Weierstrass type representation of harmonic maps into symmetric spaces^{∗}

J. Dorfmeister^{†} Mathematics Department University of Kansas Lawrence, Kansas 66045 E-mail: dorfmeister@kuhub.cc.ukans.edu

F. Pedit Department of Mathematics University of Massachusetts Amherst, MA 01003 E-mail: franz@gang.umass.edu

H. Wu Department of Mathematics Northern Illinois University DeKalb, IL 60115 E-mail: wu@math.niu.edu

1 Introduction

Over the past five years substantial progress has been made in the understanding of harmonic maps f: M → G/K of a compact Riemann surface M into a compact symmetric space G/K [29, 30, 12, 14, 17, 22, 5, 7, 10, 31]. When M is the Riemann sphere all such maps may be obtained from holomorphic curves into some associated twistor space [11, 29, 14]. In contrast, if M is a 2-torus, then every harmonic map ϕ: M → G/K (satisfying specific nondegeneracy assumptions) may be obtained from a solution to a certain family of completely integrable finite dimensional systems of Hamiltonian ODE in Lax form on a loop algebra [5, 7, 17]. Such harmonic maps are called harmonic maps of finite type and, e.g., all harmonic 2-tori in S^{n}, ℂP^{n} are accounted for in this way [4]. So far there is no comparable systematic theory when M has higher genus. Nevertheless there are examples: the harmonic maps f: M → S^{2} arising as Gauss maps of Kapouleas’ [18, 19] constant mean curvature surfaces of genus g ≥ 2 and the minimal (and hence harmonic) maps f: M → S^{3} given by Lawson’s [20] minimal surfaces of genus g ≥ 2.

The objective of the present paper is to develop a systematic scheme for the construction of all harmonic maps f: M → G/K when M is simply connected. Since the case where M is the Riemann sphere has been accounted for completely in the work of Uhlenbeck [29], we think of M as the universal cover of some compact Riemann surface of genus ≥ 1. Our method has its origins in what is known as the Symes formula for integrating Lax pair equations [28]. Analogs of this formula have been used to construct (extended solutions of) harmonic 2-spheres in Lie groups [14] and to construct (extended lifts of) finite type harmonic maps in compact symmetric spaces G/K [10, 7]. The harmonic maps so obtained have the property that the (1, 0)-part of the derivative, ∂f, takes values in a single adjoint orbit.

∗Partially supported by NSF grant DMS-9205293 and SFB 288 at Technische Universität Berlin.

†Partially supported by KITCS grant OSR-9255223.

A standard holomorphic differential argument, namely Ad-invariant polynomials on the Lie algebra 𝔤 of G evaluated on ∂f provide holomorphic differentials on M, together with a nondegeneracy assumption, ensure this orbit condition when M is the 2-sphere or a 2-torus. This is the starting point for the analysis of harmonic 2-tori in symmetric spaces in [5].

Since on a genus ≥ 2 Riemann surface M these holomorphic differentials may have isolated zeros, the derivative ∂f no longer stays in a fixed level set of Ad-invariant polynomials and thus such a harmonic map f: M → G/K cannot, in general, be of finite type. In the simplest case where G/K = SU(2)/S^{1} is the 2-sphere the Killing form ( , ) is (up to scale) the only invariant polynomial. Thus, for a harmonic map f: M → S^{2}, either (∂f, ∂f) ≡ 0, in which case f is ± holomorphic (this is always the case if M has genus 0) and ∂f is nilpotent, or (∂f, ∂f) has isolated zeros and away from these zeros ∂f is regular semisimple. If M has genus one (∂f, ∂f) is nowhere zero, i.e., f is nowhere conformal, ∂f is regular semisimple and thus of finite type by the results in [5]. Over a genus ≥ 2 Riemann surface ∂f will change orbit type at isolated points.

To account for such a behavior we use a Weierstrass-type representation for harmonic maps f: M → G/K which results from a variation of the Symes formula. We start with a holomorphic 1-form μ = ∑_{i≥−1} λ^{i}μ_{i} taking values in a subspace of a twisted loop algebra. Integrating g^{−1}dg = μ over the (simply connected) Riemann surface M gives a map g with values in a corresponding twisted loop group. Such loop groups allow decompositions of Iwasawa type into real loops and loops extending holomorphically to the interior of the λ-disc. Factorization of g according to this decomposition,

g = F b,

yields an (extended) frame of the harmonic map f = π ∘ F_{|_{λ=1}}: M → G/K whose derivative is given by

∂f = AdF_{|_{λ=1}}(μ_{−1}) .

