FILTRATIONS, FACTORIZATIONS AND EXPLICIT FORMULAE FOR HARMONIC MAPS

Transcription

FILTRATIONS, FACTORIZATIONS AND EXPLICIT FORMULAE FOR
HARMONIC MAPS
MARTIN SVENSSON AND JOHN C. WOOD
Abstract. We use filtrations of the Grassmannian model to produce explicit algebraic formulae for harmonic maps of finite uniton number from a Riemann surface to the unitary group
for general methods of factorization by unitons. We show how these specialize to give explicit
formulae for harmonic maps into the special orthogonal and symplectic groups, real, complex
and quaternionic Grassmannians, and the spaces SO(2m)/U(m) and Sp(n)/U(n), i.e., all the
classical compact Lie groups and their inner symmetric spaces. Our methods also give explicit
J2 -holomorphic lifts of harmonic maps into Grassmannians and an explicit Iwasawa decomposition.
1. Introduction
In [23], K. Uhlenbeck showed how to construct harmonic maps from a Riemann
surface into the unitary group U(n) by starting with a constant map and successively
modifying it by a process called ‘adding a uniton’, a sort of B¨acklund transform.
She showed that all harmonic maps from the 2-sphere could be obtained that way.
Various ways of making this more explicit were given by the second author and
others, e.g., [25], however, finding the unitons involved the solution of ∂-problems,
which could rarely be solved explicitly.
In [12], M. J. Ferreira, B. A. Sim˜oes and the second author showed how to solve
this problem, producing algebraic formulae for the unitons, and thus for all harmonic
maps of finite uniton number from a Riemann surface to the unitary group. They
used the factorization essentially due to G. Segal [22] which is dual to that used
by Uhlenbeck. They then related their formulae to the Grassmannian model of
Segal and indicated how to obtain harmonic maps into a complex Grassmannian
as a special case. By a completely different method, by thinking of the unitons as
stationary Ward solitons, B. Dai and C.-L. Terng [8] obtained explicit formulae for
the unitons of the Uhlenbeck factorization.
In the present paper, we use filtrations of the Grassmannian model to produce
explicit algebraic formulae for harmonic maps of finite uniton number for general
methods of factorization including not only those above as extreme cases, but also
factorizations obtained by a mixture of them, and by the method of second author
[25] which reduces to Gauss transforms in the Grassmannian case. On the way, we
establish many useful formulae relating uniton factorizations and filtrations, and
find explicit formulae for J2 -holomorphic lifts of harmonic maps into Grassmannians.
2000 Mathematics Subject Classification. 53C43, 58E20.
Key words and phrases. harmonic maps, Grassmannian model.
1
2
Finally, we show how to apply our methods to finding harmonic maps of finite
uniton number from a Riemann surface into the special orthogonal group SO(n)
± and
the real Grassmannians; we also find harmonic maps into the space SO(2m) U(m)
of orthogonal complex structures. Here we use a factorization by alternate Uhlenbeck and Segal steps. Our formulae for such mixed factorizations then gives explicit
formulae for all such harmonic maps.
The same methods apply to find all harmonic maps of finite uniton number from
a surface to the symplectic group Sp(n) and quaternionic Grassmannians; here
the factorization is that of± R. Pacheco [19]. However, we can also find harmonic
maps into the space Sp(n) U(n) of ‘quaternionic’ complex structures. In this way,
we obtain explicit formulae for all harmonic maps of finite uniton number from
Riemann surfaces to the classical compact Lie groups and their inner symmetric
spaces.
The paper is arranged as follows. In §2, we give formulae relating factorizations
and filtrations which are purely algebraic; in particular, we study the two extreme
filtrations of Segal and Uhlenbeck and characterize polynomials which are invariant
under the ‘additional S 1 -action’ of [23, §7].
Then in §3, we discuss factorizations and filtrations of harmonic maps, and see
how operators in the Grassmannian model correspond to operators on the corresponding subbundles. Our explicit formulae for harmonic maps are given in §3.4;
then we discuss how these give an explicit formula for the Iwasawa decomposition, and we examine the relationship of our work with that of F. E. Burstall and
M. A. Guest [5].
In §4, we see how our methods give harmonic maps into complex Grassmannians
and show how to get J2 -holomorphic lifts from suitable filtrations. Explicit formulae
for these are then given.
Finally, in §5, we see how our methods give harmonic maps into the groups
SO(n), Sp(n) and their inner symmetric spaces, constructing
of
± all harmonic maps
±
finite uniton number from a Riemann surface into SO(2m) U(m) and Sp(n) U(n);
a subject that, to our knowledge, has not appeared in the literature.
2. Some basic algebraic formulae
2.1. The Grassmannian model of ΩU(n). For a Lie group G, we recall that the
group of (free) loops of G is given by
ΛG = {γ : S 1 → G | γ is smooth},
and the group of (based) loops of G is given by
ΩG = {γ ∈ ΛG | γ(1) = e}.
We shall mainly consider the case when G is the unitary group U(n), or one if its
subgroups: the orthogonal group or symplectic group.
FILTRATIONS, FACTORIZATIONS AND HARMONIC MAPS
3
We denote by H = H(n) the Hilbert space L2 (S 1 , Cn ). By expanding into Fourier
series, we have
H(n) = linear closure of {λi ej | i ∈ Z, j = 1, . . . , n}
where {e1 , . . . , en } is the standard basis for Cn ; in fact, {λi ej | i ∈ Z, j = 1, . . . , n}
is an orthonormal basis.
The natural action of U(n) on Cn induces an action of ΩU(n) on H(n) which is
isometric with respect to the L2 inner product. We consider the closed subspace
H+ = linear closure of {λi ej | i ∈ N, j = 1, . . . , n}
where N = {0, 1, 2, . . .}. The action of ΩU(n) induces an action of ΩU(n) on
subspaces of H(n) ; denote by Gr(n) the orbit of H+ under this action. For a precise
description of the elements in Gr(n) we refer to [21]; here we just note that any
W ∈ Gr(n) is closed under multiplication by λ, i.e., λW ⊂ W , and we have a
bijective map
ΩU(n) 3 Φ 7→ W = ΦH+ ∈ Gr(n) ;
we shall sometimes call W the Grassmannian model of Φ.
Any λ-closed subspace W satisfying λr H+ ⊂ W ⊂ λs H+ for some r, s ∈ Z with
(r ≥ s) is in Gr(n) . Note that such a subspace can also be thought of as a subspace of
the quotient vector space λs H+ /λr H+ ; this quotient space with the inner product
induced from H(n) may be naturally identified with the finite-dimensional vector
space C(r−s)n equipped with its standard Hermitian inner product. Note that, by
multiplying by λ−s , we can assume that s = 0.
For any i ∈ Z, let Pi : H(n) → Cn be the i’th coordinate projection given by
P i
L=
λ Li 7→ Li . For any subspace α of Cn , we denote by πα and πα⊥ orthogonal
projection onto α and its orthogonal complement, respectively. The fundamental
idea behind relating uniton factorizations and filtrations is the following construction due to Segal [22], though the terminology is ours.
f = ΦH
e + ∈ Gr(n) . We say that W
f (or Φ)
e is obtained from
Let W = ΦH+ and W
W (or Φ) by a λ-step if
f⊂W ⊂W
f,
λW
f ⊂ λ−1 W.
equivalently, W ⊂ W
f = ΦH
e + where Φ, Φ
e ∈ ΩU(n). Then W
f is
Lemma 2.1. Let W = ΦH+ and W
obtained by a step from W if and only if
e α + λπα⊥ ),
Φ = Φ(π
equivalently,
e = Φ(πα + λ−1 πα⊥ ).
Φ
Further,
(2.1)
e −1 W
α = P0 Φ
and
f;
α⊥ = P0 Φ−1 λW
conversely,
e
f = Φ(α) + λW
f
(2.2) W = Φ(α)
+ λW
and
f = Φ(α
e ⊥ ) + W = λ−1 Φ(α⊥ ) + W.
W
4
f is obtained from W by a λ-step, so that λΦH
e + ⊂ ΦH+ ⊂
Proof. Assume that W
e + . Then λH+ ⊂ Φ
e −1 ΦH+ ⊂ H+ , which implies that Φ
e −1 ΦH+ = α + λH+ =
ΦH
e α + λπα⊥ ). The converse
(πα + λπα⊥ )H+ for some subspace α ⊂ Cn ; hence Φ = Φ(π
is immediate, as are (2.1) and (2.2).
¤
e = Φ and
Note that we do not exclude the extreme cases: (i) α = Cn , then Φ
f = W ; (ii) α = the zero subspace, then Φ
e = λ−1 Φ and W
f = λ−1 W . Note also
W
f ⊂ H+ for some i ∈ {1, 2, . . . , }, then W satisfies the condition
that, if λi−1 H+ ⊂ W
λi H+ ⊂ W ⊂ H+ ;
(2.3)
the converse is, in general, false.
Now let W ∈ Gr(n) be a subspace satisfying
λr H+ ⊂ W ⊂ H+
(2.4)
for some r ∈ {0, 1, 2, . . .}. As above, we shall write W = ΦH+ for some Φ ∈ ΩU(n).
Note that, if r = 0, W = H+ and Φ = I.
Definition 2.2. By a λ-filtration (Wi ) of W we mean a nested sequence
(2.5)
W = Wr ⊂ Wr−1 ⊂ · · · ⊂ W0 = H+
of λ-closed subspaces of H+ satisfying
(2.6)
λWi−1 ⊂ Wi ⊂ Wi−1
Thus Wi−1 is obtained from Wi by a λ-step.
By a simple induction starting with W0 = H+ we see that each Wi satisfies (2.3) .
We now identify the steps and loops associated to a filtration.
Proposition 2.3. Let (Wi ) be a λ-filtration of W . Define a sequence Φi ∈ ΩU(n)
(i = 0, 1, . . . , r) inductively by
Φ0 = I ,
and Φi = Φi−1 (πi + λπi⊥ )
where πi denotes orthogonal projection onto the subspace
(2.7)
αi = P0 Φ−1
i−1 Wi
and πi⊥ denotes orthogonal projection onto αi⊥ . Then
Φi H+ = Wi
(i = 0, . . . , r).
Proof. We use induction on i. For i = 0, it is trivial. For i = 1, (2.6) implies that
W1 = V + λH+ for some subspace V ⊂ Cn and (2.7) gives α1 = P0 W1 = V . Hence
W1 = π1 H+ + λH+ = (π1 + λπ1⊥ )H+ = Φ1 H+ ,
as desired.
Now suppose that Φi−1 H+ = Wi−1 for some i > 1. Then, from (2.6) and the
induction hypothesis,
λH+ ⊂ Φ−1
i−1 Wi ⊂ H+ ,
5
so that, by (2.7),
⊥
Φ−1
i−1 Wi = αi + λH+ = (πi + λπi )H+ .
(2.8)
Hence Φi H+ = Wi , completing the induction step.
¤
The proposition implies that
(2.9)
Φ0 = I
and Φi = (π1 + λπ1⊥ ) · · · (πi + λπi⊥ ) (i = 1, 2, . . . , r) ;
thus, the Φi are polynomials in λ of the form
Φi = T0i + λT1i + · · · + λi Tii
(2.10)
(λ ∈ S 1 ),
where the Tji are n × n complex matrices. In particular Φ = Φr is polynomial of
degree at most r.
The proposition shows that the choice of a λ-filtration (Wi ) of W is equivalent
to the choice of a sequence (αi ) of subspaces of Cn ; this is equivalent, in turn, to a
factorization of Φ:
Φ = (π1 + λπ1⊥ ) · · · (πr + λπr⊥ ).
(2.11)
Indeed, given (Wi ), define the sequence (αi ) by (2.7); conversely, given an arbitrary
sequence (αi ) of subspaces, define the sequence (Φi ) by (2.9) and then set Wi =
Φi H+ . From Lemma 2.1 we obtain the following formulae, with (iv) obtained by
iterating (ii).
Corollary 2.4. For i = 1, . . . , r, we have
(i) αi⊥ = P0 Φ−1
i (λWi−1 ) ;
(ii) Wi = Φi−1 (αi ) ⊕ λWi−1 = Φi (αi ) ⊕ λWi−1 ;
⊥
⊥
(iii) Wi = Φi (αi+1
) ⊕ Wi+1 = λ−1 Φi+1 (αi+1
) ⊕ Wi+1 ;
(iv) Wi = Φi−1 (αi ) ⊕ λΦi−2 (αi−1 ) ⊕ · · · ⊕ λi−1 Φ0 (α1 ) ⊕ λi H+ .
Furthermore, all the direct sums are orthogonal direct sums with respect to the L2
inner product on H+ .
¤
From (2.10) we obtain
(2.12)
Φi−1 = Φi ∗ = S0i + λ−1 S1i + · · · + λ−i Sii
(λ ∈ S 1 ),
where each Ssi is the adjoint (Tsi )∗ of Tsi . On the other hand, from (2.9) we obtain
Φi−1 = (πi + λ−1 πi⊥ ) · · · (π1 + λ−1 π1⊥ ) .
(2.13)
Comparing these, we see that Ssi is the sum of all i-fold products of the form Πi · · · Π1
where exactly s of the Πj are πj⊥ and the other i − s are πj .
