsinto G2/SO(4) as follows:

Proposition

Let F : M2 → S6 be an almost complex map and let α be aholomorphic line subbundle of F−1T 1,0S6. Set ϕ = α⊕ f ⊕ αwhere f = spanF . Then ϕ is a harmonic map from M2 toG2/SO(4).

This essentially follows from the reduction theorem of [Burstalland JCW, 1986] and the fact that α× α ⊂ f so that the image ofϕ consists of associative 3-planes.


Choosing α = G ′(f ) we obtain

Theorem (Svensson–JCW, arXiv)

There is a one-to-one correspondence between

1 almost complex maps F : M2 → S6 with image not containedin a totally geodesic S2, and

2 strongly conformal harmonic maps ϕ : M2 → G2/SO(4) with(a) ϕ⊥ strongly conformal, (b) rankG ′(ϕ) = 1 and(c) G ′(ϕ⊥)× G ′′(ϕ⊥) a non-constant subbundle,

given by

F 7→ ϕ = G ′(f )⊕ f ⊕G ′(f ) with inverse ϕ 7→ F = i L× L/|L|2,

where f = spanF and L is any non-zero (local) section of G ′(ϕ⊥).


The following additional constructions use the twistor method.

Proposition

Let F : M2 → S6 be an almost complex map with image notcontained in a totally geodesic S2. Set ψ3 = G (1)(f )×G (2)(f ).Then ϕ = ψ3 ⊕ f ⊕ ψ3 is a strongly conformal harmonic mapM2 → G2/SO(4) with rank G ′(ϕ) = 2.

TheoremLet F : M2 → S6 be an isotropic almost complex map. Thenϕ = G (−i)(f )⊕ f ⊕ G (i)(f ) is a harmonic map into G2/SO(4) fori = 1, 2, 3.

All isotropic almost complex maps M2 → S6 are given by a twistorconstruction of [Bryant, 1982], see [Bolton–Woodward, 1997] interms of superhorizontal holomorphic maps M2 → S6. Suchholomorphic maps are constructed by [Fernandez, to appear], seealso [Martins, 2013].

Maps into symmetric spaces, p. 1

F. E. Burstall and M. A. Guest, Harmonic two-spheres incompact symmetric spaces, revisited, Math. Ann. 309 (1997)541–572.

N. Correia and R. Pacheco, Harmonic maps of finite unitonnumber into G 2, Math. Z. 271 (2012), 13–32.

N. Correia and R. Pacheco, Extended solutions of the harmonicmap equation in the special unitary group, arXiv:1304.4264.

B. Dai and C.-L. Terng, Backlund transformations, Wardsolitons, and unitons, J. Differential Geom. 75 (2007) 57–108.

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Twistorial methods

F. E. Burstall, A twistor description of harmonic maps of a 2-sphereinto a Grassmannian, Math. Ann. 274 (1986), 61–74.

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M. Svensson and J. C. Wood, New constructions of twistor lifts forharmonic maps, Man. Math. (2014), DOI:10.1007/s00229-014-0659-9.

Maps into S6 and G2/SO(4) – 1

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FordyWood/contents.html for a downloadable version.

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J. Bolton and L. M. Woodward, Special submanifolds of S6 with itsG2 geometry, Geometry, topology and physics (Campinas, 1996),59–68, de Gruyter, Berlin, 1997.

Maps into S6 and G2/SO(4) – 2

R.L. Bryant, Submanifolds and special structures on theoctonians, J. Differential Geom. 17 (1982), 185–232.

L. Fernandez, The space of almost complex 2-spheres in the6-sphere, Trans. Amer. Math. Soc. (to appear).

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M. Svensson and J. C. Wood, Harmonic maps into theexceptional symmetric space G2/SO(4), arXiv:1303.7176.

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