Conversely we prove that, if M is not the Riemann sphere S^{2}, every harmonic map f: M → G/K is obtained from some holomorphic 1-form μ = ∑_{i≥−1} λ^{i}μ_{i} on the universal covering. If μ = ηdz for some suitable constant η = ∑_{k≥−1} λ^{k}η_{k} we recover the finite type harmonic maps of [5]. Finally, using a Riemann-Hilbert splitting we show that any harmonic map f: M → G/K may be obtained by the above construction from a meromorphic 1-form μ = λ^{−1}η where η takes values in the finite dimensional Lie algebra 𝔤ℂ. The set of poles of η is given as the locus where the Riemann-Hilbert factorization fails to exist. In a certain way this justifies to view the meromorphic 1-forms μ as Weierstrass-type data for harmonic maps.

We now give a brief description of the contents of the various sections in this paper: we start in section 2 by introducing the necessary loop groups of maps g: S^{1} → G where G is a compact semisimple Lie group. Using existing factorization theorems [23] we adopt these results to the twisted setting, i.e., spaces of loops g: S^{1} → G which have g(e^{2πi/k}λ) = σg(λ) for some order k automorphism σ of G. Further, we review the Grassmannian model, the determinant line bundle and its relation to the loop group factorizations. This section, containing mostly extensions to the twisted setting of material well documented, has more the flavor of an appendix. Since it also contains some basic setups and notations used throughout the paper we decided to leave it as an initial section. Crucial for the understanding of the paper is mainly the material up to theorem 2.3 and lemma 2.6. We suggest that the Reader, after familiarizing himself with the basic notational conventions, skips this section altogether and only goes back to it when prompted in the paper. The technicalities in this section are well-known to the expert and non-experts on this subject do not need to go into them to understand the geometrical constructions carried out in the paper.

In section 3 we reformulate the harmonic map equations for maps f: M → G/K of a Riemann surface M into a compact symmetric space G/K as the flatness of a certain loop algebra valued 1-form. The zero curvature equations in this case are the Yang-Mills-Higgs equations [16, 7]. Integrating this loop of 1-forms gives a loop of frames F_{λ}: M → G of harmonic maps f_{λ}: M → G/K, i.e., f_{λ} = π ∘ F_{λ}, π: G → G/K the coset projection, with f = f_{1}, which we call, following Uhlenbeck [29], an extended frame (or lift).

Section 4 deals with the construction of extended frames using the loop group factorizations discussed in section 2 together with a version of the Symes method as described above. We relate our approach to the construction of finite type harmonic maps [5], the Adler-Kostant-Symes scheme [1, 7] and the r-matrix scheme. Our construction, assigning an extended frame to a holomorphic 1-form μ = ∑_{i≥−1} λ^{i}μ_{i} is equivariant with respect to natural loop group actions by dressing on the extended frames and by gauge transformations on the 1-forms μ. We briefly discuss how this can be used to study dressing orbits of harmonic maps [6] in the spirit of Segal-Wilson’s study of the KdV equation [26]. We conclude the paper by applying the general theory to the special case of harmonic maps f: M → S^{2}, the Gauss maps of constant mean curvature surfaces in ℝ^{3}.

Parts of this work were done while various of the authors visited the SFB 288 at Technische Universität Berlin and the KITCS at the University of Kansas. We thank the members of both institutions for their interest and support in this project. We also wish to thank Fran Burstall for his many discussions and suggestions of improvements while the paper was written. The pictures at the end of the paper were produced using the “Mr. Bubble” module written by Charlie Gunn at SFB-288 TU-Berlin.

2 Loop groups, Grassmannians and the τ-function

The main technical tool for our study of harmonic maps will be the existence of various loop group factorizations. We will give a self-contained overview of the basic results, referring the reader to [23] for details and derive the factorization theorems we need from there. So let us fix the following ingredients:

(i) a compact (connected) semisimple Lie group G with Lie algebra 𝔤,

(ii) an automorphism σ: G → G of order k and fixed point set G^{σ} = K. We denote the induced automorphism on 𝔤 again by σ, so that 𝔤^{σ} = 𝔨 is the Lie algebra of K,

(iii) the primitive k-th root of unity ω = e^{2πi/k}, (iv) an Iwasawa decomposition Kℂ = K · B, K ∩ B = {e} where B ⊂ Kℂ is a

solvable subgroup, with corresponding decomposition 𝔨ℂ = 𝔨 ⊕ 𝔟.

With this at hand, we define

ΛGℂ_{σ} = {g: S^{1} → Gℂ; g(ωλ) = σg(λ), λ ∈ S^{1}}

and equip it with some H^{s}-topology for s > 1/2. Then ΛGℂ_{σ} becomes a complex Banach Lie group with Lie algebra

Λ𝔤ℂ_{σ} = {ξ: S^{1} → 𝔤ℂ; ξ(ωλ) = σξ(λ), ξ is H^{s} −smooth}.

