Harmonic maps into the orthogonal group and null curves

We find algebraic parametrizations of extended solutions of harmonic maps of finite uniton number from a surface to the orthogonal group O(n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm {O}}(n)$$\end{document} in terms of free holomorphic data which lead to formulae for all such harmonic maps. Our work reveals an interesting correspondence between certain harmonic maps and the free Weierstrass representation of null curves and minimal surfaces in 3- and 4-space.


Introduction
Harmonic maps are smooth maps between Riemannian manifolds which extremize the 'Dirichlet' energy integral (see, for example [16,37]). They include many interesting classes of mappings, including geodesics, minimal submanifolds and harmonic functions. Harmonic maps from surfaces to Lie groups and their symmetric spaces are of particular interest, as they admit an integrable systems formulation in terms of extended solutions, and they constitute the chiral or non-linear σ -model of particle physics, see for example [41].
We give an algorithm (Theorem 3.8) which determines, inductively, algebraic parametrizations of extended solutions of harmonic maps of finite uniton number from a surface to the orthogonal group O(n) in terms of free holomorphic data; this determines all such harmonic maps. In contrast to previous work, e.g. [34,Sect. 6], the holomorphic data is free. The parametrizations involves no integration: to avoid that, the algorithm replaces the initial choice of data by new data; this gives global formulae for the parametrizations. These formulae determine all harmonic maps locally by choosing the free holomorphic data to be meromorphic functions on open subsets of M. There are two important cases where all extended solutions, and so harmonic maps, are determined globally by our formulae: (i) S 1 -invariant extended solutions for harmonic maps into O(n). These relate to harmonic maps which arise from twistor constructions; these have extended solutions which are invariant under the natural S 1 -action of Terng, see [36,Sect. 7]. An early twistor construction was that of Calabi who gave [10,11] a construction of all harmonic maps from the 2-sphere to real projective spaces or spheres in terms of totally isotropic holomorphic maps. We give a correspondence (Theorem 4.3) between S 1 -invariant extended solutions for harmonic maps into O(n) of maximum uniton number and such totally isotropic holomorphic maps, and so, harmonic maps to spheres. Using our algorithm, we can give totally explicit global formulae for all these objects (Theorem 4.1).
(ii) The case n ≤ 6. In Sect. 4, by modifying our algorithm in some cases (see, for example, Sect. 4.8c), we find global formulae for all harmonic maps of finite uniton number and their extended solutions from a surface to O(n). Our formulae have the following interesting application: A null curve is a holomorphic (or meromorphic) map from a surface to C n whose derivative is null (isotropic). The real part of a null curve is a minimal surface in R n and all minimal surfaces are given that way, locally. As well as the usual Weierstrass representation involving integration, Weierstrass [38] gave a formula for such null curves in C 3 , called the free Weierstrass representation; de Montcheuil [26] gave a similar formula for C 4 , thus giving (locally) all minimal surfaces in R 3 and R 4 without integration. Our parametrizations for n = 5, 6 lead to correspondences between certain extended solutions for harmonic maps into O(n) and null curves (Theorems 5.1, 5.3), where the free Weierstrass data appear very simply in a matrix giving the extended solution.
The starting point is the seminal work of Uhlenbeck [36] who, by introducing a spectral parameter λ, showed that all harmonic maps from a surface to the unitary group U(n) can be obtained, locally at least, from certain maps into its loop group U(n), namely the extended solutions mentioned above. If there is an extended solution polynomial in λ, the harmonic map is said to be of finite uniton number; all harmonic maps from a compact Riemann surface with a globally defined extended solution, and so all harmonic maps from the 2-sphere, are of finite uniton number. Further, Uhlenbeck gave a factorization of a polynomial extended solution into certain linear factors called unitons. Using the Grassmannian model of the loop group, Segal [30] showed how to represent an extended solution by a subbundle W of a trivial bundle with fibre a Hilbert space, and showed how to find uniton factorizations from a certain natural filtration of W . This was put into a general framework in [34], which led to formulae for uniton factorizations including those of [14,22] (which had been found by different methods). The minimum number of unitons needed to obtain a given harmonic map is called its uniton number.
In [7], a different approach was taken by Burstall and Guest using a finer classification than that given by uniton number based on a Bruhat decomposition of the algebraic loop group. This reduced the problem of finding harmonic maps of finite uniton number and their extended solutions into a compact Lie group to solving a sequence of ordinary differential equations in the Lie algebra, amounting to successive integrations. They also solve the corresponding equations in the Lie group U(n) in some special cases of low dimension. Now any compact Lie group can be embedded in U(n), but this imposes conditions on the data so that it can be hard to find, cf. [34,Sect. 6]. Using the framework of [7], we solve this problem for O(n) and give an algorithm which is inductive on dimension, finding formulae for extended solutions for the group O(n) from those for O(n − 2) to end up with algebraic formulae for all harmonic maps of finite uniton number and their extended solutions from a surface to O(n) of finite uniton number in terms of free holomorphic data. Our method is to interpret the extended solution equations in the Lie group and replace the initial data of Burstall and Guest, which had to be integrated in [7], by data which gives the solution by differentiation and algebraic operations.
Note that it does not seem easy to extend our method to general compact Lie groups; however, a modification of our method has been developed for the symplectic group [27] where harmonic maps and extended solutions were found in [28], but with constrained holomorphic data.
The authors thank Fran Burstall, Joe Oliver, Rui Pacheco, Martin Svensson and the referee for some useful comments on this paper.

Harmonic maps into a Lie group
We recall the basic theory of harmonic maps from Riemann surfaces to Lie groups and symmetric spaces. Throughout this paper, all manifolds, bundles, and structures on them, will be taken to be C ∞ -smooth, and all manifolds will be without boundary. Throughout this paper Mwill denote a Riemann surface, i.e., a connected 1-dimensional complex manifold, equivalently a (smooth) oriented 2-dimensional manifold with a conformal structure. Since harmonicity of a map from a 2-dimensional manifold only depends on the conformal structure [17,Sect. 4B] (see also, for example, [40,Sect. 1.2]), the concept of harmonicity for a map from a Riemann surface is well defined.
In the case of maps from a Riemann surface M to a Lie group G, we can formulate the harmonicity equations in the following way [23,36]. For any smooth map ϕ : M → G, set A ϕ = 1 2 ϕ −1 dϕ; thus A ϕ is a 1-form with values in the Lie algebra g of G; in fact, it is half the pull-back of the Maurer-Cartan form of G. Now, any compact Lie group can be embedded in the unitary group U(n); such an embedding is totally geodesic. From the composition law [17, Sect. 5A], a smooth map into a totally geodesic submanifold N of a Riemannian manifold P is harmonic into N if and only if it is harmonic as a map into P; thus it is natural to first consider harmonic maps into U(n). Let C n denote the trivial complex bundle C n = M × C n , then D ϕ = d + A ϕ defines a unitary connection on C n . We decompose A ϕ and D ϕ into (1, 0)-and (0, 1)-parts; explicitly, in a (local complex) coordinate domain (U , z), writing dϕ = ϕ z dz + ϕ z dz, A ϕ = A the following nice formulation of harmonicity: a smooth map ϕ : M → G is harmonic if and only if, on each coordinate domain, A ϕ z is a holomorphic endomorphism of the holomorphic vector bundle (C n , D ϕ z ). We call harmonic maps ϕ and ϕ with ϕ = gϕ for some g ∈ U(n) (left-)equivalent; if ϕ is replaced by an equivalent harmonic map ϕ, then all the quantities in (2.1) are unchanged.
Let N = {0, 1, 2, . . .}. For any N ∈ N and k ∈ {0, 1, . . . , N }, let G k (C N ) denote the Grassmannian of k-dimensional subspaces of C N ; it is convenient to write G * (C N ) for the disjoint union ∪ k=0,1,...,N G k (C N ). We shall often identify, without comment, a smooth map ϕ : M → G k (C N ) with the rank k subbundle of C N = M × C N whose fibre at p ∈ M is ϕ( p); we denote this subbundle also by ϕ, not underlining this as in, for example, [9,21,22].
For a subspace V of C n we denote by π V (resp. π ⊥ V ) orthogonal projection from C n to V (resp. to its orthogonal complement V ⊥ ); we use the same notation for orthogonal projection from C n to a subbundle. The Cartan embedding [12, p. 66] of the complex Grassmannian is given by (2.2) this is totally geodesic, and isometric up to a constant factor. We shall identify V with its image ι(V ); since ι(V ⊥ ) = −ι(V ), this identifies V ⊥ with −V .