Corollary 2.5. We have the following explicit formulae for each subbundle αi :
(2.14)
(a)
αi =
i−1
X
s=0
Ssi−1 Ps Wi
,
(b)
αi⊥
=
i
X
Ssi Ps−1 Wi−1 .
s=1
Proof. The formulae are obtained by expanding (2.7) and Corollary 2.4(i).
¤
6
Note that the formula (a) gives αi in terms of the filtration and the ‘previous’
subbundles α1 , . . . , αi−1 .
2.2. Two extreme filtrations. There are two natural λ-steps, which we shall call
the Segal and Uhlenbeck steps, given on a subspace W ∈ Gr(n) satisfying (2.3) by
S
(2.15) Wi−1
= W + λi−1 H+ ,
U
Wi−1
= (λ−1 W ) ∩ H+ = (λ−1 W ) ∩ H+ + λi−1 H+ ,
respectively. Note that the Segal step depends on the choice of i.
The Segal and Uhlenbeck steps commute as shown by the following calculation:
(λ−1 W ∩ H+ ) + λi−2 H+ = λ−1 (W ∩ λH+ + λi−1 H+ )
¡
¢ ¡
¢
= λ−1 (W + λi−1 H+ ) ∩ λH+ = λ−1 (W + λi−1 H+ ) ∩ H+ .
Starting with a subspace W ∈ Gr(n) satisfying (2.4) and iterating these steps
gives λ-filtrations of W which appear in the work of Segal [22] and Uhlenbeck [23]:
(2.16) WiS = W + λi H+
(2.17)
WiU
= (λ
i−r
(i = 0, . . . , r) (the Segal filtration);
W ) ∩ H+
(i = 0, . . . , r) (the Uhlenbeck filtration).
We call the corresponding subspaces αi and factorization (2.11) the Segal (resp.
Uhlenbeck ) subspaces and factorization. The following proposition shows how these
are the two extremes of the possible filtrations of W .
Proposition 2.6. For any λ-filtration (2.5) of W , we have
WiS ⊂ Wi ⊂ WiU
(i = 0, . . . , r).
Proof. Since λi H+ ⊂ Wi and W ⊂ Wi , we see that
WiS = W + λi H+ ⊂ Wi
(i = 0, . . . , r).
To show that Wi ⊂ WiU , we use reversed induction: since Wr = WrU = W , it is true
for i = r. Assume that it is true for some i. Then we see that
Wi−1 ⊂ λ−1 Wi ⊂ λ−1 (λi−r W ) = λi−1−r W.
U
Since Wi−1 ⊂ H+ it follows that Wi−1 ⊂ (λi−1−r W )∩H+ = Wi−1
, and the induction
step is complete.
¤
⊥
Remark 2.7. Fix i ≥ 1. For W ∈ Gr(n) , set W I = λi−1 W . If W = ΦH+ ,
then clearly W I = λi ΦH+ ; it follows that W 7→ W I is an involution on the set of
f is obtained from W by a λ-step,
W ∈ Gr(n) which satisfy (2.3). Furthermore, if W
f I is obtained from W I by a λ-step; if the step W 7→ W
f is Segal (resp.
then W
f I is Uhlenbeck (resp. Segal).
Uhlenbeck) then the step W I 7→ W
This involution induces an involution on λ-filtrations: given a λ-filtration (Wi ),
⊥
WiI = λi−1 Wi is another λ-filtration. See also Example 3.6.
We now see what choices of subspace the Segal and Uhlenbeck steps corresponds
to.
7
Proposition 2.8. Let i ∈ N and let Φ ∈ ΩG be a polynomial of degree at most i :
(2.18)
Φ = T0 + T1 λ + · · · + Ti λi
so that
Φ−1 = S0 + S1 λ−1 + · · · + Si λ−i
where Sj is the adjoint of Tj (j = 1, . . . , i). Let α be a subspace of Cn . Write
e = Φ(πα + λ−1 π ⊥ ) and W
f = ΦH
e + . Then
W = ΦH+ , Φ
α
f = W + λi−1 H+ (Segal step) if and only if α = ker(Ti ) ;
(i) W
f = (λ−1 W ) ∩ H+ (Uhlenbeck step) if and only if α = Im(S0 ).
(ii) W
f = W + λi−1 H+ if and only if Φ−1 (λW
f ) = λH+ + λi Φ−1 H+ . Since
Proof. (i) W
f ⊂ W and λH+ ⊂ Φ−1 (λW
f ), this is the equivalent to P0 Φ−1 (λW
f ) = P0 λi Φ−1 H+ .
W
By Corollary 2.4(i), this holds if and only if
α⊥ = P0 λi Φ−1 H+ = Im(Si ),
equivalently, α = ker(Ti ).
⊥
⊥
f I = λi−2 W
f = λi−1 ΦH
e + , recall
(ii) Setting W I = λi−1 W = λi ΦH+ and W
−1
I
I
i−1
f = (λ W ) ∩ H+ if and only if W
f = W + λ H+ . Since
from Remark 2.7 that W
e α⊥ + λπα ), it follows from part (i) that this is equivalent to the choice
λi Φ = λi−1 Φ(π
α = P0 (Φ)−1 H+ ,
i.e., α = Im(S0 ).
¤
Note that we do not insist that Ti or S0 be non-zero in the above.
The following result shows how the particular choices of extreme filtrations correspond to ‘covering’ properties of the subspaces.
Proposition 2.9. Let (Wi ) be a λ-filtration of W and let αi be the corresponding
subspaces given by Proposition 2.3.
(i) Suppose that, for some i = 1, . . . , r − 1, we have Wi−1 = Wi + λi−1 H+ . Then
Wi = Wi+1 + λi H+ if and only if
(2.19)
πi (αi+1 ) = αi .
In particular, (Wi ) is the Segal filtration of W if and only if (2.19) holds for all
i = 1, . . . , r − 1.
(ii) Suppose that, for some i = 1, . . . , r − 1, we have Wi−1 = (λ−1 Wi ) ∩ H+ . Then
Wi = (λ−1 Wi+1 ) ∩ H+ if and only if
(2.20)
πi+1 (αi ) = αi+1 .
In particular, (Wi ) is the Uhlenbeck filtration of W if and only if (2.20) holds for
all i = 1, . . . , r − 1.
Proof. (i) By Corollary 2.4(ii), we have
Wi+1 + λi H+ = Φi (αi+1 ) + λWi + λi H+ = Φi (αi+1 ) + λ(Wi + λi−1 H+ )
¢
¡
= Φi (αi+1 ) + λWi−1 = Φi−1 (πi + λπi⊥ )(αi+1 ) + λH+
¢
¡
(2.21)
= Φi−1 πi (αi+1 ) + λH+ .
8
Now, if Wi = Wi+1 + λi H+ , then applying Φ−1
i−1 to the above gives
Φ−1
i−1 Wi = πi (αi+1 ) + λH+ .
By Proposition 2.3, P0 Φ−1
i−1 Wi = αi , so that this implies (2.19).
Conversely, if (2.19) holds, then the right-hand side of (2.21) equals
Φi−1 (αi + λH+ ) = Φi H+ = Wi ,
which establishes (i). The proof of (ii) is similar.
¤
⊥
Remark 2.10. By simple set theory, πi (αi+1 ) = αi is equivalent to πi+1
(αi⊥ ) =
⊥
αi+1 , thus transforming a Segal filtration into an Uhlenbeck filtration and conversely, see Example 3.6 .
2.3. S 1 -invariant polynomials. Recall (e.g. [23, §7]) that there is an S 1 action
on ΩU(n) given by
¡
¢
(2.22)
(µ∗ Φ)λ = Φµλ Φ−1
µ ∈ S 1 , Φ ∈ ΩU(n) .
µ
We now identify all S 1 -invariant polynomials, i.e., polynomials Φ ∈ ΩU(n) which
satisfy
(2.23)
Φλ Φµ = Φλµ
(λ, µ ∈ S 1 ).
Proposition 2.11. Let r ∈ N and let W = ΦH+ ∈ Gr(n) be a subspace satisfying
(2.4), so that Φ ∈ ΩU(n) is a polynomial in λ of degree at most r. Denote by
β1 , . . . , βr and γ1 , . . . , γr the subspaces of Cn corresponding to the Segal and the
Uhlenbeck filtrations of W , respectively, so that
(2.24)
Φ = (πβ1 + λπβ1⊥ ) · · · (πβr + λπβr⊥ ) = (πγ1 + λπγ1⊥ ) · · · (πγr + λπγr⊥ )
Then the following are equivalent.
(i) βi ⊂ βi+1 (i = 1, . . . , r − 1);
(ii) Φ is S 1 -invariant;
P
(iii) W = r−1
λi βi+1 + λr H+ ;
Pi=0
i
r
(iv) W = r−1
i=0 λ Pi W + λ H+ ;
(v) γi+1 ⊂ γi (i = 1, . . . , r − 1).
Furthermore, if any of the above hold, then γi = βr−i+1 for all i = 1, . . . , r.
Proof. The equivalence of (i), (ii) and (iii) follows easily from the treatment in [23,
§10].
Next, (iii) implies that λk Pk W ⊂ W for k = 0, . . . , r − 1, and (iv) follows.
Conversely, we show that (iv) implies (iii). We shall show by induction that
(2.25)
W + λi H+ = β1 + λβ2 + · · · + λi−1 βi + λi H+
for all i ∈ {1, . . . , r}. This clearly holds for i = 0, 1. Assume that that it holds for
all i ≤ k for some k ∈ {1, . . . , r}. Then, since W is λ-closed, we have βi−1 ⊂ βi for
all i ≤ k so that
k−1
k+1
βk ) = βk .
H+ ) ⊃ P0 Φ−1
βk+1 = P0 Φ−1
k (β1 + λβ2 + · · · + λ
k (W + λ
9
This implies that (2.25) holds for i ≤ k + 1, and (iii) follows.
We have Φ = (πγ1 + λπγ1⊥ ) · · · (πγr + λπγr⊥ ). If (v) holds, it is clear that Φ is
1
S -invariant. Conversely, if (iii) holds, then
(λi−r W ) ∩ H+ =βr−i+1 + · · · + λi−1 βr + λi H+
=(πβr−i+1 + λπβr−i+1
) · · · (πβr + λπβr⊥ )H+ ,
⊥
so that βr−i+1 = γi , and (v) follows, finishing the proof.
¤
Corollary 2.12. Let r ∈ N and let Φ be S 1 -invariant polynomial of degree at most
e + be obtained from Φ be a Segal or Uhlenbeck step. Then Φ
e is also
r. Let ΦH
S 1 -invariant.
S
Proof. By the proposition, W = ΦH+ satisfies (iv). It is easy to see that Wr−1
=
r−1
U
−1
W + λ H+ and Wi−1 = (λ W ) ∩ H+ continue to satisfy (iv).
¤
3. Harmonic maps and extended solutions
3.1. Basic facts. We review some well-known facts about harmonic maps, extended solutions, and their Grassmannian models; our main references are [23], [13]
and [5]. From now on, M will denote a Riemann surface and G a compact Lie subgroup of U(n), equipped with the natural bi-invariant metric from U(n). All maps
and sections are assumed smooth unless otherwise stated. Given a vector space V ,
we denote by V the trivial bundle M × V over M .
For any map ϕ : M → G, we define a 1-form with values in the Lie algebra g of
G as half the pull-back of the Maurer-Cartan form, i.e.,
1
Aϕ = ϕ−1 dϕ.
2
Now let V be a complex representation space for G. Then Dϕ = d + Aϕ defines a
unitary connection on the trivial bundle V . We decompose Aϕ and Dϕ into types:
for convenience we take a local complex coordinate z on an open set U of M and
write dϕ = ϕz dz + ϕz¯d¯
z , A = Aϕz dz + Aϕz¯ d¯
z , Dϕ = Dzϕ dz + Dzϕ¯ d¯
z ; then
1
1
∂
∂
Aϕz = ϕ−1 ϕz , Aϕz¯ = ϕ−1 ϕz¯ , Dzϕ =
+ Aϕz , Dzϕ¯ =
+ Aϕz¯ .
2
2
∂z
∂ z¯
The (Koszul-Malgrange) holomorphic structure induced by ϕ is the unique holomorphic structure on V with ∂-operator given locally by Dzϕ¯ ; we denote the resulting
holomorphic vector bundle by (V , Dzϕ¯ ). If ϕ is constant, Dzϕ¯ = ∂z¯ giving V the standard (product) holomorphic structure. Now [23] a map ϕ : M → G is harmonic if
and only if Aϕz is a holomorphic endomorphism of the holomorphic vector bundle
(Cn , Dzϕ¯ ). In particular its image and kernel form holomorphic subbundles of that
bundle, defined away from a discrete set of points of M where the rank of Aϕz drops;
these are independent of the local complex coordinate z. By ‘filling in holes’ as
in [7, Proposition 2.2], these subbundles can be extended smoothly to subbundles
over the whole of M , which we shall denote by Im Aϕz and ker Aϕz , respectively. The
10
technique of filling in holes applies to any holomorphic, or indeed meromorphic,
section of a holomorphic bundle, and will be used frequently in the sequel.
Let gC be the complexified Lie algebra g ⊗ C.
Definition 3.1. A smooth map Φ : M → ΩG is said to be an extended solution if,
with respect to any local holomorphic coordinate z on U ⊂ M , we have
Φ−1 Φz = (1 − λ−1 )A,
for some map A : U → gC .