ΛGℂ_{σ} has several closed Lie subgroups of importance to us: the real form

ΛG_{σ} = {g ∈ ΛGℂ_{σ}; g(λ) ∈ G, λ ∈ S^{1}} ,

the loops extending holomorphically to the interior of the unit disc D = {λ ∈ ℂ; |λ| < 1} taking values in a fixed subgroup H ⊂ Kℂ at λ = 0,

Λ_{H}^{+}Gℂ_{σ} = {g ∈ ΛGℂ_{σ}; g extends holomorphically to D, g(0) ∈ H}

and the loops extending holomorphically to the exterior of the unit disc E = {λ ∈ ℂ; λ > 1} taking values in a fixed subgroup H ⊂ Kℂ at λ = ∞,

Λ_{H}^{−}Gℂ_{σ} = {g ∈ ΛGℂ_{σ}; g extends holomorphically the E, g(∞) ∈ H}.

In case H = Kℂ we omit the subscript H and we use the subscript ∗ for H = {e}, in which case we deal with based loops. Denote the ω^{i}-eigenspace of σ on 𝔤ℂ by 𝔤_{i}. Clearly, 𝔤_{0} = 𝔨ℂ, 𝔤_{−i} = \overline{𝔤_{i}} and [𝔤_{i}, 𝔤_{j}] ⊆ 𝔤_{i+j} (with arithmetic modulo k), where conjugation is always with respect to the real form 𝔤. Then, if ξ ∈ Λ𝔤ℂ_{σ}, its Fourier decomposition is

ξ = ∑_{i∈ℤ} λ^{i}ξ_{i} , ξ_{i} ∈ 𝔤_{i}

and the (closed) Lie subalgebras of Λ𝔤ℂ_{σ} corresponding to the subgroups ΛG_{σ}, Λ_{H}^{+}Gℂ_{σ},

Λ_{H}^{−}Gℂ_{σ} are

Λ𝔤_{σ} = \{ξ ∈ Λ𝔤ℂ_{σ}; ξ_{−i} = \overline{ξ}_{i}\} ,
Λ_{𝔥}^{+}𝔤ℂ_{σ} = {ξ ∈ Λ𝔤ℂ_{σ}; ξ_{i} = 0 for i < 0, ξ_{0} ∈ 𝔥} ,
Λ_{𝔥}^{−}𝔤ℂ_{σ} = {ξ ∈ Λ𝔤ℂ_{σ}; ξ_{i} = 0 for i > 0, ξ_{0} ∈ 𝔥} ,

where 𝔥 is the Lie algebra of H. If ξ ∈ Λ𝔤ℂ_{σ} then ξ = ξ^{−} + ξ^{0} + ξ^{+} where

ξ^{−} = ∑_{i<0} λ^{i}ξ_{i}, ξ^{0} = ξ_{0} ∈ 𝔨ℂ and ξ^{+} = ∑_{i>0} λ^{i}ξ_{i} .

For η ∈ 𝔨ℂ let η_{𝔨} and η_{𝔟} denote the projections along 𝔨ℂ = 𝔨 ⊕ 𝔟. Then we have

ξ = (ξ^{−} + \overline{ξ^{−}} + (ξ^{0})_{𝔨}) + (ξ^{+} − \overline{ξ^{−}} + (ξ^{0})_{𝔟})
= ξ^{−} + (ξ^{0} + ξ^{+})
= (ξ^{−} + ξ^{0}) + ξ^{+}

where conjugation is with respect to the real form Λ𝔤_{σ}, i.e., \overline{ξ} = ∑ λ^{i}\overline{ξ}_{−i}. From this we immediately obtain

Lemma 2.1.

(i)  Λ𝔤ℂ_{σ} = Λ𝔤_{σ} ⊕ Λ_{𝔟}^{+}𝔤ℂ_{σ},

(ii)  Λ𝔤ℂ_{σ} = Λ_{∗}^{−}𝔤ℂ_{σ} ⊕ Λ^{+}𝔤ℂ_{σ} = Λ^{−}𝔤ℂ_{σ} ⊕ Λ_{∗}^{+}𝔤ℂ_{σ} .

In what follows we will need global analogs of the above decompositions.

Let us denote by ΛGℂ, Λ_{H}^{+}Gℂ, Λ_{H}^{−}Gℂ and ΛG the groups ΛGℂ_{σ} etc. for σ= id and let Gℂ = G · B̃ be an Iwasawa decomposition of Gℂ. Then Pressley-Segal [23] prove the following factorization results (Pressley and Segal consider smooth loops but remark that the result holds for H^{s}-loops, s > \frac{1}{2}, as well):

Theorem 2.2.

(i) Multiplication ΛG × Λ+ ; _{B}Gℂ → ΛGℂ is a diffeomorphism onto. In particular,

every g ∈ ΛGℂ has a unique decomposition

g = ab,

where a ∈ ΛG, b ∈ Λ+ ; _{B}Gℂ.