Extended solutions and the Grassmannian model
Let G be a compact connected Lie group with complexification G C ; denote the corresponding Lie algebras by g and g C = g ⊗ C.
For any Lie group, we define the free and based loop groups by G = {γ : S 1 → G : γ smooth} and G = {γ ∈ G : γ (1) = e}, respectively, where e denotes the identity of G; their corresponding Lie algebras g and g are similarly defined. By an extended solution [36] we mean a smooth map : M → G from a (Riemann) surface which satisfies A on each coordinate domain (U , z) for some map A : U → g C . We frequently write λ (z) = (z)(λ) (z ∈ M, λ ∈ S 1 ). Given an extended solution : M → G, for any g ∈ G, ϕ = g −1 is harmonic with the A ϕ z of (2.1) equal to the A just defined; ϕ and are said to be associated to each other. Any harmonic map on a simply connected domain has an associated extended solution. Any two extended solutions and associated to the same or equivalent harmonic map are related by a loop: = η where η ∈ G: we shall say that such extended solutions are equivalent; we are interested in finding harmonic maps and extended solutions up to equivalence.
We specialize to G = U(n) with complexification G C = GL(n, C) and corresponding Lie algebras g = u(n) and g C = gl(n, C). Define the algebraic loop group to be the subgroup alg U(n) of those γ ∈ U(n) given by finite Laurent (i.e., Fourier) series: γ = t i=s λ k S k where s ≤ t are integers and the S k are n×n complex matrices, and define alg U(n) similarly. We say that has finite uniton number if it is a map from M to alg U(n); more precisely, the uniton number is defined to be t − s assuming S s and S t are non-zero. For r ∈ N, let r U(n) denote the set of polynomials of degree at most r : Following [36] a harmonic map ϕ : M → U(n) is said to be of finite uniton number if it has an associated polynomial extended solution : M → r U(n). Then the (U(n)) (minimal) uniton number of ϕ is the minimum degree of such a . Any harmonic map from a compact surface M to U(n) which has an associated extended solution defined on the whole of M is of finite uniton number at most n − 1 [36]; in particular, this applies to any harmonic map from S 2 . Now let H = H (n) denote the Hilbert space L 2 (S 1 , C n ). By expanding into Fourier series, we have H = linear closure of span{λ i e j : i ∈ Z, j = 1, . . . , n}, The natural action of U(n) on C n induces an action on H which is isometric with respect to this L 2 inner product. We consider the closed subspace The action of U(n) on H induces an action on subspaces of H; denote by Gr = Gr (n) the orbit of H + under that action, see [29] for a description of that orbit. The action gives a bijective map We will sometimes write W λ = λ H + when we need to consider dependence on λ ∈ S 1 . Note that W = H + is 'shift-invariant', i.e., closed under multiplication by λ, indeed λW = λH + ⊂ H + = W , so that gives an isomorphism between H + /λH + ∼ = C n and W /λW . The map (2.4) restricts to a bijection from the algebraic loop group alg U(n) to the set of λ-closed subspaces W of H satisfying λ r H + ⊂ W ⊂ λ s H + for some integers r ≥ s; it further restricts to a bijection from r U(n) to the subset Gr r ⊂ Gr of those λ-closed subspaces W of H satisfying (2.5) Now let : M → U(n) be a smooth map and set W = H + : M → Gr. We can regard W as a subbundle of the trivial bundle H := M × H. Then Segal [30] showed that is an extended solution if and only if W satisfies two conditions: (2.6) Here (·) denotes the space of smooth sections. We call W = H + the Grassmannian model of the extended solution . The assignment → W = H + induces a one-to-one correspondence between polynomial extended solutions : M → r U(n) and smooth maps W : M → Gr r satisfying (2.6).

Complex extended solutions
Let + U(n) C (resp. * U(n) C ) denote the subgroup of U(n) C consisting of smooth maps S 1 → U(n) C = GL(n, C) which extend holomorphically to {λ ∈ C : |λ| < 1} (resp. {λ ∈ C : 0 < |λ| < 1}); + u(n) C = + gl(n, C) is similarly defined. Following [7], by a complex extended solution we mean a smooth map : M → * U(n) C which satisfies, on each coordinate domain (U , z), and is holomorphic with respect to the complex structure induced from U(n) C = GL(n, C), i.e., for fixed λ, the entries of M z → (z)(λ) ∈ U(n) C are holomorphic. Recall [29,Theorem 8.11] that the product map U(n) × + U(n) C → U(n) C is a diffeomorphism. This gives the Iwasawa decomposition or loop group factorization of U(n) C as the product of the two given factors. It also gives an identification between U(n) and the homogeneous space U(n) C / + U(n) C ; thus U(n) acquires the structure of a complex manifold. From [15], given a complex extended solution , its projection = [ ] onto U(n) is an extended solution; note that this is holomorphic with respect to the complex structure just defined. Further, the corresponding Grassmannian model W = H + is also given by W = H + . Conversely, as in [7,15], any extended solution is locally the projection of a complex extended solution.
More generally, we shall say that a meromorphic map : M → * U(n) C is a meromorphic complex extended solution if it is a complex extended solution away from its poles. Then we can extend W = H + , and so = [ ], smoothly over the poles: indeed the columns of give meromorphic sections of W which span W mod λW , i.e., writing Y for the span of the columns of so that Y = (C n ), then W = ∞ i=0 λ i Y . Note that Y , and so W , extend as in [34,Lemma 4.1(ii)]; in fact, the columns of form a meromorphic basis for Y , cf. [14,Sect. 7]. We will continue to write = [ ] for the projection of onto U(n) even when is meromorphic. The process of finding explicitly from can be tricky in the general case; however, in the finite uniton number case, can be found explicitly from W by the formulae in [34], see the next section. Conversely, given an extended solution : M → U(n) of finite uniton number (i.e., with values in alg U(n)), there is a meromorphic complex extended solution : M → * U(n) C with = [ ]; this follows from Proposition 2.2 below.

Uniton factorizations from extended solutions
Let ϕ : M → U(n) be a harmonic map. Uhlenbeck called a subbundle α of C n a uniton (for ϕ) if (i) α is holomorphic with respect to the Koszul-Malgrange holomorphic structure induced by ϕ, i.e., D ϕ z (σ ) ∈ (α) for all σ ∈ (α); and (ii) α is closed under the endomorphism A ϕ z , i.e., A ϕ z (σ ) ∈ (α) for all σ ∈ (α). She showed [36] that given a harmonic map ϕ and a uniton α, the product ϕ = ϕ(π α − π ⊥ α ) gives a new harmonic map, a process she called adding a uniton. If is an extended solution, we say that α is a uniton for if it is a uniton for any associated harmonic map ϕ = g −1 (g ∈ U(n)); then we have [36,Corollary 12.2]: given an extended solution : M → U(n), a subbundle α of C n is a uniton for if and only if = (π α + λπ ⊥ α ) is an extended solution. Let : M → r U(n) be a polynomial extended solution (see Sect. 2.2). By a uniton factorization of we mean a product: where each α i is a uniton for the partial product i−1 = (π α 1 + λπ ⊥ α 1 ) · · · (π α i−1 + λπ ⊥ α i−1 ); here we set 0 = I . Uhlenbeck [36] proved that any polynomial extended solution has a uniton factorization. A tool for finding uniton factorizations was proposed by Segal [30], namely that they are equivalent to certain filtrations; this was developed in [34] where the following terminology was introduced: Let H + denote the trivial bundle M × H + . By a λ-filtration (W i )of W we mean a nested sequence These are obtained by applying the following steps (called λ-steps in [34]) for i = r , r − 1, . . . , 2, 1, starting with If we apply these steps alternately, we get a filtration called an alternating filtration [34,Example 4.5]. Starting with an Uhlenbeck step on W = W r , this is given by gives a uniton factorization (2.8) with partial products given by the i ; all uniton factorizations are given this way [34,Sect. 3]. The formula (2.11) gives explicit formulae for any uniton factorization; these include the formulae of [14,22] for the Segal and Uhlenbeck factorizations. Applying (2.11) to the alternating filtration gives the alternating factorization which has the useful property in the O(n) case that adjacent unitons combine to give real quadratic factors, see [34, Sect. 6.1]. We shall use this factorization in Sect. 4.3ff.