For any map Φ : M → ΩG and λ ∈ S 1 , we define Φλ : M → G by Φλ (p) = Φ(p)(λ)
(p ∈ M ). If Φ : M → ΩG is an extended solution, the map ϕ = Φ−1 : M → G is
harmonic and ϕ−1 ϕz = 2A, so that A = Aϕz .
Conversely, given a harmonic map ϕ : M → G, an extended solution Φ : M → ΩG
satisfying
Φ−1 Φz = (1 − λ−1 )Aϕz
is said to be associated to ϕ. In this case, ϕ = gΦ−1 for some g ∈ G. Extended solutions exist locally, and globally if the domain M is simply-connected, for example
e associated to the same map ϕ
if M = S 2 ; further, any two extended solutions Φ, Φ
differ by a loop, i.e., Φ = γΦ for some γ ∈ ΩG.
In the sequel, we identify a map W : M → G(V ) into a Grassmannian of subspaces of a vector space V with the subbundle of V = M × V with fibre at p ∈ M
given by W (p); we denote this subbundle also by W .
For a smooth Φ : M → ΩU(n), set W = ΦH+ : M → Gr(n) . It is easy to see that
Φ is an extended solution if and only if W satisfies the two conditions
¡
¢
(3.1)
(a) ∂z¯σ ∈ W, (b) λ∂z σ ∈ W
σ ∈ Γ(W ) ;
here Γ(·) denotes the space of smooth sections of a vector bundle.
Conversely, if W : M → Gr(n) is a map satisfying these two conditions, then
W = ΦH+ for some extended solution Φ : M → ΩU(n). We shall therefore also
refer to such a W as an extended solution, or occasionally, as the Grassmannian
model of Φ.
An extended solution is called algebraic if it is polynomial in λ and λ−1 . An
argument of Uhlenbeck [23, Theorem 11.5] shows that, if M is compact and ϕ :
M → U(n) has an associated extended solution, then it has an algebraic extended
solution Φ. Indeed, fix a base point z0 ∈ M ; then the extended solution satisfying
the initial condition Φλ (z0 ) = I (λ ∈ S 1 ) has this property, see [18, Theorem 4.2]
where this is extended to pluriharmonic maps. In particular, any harmonic map
ϕ : S 2 → U(n) has an algebraic extended solution.
There is a one-to-one correspondence between algebraic extended solutions Φ and
extended solutions W satisfying λr H+ ⊂ W ⊂ λs H+ for some integers r ≥ s (which
depend on W ). Note that we can think of W as a subbundle of the trivial bundle
M × (λs H+ /λr H+ ), and this may be canonically identified with the trivial bundle
11
C(r−s)n with it standard holomorphic structure. Then condition (a) above says that
W is a holomorphic subbundle, and condition (b) says that it is closed under the
operator F : Γ(H(n) ) → Γ(H(n) ) given by
¡
¢
(3.2)
F = λ∂z , i.e., F (σ) = λ ∂z σ σ ∈ Γ(H(n) ) .
Let ϕ : M → U(n) be a harmonic map. Then a subbundle α of Cn is said to be
a uniton for ϕ if it is
(i) holomorphic with respect to the Koszul-Malgrange holomorphic structure
induced by ϕ, i.e.,
Dzϕ¯ (σ) ∈ Γ(α)
(σ ∈ Γ(α));
(ii) closed under the endomorphism Aϕz , i.e.,
Aϕz (σ) ∈ Γ(α)
(σ ∈ Γ(α)).
Example 3.2. Any holomorphic subbundle of (Cn , Dzϕ¯ ) contained in ker Aϕz is a
uniton for ϕ; we call such unitons basic. Any holomorphic subbundle of (Cn , Dzϕ¯ )
containing Im Aϕz is also a uniton; we call such unitons antibasic.
Let ϕ : M → U(n) be a harmonic map. Uhlenbeck showed [23] that if a subbundle
e = ϕ(πα − πα⊥ ) is harmonic. We say that ϕ is of
α ⊂ Cn is a uniton for ϕ then ϕ
finite uniton number if we can write it as
(3.3)
ϕ = ϕ0 (πα1 − πα⊥1 ) · · · (παr − πα⊥r )
where ϕ0 is constant and each αi is a uniton for the partial product
ϕi−1 = (πα1 − πα⊥1 ) · · · (παi−1 − πα⊥i−1 ) .
The minimum value of r for which (3.3) holds is called the (minimal) uniton number
of ϕ. Uhlenbeck showed that any harmonic map from a compact Riemann surface
to U(n) which has an associated extended solution, in particular, any harmonic map
from S 2 to U(n), has finite uniton number at most n − 1.
Now suppose that Φ is any extended solution associated to ϕ, then Uhlenbeck
e = Φ(πα + λπα⊥ ) is also
showed further that α is a uniton for ϕ if and only if Φ
an extended solution (associated to ϕ
e = ϕ(πα − πα⊥ ) ). We shall therefore also say
that α is a uniton for Φ. If ϕ is given by (3.3), then it has an associated extended
solution which is polynomial in λ given by
(3.4)
Φ = (πα1 + λπα⊥1 ) · · · (παr + λπα⊥r ).
We call such a product a uniton factorization of Φ if, for each i = 1, . . . , r, the
subbundle αi is a uniton for
Φi−1 = (πα1 + λπα⊥1 ) · · · (παi−1 + λπα⊥i−1 ) with Φ0 = I,
equivalently, each Φi is an extended solution.
Set W = ΦH+ . Then a uniton factorization of Φ is equivalent to a λ-filtration
(Wi ) of W where each Wi is an extended solution; the equivalence is given by
12
Wi = Φi H+ . From now on, by a λ-filtration of an extended solution W , we shall
mean a λ-filtration by subbundles of Cn where each subbundle in the filtration is an
extended solution. That such filtrations exist is shown by the following example.
Example 3.3. Given an extended solution W , it is clear that W + λi H+ and
λ−i W ∩ H+ are also extended solutions for all i ∈ N. In particular, if λr H+ ⊂ W ⊂
H+ , then all the subbundles Wi in the Segal and Uhlenbeck filtrations (2.16), (2.17)
of W are extended solutions, and the corresponding subbundles αi are unitons,
which we call the Segal and Uhlenbeck unitons, respectively. For another natural
filtration, see Example 3.20.
It follows that any polynomial extended solution Φ has a factorization into unitons, thus a harmonic map ϕ from a Riemann surface to U(n) is of finite uniton
number if and only if it has an associated polynomial extended solution. As above,
this holds when M = S 2 and when M is compact and ϕ has some (not necessarily
algebraic) associated extended solution.
Remark 3.4. If ϕ has finite uniton number, then any associated extended solution
Φ which satisfies an initial condition Φλ (z0 ) = Q(λ) for some z0 ∈ M and algebraic
function Q(λ) is algebraic. Indeed ϕ has an algebraic associated extended solution
e and Φ = QΦ(z
e 0 )−1 Φ.
e
Φ,
Note that all the algebraic formulae for filtrations and their associated factorizations apply, with the subbundles αi now unitons. In particular, the formulae for the
Segal and Uhlenbeck unitons in Proposition 2.8 give these as ker(Tii ) and Im(S0i ),
respectively; the next lemma ensures that these are well-defined by filling in holes.
Denote by (Cn , ∂z¯) the bundle Cn with its standard holomorphic structure.
Lemma 3.5. [16] Let Φ : M → ΩG be an extended solution given by (2.18). Then
(i) Tii is a holomorphic endomorphism from (Cn , Dzϕ¯ ) to (Cn , ∂z¯);
(ii) S0i is a holomorphic endomorphism from (Cn , ∂z¯) to (Cn , Dzϕ¯ ).
¤
Example 3.6. Suppose that Φ is an extended solution and a polynomial in λ of
degree r, and consider the Segal factorization
(3.5)
Φ = (πβ1 + λπβ1⊥ ) · · · (πβr + λπβr⊥ )
of Φ into unitons; recall that these satisfy the covering condition (2.19).
Define the map Ψ = Ψλ = λr Φλ−1 . This is again an extended solution, but with
respect to the opposite orientation of M ; it is associated to the same harmonic map
as Φ, in fact, Ψ−1 = (−1)r Φ−1 . From Remark 2.10 we see that the factorization
(3.5) is equivalent to the factorization
(3.6)
Ψ = (πγ1 + λπγ1⊥ ) · · · (πγr + λπγr⊥ )
13
where the γi = βi⊥ are unitons with respect to the conjugate complex structure;
note that these satisfy the covering condition (2.20), so (3.6) gives the Uhlenbeck
factorization of Ψ with respect to the conjugate complex structure.
We may also consider the map Θ = λr Φ where Φ is obtained from Φ by composition with the isometry of U(n) given by complex conjugation. This is easily seen
also to be an extended solution with respect to the original complex structure on
M , but associated to the harmonic map Φ−1 . Again, the factorization (3.5) of Φ
into unitons satisfying (2.19) is equivalent to the factorization
(3.7)
Θ = (πγ1 + λπγ1⊥ ) · · · (πγr + λπγr⊥ )
⊥
where γi = β i ; these are unitons which satisfy the covering condition (2.20), so
that (3.7) the Uhlenbeck factorization of the complex conjugate Φ of Φ.
3.2. Correspondence of operators under extended solutions. As usual, let
Φ : M → ΩU(n) be an extended solution associated to a harmonic map ϕ and
W = ΦH+ its Grassmannian model; note that Φ gives a linear bundle-isomorphism
from H+ to W , and this induces a linear isomorphism between the spaces of sections
Γ(H+ ) and Γ(W ) which we continue to denote by Φ.
Consider the following three operators on Γ(W ): (i) λ induced by
¡ the linear¢ map
w 7→ λw (w ∈ W ), (ii) ∂z¯ : Γ(W ) → Γ(W ) defined by σ 7→ ∂z¯σ σ ∈ Γ(W ) , and
(iii) F = λ∂z : Γ(W ) → Γ(W ) as in (3.2). In the next result, we see how these give
operators on Γ(H+ ), with (iii) illustrated by the following commutative diagram.
W
F
-W
Φ−1
Φ−1
? λD ϕ − Aϕ
?
z
zH+
H+
P0
P0
?
?
n
C
n
−Aϕz
- C
Proposition 3.7. Under the isomorphism Φ, the operators λ, ∂z¯ and F on Γ(W )
correspond to the following operators on Γ(H+ ) :
(i) Φ−1 ◦ λ ◦ Φ = λ ;
(ii) Φ−1 ◦ ∂z¯ ◦ Φ = Dzϕ¯ − λAϕz¯ ;
(iii) Φ−1 ◦ F ◦ Φ = λDzϕ − Aϕz .
In particular, the three operators induce the following operators on Γ(Cn ) :
(i) P0 ◦ Φ−1 ◦ λ ◦ Φ = 0 ;
(ii) P0 ◦ Φ−1 ◦ ∂z¯ ◦ Φ = Dzϕ¯ ;
(iii) P0 ◦ Φ−1 ◦ F ◦ Φ = −Aϕz .
Proof. (i) Trivial.
14
(ii) For a section f ∈ Γ(H+ ) we have
¡
¢
(Φ−1 ◦ ∂z¯ ◦ Φ)(f ) = Φ−1 (∂z¯Φ)(f ) + Φ(∂z¯f ) = (Φ−1 ∂z¯Φ)(f ) + ∂z¯f
= (1 − λ)Aϕz¯ f + ∂z¯f = Dzϕ¯ f − λAϕz¯ f .
(iii) Similar.
¤
Note that (ii) and (iii) express the well-known fact that Φ gauges the flat connection induced by Φλ to the standard connection.
In the sequel, for subspaces A, B of an inner product space with B ⊂ A, we write
A ª B for A ∩ B ⊥ . Note that this can be canonically identified with the quotient
space A/B.
Remark 3.8. By (i) the isomorphism Φ restricts to an isomorphism H+ /λH+ ∼
=
H+ ªλH+ → W ªλW ∼
= W/λW , which we continue to denote by Φ; also the natural
projection P0 : H+ → Cn restricts to an isomorphism H+ /λH+ ∼
= H+ ª λH+ ∼
= Cn .
Clearly P0 ◦ Φ−1 = Φ−1 ◦ pr where pr : W → W/λW ∼
= W ª λW is the natural
projection, and the last commutative diagram induces the following one.
F W ª λW ∼
= W/λW
W/λW ∼
= W ª λW
Φ−1
?
ϕ
−Az
H+ /λH+ ∼
H+ ª λH+
= H+ ª λH+
∼
=
?
Cn
Φ−1
∼
= H+ /λH+
?
∼
=
?
- Cn
−Aϕz
Corollary 3.9. Let (Wi ) be a λ-filtration of W . Then the map P0 ◦Φ−1
i : (Wi , ∂z¯) →
ϕi
n
(C , Dz¯ ) is holomorphic and sends
(i) Wi onto Cn with kernel λWi ;
(ii) λWi−1 onto αi⊥ with kernel λWi ;
(iii) Wi+1 onto αi+1 with kernel λWi .
The corollary is illustrated by the following diagram.
λWi
?
0
⊂
λWi−1
?
⊂
αi⊥
⊂
P0 ◦
⊂
Wi
Φ−1
i
⊃
Wi+1
⊃
αi+1
?
Cn
?
⊃
λWi
⊃
0
?
