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 MARTIN SVENSSON AND JOHN C. WOOD 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 MARTIN SVENSSON AND JOHN C. WOOD 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+ , FILTRATIONS, FACTORIZATIONS AND HARMONIC MAPS 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 MARTIN SVENSSON AND JOHN C. WOOD 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. FILTRATIONS, FACTORIZATIONS AND HARMONIC MAPS 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 MARTIN SVENSSON AND JOHN C. WOOD 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 FILTRATIONS, FACTORIZATIONS AND HARMONIC MAPS 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 MARTIN SVENSSON AND JOHN C. WOOD 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 FILTRATIONS, FACTORIZATIONS AND HARMONIC MAPS 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 MARTIN SVENSSON AND JOHN C. WOOD 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⊥ ) FILTRATIONS, FACTORIZATIONS AND HARMONIC MAPS 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 MARTIN SVENSSON AND JOHN C. WOOD (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+ . FILTRATIONS, FACTORIZATIONS AND HARMONIC MAPS 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 MARTIN SVENSSON AND JOHN C. WOOD 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 . FILTRATIONS, FACTORIZATIONS AND HARMONIC MAPS 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 MARTIN SVENSSON AND JOHN C. WOOD 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 FILTRATIONS, FACTORIZATIONS AND HARMONIC MAPS 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 MARTIN SVENSSON AND JOHN C. WOOD 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. FILTRATIONS, FACTORIZATIONS AND HARMONIC MAPS 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 MARTIN SVENSSON AND JOHN C. WOOD 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}, FILTRATIONS, FACTORIZATIONS AND HARMONIC MAPS 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 MARTIN SVENSSON AND JOHN C. WOOD 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. FILTRATIONS, FACTORIZATIONS AND HARMONIC MAPS 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 MARTIN SVENSSON AND JOHN C. WOOD 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 FILTRATIONS, FACTORIZATIONS AND HARMONIC MAPS 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 MARTIN SVENSSON AND JOHN C. WOOD 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) , FILTRATIONS, FACTORIZATIONS AND HARMONIC MAPS 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 MARTIN SVENSSON AND JOHN C. WOOD 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. FILTRATIONS, FACTORIZATIONS AND HARMONIC MAPS 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 MARTIN SVENSSON AND JOHN C. WOOD 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 FILTRATIONS, FACTORIZATIONS AND HARMONIC MAPS 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 MARTIN SVENSSON AND JOHN C. WOOD 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). FILTRATIONS, FACTORIZATIONS AND HARMONIC MAPS 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. References [1] A. Bahy-El-Dien and J. C. Wood The explicit construction of all harmonic two-spheres in G2 (Rn ), J. Reine u. Angew. Math. 398 (1989), 36–66. [2] A. Bahy-El-Dien and J. C. Wood The explicit construction of all harmonic two-spheres in quaternionic projective spaces, Proc. London Math. Soc. (3) 62 (1991) 202–224. [3] J Bolton and L. M. Woodward The affine Toda equations and minimal surfaces, in: Harmonic maps and integrable systems, 59–82, Vieweg, Braunschweig, 1994. [4] F. E. Burstall, A twistor description of harmonic maps of a 2-sphere into a Grassmannian, Math. Ann. 274 (1986), 61–74. [5] F. E. Burstall and M. A Guest, Harmonic two-spheres in compact symmetric spaces, revisited, Math. Ann. 309 (1997), 541–572. [6] F. E. Burstall and J. H. Rawnsley, Twistor theory for Riemannian symmetric spaces, Lecture Notes in Mathematics, vol. 1424, Springer-Verlag, Berlin, Heidelberg, 1990. [7] F. E. Burstall and J. C. Wood, The construction of harmonic maps into complex Grassmannians, J. Diff. Geom. 23 (1986), 255–298. [8] B. Dai and C.-L. Terng, B¨ acklund transformations, Ward solitons, and unitons, J. Differential Geom. 75 (2007), 57–108. [9] J. Davidov and A. G. Sergeev, Twistor spaces and harmonic maps, Russian Math. Surveys 48 (1993), 1–91. [10] Y. Dong and Y. Shen, Factorization and uniton numbers for harmonic maps into the unitary group U(n), Sci. China Ser. A 39 (1996), 589–597. [11] J. Dorfmeister, F. Pedit and H. Wu, Weierstrass type representation of harmonic maps into symmetric spaces, Comm. Anal. Geom. (1998), 633–668. [12] M. J. Ferreira, B. A. Sim˜ oes and J. C. Wood, All harmonic 2-spheres in the unitary group, completely explicitly, Math. Z., to appear. [13] M. A. Guest, Harmonic maps, loop groups, and integrable systems, London Mathematical Society Student Texts, 38, Cambridge University Press, Cambridge, 1997. [14] M. A. Guest, An update on harmonic maps of finite uniton number, via the zero curvature equation, Integrable systems, topology, and physics (Tokyo, 2000), 85–113, Contemp. Math. 309, Amer. Math. Soc., Providence, RI, 2002. [15] Q. He and Y. Shen, Factorization and symplectic uniton numbers for harmonic maps into symplectic groups, Sci. China Ser. A 44 (2001), 1225–1235. [16] Q. He and Y. B. Shen, Explicit construction for harmonic surfaces in U(N ) via adding unitons, Chinese Ann. Math. Ser. B 25 (2004), 119–128. [17] J. L. Koszul and B. Malgrange, Sur certaines structures fibr´ ees complexes, Arch. Math. 9 (1958), 102–109. 36 MARTIN SVENSSON AND JOHN C. WOOD [18] Y. Ohnita and G. Valli, Pluriharmonic maps into compact Lie groups and factorization into unitons, Proc. London Math. Soc. (3) 61 (1990), 546–570. [19] R. Pacheco, Harmonic two-spheres in the symplectic group Sp(n), Internat. J. Math. 17 (2006), 295–311. [20] R. Pacheco, On harmonic tori in compact rank one symmetric spaces, Diff. Geom. Appl. 27 (2009), 352–361. [21] A. Pressley and G. Segal, Loop groups, Oxford Mathematical Monographs, Oxford Science Publications, The Clarendon Press, Oxford University Press, Oxford, 1986. [22] G. Segal, Loop groups and harmonic maps, Advances in homotopy theory (Cortona, 1988), 153–164, London Math. Soc. Lecture Notes Ser., 139, Cambridge Univ. Press, Cambridge, 1989. [23] K. Uhlenbeck, Harmonic maps into Lie groups: classical solutions of the chiral model, J. Differential Geom. 30 (1989), 1–50. [24] J. C. Wood, The explicit construction and parametrization of all harmonic maps from the two-sphere to a complex Grassmannian, J. Reine Angew. Math. 386 (1988), 1?–31. [25] J. C. Wood, Explicit construction and parametrization of harmonic two-spheres in the unitary group, Proc. London Math. Soc. (3) 58 (1989), 608–624. 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]

© Copyright 2019