Maps into complex Grassmannians and S 1 -invariant maps
Recall the Cartan embedding (2.2). Let be an extended solution and set W = H + . Then satisfies the symmetry condition: if and only if W −λ = W λ (λ ∈ S 1 ) . In this case, the corresponding harmonic map ϕ = −1 satisfies ϕ 2 = I and so is a (harmonic) map into a complex Grassmannian G * (C n ); conversely, it follows from [36,Sect. 15] that any harmonic map ϕ : M → G * (C n ) of finite uniton number is of the form ϕ = −1 for some polynomial extended solution satisfying (2.12), see [34,Sect. 5.1] where bounds on the degree of are given. See [21] for more information and explicit formulae.
As a special case of the above, an extended solution for some r . Further, the α i are holomorphic subbundles of C n which form a superhorizontal sequence (see, for example [34, Definition 3.13]), i.e., for all i ∈ {0, 1, . . . , r }, ∂ z (s) ∈ (α i+1 ) for all s ∈ (α i ). The corresponding Grassmannian model W = H + is given by 15) and the corresponding harmonic map ϕ = −1 is the map into a complex Grassmannian given by The map (ψ i ) → ϕ can be interpreted as a twistor fibration, see [7,Sect. 3] and [8] for the general theory, [35] for further constructions, and Sect. 3.1 for the real case.
An example of an S 1 -invariant extended solution with r = n − 1 is given by setting

The method of Burstall and Guest for U(n)
The starting point for the theory in [7] is a finer classification than that provided by uniton number by using 'canonical elements': Let G be a compact connected semisimple Lie group with complexification G C ; denote the corresponding Lie algebras by g and g C = g ⊗ C.
Let δ 1 , . . . , δ be a choice of simple roots for some Cartan subalgebra t. Then a canonical element (for g) [7,8] is an element ξ ∈ t such that δ j (ξ ) = 0 or i ( = √ −1) for all j. The eigenvalues of ad ξ are of the form ik where k is an integer with −r ≤ k ≤ r where r = r (ξ ) = max{k : g k (ξ ) = 0}; we define g k = g k (ξ ) to be the corresponding eigenspace; we then have g C = r k=−r g k . We now apply this to u(n): we shall denote the eigenspace g k (ξ ) of ad ξ in u(n) C = gl(n, C) by g C k = g C k (ξ ) to distinguish it from the o(n) case in Sect. 3.2. According to [6, Proposition A1], the canonical elements of u(n) are of the form ξ = i diag(ξ 1 + λ 0 , . . . , ξ n + λ 0 ) where λ 0 ∈ R and the ξ i are non-negative integers satisfying Note that this implies that ξ 1 = r (ξ ). As in [7, p. 562], essentially by considering the centreless group U(n)/Z (U(n)), with Lie algebra su(n), we may take λ 0 = 0, so that by a canonical element of r U(n) we mean a diagonal matrix ξ = i diag(ξ 1 , . . . , ξ n ) where the ξ i are non-negative integers satisfying (2.17). We have a corresponding canonical geodesic γ ξ : S 1 → U(n) defined by γ ξ (λ) = diag(λ ξ 1 , . . . , λ ξ n ), thus γ ξ ∈ r U(n). The canonical element ξ is determined by the (r + 1)-tuple (t 0 , t 1 , . . . , t r ) of positive integers where t j := #{i : ξ i = j}; we call (t 0 , t 1 , . . . , t r ) the type of ξ . Note that r j=0 t j = n; we shall see that the type determines the block structure of the n × n-matrices below. In particular, i.e., g C k consists of matrices with entries zero unless they are on the kth block superdiagonal: ξ i − ξ j = k (if k is negative this is below the diagonal). As in [36,Corollary 14.4], r ≤ n − 1; equality is attained by type (1, 1, . . . , 1), in which case ξ i = r + 1 − i and ξ i − ξ j = j − i. The example at the end of Sect. 2.5 is of this type.
Write + alg U(n) C = + alg GL(n, C) := + U(n) C ∩ alg U(n) C and similarly for + alg u(n) C = + alg gl(n, C). To apply the above to find polynomial extended solutions, and so harmonic maps of finite uniton number into U(n), we need Definition 2.1 Define a finite-dimensional Lie subgroup A C ξ of + alg GL(n, C) by In the sequel, [ ] denotes the projection U(n) C → U(n) onto the first factor in the Iwasawa decomposition of Sect. 2.3. Proof Define a finite-dimensional Lie subalgebra a C ξ of + alg gl(n, C) by (2.18) this is the u 0 ξ of [7, Proposition 2.7] for g = u(n). It is the Lie algebra of A C ξ and the exponential map is the product of block upper-triangular matrices, so is block upper-triangular, i.e. b i j = δ i j (ξ i ≤ ξ j ). On the other hand, the entries of B below the block diagonal are given by b i j = λ ξ j −ξ i b i j (ξ i > ξ j ) which, since B ∈ A C ξ , has degree at most (ξ j − ξ i ) + (ξ i − ξ j − 1) = −1, a contradiction to B having values in + U(n) C unless b i j = 0. Hence B = I and uniqueness is established.
Since : M → r U(n) is holomorphic map to a projective algebraic variety, B, and so A and = Aγ ξ , are meromorphic on M as in [7, p. 560].
All harmonic maps of finite uniton number have a polynomial associated extended solution : M → r U(n), and so an associated extended solution : M → r U(n) given as described.

Remark 2.3 (i)
The method of Burstall and Guest applies to centreless groups, see [13] for a study of extended solutions into groups with centre, using a related notion of 'I -canonical element'. (ii) The matrices B in a C ξ are nilpotent, and the matrices A in A C ξ are block unitriangular by which we mean upper block-triangular with identity matrices on the block diagonal; in particular A − I is nilpotent. The exponential map B → A = exp B is given by a finite power series in B; further, it is surjective with inverse given A → log A, a finite power series in A − I . (iii) We exemplify the form of A by showing it for types (1, 1, 1, 1, 1, 1) (so r = 5) and (1, 2, 2, 1) (so r = 3), respectively: the superscript in the notation a [k] i j show the maximum degree ξ i − ξ j − 1 of the polynomial a i j ; observe that this equals k − 1 on the kth block superdiagonal (k = 1, 2, . . . , r ): (iv) The Grassmannian model W = H + is given by W = Aγ ξ H + and so by (2.15) where α i is the span of the columns c j of A with ξ j < i (these α i are functions of λ as well as of points of M); clearly, the α i are nested. The columns of the matrix A provide a canonical (a sort of 'reduced echelon form') meromorphic basis for Y = Aγ ξ C n (and so for W ), adapted to the nested sequence (α i ). In the S 1 -invariant case, the α i do not depend on λ and are the subbundles (2.14). (v) = [Aγ ξ ] satisfies the symmetry condition (2.12), and so −1 is a harmonic map into a Grassmannian, if and only if A is a function of λ 2 , i.e., its entries only involve polynomials with even powers of λ. Further is S 1 -invariant if and only A is independent of λ. Both statements follow from (iv), Sect. 2.5, and the uniqueness of A.
We now give a converse to Proposition 2.2. As above, denote the columns of A by c 1 , . . . , c n so that c j = (a 1 j , . . . , a nj ) T . We write j: P( j) to mean the sum over all j satisfying the condition P( j); for example, j: ξ j >ξ k means the sum over all columns c j in the blocks to the left of the block containing c k . Primes denote derivatives with respect to any local complex coordinate on M. Recall the concept of 'complex extended solution' from Sect. 2.3.
where ρ jk is the coefficient of the term of degree ξ j − ξ k − 1 in a jk . This equation is equivalent to We shall call any of the above three equations the extended solution equation (for A).
Proof On a coordinate domain (U , z), set Then P is algebraic, i.e., its entries p jk are polynomial in λ and λ −1 (with coefficients holomorphic in z); further, from the block structure of A, P is strictly upper block-triangular, i.e., p jk = 0 for ξ j ≤ ξ k , so (2.23) reads Suppose that is a complex extended solution. Then, from (2.7), each p ik is polynomial in λ (with no λ −1 ). We prove by induction on ξ i − ξ k that (*): each p ik is of degree 0 and equals ρ ik .
First, if ξ i − ξ k = 1, since a i j = δ i j when ξ j = ξ i , (2.25) reads a ik = p ik , which establishes (*) since a ik has degree 0. Now suppose that (*) holds for ξ i − ξ k ≤ s for some s ≥ 1. Then for ξ i − ξ k = s + 1, (2.25) reads By the induction hypothesis, all the terms in the sum have degree at most Then equating coefficients of degree ≥ s establishes (*) for ξ i − ξ k = s + 1, and the induction step is complete.
Equation (2.20) follows. Equation (2.21) is the ith row of (2.20) and so is equivalent to it. Now, by definition of ρ ik , the term of maximum possible degree ξ i −ξ k −1 on the left-hand side of (2.21) equals the term of that degree, Hence (2.21) holds if and only if it holds mod λ ξ i −ξ k −1 , and we can miss out terms of degree ξ i − ξ k − 1, i.e., those with with ξ i = ξ j , in the summation. In particular, (2.21) is equivalent to (2.22).
Conversely, suppose that (2.20) holds. Then (2.23) holds with each p jk polynomial in λ, so that (2.7) holds and is a complex extended solution.
Let ξ be a canonical element of r U(n) and let (A C ξ ) 0 = A C ξ ∩ U(n), the group of block unitriangular n × n matrices with complex entries.
In the above deformation, these α i tend to the unitons (2.14) of the S 1 -invariant limit.
Equation (2.20) for U(n) are easy to solve, see [7,Sect. 4] and [23,Ch. 22]. However, finding all solutions in O(n) is not so easy: we turn to that problem now.