Proof. Holomorphicity follows from Proposition 3.7(iii); the rest follows from (2.7)
and Corollary 2.4(i) with kernels λW as in Remark 3.8.
¤
Proposition 3.10. Suppose that W is an extended solution with λi H+ ⊂ W ⊂ H+
U
S
by the Segal and Uhlenbeck steps (cf. (2.15)),
and Wi−1
for some i ≥ 1 . Define Wi−1
respectively:
S
Wi−1
= W + λi−1 H+
and
U
Wi−1
= (λ−1 W ) ∩ H+ .
15
Then
S
S
F Wi−1
⊂ W ⊂ Wi−1
,
(a)
(b)
U
F W ⊂ λWi−1
⊂ W.
Proof. We have
U
F W ⊂ W ∩ λH+ = λ(λ−1 W ∩ H+ ) = λWi−1
= W ∩ λH+ ⊂ W,
which proves (a). Furthermore,
S
S
,
⊂ W + λi H+ = W ⊂ W + λi−1 H+ = Wi−1
F Wi−1
which proves (b).
¤
The last result is illustrated by the following two commutative diagrams.
S
Wi−1
Φ−1
i−1
F W
?
Cn
ϕS
Az i−1
?
-α
W
Φ−1
i−1
F-
?
γ
ϕU
U
λWi−1
?
- 0
Az i−1
It implies some properties of the unitons which give the Segal and Uhlenbeck steps.
S
U
Corollary 3.11. Write W = ΦH+ , Wi−1
= ΦSi−1 H+ and Wi−1
= ΦUi−1 H+ , so that
Φ = ΦSi−1 (πβ + λπβ ⊥ ) = ΦUi−1 (πγ + λπγ ⊥ ) for some unitons β and γ. Then β is
antibasic and γ is basic.
Proof. By Propositions 3.7 and 3.10 we have
¡
¢
¡
¢
¡
¢
ϕS
S
⊂ P0 (ΦSi−1 )−1 W = β,
Az i−1 (Cn ) = P0 (ΦSi−1 )−1 F ΦSi−1 Cn = P0 (ΦSi−1 )−1 F Wi−1
hence β is antibasic.
By Corollary 2.4(ii), ΦUi−1 (γ) ⊂ Wi , so
¡
¢
¡
¢
¡
¢
ϕU
U
Az i−1 (γ) = P0 (ΦUi−1 )−1 F ΦUi−1 γ ⊂ P0 (ΦUi−1 )−1 F W ⊂ P0 (ΦUi−1 )−1 λWi−1
= 0,
hence γ is basic.
¤
Proposition 3.12. Let W be an extended solution satisfying λr H+ ⊂ W ⊂ H+ for
some r > 0, and let (WiS ) be the Segal filtration of W with unitons β1 , . . . , βr . If
β1 is full and βi 6= Cn for all i = 1, 2, . . . , r, then 0 < dim βi < dim βi+1 < n for all
i = 1, 2, . . . , r − 1.
Proof. By the covering condition: πβi (βi+1 ) = βi , the map πβi : βi+1 → βi is
surjective. Suppose that dim βi = dim βi+1 for some i. Then πβi is also injective so
that βi+1 ∩ βi⊥ = {0}. By Proposition 2.8(i), we have βi⊥ = P0 (λi (ΦSi )−1 H+ ); hence
S
∩λi H+ = (λWiS )∩λi H+ .
(βi+1 +λH+ )∩(λi (ΦSi )−1 H+ ) ⊂ λH+ , or, equivalently, Wi+1
Consider now the λ-closed holomorphic bundle
S
Vi = (λ1−i WiS ) ∩ H+ = (λ−i Wi+1
) ∩ H+ .
16
S
By Proposition 3.10, F WiS ⊂ Wi+1
, and so
S
∂z Vi ⊂ (λ1−i ∂z WiS ) ∩ H+ ⊂ (λ−i Wi+1
) ∩ H+ = Vi .
Thus Vi is both holomorphic and antiholomorphic, and hence constant. But by
Corollary 2.4(iv), we see that β1 ⊂ P0 Vi , so the fullness of β1 implies that P0 Vi = Cn .
However, since Vi is closed under multiplication by λ, this implies that Vi = H+ ,
S
= WiS , i.e., ΦSi−1 H+ = ΦSi H+ , which implies that
i.e., λi−1 H+ ⊂ WiS . Hence Wi−1
βi + λH+ = H+ , i.e., βi = Cn , in contradiction to the hypotheses.
¤
3.3. Normalized extended solutions. Again, let W : M → Gr(n) be an extended
solution satisfying (2.4) for some r ≥ 0. Consider the filtration
W ⊃ W ∩ λH+ ⊃ W ∩ λ2 H+ ⊃ · · · ⊃ W ∩ λr H+ ⊃ W ∩ λr+1 H+ .
±
This induces a filtration of W λW :
±
(3.9)
W λW = Yb0 ⊃ Yb1 ⊃ · · · ⊃ Ybr ⊃ Ybr+1 = 0
(3.8)
where
±
±
Ybi = (W ∩ λi H+ + λW ) (λW ) ∼
= (W ∩ λi H+ ) (λW ∩ λi H+ ),
or, equivalently, a direct sum decomposition:
(3.10)
W ª λW = A0 ⊕ A1 ⊕ · · · ⊕ Ar
where
Ai = (W ∩ λi H+ ) ª (λW ∩ λi H+ + W ∩ λi+1 H+ ).
±
Note that Ai ∼
of natural
= Pi (W ∩λi H+ ) Pi−1 (W ∩λi−1 H+ ); indeed the composition
±
i
i
i
projections Pi : W ∩ λ H+ → Pi (W ∩ λ H+ ) → Pi (W ∩ λ H+ ) Pi−1 (W ∩ λi−1 H+ )
is surjective and has kernel λW ∩ λi H+ + W ∩ λi+1 H+ . In particular, denoting the
P
rank of Ai by ni , we have ri=0 ni = n.
We say that W is normalized if ni 6= 0, i.e., Ai 6= 0 for all i = 0, . . . , r,, cf. [5,
Theorem 4.5].
Lemma 3.13. Suppose that W is an extended solution satisfying (2.4). If Ai = 0 for
some i ∈ {0, . . . , r}, then there is an η ∈ ΩU(n), such that λr−1 H+ ⊂ ηW ⊂ H+ .
Proof. Suppose that Ai = 0; then
(3.11)
W ∩ λi H+ = λW ∩ λi H+ + W ∩ λi+1 H+ .
As in the proof of Proposition 3.12, we consider the λ-closed holomorphic bundle
Vi = (λ1−i W ) ∩ H+ + λH+ = λ1−i (W ∩ λi−1 H+ ) + λH+ . Using (3.11) gives
∂z Vi = λ1−i (Wz ∩ λi−1 H+ ) + λH+ ⊂ λ−i (W ∩ λi H+ ) + λH+
= λ−i (λW ∩ λi H+ + W ∩ λi+1 H+ ) + λH+
=λ1−i (W ∩ λi−1 H+ ) + λH+ = Vi .
17
Hence Vi is also antiholomorphic, and thus constant. Since λW ⊂ W , it is easy
to see that λr−1 Vi ⊂ W ⊂ Vi . Choose γ ∈ ΩU(n) such that γH+ = Vi . Then,
applying γ −1 to the last equation gives λr−1 H+ ⊂ γ −1 W ⊂ H+ .
¤
Continuing in this way, we see that we can always normalize a given complex
extended solution, as follows.
Proposition 3.14. Given an extended solution W satisfying (2.4), there exists an
integer s with 0 ≤ s ≤ r and η ∈ ΩU(n) with λs H+ ⊂ ηW ⊂ H+ such that ηW is
normalized. In particular, s ≤ n − 1.
P
Definition 3.15. A polynomial extended solution Φ = ri=0 λi Ti : M → ΩU(n) is
said to be of type one if the linear span of {Im T0 (q) | q ∈ M } equals Cn .
Recalling that, by Lemma 3.5 the adjoint S0 : (Cn , ∂z¯) → (Cn , Dzϕ¯ ) of T0 is a
holomorphic endomorphism, filling in holes gives a well-defined subbundle Im T0
and the type one condition is equivalent to fullness of this bundle. We note that
this is equivalent to the fullness of the first Segal uniton β1 of Φ.
Uhlenbeck proved in [23] that to any harmonic map ϕ : M → U(n) of finite
uniton number, there is a unique type one associated polynomial extended solution
Φ, and its degree equals the minimal uniton number of ϕ. By Lemma 3.13, we see
that the corresponding W must be normalized. However, it is easy to construct
polynomial extended solutions which are normalized but not of type one.
Proposition 3.16. Let W is a normalized extended solution satisfying (2.4), and
let β1 , . . . , βr be the Segal unitons of W . Then 0 < dim βi < dim βi+1 < n for all
i = 1, 2, . . . , r − 1.
Proof. If β1 = 0 or βr = Cn , it is clear that W is not normalized, establishing the
inequalities 0 < dim β1 and dim βr < n.
Now suppose that dim βi = dim βi+1 for some i ∈ {1, 2 . . . , r − 1}. Then the proof
of Proposition 3.12 shows that Vi is constant; the proof of Lemma 3.13 then shows
that W is not normalized.
¤
A similar result is true for the Uhlenbeck unitons.
3.4. Explicit formulae for harmonic maps. For a complex vector space V ,
let G∗ (V ) denote the disjoint union of the complex Grassmannians Gk (V ) over
k ∈ {0, 1, . . . , dim V }. Let W be an extended solution satisfying (2.4). Guest [14]
noted that all such W are given by taking an arbitrary holomorphic map X : M →
G∗ (H+ /λr H+ ) ∼
= G∗ (Crn ), equivalently, holomorphic subbundle X of (Crn , ∂z¯), and
setting W equal to the coset
(3.12)
W = X + λX(1) + λ2 X(2) + · · · + λr−1 X(r−1) + λr H+
where X(i) denotes the i’th osculating subbundle of X spanned by the local holomorphic sections of X and their derivatives of order up to i.
18
More explicitly, choose a meromorphic spanning set {Lj } for X. Then, since
λr+1 H+ ⊂ λW , the set {λk (Lj )(k) | 0 ≤ k ≤ r} gives a meromorphic spanning set
for W mod λW , by which± we mean that the λk (Lj )(k) are meromorphic sections
of W whose cosets span W λW .
From Corollary 2.5, given a filtration Wi of W , the unitons are given explicitly
by (2.7), or equivalently (2.14), furthermore by Corollary 3.9, given a meromorphic
spanning set of each Wi , (2.7) and (2.14) give meromorphic spanning sets for each
αi . If we now specify how to get a basis for Wi from a basis for W , this leads to
explicit formulae for the unitons, and so for the extended solution Φ with W =
i
ΦH+ . Since P0 Φ−1
i−1 (λWi−1 + λ H+ ) = 0, it suffices to know a spanning set for Wi
mod (λWi−1 + λi H+ ).
We now see how this works for the Segal filtration.
Example 3.17. Given an extended solution W satisfying (2.4), for the Segal filtration (2.16), formulae (2.7) and (2.14) simplify to
(3.13)
βi =
S
P0 Φ−1
i−1 Wi
=
P0 Φ−1
i−1 (W
i
+ λ H+ ) =
P0 Φ−1
i−1 W
=
i−1
X
Ssi−1 Ps W.
s=0
More explicitly, let X be a holomorphic bundle generating W as in (3.12). Choose
a meromorphic spanning set {Lj } of X. Then, from the above, {λk (Lj )(k) | 0 ≤ k ≤
r − 1} (note the r − 1) gives a meromorphic spanning set of W mod λW + λr H+ .
For every i, j, write Pi Lj = Lji . Note that Lji = 0 for i < 0 and for i ≥ r so that
P
i j
Lj = r−1
i=0 λ Li . Then the formula (3.13) becomes
(3.14)
βi = span
i−1
nX
o
Ssi−1 (Ljs−k )(k) | 0 ≤ k ≤ i − 1 .
s=0
Note that the sum can be taken from s = k; this is then the formula in [12,
Proposition 4.4]; it gives the first three unitons as:
β1 = span{Lj0 } ;
β2 = span{π1 Lj0 + π1⊥ Lj1 , π1⊥ (Lj0 )(1) } ;
β3 = span{π2 π1 Lj0 + (π2 π1⊥ + π2⊥ π1 )Lj1 + π2⊥ π1⊥ Lj2 ,
(π1⊥ + π2⊥ )(Lj0 )(1) + π2⊥ π1⊥ (Lj1 )(1) , π2⊥ π1⊥ (Lj0 )(2) } .
As shown in [12, Lemma 4.2], the linear transformation E : H+ → H+ , given by
P i
P i
H=
λ Hi 7→ L =
λ Li where
i µ ¶
X
i
(3.15)
Li =
H` (i ≥ 0)
`
`=0
converts (3.13) to the formula in [12, Theorem 1.1]:
(3.16)
βi = span
i−1
nX
s=k
o
j
Csi−1 (Hs−k
)(k) | 0 ≤ k ≤ i − 1
where Csi =
P
1≤i1 <···<is ≤i
19
πi⊥s · · · πi⊥1 .