(ii) Multiplication Λ_{∗}^{−}Gℂ × Λ^{+}Gℂ → ΛGℂ is a diffeomorphism onto the open and dense subset Λ_{∗}^{−}Gℂ ·Λ^{+}Gℂ, called the “big cell”. In particular, if g ∈ ΛGℂ is contained in the big cell then g has a unique decomposition

g = g_{−}g_{+},

where g_{−} ∈ Λ_{∗}^{−}G and g_{+} ∈ Λ^{+}Gℂ. The analogous result holds for the multiplication map Λ^{−}Gℂ × Λ_{∗}^{+}Gℂ → ΛGℂ.

From this we can relatively easily deduce the twisted loop group factorizations corresponding to the Lie algebra decompositions in Lemma 2.1.

Theorem 2.3.

(i) Multiplication ΛG_{σ} × Λ_{B}^{+}Gℂ_{σ} → ΛGℂ_{σ} is a diffeomorphism onto.

(ii) Multiplication Λ_{∗}^{−}Gℂ_{σ} × Λ^{+}Gℂ_{σ} → ΛGℂ_{σ} is a diffeomorphism onto the open and dense set Λ_{∗}^{−}Gℂ_{σ} · Λ^{+}Gℂ_{σ}, again called the big cell. The analogous result holds for the multiplication map Λ^{−}Gℂ_{σ} × Λ_{∗}^{+}Gℂ_{σ} → ΛGℂ_{σ}.

Proof. (i) Since the multiplication map of Theorem 2.2 (i) restricts to a diffeomorphism ΛG_{σ}×Λ^{+}Gℂ_{σ} → ΛGℂ_{σ} into, it remains to verify the surjectivity of this map. If g ∈ ΛGℂ_{σ} then by Theorem 2.2 (i) g = ab with a ∈ ΛG and b ∈ Λ+ ; _{B}Gℂ. Thus, letting (σg)(λ) = σ(g(ω^{−1}λ)),

ab = g = σ(g) = σ(a)σ(b)

and, since σ(ΛG) ⊆ ΛG, σ(Λ^{+}Gℂ) ⊆ Λ^{+}Gℂ and ΛG ∩ Λ^{+}Gℂ = G, we obtain

a^{−1}σ(a) = bσ(b)^{−1} = u ∈ G

so that

σ(a) = au, σ(b) = u^{−1}b .  (2.1)

Using the unique decomposition Λ^{+}Gℂ = Gℂ · Λ_{∗}^{+}Gℂ we write

b = b_{0}h

and get from (2.1)

σ(b) = σ(b_{0})σ(h) = u^{−1}b = (u^{−1}b_{0})h

and thus, since σ(Λ_{∗}^{+}Gℂ) ⊆ Λ_{∗}^{+}Gℂ,

σ(b_{0}) = u^{−1}b_{0}, σ(h) = h .  (2.2)

Now we express b_{0} ∈ Gℂ in its polar decomposition [15] b_{0} = r exp(ix), where r ∈ G and x ∈ 𝔤. Applying σ we get

σ(b_{0}) = u^{−1}b_{0} = σ(r) exp(iσ(x)) = (u^{−1}r) exp(ix)

which implies

σ(r) = u^{−1}r, exp(iσ(x)) = exp(ix) .  (2.3)

Now let Kℂ = K · B be the Iwasawa decomposition for Kℂ and decompose

exp(ix) = kv ∈ K · B .

Define

ã = ark, b̃ = vh

then, by (2.1), (2.2), (2.3) and the fact that Kℂ = (Gℂ)^{σ},

\begin{matrix}σ(ã) = σ(a)σ(r)σ(k) = (au)(u^{−1}r)k = ark = ã, \\ σ(b̃) = σ(v)σ(h) = vh = b̃, \end{matrix}

which shows that ã ∈ ΛG_{σ} and, since v ∈ B, h ∈ Λ_{∗}^{+}Gℂ, that b̃ ∈ Λ_{B}^{+}Gℂ_{σ}. Moreover

ãb̃ = (ark)(vh) = ar exp(ix)h = a(b_{0}h) = ab = g,

which proves surjectivity of the multiplication map ΛG_{σ} × Λ_{B}^{+}Gℂ_{σ} → ΛGℂ_{σ}.

(ii) This follows from Theorem 2.2.(ii) and the fact that

Λ_{∗}^{−}Gℂ_{σ} · Λ^{+}Gℂ_{σ} = (Λ_{∗}^{−}Gℂ · Λ^{+}Gℂ) ∩ ΛGℂ_{σ} .