Generalities on harmonic maps into O(n) and its symmetric spaces
Let z = x +iy → z = x −iy denote standard complex conjugation on C. To adapt the theory of the last section to O(n), we include R n in C n so that However, to deal with polynomial extended solutions, as in [34] we define for each r ∈ N the following subset of r U(n) (cf. 2.3):  U(n) R with r even, and ϕ = ± −1 -note that the (minimal) uniton number of ϕ may be less than r and may be even or odd. If r is odd, then, following [34, Sect. 6.3], n must be even, say n = 2m, and −1 = − −1 so that ϕ = ±i −1 are maps into O(2m). In all cases, the alternating factorization [34, Sect. 6.1] of , which can be calculated from W by (2.10), (2.11) and (2.8), gives an explicit factorization into unitons.
The symmetric spaces of O(n) and SO(n) are the real Grassmannians , and, when n = 2m, the space O(2m)/U(m) of orthogonal complex structures J on R 2m and its identity component SO(2m)/U(m). Note that mapping each J to its i-eigenspace identifies O(2m)/U(m) with the space of all maximally isotropic subspaces of C 2m . Let : M → r U(n) R be an extended solution which satisfies the symmetry condition (2.12). If r is even, ϕ = ± −1 are harmonic maps of finite uniton number into a real Grassmannian G * (R n ), all such harmonic maps can be obtained this way [34, Lemma 6.6]; note that −ϕ = ϕ ⊥ . If r is odd, then n is even, and ± −1 define harmonic maps of finite uniton number into O(2m)/U(m) for m = n/2; all such harmonic maps are obtained this way [34, Lemma 6.9].
Lastly, let : M → r U(n) R be an extended solution which is S 1 -invariant, i.e., satisfies (2.13). Then is given by (2.8) for some superhorizontal sequence (2.14) of holomorphic subbundles of C n which is real in the sense that the polar α The corresponding harmonic map ϕ := −1 is given by (2.16); it defines a map into a real Grassmannian (resp. O(2m)/U(m) with n = 2m) according as r is even (resp. odd).

Analysis of harmonic maps into O(n)
To analyse further harmonic maps into O(n), we equip C n with its standard symmetric inner With respect to the standard basis {e 1 = (1, 0, 0, . . . , 0), e 2 = (0, 1, 0, . . . , 0), . . . , e n = (0, 0, . . . , 0, 1)}, the matrix for A T is the usual transpose (a ji ) obtained from the matrix A = (a i j ) by reflection in the principal diagonal i = j. However, calculations are aided by taking a null basis { e i } for C n , i.e., one with ( e i , e j ) = δ ij where, for any j ∈ {1, . . . , n} we writej = n + 1 − j. Such a basis is given by From now on, we shall write all vectors and matrices with respect to this null basis; then the standard symmetric bilinear inner product on C n of v = j v j e j and w = j w j e j is given by (v, w) = n j=1 v j w¯j . In this null basis the transpose A T is represented by the matrix A T with entries (A T ) i j = a¯j¯i ; we shall call this the second transpose of A. This definition makes sense for any (rectangular) matrix; for a square matrix A, A T is obtained from A by reflection in the second diagonal i = j.
As before, denote the ith column of A by c i . Then A ∈ O(n, C) if and only if Now, according to [5], the canonical elements of o(n) are of the form ξ = i diag(ξ 1 , . . . , ξ n ) where ξ i are integers or half-integers with ξ i −ξ i+1 = 0 or 1, ξ 1 = r /2 for some r = r (ξ ) ∈ N and ξ¯i = −ξ i ∀i, which satisfy the rider (R): if r is odd, #{i : ξ i = 1/2} ≥ 2. This corrects [6, Proposition A.2] which omits the rider and gives a condition (C2) which is incorrect in the o(n) case. The corresponding eigenspaces of ad ξ , which we shall denote by . When the ξ i are half-integers, the canonical elements above do not exponentiate to geodesics in O(n). However, we can work in r U(n) R by adding the constant matrix (r /2)I on to each canonical element (cf. Sect. 2.6) to give the following definition.
Recall that, if r is odd, n is even. In this case, the rider (R) says ξ n/2−1 = ξ n/2 = (r + 1)/2 and ξ n/2+1 = ξ n/2+2 = (r − 1)/2. Noting that the canonical elements of r U(n) R form a subset of those in r U(n), we may define 'type' as in Sect. 2.6. Then the possible types of canonical elements for r U(n) R are (t 0 , t 1 , . . . , t r ) where the t i are positive integers such that t i = t r −i for all i, and (by the rider (R)) if r is odd, the two middle entries t (r −1)/2 = t (r +1)/2 are at least 2. Let ξ be a canonical element of r U(n) R . Recall the space A C ξ from Definition 2.1, and

Remark 3.2 (i) When the type is
By definition of A C ξ , each entry a i j of A above the block diagonal, i.e., with ξ i − ξ j ≥ 1 , is polynomial of degree at most ξ i − ξ j − 1. We now show that when A ∈ A R ξ , the degrees of the entries a iī on the second diagonal which lie above the block superdiagonal, i.e. with ξ i − ξ¯i ≥ 2, are at most one less than this.
, which gives the stated bound.
We now give a version of Proposition 2.  : Further, A and ξ are uniquely determined by , in fact, the assignment A → = [Aγ ξ ] defines a one-to-one correspondence between Sol R ξ and the space of extended solutions It restricts to a one-to-one correspondence between (Sol R ξ ) 0 and the space of S 1 -invariant extended solutions : M → r U(n) R of type ξ .
All harmonic maps of finite uniton number ϕ : M → O(n) have an associated extended solution ∈ Sol R ξ for some canonical element ξ .
Proof Let C be the centre of SO(n), this is trivial if n is odd and {±I } if n is even; let π : SO(n) → SO(n)/C be the natural projection. Recall that, if r is odd, then n is even [34,Sect. 6.3]. Then, in all cases, λ − r /2 π • : M → (SO(n)/C) is an extended solution. We apply [7,Theorem 4.5] to the centreless group SO(n)/C which has Lie algebra o(n). We set a R ξ equal to the intersection of the set a C ξ defined by (2.18) with o(n, C), then a R ξ is the u 0 ξ of [7, Proposition 2.7] for g = o(n), and the exponential map sends a R ξ to A R ξ . By [7, p. 560] there is an associated extended solutionˇ : , is a loop in U(n), so that is equivalent to and we are done. In the S 1invariant case, we have the simpler procedure: set α i = the span of the columns c j of A with ξ j < i; then the corresponding factors π α i + λπ ⊥ α i are unitons which commute and give the Segal, Uhlenbeck and alternating factorizations depending on the order in which they are written.