These formulae give all harmonic maps of finite uniton number from any Riemann
surface M to U(n) as follows. For any integer r ∈ {0, 1, . . . , n − 1}, choose an
arbitrary matrix (Lji )0≤i≤r−1, 1≤j≤n or (Hij )0≤i≤r−1, 1≤j≤n of Cn -valued meromorphic
functions on M and an arbitrary element ϕ0 ∈ U(n), compute the βi in turn for
i = 1, 2, . . . , r from (3.14) or (3.16); then compute ϕ from (3.3); an associated
extended solution Φ is given by (2.11). This gives all harmonic maps and associated
extended solutions of finite uniton number at most r explicitly in terms of the
starting data using only algebraic operations.
More generally, we can use a mixture of Segal and Uhlenbeck steps to get new
formulae, as follows; this will be vital in Section 5. Let L be a meromorphic section
of H+ . By the order o(L) of L we mean the least i such that Pi L 6= 0, equivalently
b for some L
b = P λi L
bi with L
b0 non-zero.
L = λo(L) L
i≥0
Proposition 3.18. Let W be an extended solution satisfying (2.4). Let {Lj } be a
meromorphic spanning set for X; denote the order of Lj by o(j). Let Wi be obtained
from W by u Uhlenbeck steps and r − i − u Segal steps, in any order. Then
(3.17) αi = span
i−1
nX
Ssi−1 (Ljs−k+u )(k)
o
| 0 ≤ o(j) + k − u ≤ i − 1
s=0
+ span
i−1
nX
o
bj )(k) | o(j) + k − u < 0 .
Ssi−1 (L
s
s=0
Proof. We have
αi = P0 Φ−1
i−1 Wi =
i−1
X
Ssi−1 Ps (Wi ) where Wi = λ−u W ∩ H+ + λi H+ .
s=0
Now, a meromorphic spanning set for W mod λr H+ is {λ` (Lj )(k) | k ≤ ` ≤ r − 1 −
o(j)}, hence a meromorphic spanning set for W ∩λu H+ mod λ(W ∩λu H+ )+λr H+
is
{λk (Lj )(k) | u ≤ o(j) + k ≤ r − 1} ∪ {λu−o(j) (Lj )(k) | o(j) + k < u}.
It follows that a meromorphic spanning set for Wi = λ−u W ∩H+ mod λWi +λi H+
is
{λk−u (Lj )(k) | 0 ≤ o(j) + k − u ≤ i − 1} ∪ {λ−o(j) (Lj )(k) | o(j) + k − u < 0}.
bj )(k) , the proposition follows.
Noting that λ−o(j) (Lj )(k) = (L
¤
Note that (3.17) reduces to (3.13) when u = 0. We now consider the other
extreme case.
Example 3.19. Given an extended solution W satisfying (2.4), if we take the
Uhlenbeck filtration, then u = r − i, so that (3.17) reduces to the following formula
20
for the Uhlenbeck unitons.
(3.18)
i−1
X
bj )(k) | o(j) + k ≤ r − i}.
γi = span{
Ssi−1 (L
s
s=0
b = E(H)
b where E : H+ → H+ is defined above. Then, by the same
Now set L
calculations as in [12, Lemma 4.2], this gives
(3.19)
i−1
X
b j )(k) | o(j) + k ≤ r − i},
γi = span{
Csi−1 (H
s
s=0
which is equivalent to the formulae in [8]. As a specific example, suppose that r = 3
b2 . Then the formula
and that X is spanned by L1 = L10 + λL11 + λ2 L12 and L2 = λ2 L
0
(3.19) gives
b 02 , (H01 )(k) | k ≤ 2} ,
γ1 = span{H
γ2 = span{(H01 )(k) + π1⊥ (H11 )(k) | k ≤ 1} ,
γ3 = span{H01 + (π1⊥ + π2⊥ )H11 + π2⊥ π1⊥ H21 } .
These are the formulae of [8, Example 9.4].
There are many other natural factorizations for which we can give explicit formulae, we mention just one.
Example 3.20. Let W ⊂ H+ be an extended solution. For i = 0, 1, 2, . . ., let W(i)
denote the ith osculating subbundle of W (see above). From (3.1), it follows that
f = W(1) is a λ-step (see Lemma 2.1), which we shall call a Gauss step, and
W 7→ W
each W(i) is an extended solution. Suppose that P0 W is full; by Example 4.12, this
is certainly the case if Φ is the type one extended solution associated to a harmonic
map of finite uniton number. Then (P0 W )(r) = Cn for some r ≤ n; it follows that
W(r) = H+ .
For i = 0, 1, . . . , r, set Wi = W(r−i) . Then (Wi ) is a λ-filtration by extended
solutions. In particular, Φ has finite uniton number at most r; however, the uniton
number may be less than r, see below.
We claim that the corresponding unitons are given by αi⊥ = Im Aϕz i . Indeed, from
Corollary 2.4(i),
−1
αi⊥ = P0 Φ−1
i λWi−1 = P0 Φi λ(Wi )(1)
Now λ(Wi )(1) = λWi + F Wi so, on using Proposition 3.7, we obtain
−1
ϕi
n
ϕ
αi⊥ = P0 Φ−1
i F Wi = Az P0 Φi Wi = Az (C ),
as desired. The resulting factorization (3.3) is the factorization by Az -images considered by the second author [25]; for maps into Grassmannians, this reduces to the
factorization by Gauss transforms in [24]. Note that, if Φ is the type one extended
solution associated to a non-constant harmonic map ϕ : S 2 → CP n , then the uniton
number of ϕ is one or two, but r may be anything between 1 and n, and ϕi is the
(r − i)th ∂ 0 -Gauss transform of ϕ.
21
We can now give explicit formulae for the unitons in this factorization in terms
of a meromorphic spanning set Lj of X. Indeed Wi has a meromorphic spanning
set {λk (Lj )(k+`) | k, ` ∈ N, 0 ≤ o(j) + k ≤ i − 1, 0 ≤ ` ≤ r − i} mod λWi + λi H+ ,
so that Corollary 2.5 gives
αi = span
i−1
©X
ª
Ssi−1 (Ljs−k )(k+`) | k, ` ∈ N, 0 ≤ o(j) + k ≤ i − 1, 0 ≤ ` ≤ r − i .
s=k
3.5. Complex extended solutions and an explicit Iwasawa factorization.
For a compact Lie subgroup G of U(n) with complexification Gc , let ΛGc = {γ :
S 1 → Gc | γ is smooth}. With notation as in [5], let Λ+ Gc (resp. Λ∗ Gc ) denote the
subgroup of ΛGc consisting of maps S 1 → G which extend holomorphically to the
region {λ ∈ C | |λ| < 1} (resp. {λ ∈ C | 0 < |λ| < 1}).
Then by a complex extended solution Ψ : M → ΛGc we mean a holomorphic map
M → Λ∗ Gc which satisfies
λΨ−1 Ψz ∈ Λ+ gc
where Λ+ gc is the Lie algebra of Λ+ Gc .
Set W = ΨH+ for some holomorphic map Ψ : M → ΛGc , then W is an extended
solution if and only if Ψ is a complex extended solution. Explicitly, {F1 , . . . , Fn } is
a meromorphic basis for W mod λW if and only if the matrix Ψ with columns Fi
is a complex extended solution.
Now, recall the Iwasawa decomposition: ΛGc = ΩG·Λ+ Gc and ΩG∩Λ+ Gc = {e},
i.e., any γ ∈ ΛGc can be written uniquely as γ = γu γ+ where γu ∈ ΩG and
γ+ ∈ Λ+ Gc . Then given a complex extended solution Ψ : M → ΛGc , setting
Φ = Ψu gives an extended solution and all extended solutions arise this way, at
least locally, see [11]. We can find Φ from Ψ explicitly as follows: The columns of
Ψ give a holomorphic basis for W mod λW ; use any of the formulae in §3.4 to
obtain unitons αi , then Φ is given by (2.11). Note that if Ψ is polynomial, then this
gives Φ polynomial of the same or lesser degree.
3.6. Relationship with work of Burstall and Guest. As in the last section,
let W be an extended solution which satisfies (2.4); recall that dim W/λW = n.
Choose a meromorphic spanning set {F1 , . . . , Fn } for W mod λW adapted to the
filtration (3.9) of W/λW . Recalling the associated decomposition (3.10) and the
notation ni = the rank of Ai , we see that the first nr of the Fi , i.e., F1 , . . . , Fnr , have
order r, the next nr−1 of the Fi , i.e., Fnr +1 , . . . , Fnr +nr−1 , have order r − 1, etc. In
P
P
general, for each j ∈ {0, 1, . . . , r} , when r`=j+1 n` + 1 ≤ i ≤ r`=j n` we have (i)
o(Fi ) = j, (ii) Fi = λj Fbi for some non-zero meromorphic section Fbi of H+ of order
zero, (iii) under the isomorphism W/λW ∼
= W ª λW , these Fi give a basis for Aj .
22
We use this data to form the following n × n matrices:
¡
¢
Ψ = F1 · · · Fnr |Fnr +1 · · · Fnr +nr−1 | · · · |Fn−n0 +1 · · · Fn ,
¡
¢
(3.20)
A = Fb1 · · · Fbnr |Fbnr +1 · · · Fbnr +nr−1 | · · · |Fbn−n0 +1 · · · Fbn ,
Λ = diagonal(λr , . . . , λr , λr−1 , . . . , λr−1 , . . . , 1, . . . , 1).
| {z } |
{z
}
| {z }
nr
nr−1
n0
Then Ψ = AΛ and W = ΨH+ = AΛH+ . Note that Ψ is invertible away from a
discrete set D since its columns give a basis for W/λW except on the discrete set
where the Fi have poles or the determinant of W is zero. Further, Ψ is a complex
extended solution on M \ D. Note that the columns of all such matrices A can be
given explicitly in terms of a spanning set for X as in the last subsection.
Lemma 3.21. A : M → Λ+ GLn (C) away from a discrete set.
Proof. Since Λ is invertible for λ ∈ S 1 , A is invertible if and only if Ψ is. Thus,
away from some discrete subset, A takes values in ΛGLn (C). That A takes values
in Λ+ GLn (C), can be easily deduced from [5, Proposition 4.1].
¤
S
Example 3.22. (i) We show that Wr−1
= AΛSr−1 H+ , where
ΛSr−1 = diagonal(λr−1 , . . . , λr−1 , λr−2 , . . . , λr−2 , . . . , 1, . . . , 1).
{z
} |
{z
}
|
| {z }
nr +nr−1
nr−2
n0
S
Indeed, since A−1 H+ = H+ away from a discrete set, we have Wr−1
= W +λr−1 H+ =
AΛ(H+ + λr−1 Λ−1 H+ ), and it is easily seen that H+ + λr−1 Λ−1 H+ = Λ−1 ΛSr−1 H+ .
U
(ii) Similarly, it can be shown that Wr−1
= AΛUr−1 H+ , where
ΛUr−1 = diagonal(λr−1 , . . . , λr−1 , λr−2 , . . . , λr−2 , . . . , 1, . . . , 1).
{z
} |
{z
}
| {z }
|
nr
nr−1
n1 +n0
Remark 3.23. In [5], the authors obtain all harmonic maps from the 2-sphere by
deforming them to S 1 -invariant maps. It can easily be seen that this S 1 -invariant
map Φ0 has complex extended solution Ψ0 = A0 Λ where A0 is made up of the
leading terms of A, i.e.,
¡
¢
A0 = P0 (Fb1 ) · · · P0 (Fbnr )|P0 (Fbnr +1 ) · · · P0 (Fbnr +nr−1 )| · · · |P0 (Fbn−n0 +1 ) · · · P0 (Fbn ) .
See also §4.3.
4. Maps into complex Grassmannians and J2 -holomorphic lifts
4.1. Harmonic maps into complex Grassmannians. It is well known (see [5])
that any compact connected inner symmetric space can be immersed in a Lie group
G as a component of
√
e = {g ∈ G | g 2 = e},
23
√
and the immersion is totally geodesic. For example, when G = U(n), then e =
{g ∈ G | g 2 = e} is the disjoint union G∗ (Cn ) of the complex Grassmannians Gk (Cn )
k ∈ {0, 1, . . . , n}. For each k, we have the totally geodesic Cartan embedding
(4.1)
ι : Gk (Cn ) → U(n),
ι(V ) = πV − πV ⊥ .
Consider the involution on ΩU(n) given by I(η)(λ) = η(−λ)η(−1)−1 and write
ΩU(n)I = {η ∈ ΩU(n) | I(η) = η}.
Then an extended solution Φ : M → ΩU(n) lies in ΩU(n)I if and only if it satisfies
the symmetry condition
(4.2)
Φλ Φ−1 = Φ−λ .
Clearly, if an extended solution Φ satisfies this condition, the harmonic map Φ−1
satisfies Φ−12 = I, and so takes values in G∗ (Cn ). Conversely [5], given a harmonic
map ϕ : M → G∗ (Cn ) which has an associated extended solution Φ : M → ΩU(n)
e : M → ΩU(n)I with
with ϕ = Φ−1 , then there is also an extended solution Φ
e −1 = ϕ. Furthermore, if ϕ has finite uniton number, then Φ
e can also be chosen
Φ
to be polynomial; indeed, by [23, Lemma 15.1] the type one extended solution Φ
associated to ϕ satisfies Φλ = QΦ−λ Φ−1
−1 Q for some Q = πV − πV ⊥ , and Φ−1 = Qϕ.
e
Setting Qλ = πV + λπV ⊥ , we see that Φλ = Qλ Φλ is a polynomial extended solution
e −1 = ϕ and satisfies the symmetry condition (4.2), i.e., is a map Φ
e :
which has Φ
M → ΩU(n)I .