□

Another ingredient necessary for our studies of harmonic maps is the Grassmannian model for the loop group and its determinant line bundle [23], [26] which we review briefly in a language suitable for our developments. Since a compact semi-simple Lie group can be represented as a subgroup of the unitary group via the adjoint representation we begin with the loop groups ΛU(n) and its complexification ΛGL(n, ℂ). Let H = L^{2}(S^{1}, ℂ) with the “polarization”

H = H_{+} ⊕ H_{−} ,  (2.4)

where H_{+} is the subspace generated by {λ^{k}; k ≥ 0} and H_{−} is generated by {λ^{k}; k < 0}. Note that ΛGL(n, ℂ) acts on L^{2}(S^{1}, ℂ^{n}) by bounded linear operators so that, using an appropriate isometry between L^{2}(S^{1}, ℂ^{n}) and H, we obtain an injective holomorphic group homomorphism

ΛGL(n, ℂ) ↪ GL(H): g ↦ A_{g}  (2.5)

of the loop group into the space of bounded linear operators on H. In terms of the natural basis …, λ^{2}, λ, 1, λ^{−1}, λ^{−2}, … of H_{+} ⊕ H_{−} this inclusion is expressed by infinite matrices: let

g = ∑_{k∈ℤ} g_{k}λ^{k}

be the Fourier decomposition of g ∈ ΛGL(n, ℂ), where g_{k} ∈ M_{n}(ℂ), then the (i, j)-th n × n-block in A_{g} is given by

(A_{g})_{i,j} = g_{i−j} .  (2.6)

Writing elements A ∈ GL(H) in block form with respect to the polarization H = H_{+} ⊕ H_{−},

A = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ,  (2.7)

it follows from [23], [26] that the inclusion (2.5) takes values in

GL_{res}(H) = {A ∈ GL(H); a, d Fredholm , b, c Hilbert-Schmidt} .  (2.8)

The identity component of ΛGL(n, ℂ) is mapped into the identity component of GL_{res}(H) characterized by the condition that the Fredholm operators a, d have index zero. Without further mentioning, we will work in these connected components. The Grassmannian of subspaces of H, comparable to H_{+}, is the orbit

Gr(H) = GL_{res}(H) · H_{+} ,  (2.9)

which is a complex homogeneous space on which GL_{res}(H) acts holomorphically. One can make sense of the “top exterior power” of W ∈ Gr(H) [23], [26] and thus obtain a holomorphic line bundle

Det → Gr(H) ,

the determinant line bundle. Its dual Det^{∗} → Gr(H) has a non-trivial canonical holomorphic section

τ: Gr(H) → Det^{∗}

given by

τ(W) = (w, det w_{+})  (2.10)

where

w = ( \begin{matrix}w_{+} \\ w_{−} \end{matrix} ): H_{+} → H  (2.11)

is the ℤ × ℕ-matrix consisting of the columns of an admissible basis [26] for W. Consider the composition of holomorphic maps

ΛGL(n, ℂ) ↪ GL_{res}(H) → Gr(H): g ↦ A_{g} · H_{+} .

Lemma 2.4. τ(A_{g} · H_{+}) ≠ 0 if and only if g ∈ ΛGL(n, ℂ) is contained in the big cell (c.f. Theorem 2.2(ii)), i.e., g = g_{−}g_{+} with g_{−} ∈ Λ_{∗}^{−}GL(n, ℂ), g_{+} ∈ Λ^{+}GL(n, ℂ).

Proof. From Birkhoff’s factorization theorem [23] we know that

g = g_{−}dg_{+}

with g_{−} ∈ Λ_{∗}^{−}GL(n, ℂ) and d = diag(λ^{a_{1}}, …, λ^{a_{n}}), where a_{i} ∈ ℤ satisfy a_{1} ≥ ⋯ ≥ a_{n} and ∑_{i=1}^{n} a_{i} = 0. Then,

τ(A_{g} · H_{+}) = τ(A_{g_{−}}A_{d}A_{g_{+}} · H_{+}) = τ(A_{g_{−}}A_{d} · H_{+}),

since A_{g_{+}} · H_{+} = H_{+}. Because g_{−} ∈ Λ_{∗}^{−}GL(n, ℂ) we have by (2.6), (2.7)

A_{g_{−}} = ( \begin{matrix}a_{−} & 0 \\ b_{−} & c_{−} \end{matrix} ) ,

where a_{−}, c_{−} are lower diagonal with I_{n}-blocks in the diagonal. Let w = ( \begin{matrix}w_{+} \\ w_{−} \end{matrix} ) be an

admissible basis for A_{d} · H_{+}. Then

v = g_{−} ( \begin{matrix}w_{+} \\ w_{−} \end{matrix} ) a_{−} = ( a_{−}w_{+}a_{−}^{−1}_{∗} )

is an admissible basis [26] of A_{g_{−}}A_{d}H_{+} so that by (2.10)

τ(A_{g} · H_{+}) = (v, det(a_{−}w_{+}a_{−}^{−1})) = (v, det w_{+}) .