Adding a border to increase dimension
We will give a method of finding parametrizations of complex extended solutions of finite uniton number from a Riemann surface M to O(n) by induction on the dimension n. Our starting point is Proposition 3.4 which reduces the problem to finding, for each canonical element ξ , all meromorphic maps A : M → A R ξ satisfying the extended solution Eq. (2.20). We shall give an algorithm for parametrizing such A.
Given A : M → O(n, C) with values in A R ξ , the matrix A obtained by removing the border, i.e., A = (a i j ) i, j=2,...,n−1 defines a map by a process of adding a border.  This consists of adding a new top row (a 12 , . . . , a 1,n−1 ), new last column (a 2n , . . . , a n−1,n ) T and new top-right element a 1n , and then completing the border by setting a i1 = δ i1 and a nj = δ nj for i, j = 1, . . . , n. Note that our definitions of 'new top row' and 'new last column' exclude the new top-right element a 1n . Note also that, given A and either the new top row or the new last column, we can find the rest of the matrix by imposing the complex-orthogonality (3.2) of the columns c i of A; in fact, using (c i , c n ) = 0 for i = 2, . . . , n − 1 in turn gives the new top row from the new last column or vice-versa, and then using (c n , c n ) = 0 gives the new top-right element. We refer to this as completing the matrix by algebra. Note that, although removing the border preserves symmetry and S 1 -invariance (by Remark 2.3(v)), adding a border may destroy these, depending on the data chosen.
The following lemma underpins the induction step. For a canonical element ξ of type (t 0 , . . . , t r ), define integers 0 = T r +1 < T r < · · · < T 0 = n by T k = r j=k t j . Note that By (3.5), the second term on the right-hand side of (3.6) is ( c j , c n ) = i: ξ i >0 λ ξ i −1 ρ in ( c j , c i ) = λ ξ¯j −1 ρ j n = 0 mod λ r −ξ j −1 using complex-orthogonality for A and ξ¯j = r − ξ j . As for the first term on the right-hand side of (3.6), by the extended solution Eq. (2.20) for A, we have c j = i≥2: , and then, adding in the second term calculated above, (3.6) gives a 1 j = i≥2:ξ i >ξ j λ ξ i −ξ j −1 ρ i j a 1i mod λ r −ξ j −1 , which is equivalent to (b) by Remark 2.6(i).
We also see that (2.20) holds for the top-right entry, indeed, expanding (c n , c n ) = 0 gives a 1n = − 1 2 ( c n , c n ). Differentiating this and using (3.5) gives a 1n = − i: (2.20) holds for all columns of A including the last, i.e., (a) holds.
Next, assume that (b) holds. We prove that (c) holds by downward induction on i ∈ [2, T 1 ]. For T 2 < i ≤ T 1 so that ξ i = 1, (3.4) is trivially true as it says a in = ρ in . We may thus use I = T 2 + 1 as the starting point of our induction.
We now use the notationsĉ i k = (a 1k , . . . , a i−1,k ) T for the part of c k 'above' a ik anď c¯i k = (a¯i +1,k , . . . , a nk ) T for the part of c k 'below' a¯i k ; note these are both columns of length i − 1. Suppose (3.4) holds for i > I for some I ∈ {2, . . . , T 2 }. We show that it holds for i = I , i.e., that a I n = j: ξ I ≥ξ j >0 λ ξ j −1 ρ jn a I j mod λ ξ I −1 . (3.7) Clearly, a I n + (ĉĪĪ ,č I n ) = (cĪ , c n ) which is zero sinceĪ > 1. Differentiating this gives (a I n ) = − (ĉĪĪ ) ,č I n − ĉĪĪ , (č I n ) . (3.8) By (2.20), the first term on the right-hand side of (3.8) is (ĉĪĪ ) ,č I n = j: ξ j >ξĪ λ ξ j −ξĪ −1 ρ jĪ (ĉĪ j ,č I n ). But (ĉĪ j ,č I n ) = (c j , c n ) = δ¯j n by (3.2), so that (ĉĪĪ ) ,č I n = λ ξ I −1 ρ 1Ī . By the induction hypothesis, the second term on the right-hand side of (3.8) is We show the general term in the sum on the right-hand side of (3.9) is given by First, (ĉĪĪ ,č I j ) + a I j = (cĪ , c j ), which is zero for j = I by (3.2), so (3.10) holds for this case. On the other hand, if j = I , then the left-hand side of (3.10) is zero sinceč I I is a zero column and the right-hand side is a multiple of λ ξ I −1 , so the two sides are equal mod λ ξ I −1 as required.
Substituting (3.10) into (3.9) and then into (3.8) we obtain (3.4) for i = I completing the induction step, and so (c) holds. This completes the proof of the lemma.

Parametrization of extended solutions for O(n)
By a generalized derivative of a meromorphic function ν on M we mean a quotient ν /e, where denotes derivative with respect to some local complex coordinate z on M and e = β j b j is a finite sum which is not identically zero. Here β j and b j are meromorphic functions on M; note that the quotient ν /e is independent of the choice of complex coordinate z on M. In particular, we shall call a generalized derivative of the form ν /β with β meromorphic and non-constant the generalized derivative of ν with respect to β; all generalized derivatives are locally of this form. Away from points where β has a pole or β is zero, β gives an alternative complex coordinate to z and ν /β is the derivative of ν with respect to that complex coordinate. When the denominator is unimportant, we shall often denote a generalized derivative by ν (1) and higher generalized derivatives by ν (2) , ν (3) , . . ., and we set ν (0) = ν; thus for any d ≥ 1, ν (d) is the generalized derivative of ν (d−1) given by Proof We first give the algorithm which defines h 0 : M(M) p 1 → (Sol R ξ ) 0 for any ξ . This is trivial when n = 1, 2, as O(n, C) = {I } so ξ = iI and Sol R ξ = (Sol R ξ ) 0 = {I }. We use these as a base for an induction on the dimension n: in the induction step n is increased by 2.
To find our parametrization, we initially parametrize the above entries by meromorphic functions ν = (ν 0,1 , . . . , ν 0,t r−1 ), setting a 1,t r +i = ν 0,i (i = 1, . . . , t r −1 ). These are essentially the parameters used in [7], however, they will not usually be our final choice of parameters. For the next entry, (3.12) reads (3.13) By the inductive hypothesis, the ρ jk are known functions of the parameters μ for A. We now replace our initial choice of parameters ν 0,i by a new choice ν 1,i of parameters where the 'old' parameters ν 0,i are given in terms of the new ones by ν 0,i = ν 1,i if ρ i+t r ,k is identically zero, and ν 0,i = the generalized derivative ν (1) 1,i := (ν 1,i ) /ρ i+t r ,k , otherwise. Then integrating (3.13) gives where b ik (μ) = 0 when ρ i+t r ,k is identically zero, and b ik (μ) = 1 otherwise; thus the value of b ik (μ) depends on μ. Note that the previous entries a 1k can now be written in terms of the new parameters, in fact, for some functions b ikp (μ) (which are here just 0 or 1).; We prove by induction that, for each K = 1, . . . , n − 1 − T r −1 , there are parameters ν = (ν K ,1 , . . . , ν K ,t r−1 ) with each ν K−1,i equal either to ν K ,i , or to a generalized derivative ν ( p) K ,i of ν K ,i with respect to a function of μ, such that Here each b ikp is now a rational function of the parameters μ for A and the derivatives of those parameters, and { } ( p) denotes a pth generalized derivative as explained above. This is established for K = 1 by (3.14). Suppose we know that, for some K with 2 ≤ K ≤ n − 1 − T r −1 , (3.15) holds with K replaced by K− 1, i.e., then we shall deduce that (3.15) holds. From (3.12) we have a 1K = K−1 j=1 ρ j K a 1 j . Using the induction hypothesis (3.16) for each a 1 j gives us We integrate by parts each term in the last sum, first interpreting ν ( p) K−1,i using (3.11), as follows: for some functions c i p (μ). Repeating the procedure p times gives for some functions d i p (μ), f i (μ) and constant of integration c K . Here, for each i = 1, . . . , t r −1 , e i = K−1 j=1 ρ j K b i j0 + f i . We now replace the parameters ν K−1,i by 'new' parameters ν K ,i where the 'old' parameters ν K−1,i are given in terms of the new ones as follows. If e i is identically zero, ν K−1,i = ν K ,i ; we call this a degenerate step and say that the algorithm is degenerate if this ever occurs. Otherwise, ν K−1,i is equal to the generalized derivative (ν K ,i ) /e i , so that the integral in (3.17) evaluates to ν K ,i . If not all e i are identically zero, we may absorb the constant c K of integration into one of the new parameters ν K ,i ; however, if all e i are identically zero, then we cannot. In this case, we remove c K by premultiplying A by a matrix E = (e i j ) ∈ O(n, C) which is the identity matrix except that e 1K = −c, e n+1−K ,n = c and, if n is odd and K = (n + 1)/2, e 1n = − 1 2 c 2 . This does not alter A or any previous entries a 1k (k < K ) of the new first row. This establishes (3.15) for k = K .
Finally, for k < K we replace the ν K−1,i in (3.16) by the expressions in terms of ν K ,i just given, and the induction step is complete. This gives the new first row (a 12 , . . . , a 1,n−1 ); we complete the matrix finding the new last column (a 2n , . . . , a n−1,n ) T and new top-right element a 1n by algebra, i.e., imposing that A has values in O(n, C) by using (3.2), see Sect. 3.3. We have now given an algorithm for finding h 0 : M(M) p 1 → (Sol R ξ ) 0 from h 0 which completes the induction on dimension.
Note that the subset of data where the algorithm is degenerate at some stage in the induction forms an algebraic subvariety of M(M) p 1 ; define M(M) p 1 ND to be its complement. We now extend the algorithm to define a map h : M(M) p → Sol R ξ . We follow the same method of adding a border, then the equations to satisfy for the first row are again (3.3) but now each element is a polynomial in λ; we write a q i j for the coefficient of λ q in a i j . When i = 1, for each j, a 1 j is a polynomial of degree at most ξ i − ξ j − 1 = r − ξ j − 1. We now equate coefficients of λ q in Eq. (3.12). For the highest possible degree on the left-hand side, q = r − ξ k − 1, there is no equation to satisfy since we are working mod λ r −ξ k −1 . Thus our initial choice of data for the first row will be {a r −ξ k −1 1k : t r + 1 ≤ k ≤ n − 1}; note that this does not include a 1n , which is determined by algebra, see Sect. 3.3. We set a r −ξ k −1 1k = ν 0 k−t r (k = t r + 1, . . . , n − 1) giving our initial choice of parameters ν 0 = (ν 0 1 , . . . , ν 0 n−1−t r ). For q < r − ξ k − 1, by equating coefficients of λ q we obtain the equations: (3.18) Note that the sum is over the q +1 blocks preceding that containing a 1k : since q +ξ k +1 < r , this never includes the entries {a 1 j : ξ j = r } in the left-most block. Note also that, for each j the sum concerns the coefficient of λ q−(ξ j −ξ k −1) of a 1 j ; since q < r − ξ k − 1, this is at most r − ξ j − 1, the maximum possible power for a 1 j . Finally note that the condition ξ k ≤ r − q − 1 is saying that a 1k is in the block where ξ k = r − q − 1 or in a block to the right of that. For clarity, we write out the first three equations of (3.18): We solve (3.18) for each k by induction on q with initial data ν 0 as above; we omit the details.
Putting the initial data for each new first row together shows that our initial data for i.e., the λ q -coefficient of each entry of A on the part of the (q + 1)st block superdiagonal of A above the second diagonal, for q = 0, 1, 2, . . . , r − 1. Note that this initial data is related to that in [7] by the exponential map; it is, however, our final data which forms μ ∈ M(M) p .
Again, the subset of (final) data where the algorithm is degenerate at some stage in the induction forms an algebraic subvariety of M(M) p ; define M(M) p ND to be its complement.
We now see how the above algorithm gives parametrizations of extended solutions of canonical type: recall that by Proposition 3.4, any extended solution is equivalent to one of canonical type.