P
Let ν : H+ → H+ be the involution induced by λ 7→ −λ; thus if T = i Ti λi
P
then ν(T ) = i (−1)i Ti λi . This induces the involution ν : Gr(n) → Gr(n) given by
Wλ 7→ W−λ . Under the identification of ΩU(n) with Gr(n) , this corresponds to the
(n)
involution I on ΩU(n). Denote by Grν the fixed point set of ν.
Most of the following is in [23, Theorem 15.3].
(n)
Lemma 4.1. Let W = ΦH+ ∈ Grν be an extended solution satisfying (2.3), and
f = ΦH
e + by Φ
e = Φ(πα + λ−1 π ⊥ ) for some subbundle α of Cn . Then
define W
α
(n)
f
e −1 ;
(i) W ∈ Grν if and only if πα commutes with Φ−1 , equivalently with Φ
f by a Segal or Uhlenbeck step
(ii) this condition holds if W is obtained from W
(§2.2), or a Gauss step (Example 3.20).
¤
Thus any of the factorizations in §3.4 give sequences of extended harmonic maps
Φi in ΩU(n)I .
Conversely, starting with data Lj with each polynomial Lj having only even
or odd powers of λ, we obtain all harmonic maps into a complex Grassmannian
explicitly.
Suppose that ϕ : M → G∗ (Cn ) ⊂ U(n) is a harmonic map with an associated
(n)
polynomial extended solution Φ : M → ΩU(n)I , so that W = ΦH+ ∈ Grν . Recall
that Φ−1 restricts to an isomorphism from W ª λW to Cn ; clearly ν restricts to an
24
involution on W ª λW , and by (4.2), we have a commutative diagram
ν-
W ª λW
Φ−1
W ª λW
Φ−1
?
Cn
Φ−1
?
- Cn
so that the involution ν on W ª λW corresponds to the involution Φ−1 = ι(ϕ) on
Cn . As before, let
Ai = (W ∩ λi H+ ) ª ((λW ) ∩ λi H+ + W ∩ λi+1 H+ ) ∼
=
Pi (W ∩ λi H+ )
.
Pi−1 (W ∩ λi−1 H+ )
Considering Ai as a subbundle of W ª λW , it is clear that ν maps each Ai to itself,
and hence the maps
(
1
(x + ν(x)) (i even),
Ai 3 x 7→ 21
(x − ν(x)) (i odd)
2
give isomorphisms on each
a basis for Ai in the image of this
¯ Ai . By choosing
i
map, we conclude that ν ¯Ai = (−1) . We also see that, under the isomorphism
Φ : Cn → W ª λW , the decomposition
X
X
W ª λW ∼
A2i ) ⊕ (
A2i−1 )
=(
i
i
n
corresponds to the decomposition of C into the (±1)-eigenspaces of the involution
Φ−1 = ι(ϕ) = (πϕ − πϕ⊥ ).
(n)
Lemma 4.2. Suppose that W : M → Grν is an extended solution satisfying (2.4).
If Ai = 0 for some i ∈ {1, . . . , r}, then there is a γ ∈ ΩU(n)I with γ(−1) = I such
that λr−1 H+ ⊂ γW ⊂ H+ .
Proof. Set V = λ1−i (W ∩ λi−1 H+ ) + λ2 H+ . Then V is holomorphic, λ-closed and
(n)
defines a map V : M → Grν . Furthermore, λr−1 V ⊂ W ⊂ V .
As V is invariant under ν, we have a decomposition V = V+ ⊕ V− where V± are
the (±1)-eigenspaces of ν|V . Since Pi−1 W ∩ λi−1 H+ = Pi W ∩ λi H+ , it follows easily
that λV+ = V− . We also have
Vz = λ1−i (Wz ∩ λi−1 H+ ) + λ2 H+ ⊂ λ−i ((λW ) ∩ λi H+ ) + λ−i (W ∩ λi+1 H+ ) + λ2 H+
⊂ λ1−i (W ∩ λi−1 H+ ) + λ−i (W ∩ λi−1 H+ ) + λ2 H+ ⊂ λ−1 V.
Hence (V+ )z ⊂ λ−1 V− = V+ and (V− )z = λ(V+ )z ⊂ V− , so Vz ⊂ V , and hence V
is constant, so that V = γH+ for some γ ∈ ΩU(n)I . Then λr−1 H+ ⊂ γW ⊂ H+ .
Further, since (V ª λV )odd = V− ª λV+ = 0, we get γ(−1) = I.
¤
Note that if A0 = 0 then λr−1 H+ ⊂ λ−1 W ⊂ H+ . By repeating this and applying
the lemma above, we obtain the following result.
25
(n)
Proposition 4.3. Given an extended solution W : M → Grν with λr H+ ⊂ W ⊂
H+ , then there is an integer s with 0 ≤ s ≤ r, and η ∈ ΩU(n)I with η(−1) = ±I
(n)
and λs H+ ⊂ ηW ⊂ H+ , such that ηW : M → Grν is normalized.
¤
Corollary 4.4. Let ϕ : M → Gk (Cn ) be a harmonic map into a complex Grasmannian which is of finite uniton number as a map into U(n). Then there is a
polynomial extended solution Φ : M → ΩU(n)I of degree r ≤ 2 min{k, n − k}, with
Φ−1 = ±ι(ϕ). If 2k = n, then Φ can be chosen of degree r ≤ 2k − 1.
e be any
Proof. Without loss of generality, we may assume that k ≤ n − k. Let Φ
e =
algebraic extended solution associated to ϕ. Fix a point z0 ∈ M and set Θ
e 0 )−1 Φ.
e Then clearly, Θ
e is still algebraic. Let ϕ(z0 ) = V0 and Qλ = πV0 + λπV ⊥ .
Φ(z
0
e satisfies Θ−1 (z0 ) = ϕ(z0 ), so by uniqueness, Θ−1 = ϕ. Furthermore,
Then Θ = QΘ
since Q = Θ(z0 ) ∈ ΩU(n)I , it follows again by uniqueness that Θ : M → ΩU(n)I .
Finally, let Φ = λj Θ, where j is chosen to make Φ is polynomial of degree, say,
r. By the above proposition we may assume that W = ΦH+ is normalized; this
implies that r ≤ 2k and, when 2k = n, that r ≤ 2k − 1.
¤
Remark 4.5. (i) The corollary applies any harmonic map from a compact Riemann
surface which has an extended solution since, as noted above, this is of finite uniton
e the extended solution
number. Indeed for any associated extended solution Φ,
e = Φ(z
e 0 )−1 Φ
e satisfies Θ(z
e 0 ) = I, and so is algebraic by [18].
Θ
(ii) This bound on the minimal uniton number was originally conjectured by
Uhlenbeck [23], and later proved using different methods by Dong and Shen [10].
4.2. J2 -holomorphic lifts. We begin this section by reviewing some facts from
the twistor theory of harmonic maps. Let N be a Riemannian manifold of even
dimension, say 2n, and let J(N ) denote its bundle of almost Hermitian structures
with fibre at a point q ∈ N given by
J(N )q = {J ∈ End(Tq N ) | J is an isometry and J 2 = −I}.
Then J(N ) is associated to the orthogonal frame bundle of N with fibre the Hermitian symmetric space O(2n)/U(n), so it inherits a metric for which the projection π : J(N ) → N is Riemannian submersion. We define two almost Hermitian
structures J1 and J2 on J(N ) as follows. The Levi-Civita connection gives a decomposition of the tangent bundle T J(N ) = H ⊕ V into horizontal and vertical
subbundles; the vertical spaces are tangent to the fibres of π and so inherit a canonical almost Hermitian structure J V ; whilst the horizontal spaces are given the almost
Hermitian structure J H by lifting that from the tangent spaces of N . We then set
J1 = (J H , J V ) and J2 = (J H , −J V ). Whilst J1 is integrable if and only if N is
conformally flat, J2 is never integrable, see for example [9]. Nevertheless, the projection π : (J(N ), J2 ) → N is a twistor fibration, in the sense that, for any Riemann
surface (or, more generally, cosymplectic manifold) M and any holomorphic map
26
ψ : M → (J(N ), J2 ), the map ϕ = π ◦ ψ : M → N is harmonic. In this case, ψ is
said to be a J2 -holomorphic lift of ϕ.
It is proved in [6] that, when N = G/K is a simply connected inner Riemannian symmetric space, any flag manifold of G can be realized as a submanifold of
J(N ), invariant under both J1 and J2 . In particular, flag manifolds of U(n) are
twistor fibrations of Grassmannian manifolds. In fact, it is known from [4] that any
harmonic map from S 2 into a Grassmannian Gk (Cn ) has a J2 -holomorphic lift into
some flag manifold of U(n). In this section, we shall show how to construct such
J2 -holomorphic lifts explicitly from their extended solutions W .
Let ϕ : M → U(n) be harmonic of finite uniton number, and let W = ΦH+ for
some polynomial extended solution Φ associated to ϕ of degree r. Suppose that we
have a filtration of W by λ-closed subbundles:
(4.3)
W = Y0 ⊃ Y1 ⊃ · · · ⊃ Ys ⊃ Ys+1
with Ys+1 ⊂ λW . Then, denoting by pr : W → W/λW the natural projection,
and recalling from Remark 3.8 that P0 ◦ Φ−1 = Φ−1 ◦ pr, we obtain the following
commutative diagram:
W
pr
?
= Y0 ⊃
pr
W/λW = Ye0 ⊃
Φ−1
?
Y1
?
Cn = Z0 ⊃ Z1
Ys
pr
?
Ye1
Φ−1
⊃ ... ⊃
⊃ ... ⊃
pr
?
Yes
Φ−1
⊃ Ys+1
?
⊃ . . . ⊃ Zs
⊃
?
0
Φ−1
⊃
?
0
where
Zi = Φ−1 Yei = P0 ◦ Φ−1 Yi .
Call (Yi ) an F -filtration if, for each i, Yi holomorphic, i.e., closed under ∂z¯, and
Fi maps Yi into the smaller subbundle Yi+1 ; similarly for (Yei ). Call (Zi ) an Aϕz filtration if, for each i, Zi holomorphic, i.e., closed under Dzϕ¯ , and Aϕz maps Zi into
the smaller subbundle Zi+1 .
From Proposition 3.7 we see that, (i) if (Yi ) is an F -filtration, then so is (Yei ); (ii)
(Yei ) is an F -filtration if and only if (Zi ) is an Aϕz -filtration.
We now see how such sequences give rise to J2 -holomorphic maps.
Proposition 4.6. Let ϕ : M → G∗ (Cn ) be a harmonic map into a Grassmannian
and let Φ : M → ΩU(n)I be an extended solution with Φ−1 = ϕ. Let (Yi ) be a finite
Aϕz -sequence which is invariant under ν. Then (Yi ) defines a J2 -holomorphic lift of
ϕ into a flag manifold.
Proof. We saw above that the involution ν : λ 7→ −λ on H+ restricts to an involution
on W and, via Φ−1 , this corresponds to the involution Φ−1 = πϕ − πϕ⊥ on Cn .
Thus, each Zi splits, i.e., is the direct sum of subbundles Vi and Wi with Vi ⊂ ϕ
27
and Wi ⊂ ϕ⊥ , if and only if each Yei is invariant under ν, and this is certainly the
case if each Yi is invariant under ν.
Then the Vi (resp. Wi ) give a holomorphic filtration of ϕ (resp. ϕ⊥ ), and Aϕz
maps Vi into Wi+1 and Wi into Vi+1 . Thus (i) ϕ = V0 ⊃ V2 · · · and ϕ⊥ ⊃ W1 ⊃ W3
and (ii) ϕ⊥ = W0 ⊃ W2 ⊃ . . . and ϕ ⊃ V1 ⊃ V3 · · · give a filtrations of ϕ and ϕ⊥
P
(which could be the same). In case (i), set ψi = Vi−2 ª Vi so that ϕ = i even ψi ,
P
and ϕ¡⊥ =
ψ = (ψ1 , . . . , ψs ) defines a map into a flag manifold
i odd ψi then ¢
U(n)/ U(k1 ) × · · · × U(ks ) where ki is the rank of ψi (which may be zero; we
interpret U(0) as {±I}); we call the ψi the legs of ψ. Further, by [4, Lemma, §2],
this map is holomorphic with respect to the non-integrable complex structure J2 on
the flag manifold, and so is a J2 -holomorphic lift of ϕ.
¤
In the following examples, ϕ : M → G∗ (Cn ) is a harmonic map into a Grassmannian, Φ : M → ΩU(n)I is an extended solution with Φ−1 = ϕ and W = ΦH+ .
Example 4.7. Let W satisfy (2.4). Set Yi = W ∩ λi H+ ; this gives the filtration
U
U
(3.8). Note that Yi = λi Wr−i
where (Wr−i
) is the Uhlenbeck filtration (2.17) of W ,
and s = r. Clearly F (Yi ) ⊂ Yi+1 so that we obtain an F -filtration (4.3) with s = r.