Since d = λ^{a_{1}}E_{a_{1}} + ⋯ + λ^{a_{n}}E_{a_{n}}, where E_{a_{k}} = diag(0, …, 1, … 0) with 1 in the a_{k}-th place, we get from (2.6) that the (i, j)-th block in A_{d} is given by

(A_{d})_{i,j} = \{ \begin{matrix}E_{a_{k}} & for & i − j = a_{k} \\ 0 & else . \end{matrix}  (2.12)

Writing

A_{d} = ( \begin{matrix}α & β \\ γ & δ \end{matrix} )

according to (2.7) we see that det w_{+} ≠ 0 if and only if α is invertible. Assuming a_{n} < 0 we see that α is not invertible so that det w_{+} = 0. Conversely, if d = I_{n}, then w_{+} = Id for the natural choice of w, and so det w_{+} ≠ 0. □

If G is a semisimple compact Lie group then G ↪ U(n) and hence Gℂ ↪ GL(n, ℂ), so that we obtain a holomorphic inclusion ΛGℂ_{σ} ↪ ΛGL(n, ℂ).

Corollary 2.5. Let g ∈ ΛGℂ_{σ}. Then g = g_{−}g_{+} is in the big cell (c.f. Theorem 2.3(ii)) if and only if τ(A_{g}H_{+}) ≠ 0.

As a further application, which will be needed in section 4, we show a certain meromorphic behavior in the factorization of a holomorphic ΛGℂ_{σ}-valued curve across the boundary of the big cell. Let D ⊂ ℂ be some domain containing 0 ∈ ℂ and

g: D → ΛGℂ_{σ}

a holomorphic (i.e., \overline{∂}g = 0) curve with g(0) = e.

Since τ: ΛGℂ_{σ} → Det^{∗} and g: D → ΛGℂ_{σ} are holomorphic, we obtain from Corollary 2.5 that the set

S = (τ ∘ g)^{−1}(0) = {p ∈ D; g(p) not contained in the big cell} ⊂ D

is discrete. Let p_{0} ∈ S and U ⊂ D be a neighborhood of p_{0} so that (c.f. Theorem 2.3(ii))

g(p) = g_{−}(p)g_{+}(p)

for p ∈ U \ {p_{0}} where g_{±}: U \ {p_{0}} → Λ^{±}Gℂ_{σ} are holomorphic. Then we have

Lemma 2.6. g_{−} extends meromorphically into p_{0}, i.e., g_{−}(λ): U \{p_{0}} → Gℂ has a pole of at most order N independent of λ ∈ S^{1} at p_{0}. Since g is holomorphic the same statement holds for g_{+}.

Proof. Throughout the proof we will not distinguish between operators on H = H_{−} ⊕ H_{+} and their representation as infinite matrices with respect to the standard basis …, λ^{2}, λ, 1, λ^{−1}, λ^{−2}, . . .. The map g: D → ΛGℂ_{σ} gives rise (2.9), (2.5) to a holomorphic map

W = g · H_{+}: D → Gr(H) ,

which, on U \ {p_{0}}, is given by

W = g_{−} · H_{+} .

If g_{−} = e + ∑_{k≤−1}(g_{−})_{k}λ^{k} then (2.7), (2.6)

A_{g_{−}} = ( \begin{matrix}a_{−} & 0 \\ c_{−} & d_{−} \end{matrix} )

with a_{−} lower diagonal and (a_{−})_{ii} = I_{n}. In particular, a_{−}: H_{+} → H_{+} is invertible. Thus,

W (p) = [ \begin{matrix}a_{−}(p) \\ c_{−}(p) \end{matrix} ] = [ \begin{matrix}Id \\ c_{−}(p)a_{−}^{−1}(p) \end{matrix} ] ,  (2.13)

where [ ] denotes the subspace generated by the columns. Using (2.6) and the special form of a_{−} we obtain

(c_{−}a_{−}^{−1})_{0,k} = (c_{−})_{0,k} = (g_{−})_{k}, k ≤ −1 .  (2.14)

We now choose a chart around W_{0} = A_{g(p_{0})} · H_{+}: since by assumption τ(W_{0}) = 0, a chart is given by [9]

\{[ \begin{matrix}Id − yx \\ −x \end{matrix} ]; x: H_{+} → H_{−} Hilbert-Schmidt\},

where y: H_{−} → H_{+} is a “finite” matrix, i.e.,

y = ( \begin{matrix}0 & 0 \\ f & 0 \end{matrix} )

for some f ∈ M_{m}(ℂ), m ∈ ℕ. In this chart we express W as

\begin{aligned}W (p) = A_{g_{−}(p)} · H_{+} & = [ \begin{matrix}Id − yx(p) \\ −x(p) \end{matrix} ] \\ & = [ \begin{matrix}Id \\ −x(p)(Id − yx(p))^{−1} \end{matrix} ] , p ∈ U \ \{p_{0}\} , \end{aligned}  (2.15)

where x depends holomorphically on p ∈ U. Since y is a finite matrix we have

Id − yx(p) = ( \begin{matrix}Id & 0 \\ α(p) & β(p) \end{matrix} ): H_{+} → H_{+}

with β: U → GL(m, ℂ) holomorphic and thus

(Id − yx(p))^{−1} = ( \begin{matrix}Id & 0 \\ −β^{−1}(p)α(p) & β^{−1}(p) \end{matrix} ) .