S 1 -invariant solutions of type
Note that the ρ k−1,k = a k−1,k are the entries of A 'in the g 1 -position', i.e., on the superdiagonal; we shall say that A and the corresponding extended solution = [Aγ ξ 0 ] are non-degenerate if the superdiagonal elements a k−1,k of A are non-constant, equivalently their derivatives are not identically zero. The weaker condition of non-degeneracy of A is also important; the development below shows that it is equivalent to our algorithm being non-degenerate. In either case, we get the following more precise version of Theorem 3.8.   (μ 1 , . . . , μ m−1 ) in the fashion described by the theorem. Following our algorithm, we add a border to give a square matrix A of size n. As usual, it suffices to find the new first row (a 11 , . . . , a 1,n−1 ) by solving (4.1) for i = 1. Of course, a 11 = 1, and the next entry a 12 satisfies no equation so we initially parametrize it by ν 0 = a 12 . If n = 3, there are no equations to satisfy and we complete the matrix by algebra, i.e. by using (3.2), see Example 4.2 below.
In degenerate cases, different formulae are obtained. For example, if μ 1 is constant, then by premultiplying by a suitable matrix E as in the algorithm, we can make it 0 and we obtain the middle 5 × 5 matrix in the right-hand matrix below. Then, if μ 2 is constant, again we can make it zero and we obtain the left-hand 7 × 7 matrix; if μ 2 is not constant, we obtain the right-hand matrix.
Here μ 2 and μ 3 are arbitrary meromorphic functions and, in the right-hand matrix, μ

Totally isotropic maps and extended solutions
We now see how the extended solutions constructed in the last section relate to other interesting maps. Recall ( [39], see also [34,Example 4.7]) that a harmonic map f : M → G * (C n ) generates a harmonic sequence G (i) ( f ) (i ∈ Z) of Gauss bundles or transforms, all harmonic maps. By the (complex) isotropy order of a harmonic map f : M → CP n−1 , we mean the maximum r such that f is perpendicular to On the other hand, by the real isotropy order of a full holomorphic map f = [F] : M → CP n−1 we mean the maximum integer t ≥ −1 such that (4.5) Here F : U → C n denotes a local holomorphic representative of f and F (i) denotes the ith derivative with respect to a local complex coordinate: the definition is independent of choice of F and of local coordinate. Differentiation shows that, if (F (s) , F (s) ) = 0 for some s, then also (F (s+1) , F (s) ) = 0. It follows that t is odd, i.e. t = 2s + 1 for some s ≥ −1; note that (F, F) = 0 ⇐⇒ s ≥ 0. The largest possible value of s is [(n − 3)/2]: in that case fullness implies that n is odd and t = n − 2, and we say that f is totally isotropic [18]. Note that the real isotropy order t is not the same as the complex isotropy order: indeed, the latter is infinite for a holomorphic map. However, if f is a holomorphic map of real isotropy order t ≥ 0, the map f ⊕ f : M → G 2 (R n ) is a harmonic map called a real mixed pair; by [2, Lemma 2.14] this has complex isotropy order t. In [10,11], Calabi showed how that all harmonic maps into RP 2m or S 2m can be obtained from totally isotropic holomorphic maps, giving the bijections between (ii), (iii) and (iv) below; in particular, the bijection from (ii) to (iii) is given by f → G (m) ( f ). We now explain how these relate to polynomial extended solutions of harmonic maps into O(2m + 1) of type (1, 1, . . . , 1), and so of the maximum possible uniton number 2m. The corresponding canonical element is ξ 0 = i diag (2m, 2m − 1, . . . , 1, 0). In particular, we obtain an explicit algebraic parametrization of sets (i)-(iv) by m- tuples  (μ 1 , . . . , μ m ) of meromorphic functions satisfying the non-degeneracy condition (4.3).
Proof By Proposition 3.4, the map = [Aγ ξ 0 ] defines a bijection between (i) and (i) . Given A in (i) , its last column gives a full totally isotropic holomorphic map f ; indeed, each associated curve f (i) is the span of the last i + 1 columns of A, so that f is full and (4.5) holds for t = n − 2; thus f is in set (ii).
The last statement follows by parametrizing set (i) as in Theorem 4.1.

Uniton number at most 2
In this case, we find all harmonic maps completely explicitly, as follows. In the sequel, all uniton factorizations will be the alternating factorization, see Sect. 2.4.
where V is a maximally isotropic holomorphic subbundle of C n ; ϕ has associated extended solution Proof (i) Evident, since it has a polynomial associated extended solution of degree 0, which must equal the identity matrix. When n ≤ 2 this is the only element of O(n, C).