Note that, by Proposition 3.14, we may assume that W is normalized, which implies
that the inclusion Zi+1 ⊂ Zi is strict for all i. In any case, each Yi is invariant under
I, and hence Zi splits. In fact, we can calculate Zi as follows.
i
U
Zi = P0 Φ−1 (Yi ) = P0 (πγr + λ−1 πγr⊥ ) · · · (πγr−i+1 + λ−1 πγr−i+1
)Φ−1
⊥
r−i λ Wr−i
= P0 λi (πγr + λ−1 πγr⊥ ) · · · (πγr−i+1 + λ−1 πγr−i+1
)H+
⊥
(Cn ).
= πγr⊥ · · · πγr−i+1
⊥
In particular, Z1 = γr⊥ where γr is the last uniton in the Uhlenbeck factorization.
Note that this may be applied to W = ΦH+ where Φ is the type one extended
solution associated to a harmonic map M → G∗ (Cn ) of finite uniton number; as
remarked above, this is already normalized.
Example 4.8. Set Y0 = W and Yi = F Yi−1 (i = 1, 2, . . .); then Zi = (Aϕz )i Cn . If
ϕ is Grassmannian, this clearly splits and gives the filtration due to F. Burstall [4,
Section 3]. This filtration is finite provided Aϕz is nilpotent; such maps are called
nilconformal in [4]. For such a map, Proposition 4.6 gives a J2 -holomorphic lift.
Note that any harmonic map of finite uniton number, say r, is nilconformal since
Yr+1 ∈ λr+1 H+ ⊂ λW . However, there are harmonic maps which are nilconformal
but not of finite √
uniton number, for example the isometric minimal immersion of
the torus C/h2π/ 3, 2πii into CP 2 given by
£
2
2 ¤
(4.4)
z 7→ ez−z , eζz−ζz , eζ z−ζ z ,
where ζ = e2πi/3 , is superconformal and so of finite type, see [3, Corollary 2.7]; such
a map cannot be of finite uniton number by a result of Pacheco [20].
28
U
Example 4.9. Consider, instead the F -filtration Y0 = W Y1 = W ∩ λH+ = λWr−1
and Yi+1 = F Yi , i = 1, 2, . . .. Again, Z1 = γr⊥ where γr is the last uniton in the
Uhlenbeck factorization, and the subsequent Zi are successive images under Aϕz .
Then, if ϕ is Grassmannian, all Zi split. If the sequence is finite, certainly the case
if ϕ is nilconformal, we obtain a J2 -holomorphic lift of ϕ with γr appearing as one
leg.
More generally, we can take Z1 to be any antibasic uniton α⊥ for ϕ, and then
Zi = (Aϕz )i−1 Z1 . If ϕ is Grassmannian and we insist that α commute with ϕ, then
again we obtain a J2 -holomorphic lift of ϕ with α⊥ appearing as one leg.
Example 4.10. Let Yi = F i W + λr H+ , (i = 0, 1, 2, . . . , r) and Yr+1 = λr+1 H+ .
Note that Yr = λr H+ and Yr+1 ⊂ λW . By Proposition 3.7(iii), we have for i ≤ r,
Zi = P0 Φ−1 (F i W ) + P0 Φ−1 (λr H+ ) = (Aϕz )i (Cn ) + βr⊥ = (Aϕz )i (βr ) + βr⊥ ,
where βr is the last uniton in the Segal factorization of Φ, and Zr+1 = 0. Note that
Zr = βr⊥ . It follows that Yi is an F -filtration; if ϕ is Grassmannian, then since Yi
is invariant under I, Zi splits. More generally, we can take α⊥ be any basic uniton
for ϕ, equivalently, α is an antibasic uniton for ϕ
e = ϕ(πα − πα⊥ ). For i = 0, 1, . . . set
ϕ
e i
⊥
(α)(i) = (Az ) (α) and set Zi = (α)(i) + α . Then Zi is an Az -filtration. Indeed α
is closed under Aϕze, and, by the formula Aϕz = Aϕze − ∂πα [23, Theorem 12.6], we see
that Aϕz maps Zi into Zi+1 .
If ϕ is Grassmannian and α commutes with it, then each Zi splits. If the sequence
is finite — as before this holds if ϕ is nilconformal — we get a J2 -holomorphic lift
of ϕ.
4.3. S 1 -invariant maps and superhorizontal lifts. An important special case
of the above constructions is when the unitons are nested. We saw in §2.3 that,
algebraically, this corresponds to maps Φ invariant under the S 1 -action. In fact,
the sequence of unitons satisfies the following property.
Definition 4.11. Let δ0 ⊂ δ1 ⊂ · · · be a nested sequence of subbundles of a trivial
bundle V . Say that the sequence is superhorizontal if
(i) each subbundle is holomorphic with respect to the standard complex structure;
i.e. ∂z¯σ ∈ Γ(δi ) for all σ ∈ Γ(δi );
(ii) the operator ∂z maps smooth sections of δi into sections of δi+1 , i.e., ∂z σ ∈
Γ(δi+1 ) for all σ ∈ Γ(δi ).
A superhorizontal sequence is equivalent to a superhorizontal holomorphic map
from M to a flag manifold of U(n), see [6, Chapter 4].
Example 4.12. Let r ∈ N and let Φ : M → ΩU(n) be a polynomial extended
solution of degree at most r; set W = ΦH+ . Let
δi = Pi (W ∩ λi H+ ) = P0 (λ−i W ∩ H+ )
(i = 0, . . . , r) ,
29
so that we have a filtration
0 ⊂ δ0 ⊂ δ1 ⊂ · · · ⊂ δr = Cn .
(4.5)
U
Note that δi is the first Segal uniton of Wr−i
; in particular, it is a holomorphic
n
subbundle of C . Further, on setting δ−1 = 0, we have δi /δi−1 ∼
= Ai (i = 0, 1, . . . , r) .
Note also that the sequence (4.5) is superhorizontal; in particular, if δi = δi+1 for
some i, then δi is constant, In fact, this is the sequence of Segal unitons for the
S 1 -invariant map Φ0 in Remark 3.23.
Using this sequence, we see again that, if Φ is of type one, then it is normalized.
Indeed, if Ai = 0 for some i > 0, then δi−1 is constant, in which case none of the δj
with j < i are full, so that W is not of type one.
For the next result we recall from Proposition 2.11 that given an S 1 -invariant
polynomial extended solution Φ of some degree r, the Segal unitons β1 , . . . , βr are
nested, in the sense that βi ⊂ βi+1 for all i, and the Uhlenbeck unitons γ1 , . . . , γr
are given by γi = βr+1−i .
Proposition 4.13. Let r ∈ N and let Φ : M → ΩU(n) be an S 1 -invariant polynomial extended solution of degree at most r; set W = ΦH+ . Write Φ−1 = ϕ. Let
β1 , . . . , βr be the corresponding Segal unitons. Then,
(i) the sequence β1 ⊂ · · · ⊂ βr is superhorizontal (Definition 4.11);
(ii) the Uhlenbeck unitons satisfy γi = βr+1−i ;
(iii) Φ satisfies the symmetry condition (4.2) so that ϕ maps into a Grassmannian,
and the following formula gives ϕ if r is odd or ϕ⊥ if r is even:
[(r−1)/2]
(4.6)
X
⊥
⊥
⊥
βr−1−2k
∩ βr−2k = βr−1
∩ βr ⊕ βr−3
∩ βr−2 ⊕ · · · ,
k=0
where we set β0 equal to the zero subbundle;
⊥
(iv) for the filtration (Yi ) of Example 4.7, we have Zi = P0 Φ−1 (Yi ) = γr−i+1
= βi⊥ ,
further, the sequence Cn ⊃ β1⊥ ⊃ · · · ⊃ βr⊥ ⊃ 0 is an Aϕz -sequence.
Proof. (i) This follows from Proposition 2.11(iii).
(ii) and (iii) This follows from Proposition 2.11.
(iv) From Example 4.7 and the nesting, we have Zi = πγr⊥ · · · πγr−i+1
(Cn ) =
⊥
⊥
¤
= βi⊥ .
γr−i+1
Remark 4.14. Superhorizontality of (βi ) is equivalent to the condition that the
sequence Cn ⊃ β1⊥ ⊃ · · · ⊃ βr⊥ ⊃ 0 be superhorizontal with respect to the conjugate
complex structure. This does not contradict the fact that (βi⊥ ) is an Aϕz -sequence
with respect to the original complex structure; indeed, the two properties of the
sequence (βi⊥ ) are equivalent as follows: that βi⊥ is holomorphic with respect to the
⊥
∩ βi is either in
conjugate complex structure means ∂z βi⊥ lies in βi⊥ , and since βi−1
⊥
⊥
⊥
ϕ
ϕ or in ϕ , this implies that Az maps βi to βi+1 . The converse is similar, as is the
equivalence of the ∂z¯ conditions.
30
4.4. Explicit formulae for J2 -holomorphic lifts. Explicit formulae for the J2 holomorphic lifts can be given for all the examples in this section; it suffices to
find a meromorphic spanning set for each Yi in terms of W , and so in terms of
a meromorphic spanning set for the freely chosen holomorphic subbundle X; we
do this for the first two examples, the others are similar. As before, let {Lj } be
a meromorphic spanning set for X. Then a meromorphic spanning set for W
mod λW , is {λk (Lj )(k) : 0 ≤ k ≤ r} (note that we now require k ≤ r, rather than
k ≤ r − 1 which we used when we were working mod λW + λr W ).
For Example 4.7, a meromorphic spanning set for Yi mod λW is given by
k
λ (Lj )(k) with i ≤ o(j) + k ≤ r. Hence a spanning set for Zi by meromorphic
sections of (Cn , Dzϕ¯ ) is given by
Zi = span
r
©X
ª
(k)
Ssr Ls−k,j | i ≤ o(j) + k ≤ r .
s=k
For Example 4.8, a meromorphic spanning set for Yi mod λYi is given by
λ (Lj )(k) with i ≤ k ≤ r. Hence a spanning set for Zi by meromorphic sections
of (Cn , Dzϕ¯ ) is given by
k
Zi = span
r
©X
ª
(k)
Ssr Ls−k,j | i ≤ k ≤ r .
s=k
5. Harmonic maps into SO(n), Sp(n) and their inner symmetric spaces
5.1. Harmonic maps into SO(n). As usual, let M be any Riemann surface. We
consider SO(n) as a subgroup of U(n) and look for uniton factorizations which give
harmonic maps from M into this subgroup.
Definition 5.1. Let W ∈ Gr(n) and let i ∈ N. We say that W is real (of degree i)
⊥
if it satisfies (2.3) and W = λ1−i W .
Writing W = ΦH+ for some Φ ∈ ΩU(n), then W is real if and only if Φ = λ−i Φ.
If i = 2j, then this condition is equivalent to Ψ = λ−j Φ being real in the usual
sense, i.e., Ψ ∈ ΩSO(n), equivalently,
Ψ=
j
X
λ ` T`
with T−` = T`
∀` .
`=−j
Note that this implies that Ψ−1 = ±Φ−1 is a harmonic map into SO(n).
Now let Wi ∈ Gr(n) be real of some degree i ≥ 2. Consider the subspace given by
Wi−2 = (λ−1 Wi ∩ H+ ) + λi−2 H+ .
Thus Wi−2 is obtained from Wi by first doing an Uhlenbeck step and then a Segal
step; since since these two operations commute (see §2.2), we could equally well
first do a Segal step and then an Uhlenbeck step.
31
Remark 5.2. Write Wi−2 = Φi (πγ + λ−1 πγ ⊥ )(πβ + λ−1 πβ ⊥ )H+ for two unitons β, γ
with γ Uhlenbeck. Then β ⊥ and γ are isotropic.
U
U
To see this, let Wi−1
and Wi−2
be obtained from Wi by one and two Uhlenbeck steps, respectively, and denote by γi and γi−1 the corresponding Uhlenbeck
U
unitons. Then γi = γ and since by Proposition 2.6, Wi−2 ⊂ Wi−2
, we obtain
−1
−1
(πβ + λ πβ ⊥ )H+ ⊂ (πγi−1 + λ πγi−1
Thus
⊥ )H+ , which implies that γi−1 ⊂ β.
γ = πγ (γi−1 ) = πγ (β).
Since both Φi and Φi−2 are real, the factor (πβ + λπβ ⊥ )(λ−1 πγ + πγ ⊥ ) takes values
in ΩSO(n). Hence πβ πγ = πβ ⊥ πγ ⊥ ; taking images we get β ⊥ = πβ (γ) ⊂ β, and
hence β ⊥ is isotropic. On the other hand, taking the adjoint gives πγ πβ = πγ ⊥ πβ ⊥ ;
hence γ = πγ ⊥ (β ⊥ ) ⊂ γ ⊥ , and thus γ is also isotropic.
We would get similar results if we did the Segal step first.
Proposition 5.3. If Wi is real, then so is Wi−2 . Further, if Wi is an extended
solution, then so is Wi−2 .
Proof. The first statement follows from:
⊥
⊥
W i−2 = (λW + λH+ ) ∩ λ3−i H+ = λ2−i W ∩ λ3−i H+ + λH+ = λ3−i Wi−2 .
The second statement is clear.
¤
When we start with W of even degree r, iterating this process leads to a factorization of a real algebraic extended solution into real quadratic factors, as follows.