Comparing (2.13) with (2.15) and using (2.14) we obtain

(g_{−})_{k} = (−x(Id − yx)^{−1})_{0,k}, k ≤ −1 ,

whose pole behavior at p_{0} is determined by the behavior of β^{−1} at p_{0}. But β^{−1}: U \ {p_{0}} → GL(m, ℂ) is holomorphic with pole at p_{0} given by \frac{1}{det β}. □

3 Loop group formulation of harmonic maps into symmetric spaces

In this section we review what may be called the “Yang-Mills-Higgs” formulation of harmonic maps into a symmetric space [16, 7], which will lead to a loop group formulation similar to Uhlenbeck’s [29] for harmonic maps into a Lie group.

Let N = G/K be a Riemannian symmetric space with symmetric involution σ: G → G so that G^{σ} ⊃ K ⊃ (G^{σ})_{0}. We denote by 𝔤 and 𝔨 the Lie algebras of G and K respectively. The decomposition of 𝔤 into ±1-eigenspaces of the derivative of σ gives the Cartan decomposition

𝔤 = 𝔨 ⊕ 𝔭  (3.1)

with the familiar commutation relations [𝔨, 𝔨] ⊆ 𝔨, [𝔨, 𝔭] ⊆ 𝔭 and [𝔭, 𝔭] ⊆ 𝔨. Recall that there is a canonical identification of the tangent bundle T N of N with a subbundle of the trivial bundle N × 𝔤: for x = gK ∈ N the map 𝔤 → T_{x}N given by η ⟼ \frac{d}{dt}_{|_{t=0}} exp(tη)x is a linear isomorphism on 𝔭_{x} = Adg(𝔭). Inverting this at each point x ∈ N gives a 𝔤-valued 1-form β: T N → 𝔤 which identifies T N with the adjoint bundle [𝔭] ⊂ N × 𝔤 whose fibre over x = gK ∈ N is 𝔭_{x}. Under this identification the Levi-Civita connection of N is just flat differentiation in N × 𝔤 followed by projection along [𝔨] (defined in the same way as [𝔭]) onto [𝔭], [8], i.e.,

β(∇^{N}_{X} Y) = (d_{X}β(Y))^{[𝔭]} .  (3.2)

Let π: G → G/K be the canonical projection and denote by θ: T G → 𝔤 the (left) Maurer-Cartan form of G. Then, according to (3.1),

θ = θ_{0} + θ_{1}  (3.3)

decomposes into a 𝔨-valued 1-form θ_{0} and a 𝔭-valued 1-form θ_{1}. θ_{0} is the canonical connection of the principal K-bundle G → G/K and θ_{1} is related to β by

(π^{∗}β)_{g} = Adg(θ_{1})_{g} .  (3.4)

Now let M be a Riemann surface and f: M → G/K a smooth map with lift F: M → G so that π ∘ F = f. Then the pullback

α = F^{−1}dF = F^{∗}θ = F^{∗}θ_{0} + F^{∗}θ_{1} = α_{0} + α_{1}  (3.5)

splits into a 𝔨 resp. 𝔭-valued 1-form on M and so does the Maurer-Cartan equation dα + 1/2[α ∧ α] = 0:

dα_{0} + 1/2[α_{0} ∧ α_{0}] = −1/2[α_{1} ∧ α_{1}] ,  (3.6a)
dα_{1} + [α_{0} ∧ α_{1}] = 0 .  (3.6b)

Conversely, every pair of 𝔨 resp. 𝔭-valued 1-forms α_{0}, α_{1} solving (3.6) integrate, if M is simply connected, to a map F: M → G which is unique up to left translation by G.

Recall that f: M → G/K = N is harmonic if and only if

∇′′∂f = 0 ,  (3.7)

where T Mℂ = T′M ⊕ T′′M is the decomposition into (1, 0) and (0, 1) tangent spaces,

d = ∂ + \overline{∂} and ∇^{N}= ∇′ + ∇′′. Using (3.2) this becomes

(\overline{∂}β′)^{[𝔭]} = 0  (3.8)

and thus, for a lift F: M → G of f, we obtain due to (3.4), (3.5)

(\overline{∂}(AdF α′_{1}))^{[𝔭]} = 0 ,  (3.9)

which unravels to

\overline{∂}α′_{1} + [α′′_{0} ∧ α′_{1}] = \overline{∂}α′_{1} + [α_{0} ∧ α′_{1}] = 0 .  (3.10)

Since the sum of (3.10) and its complex conjugate give (3.6b) we arrive at the following characterization of a harmonic map f: M → G/K in terms of a lift F: M → G.