Proposition 4.5 (i) Any extended solution
: M → 2 U(n) R of canonical type has a uniton factorization of the form where X and V are holomorphic subbundles of C n with X ⊥ and V isotropic and π V X = V . This is S 1 -invariant if and only if X is the polar V • = V ⊥ of V , in which case X , V and V all commute and (4.7) reads The corresponding harmonic map is then , which is a (higher dimensional) real mixed pair [2], and has (minimal) uniton number 2 unless V is constant. Proof (i) Write = [Aγ ξ ]; note that the type must be (t 0 , t 1 , t 0 ) for some t 0 , t 1 with 2t 0 + t 1 = n. For each j, write the jth column of A as c j = c 0 j + λc 1 j , note c 1 j = 0 for all j ≤ t 0 + t 1 . Set X = span{c 0 j : t 0 < j ≤ n} and V = span{c j = c 0 j + λc 1 j : t 0 + t 1 < j ≤ n}. Then the Grassmannian model W = Aγ ξ H + is W = V + λX + λ 2 H + so, from (2.10), (2.11) and (2.8), the alternating uniton factorization is given by (4.7) where V = span{c 0 j + π ⊥ X c 1 j : t 0 + t 1 < j ≤ n}. This is S 1 -invariant if and only if c 1 j = 0 for all t 0 +t 1 < j ≤ n, equivalently X is the polar of V . Then (V , X ) = (V , V • ) is a ∂ -pair in the sense of [20]. Thus the Grassmannian model W = Aγ ξ H + is W = V + λV • + λ 2 H + , giving extended solution (4.8).
(ii) By Lemma 3.3, the maximum degree of any term of A is 0, giving an S 1 -invariant extended solution.

All extended solutions for n at most 6
We will now find all extended solutions of canonical type for n ≤ 6. To do this we find all solutions A : M → A R ξ to (2.20) by our algorithm; we can then compute the corresponding extended solutions = [Aγ ξ ] using the formulae in Sect. 2.4, or Sect. 2.5 in the S 1 -invariant case. By modifying our algorithm, and so the mappings h and h 0 in some cases, we obtain the following improvement of Theorem 3.8 where 'locally surjective' is replaced by 'surjective', or even, 'bijective'. We shall show this for each dimension in turn, concentrating on non-degenerate cases; the reader can easily calculate degenerate cases as in Example 4.2. Dimensions n = 1 and 2 are trivial, see Proposition 4.4, so we start with n = 3.

Dimension n = 3
All solutions are obtained from the unique n = 1 case A = (1) by adding a border. This gives one non-trivial type, (1, 1, 1 Here and in the rest of the paper, ≡ denotes a standard identification. The real line (h ⊕ h) ⊥ is given a canonical orientation so that it gives a point of S 2 ; the composition C ∪ ∞ → CP 1 → Q 1 → S 2 of the maps above is stereographic projection. Note that, if g is constant, then ϕ is constant and has (minimal) uniton number 0, otherwise it has uniton number 2.

Dimension n = 4
There are two non-trivial types, as follows.
(a) Type (2,2). Here r = 1 and ξ = i diag (1, 1, 0, 0), and, as in Proposition 4.4(ii), for some arbitrary meromorphic function g on M. Then W = V ⊕ λH + , where V is the maximally isotropic subbundle of C 4 spanned by the last two columns c 3 and c 4 of A and the extended solution = [Aγ ξ ] is = π V + λπ ⊥ V . The corresponding harmonic map −1 is the holomorphic map V : M → O(4)/U(2). More explicitly, it is the composition: (1,2,1). Here r = 2, the maximum possible for n = 4, and ξ = i diag (2, 1, 1, 0). We obtain the solution by adding a border to the unique solution A = I of type (2). Then we have two new entries a 12 , a 13 in the g 1 -position (i.e., on the block superdiagonal), we set a 12 = −g 1 , a 13 = −g 2 where g 1 , g 2 are arbitrary meromorphic functions. Filling in the last column by algebra (see Sect. 3

.3) gives
Let h denote the span of the last column c 4 , thus h = [1, g 1 , g 2 , −g 1 g 2 ] T where T denotes the second transpose as in Sect. 3.2; by (3.2) the polar h • = h ⊥ of h is the span of the last three columns. The above A gives W = h ⊕ λh • ⊕ λ 2 H + , and as in Proposition 4.5, the corresponding extended solution is . This is an extended solution of the real mixed pair h ⊕ h : M → G 2 (R 4 ) (or its orthogonal complement). More explicitly, it is the composition: Up to now, there have been no equations to satisfy and no terms in λ; this shows the following, which is a consequence of [34,Proposition 6.20]. Note that the proposition together with the last statement of Proposition 3.4 shows that every harmonic map of finite uniton number from a surface to O(n) with n ≤ 4 has an associated extended solution which is S 1 -invariant.

Dimension n = 5
All solutions are obtained from one of the two n = 3 cases of Sect. 4.5, i.e., type (3) or type (1,1,1), by adding a border. This gives three non-trivial types, as follows.
(a) Type (2,1,2). Here r = 2 and ξ = i diag(2, 2, 1, 0, 0). We apply the algorithm in the proof of Theorem 3.8 to obtain this case from the (1, 1, 1) case (4.9); we shall give the details in the non-degenerate case, i.e., when g is non-constant. We have one new entry a 13 in the g 1 -position; we initially set this equal to an arbitrary meromorphic function ν 1 . Write a 14 = a 0 14 + λa 1 14 . Then a 1 14 is arbitrary, say σ , and a 0 14 satisfies (a 0 14 ) = −g a 13 mod λ. According to the algorithm, to integrate this, we replace our initial choice ν 1 of parameter by a new parameter ν = a 0 14 so that ν 1 = ν (1) , where generalized derivatives ν (d) are taken with respect to g. As no further integrations are necessary, ν is our final parameter. Then, filling in the last column by algebra, i.e., using (c i , c 5 ) = 0 for i = 3, 4, 5 (see Sect. 3.3), we obtain By Remark 2.3(v), this gives an S 1 -invariant extended solution = [Aγ ξ ] if and only if σ ≡ 0, i.e., σ is identically zero; in which case it has corresponding harmonic map ϕ = α 1 ⊕ α 1 where α 1 is the span of the last two columns. Define h : M → CP 4 as the span of c 5 + ν (2) c 4 . When ν (3) ≡ 0, the last two columns c 4 , c 5 are spanned by h and its derivative, thus ϕ = h (1) ⊕ h (1) : M → G 4 (R 5 ). Its orthogonal complement is the harmonic map ϕ ⊥ : M → RP 4 given by the middle vertex of the following harmonic sequence-by being careful with orientations ϕ ⊥ actually defines a map into S 4 .
If σ is not identically zero, then the harmonic map −1 does not lie in a Grassmannian.

Remark 4.8
This example is equivalent to that of [34,Example 6.21]. The reality conditions (i)-(iii) of that example, which were hard to solve using the methods of [34], are automatically satisfied by our method.
(b) Type (1,3,1), so r = 2. This is obtained from n = 3, type (3), i.e., the identity matrix, by adding a border giving The resulting extended solution and harmonic map are described by Proposition 4.5(ii). (c) Type (1,1,1,1,1). Here r = 4 and ξ = i diag(4, 3, 2, 1, 0). As in the (2, 1, 2) case above, we apply the algorithm in the proof of Theorem 3.8 to obtain this case from the (1, 1, 1) case (4.9). As in Theorem 4.1, this shows that any S 1 -invariant extended solution with middle 3 × 3 matrix A non-degenerate, i.e., g non-constant, has a complex extended for arbitrary meromorphic functions g and ν 1 with g non-constant, and generalized derivatives are taken with respect to g. When A is degenerate, i.e. g is constant, we obtain a simpler formula, see Example 4.2.
Note that A itself is non-degenerate if and only if both g and ν (2) 1 are non-constant; equivalently, the last column spans a full holomorphic map h : M → CP n . Then ϕ = −1 is the harmonic map ϕ = h ⊕ G (2) (h) ⊕ G (4) (h); as in Theorem 4.3, h totally isotropic, i.e., G (4) (h) = h, so that ϕ is a harmonic map into the real Grassmannian G 3 (R 5 ). Also, G (2) (h) defines a harmonic map into RP 4 and into its double cover S 4 . Finally note that the middle three components of h give a 'null curve' in C 3 , see Sect. 5.1.