Proposition 5.4. Let r = 2s where s ∈ N. Any extended solution Ψ : M →
ΩSO(n) of the form
(5.1)
Ψ=
s
X
λ ` T`
(T−` = T`
∀`)
`=−s
−s
has a factorization Ψ = λ η1 · · · ηr with ηi = παi + λπα⊥i where
(i) each quadratic subfactor λ−1 η2j−1 η2j takes values in ΩSO(n). In particular,
the ‘even’ partial product λ−j η1 · · · η2j takes values in ΩSO(n) (j = 1, . . . , s).
(ii) each partial product η1 · · · ηi : M → ΩU(n) (i = 1, . . . , r) is an extended
solution.
Note that (ii) is equivalent to saying that each αi is a U(n)-uniton for the previous
partial product η1 · · · ηi−1 .
Lemma 5.5. Let ϕ : M → SO(n) be a harmonic map of uniton number at most s
as a map into U(n). Then ϕ has an associated extended solution Ψ : M → SO(n)
of the form (5.1).
e : M → ΩU(n) be a polynomial extended solution of degree at most
Proof. Let Φ
e Then Ψ(z0 ) = I, and
e 0 )−1 Φ.
s associated to ϕ. Choose z0 ∈ M and set Ψ = Φ(z
Ψ = Ψ by uniqueness. It follows that Ψ is of the form (5.1).
¤
32
In particular, Proposition 5.4 gives all harmonic maps from a Riemann surface of
finite uniton number into SO(n), see §5.4.
5.2. Harmonic maps into real Grassmannians. The Cartan embedding (4.1)
restricts to an identification of the union G∗ (Rn ) = ∪k Gk (Rn ) of real Grassmannians
with the totally geodesic submanifold {g ∈ U(n) : g 2 = I and g = g} = {g ∈ U(n) :
g = g ∗ = g}. Recall from §4.1 the involutions I and ν on ΩU(n) and Gr(n) , and their
(n)
fixed point sets ΩU(n)I and Grν . Clearly I restricts to an involution of ΩSO(n);
denote its fixed point set by ΩSO(n)I = ΩSO(n) ∩ ΩU(n)I .
Let r = 2s for some s ∈ N and let W = λs ΨH+ : M → Gr(n) be an extended
solution where Ψ : M → ΩU(n). Then W is real of degree r if and only if Ψ maps
M into ΩSO(n)I . In this case, Ψ−1 is a harmonic map into a real Grassmannian
G∗ (Rn ).
By Lemma 4.1, Proposition 5.4 gives a uniton factorization of any algebraic extended solution M → ΩSO(n)I , with each partial product an extended solution
η1 · · · ηi : M → ΩU(n)I , and each even partial product η1 · · · η2j an extended solution M → ΩSO(n)I .
To apply this to harmonic maps we need the following result.
Lemma 5.6. Let ϕ : M → Gk (Rn ) be a harmonic map to a real Grassmannian,
which is of finite uniton number as a map into U(n). Then ϕ has an extended
solution Ψ : M → ΩSO(n)I of the form (5.1) for some s ∈ N, and with Ψ−1 = ϕ.
Proof. If k is odd, embed Gk (Rn ) in Gk+1 (Rn+1 ). Thus we can assume that k is
even. Let z0 ∈ M and write ϕ(z0 ) = δ + δ, where δ ⊂ Cn is an isotropic subspace.
Set
QRλ = λ−1 πδ + π(δ+δ)⊥ + λπδ ∈ ΩSO(n)I ;
then QR−1 = ϕ(z0 ). There is a unique extended solution Ψ : M → ΩU(n) associated
to ϕ with initial condition Ψ(z0 ) = QR . Since QR is algebraic, by Remark 3.4, so
is Ψ. By uniqueness, Ψ = Ψ and Ψλ = Ψ−λ Ψ−1
−1 , i.e., Ψ : M → ΩSO(n)I . Now
Ψ−1 = gϕ for some g ∈ U(n). Evaluating at z0 shows that g = I, i.e., Ψ−1 = ϕ. ¤
Example 5.7. Let W = λs ΨH+ is an extended solution with Ψ : M → ΩSO(n),
so that ϕ = Ψ−1 is Grassmannian.
(i) Suppose that r = 2s with s = 1. By Remark 5.2, we have γ = πγ (β), but
since ϕ is Grassmannian, γ must be a direct sum of a subbundle of β and of β ⊥ .
This implies that γ ⊂ β. Thus Ψ is S 1 -invariant and β ⊥ = γ. Note that γ and β
are holomorphic and γ differentiates into β. We have Ψ−1 = ι(γ + γ) and, when
the rank of β is one more than that of γ, this construction gives all harmonic maps
from S 2 to real projective n − 1-space or S n−1 .
(ii) Suppose instead that Φ = λs Ψ is S 1 -invariant with r = 2s arbitrary. Then
we know from Proposition 2.11 that γr ⊂ γr−1 , so that γ ⊂ β, and hence from the
above calculation we get γ = πγ ⊥ (β ⊥ ) = β ⊥ . Furthermore, by Proposition 2.11(iv),
it follows easily that Φr−2 is also S 1 -invariant, from which we see that Φr−2 and
33
(πβ + λπβ ⊥ )(πγ + λπγ ⊥ ) commute. Continuing this argument, we see that (i) each
pair η2j−1 and η2j in Proposition 5.4 commute, (ii) each quadratic subfactor η2j−1 η2j
commutes with the partial product η1 η2 · · · η2j−3 η2j−2 .
Example 5.8. Let W be an extended solution which is real of some degree r, it is
easy to see that
(5.2)
⊥
Pi (W ∩ λi H+ ) = Pr−i−1 (W ∩ λr−i−1 H+ )
(i = 0, . . . , r).
As in Example 4.12, consider the superhorizontal sequence 0 ⊂ δ0 ⊂ δ2 ⊂ · · · ⊂
⊥
δr = Cn , where δi = Pi (W ∩ λi H+ ). Since δ i = δr−i−1 by (5.2), this defines a
superhorizontal holomorphic map into a flag manifold of SO(n), see [6, Chapter 4].
When r = 2s it also follows that λ−s Φ0 takes values in ΩSO(n), where Φ0 is the
limiting S 1 -invariant map considered in Remark 3.23 .
5.3. Harmonic maps into the space of orthogonal complex structures. By
an orthogonal complex structure on C2m we mean a skew-symmetric endomorphism
J on R2m with J 2 = −I; such a J is called positive if there is a positively oriented orthonormal basis {e1 , . . . , e2m } of R2m with e2j = Je2j−1 (j = 1, . . . , m).
Clearly, the space of orthogonal complex structures is the Hermitian symmetric
space O(2m)/U(m) with SO(2m)/U(m) representing the positive ones. The space
O(2m)/U(m) can be embedded as a totally geodesic submanifold of Gm (C2m ) by
sending an orthogonal complex structure J to its maximally isotropic (0, 1)-space V
in C2m . With the standard conventions this embedding is holomorphic. Composing
it with the Cartan embedding of Gm (C2m ) into U(2m) gives the totally geodesic
embedding J 7→ πV − πV⊥ = πV − πV with image {g ∈ U(2m) : g 2 = I and g =
−g} = {g ∈ U(2m) : g = g ∗ = −g}.
Alternatively, we have the Cartan embedding of O(2m)/U(m) into O(2m) which
sends J to the endomorphism J ∈ O(2m); note that J = i(πV − πV ). On composing
this with the totally geodesic embedding of O(2m) into U(2m) we obtain another
totally geodesic embedding of O(2m)/U(m) into U(2m), given by J 7→ i(πV − πV );
note this is equal to the first one up to a factor i.
In §5.1, we started with W ∈ Gr(n) which was real and of even degree. Let us
now consider a subspace W which is real of odd degree r.
Example 5.9. Suppose that r = 1, so that W = V + λH+ . Then W is real if
and only if V ⊥ = V , i.e., V ⊂ Cn is maximal isotropic. In particular we must have
n = 2m for some m.
If, now W is an extended solution, then V must be holomorphic as a map into
Gm (C2m ); it follows that an extended solution of degree one corresponds to a holomorphic map into O(2m)/U(m).
In general, if W is real of odd degree r, then after r − 1 double Segal–Uhlenbeck
steps as described in §5.1, we are left with a space which is real of degree one, and
hence n must be even. We have proved the following result.
34
Proposition 5.10. Let Φ : M → ΩU(n) be a polynomial extended solution of odd
degree r, satisfying Φ = λ−r Φ. Then n = 2m and Φ has a factorization Φ =
η0 η1 · · · ηr−1 with ηi = παi + λπα⊥i where
(i) each quadratic subfactor λ−1 η2i−1 η2i : M → ΩSO(n);
(ii) each partial product η0 · · · ηk : M → ΩU(n) is an extended solution;
(iii) η0 = πV + λπV , where V is maximally isotropic and corresponds to a holomorphic map M → O(2m)/U(m).
Furthermore, if Φ : M → ΩU(n)I , then each subfactor η0 · · · ηk : M → ΩU(n)I .
To apply this to harmonic maps, note that if W = ΦH+ : M → Gr(n) is an
extended solution which is real of odd degree r, then iΦ−1 lies in O(n), so this
(n)
would seem to be little use. However, if additionally, W lies in Grν , then Φ−1 is a
harmonic map into O(2m)/U(m). We give a converse.
Lemma 5.11. Let ϕ : M → O(2m)/U(m) be a harmonic map to a real Grassmannian, which is of finite uniton number as a map into U(n). Then there is a
polynomial extended solution Φ : S 2 → ΩU(2m)I of odd degree r with Φ−1 equal to
either ϕ or ϕ. Furthermore, Φ = λ−r Φ.
Proof. Let z0 ∈ M and write ϕ(z0 ) = δ0 , where δ0 ⊂ C2m is maximally isotropic.
Then Qλ = πδ + λπδ ∈ ΩU(2m)I satisfies Q−1 = ϕ(z0 ). There is a unique extended
solution Ψ : S 2 → ΩU(2m) associated to ϕ with Ψ(z0 ) = Q; by Remark 3.4, since
Q is algebraic, so is Ψ, and by uniqueness we see that Ψ : M → ΩU(2m)I . Now
P
Q = λ−1 Q; clearly, Ψ satisfies the same relation. Thus, if Ψ = t`=−s λ` T` with
T−s , Tt 6= 0, then from Ψ = λ−1 Ψ we see that s = t − 1. Hence Φ = λs Ψ : M →
ΩU(2m)I is polynomial of odd degree and Φ−1 is equal to either ϕ or ϕ.
¤
5.4. Explicit formulae for real harmonic maps. As in §3.4, all extended solutions W satisfying (2.4) are generated from a holomorphic subbundle X of H+ /λr H+
by (3.12). As before, let {Lj } be a meromorphic spanning set for X. Then
{λi+k (Lj )(k) : i + k + o(j) ≤ r − 1} is a spanning set for W .
Define a complex-symmetric inner product on H+ /λr H+ by (v, w) = the HermitP
P
i
λi vi and w = r−1
ian inner product of λ1−r v and w, then if v = r−1
i=0
i=0 λ wi we
Pr−1
have (v, w) = i=0 vi wr−i . Then W is real of degree r if and only if it is isotropic
with respect to this inner
i.e., (v, w) ¢= 0 for all v, w ∈ W , explicitly,
¡ i+kproduct,
0
0
j (k)
i0 +k0
the data must satisfy λ (L ) , λ
(Lj )(k ) = 0 for all j, j 0 ∈ {1, 2, . . .} and
i, k, i0 , k 0 ∈ N with i + k + o(j) ≤ r − 1 and i0 + k 0 + o(j 0 ) ≤ r − 1. In the case
M = S 2 , this gives quadratic equations on the coefficients of the polynomials Lji ;
solutions can be found by a generalization of the method in [1, §5(B)].
Then all harmonic maps ϕ : M → O(n) of finite uniton number are determined
from the {Lj } as in Proposition 3.18; explicitly, for j = 1, . . . , r and i = 2j − 1 or
2j, αi is given by (3.17) with u = s − j, then an extended solution associated to ϕ
is given by (3.4).
35
To obtain all harmonic maps of finite uniton number into a real Grassmannian
(resp. O(2m)/U(m)), it suffices to take the Lj to have all odd or even coefficients
zero and r even (resp. odd).
5.5. Harmonic maps to the symplectic group and its quotients. In a similar
way, we can say that W ∈ Gr(2n) is symplectic of degree r if W satisfies (2.4)
and J W = λ1−r W , where J is left multiplication by the unit quaternion j on
C2n ∼
= Hn . It follows easily that the results described in §§5.1 — 5.3 are still true
with O(n) replaced by Sp(n). For the first two subsections, where r is even, this
was done in [19]. Regarding the new results in §5.3 for r odd, the first term in the
factorization described in Proposition 5.10 will correspond to a holomorphic map
into the Hermitian symmetric space Sp(n)/U(n). In all cases, §5.4 gives explicit
formulae for the harmonic maps and their extended solutions, this time finding
initial data Lj by a generalization of the method in [2, §5(B)].
We have thus determined all harmonic maps of finite uniton number from a
Riemann surface to a classical compact Lie group or inner symmetric space of it.
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Department of Mathematics & Computer Science, University of Southern Denmark, Campusvej 55,
DK-5230 Odense M, Denmark
E-mail address: [email protected]
Department of Pure Mathematics, University of Leeds, Leeds LS2 9JT, Great Britain
E-mail address: [email protected]

FILTRATIONS, FACTORIZATIONS AND EXPLICIT FORMULAE FOR HARMONIC MAPS

Transcription

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