Dimension n = 6
All solutions are obtained from one of the three n = 4 cases in Sect. 4.6 by adding a border. This gives five non-trivial types, as follows.
(a) Type (1,4,1), so r = 2. This is similar to n = 5, type (1, 3, 1) above. (3,3). This has r = 1 and is obtained from type (2, 2) by adding a border; there are two new parameters ν 1 , ν 2 in the g 1 -position, call these −h and −k giving the S 1 -invariant solution depending on three arbitrary meromorphic functions:  (2,2,2). This has r = 2 and is obtained from type (1,2,1) in Sect. 4.6 above by adding a border. The entries in the first row in the g 1 -position are a 13 and a 14 , giving two new parameters, and the λ-term of a 15 gives a further parameter. Carrying out our algorithm in the case that g 1 and g 2 are non-constant gives Here (ν 1 ) (1) = ν 1 /g 2 and (ν 2 ) (1) = ν 2 /g 1 , and our final new parameters are ν 1 , ν 2 and ν 3 , together with the existing parameters g 1 , g 2 . The remaining entries a in are given by algebra, i.e., using (c i , c 6 ) = 0 for i = 3, 4, 5, 6. This illustrates that our algorithm does not always give an injective map, indeed we may replace ν 1 and ν 2 by ν 1 + c and ν 2 − c for any constant c. Also, although it is surjective locally as ν 1 and ν 2 can be found by integration from a 13 and a 14 , it is not globally surjective. For example, if M = S 2 , g 1 = g 2 = z and a 13 = −a 14 = 1/z, then ν 1 = −ν 2 = (1/z)dz = log z which is not globally defined, though a 15 = 0 is. However, we can modify our algorithm for this case as follows. Replace the final new parameters ν 1 and ν 2 by ν 1 , ν 2 with a 13 = ν 1 and a 15 = ν 2 + λν 3 , then we obtain (which holds even if g 2 is constant) where the remaining entries a i j are calculated by algebra, as usual. The resulting harmonic maps are described by Proposition 4.5(i). (d) Type (1,2,2,1). This has r = 3 and is obtained from type (2, 2) by adding a border; it has two new initial parameters ν 1 1 , ν 1 2 in the g 1 , i.e., block superdiagonal positions a 12 , a 13 , and two further parameters ν 3 , ν 4 on the second block superdiagonal. Carrying out our algorithm in the non-degenerate case when g is non-constant replaces ν 1 1 , ν 1 2 by ν 1 , ν 2 giving Here our final parameters g, ν 1 , ν 2 , ν 3 , ν 4 are arbitrary meromorphic functions, and all generalized derivatives are taken with respect to g. The top-right entry ζ 0 + λζ 1 is determined by algebra from (c 6 , c 6 ) = 0, in fact, ζ 0 = ν (1) This is the extended solution of a map into a Grassmannian if and only if ν 3 ≡ ν 4 ≡ 0; in that case we have an S 1 -invariant extended solution: are the subbundles given by δ 1 = span{c 0 6 }, δ 2 = span{c 0 6 , c 0 5 , c 0 4 } and δ 3 = span{c 0 6 , c 0 5 , c 0 4 , c 0 3 , c 0 2 }. Note that δ 3 is the polar of δ 1 and δ 2 is self-polar, i.e., maximally isotropic. As in Sect. 3.1, the corresponding harmonic map ϕ 0 is ψ 0 ⊕ ψ 2 where ψ i = δ ⊥ i ∩ δ i+1 , or its orthogonal complement ψ 1 ⊕ ψ 3 . Since these are conjugates of each other, ϕ 0 is a harmonic map into O(6)/U(3).
(e) Type (1,1,2,1,1) This has r = 4 and, like type (2,2,2) above, is obtained from n = 4, type (1, 2, 1) by adding a border. However, due to the special nature of SO(4) as being double-covered by the product of SU(2) with itself, there is an easier way which involves first finding the new last column of A then filling in the top-right element and new first row by algebra (see Sect. 3.3); for the S 1 -invariant case this is as follows, with all generalized derivatives with respect to g 1 : Write the last column as [1, χ 1 , χ 2 , χ 3 , χ 4 , ζ ] T . From the extended solution equation (2.20) we have the following, assuming that g 1 is non-constant.
Thus we obtain, with generalized derivatives taken with respect to g 1 , Here g 1 , h 1 , h 2 are arbitrary meromorphic functions. If h (1) 1 is non-constant, then g 2 = h (2) 2 /h (2) 1 . Note how this departs from our usual algorithm by replacing a parameter in the middle 4 × 4 matrix A, in this case g 2 by h 2 . Note that the parameters g 1 , h 1 , h 2 can be read off from the matrix A as entries, or combinations of entries. Note also that the middle four entries of the last column give the standard formula for null curves in C 4 , see Sect. 5.2. Proof As in Sect. 3.1, ϕ has a symmetric extended solution = [Aγ ξ ] with r odd. By Remark 3.5, if is not S 1 -invariant then A must contain a term in λ 2 . By Lemma 3.3 this means that, either r = 3 with t 1 > 1, or r ≥ 5. Given that r i=1 t i = 2m, neither of these is possible with m ≤ 3.
That this result is sharp is shown by the following example which is a particular case of [34,Example 6.26]. In that paper, reality conditions had to be solved: this was only done for m ≥ 5; by using our approach, the reality conditions in that example are automatic and give us an example for m = 4. Explicitly, take ξ of type (2, 2, 2, 2). By our method we may construct a solution A : M → A R 3 in the form A = A 0 + λ 2 A 2 where the penultimate entry of the top row of A 2 is a freely chosen parameter ν. Completing the matrix A by algebra and setting = [Aγ ξ ] gives an extended solution which is S 1 -invariant if and only if ν ≡ 0.

Null curves, extended solutions and the Weierstrass representation
By a (generalized) minimal surface in R n we mean a non-constant weakly conformal map from a Riemann surface M to R n whose image is minimal away from branch points, equivalently, a weakly conformal harmonic map. Such a map is, on a simply connected domain, the real part of a null holomorphic curve by which is meant (somewhat confusingly) a holomorphic map χ : M → C n with (χ , χ ) = 0 and χ not identically zero. We extend this definition to null meromorphic curve: note that for such a curve, [χ ] : M → Q n−2 is a welldefined holomorphic map to the complex quadric and gives the Gauss map of the minimal surface. The usual Weierstrass representation parametrizes all such χ so that χ is given by an integral with real part the minimal surface. In contrast, in the Weierstrass representation in free form, the null curve itself is parametrized and no integral is necessary. We see how this is related to our work.
Composing the above bijections we deduce the Weierstrass representation in free form of null meromorphic curves:

Corollary 5.2 There is a bijection between the following sets:
(i) the set of pairs of meromorphic functions (g, ν) on M with g and ν (2) non-constant, (ii) the set of null meromorphic curves χ : M → C 3 with [χ ] : M → Q 1 non-constant, given by χ = (ν (2) , −ν (1) + gν (2) , −ν + gν (1) − 1 2 g 2 ν (2) ). (5.3) Recall that minimal surfaces in R 3 appear as the real part of such curves χ. The representation (5.3) seems to have been first given by Weierstrass [38]; explanations are given by Hitchin [24] and Small [32]. The new feature in our work is the correspondence with extended solutions for harmonic maps, specifically, the free Weierstrass data (g, ν) of χ is given simply by the two entries (5.1) of the matrix A associated to χ by (5.2), and this matrix defines an extended solution = [Aγ ξ 0 ] for a harmonic map M → O(5).

Null curves in C 4 and extended solutions
Theorem 5.1 has an analogue in C 4 as follows. For a null curve χ = (χ 1 , χ 2 , χ 3 , χ 4 ) : M → C 4 , by definition, χ is not identically zero, so by permuting coordinates if necessary, we can assume that χ 1 is non-constant. Then we can set g 1 = χ 2 /χ 1 and g 2 = χ 3 /χ 1 so that [χ ] = [1, g 1 , g 2 , −g 1 g 2 ] and [χ ] is non-constant if and only if at least one of the Gauss maps g 1 or g 2 is non-constant; again, after permuting coordinates, if necessary, we can assume that g 1 is non-constant. By A non-degenerate we shall now mean that a i,i+1 is non-constant for i = 3. The extended solutions in (ii) below are polynomial extended solutions for harmonic maps into O (6), and, as in the C 3 case, are of type (1, 1, 2, 1, 1), and so of the maximum possible uniton number, 4; the corresponding canonical element is ξ = i diag(4, 3, 2, 2, 1, 0).

Corollary 5.4
There is a bijection between the following sets: Again minimal surfaces in R 4 appear as the real part of such χ. This seems to have been first given by de Montcheuil [26], see also Eisenhart [19]; explanations are given by Small [33] and Shaw [31]. As before, the free Weierstrass data (g 1 , h 1 , h 2 ) of χ are given very simply by (5.4) from the entries of the matrix A associated to χ by (5.5), and this matrix defines an extended solution = [Aγ ξ ] for a harmonic map M → O(6).
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