Lemma 3.1. Let f: M → G/K be a smooth map with lift F: M → G and induced Maurer-Cartan form α = F^{−1}dF = α_{0} + α_{1}. Then f is harmonic iff

\begin{matrix}dα_{0} + 1/2[α_{0} ∧ α_{0}] = −[α′_{1} ∧ α′′_{1}] , \\ \overline{∂}α′_{1} + [α_{0} ∧ α′_{1}] = 0 . \end{matrix}  (3.11)

Conversely, if α_{0}: T M → 𝔨 and α_{1}: T M → 𝔭 are 𝔨-resp. 𝔭-valued 1-forms solving (3.11) and M is simply connected, then there exists a unique (up to G-translation) harmonic map f = π ∘ F: M → G/K where F: M → G integrates F^{−1}dF = α_{0} + α_{1}.

Remark 3.2. Equations (3.11) are sometimes referred to as the Yang-Mills-Higgs equations for the K-connection α_{0} and the Higgs-field α′_{1}. They are clearly invariant under the action of the Gauge group C^{∞}(M, K).

Similar to [29] we introduce a spectral parameter and associate to a harmonic map f: M → G/K an extended harmonic map into a certain loop space. Let α_{0}: T M → 𝔨 and α_{1}: T M → 𝔭 be 𝔨 resp. 𝔭-valued 1-forms on M. Then we have the following equivalent statements which follow immediately from a comparison of coefficients at like powers of λ:

(i) α_{0} and α_{1} solve (3.11).

(ii) α_{0} and λ^{−1}α′_{1} + λα′′_{1} solve (3.11) for any λ ∈ S^{1}.

(iii) The Λ𝔤_{σ}-valued 1-form (3.12)

A = λ^{−1}α′_{1} + α_{0} + λα′′_{1}

solves the Maurer-Cartan equation dA + \frac{1}{2}[A ∧ A] = 0.

In case M is simply connected, (3.12 (iii)) can be integrated to a map

F̃: M → ΛG_{σ}, F̃^{−1}dF̃ = A  (3.13)

and Lemma 3.1 and (3.12 (ii)) imply that

f_{λ} = π ∘ F̃_{λ}: M → G/K  (3.14)

is harmonic for every λ ∈ S^{1}. Here F̃_{λ} denotes the composition ev_{λ} ∘ F̃ of F̃ with the evaluation map ev_{λ}: ΛG_{σ} → G, ev_{λ}(g) = g(λ). This yields the following characterization of harmonic maps f: M → G/K when M is simply connected.

Proposition 3.3. Assume that M is a simply connected Riemann surface. Then a map f: M → G/K is harmonic if and only if there exists a map F̃: M → ΛG_{σ} with F̃^{−1}dF̃ = λ^{−1}α′_{1} + α_{0} + λα′′_{1} and π ∘ F̃_{1} = f, where α_{0}: T M → 𝔨 and α_{1}: T M → 𝔭 are 𝔨 resp. 𝔭-valued 1-forms on M.

Definition 3.4. Let f: M → G/K be harmonic with f(p_{0}) = eK and let F: M → G be a lift of f with F^{−1}dF = α_{0} +α_{1}. Then F̃: M → ΛG_{σ} integrating F̃^{−1}dF̃ = λ^{−1}α′_{1} +α_{0} +λα′′_{1} with initial condition F̃(p_{0}) = k ∈ K is called an extended lift of f.

Remark 3.5. Given a harmonic map f: M → G/K an extended lift F̃: M → ΛG_{σ} is determined only up to a gauge transformation H: M → K, that is to say, F̃H: M → ΛG_{σ} is also an extended lift. This freedom will be useful for the discussion in section 4. A gauge invariant description may be obtained by considering the homogeneous space ΛG_{σ}/K which has a ΛG_{σ}-invariant complex structure given by the decomposition of the complexified tangent space at the base point o = eK

T_{o}(ΛG_{σ}/K)ℂ = Λ𝔤ℂ_{σ}/𝔨ℂ = Λ_{σ}^{−} ⊕ Λ_{σ}^{+} ,

where Λ_{σ}^{∓} = {ξ ∈ Λ𝔤ℂ_{σ} | ξ = ∑_{k_{>}^{<}0} λ^{k}ξ_{k}}. Further we call the finite dimensional ΛG_{σ}- invariant subbundle defined by ΛG_{σ}-translation of

\{λ^{−1}η + λ\overline{η} | η ∈ 𝔭ℂ\}

the horizontal distribution. Then Proposition 3.3 gives the following familiar twistorial reformulation of the harmonicity condition.

Proposition 3.6. Assume that U is a simply connected Riemann surface. Then a map f: M → G/K is harmonic if and only if there exists a holomorphic and horizontal map f̃: M → ΛG_{σ}/K with f̃_{1} = f.

Note, that unlike in Uhlenbeck’s [29] setup for harmonic maps into Lie groups, f̃_{λ}: M → G/K is harmonic for any λ ∈ S^{1}. Since any Lie group G can be considered as the symmetric space (G × G)/ΔG the present setup applies to this case as well. In analogy to the theory of minimal surfaces, f̃ is usually called the associated family of harmonic maps to f.

4 Weierstrass-type representation of harmonic maps

In this section we will discuss how every harmonic map f: M → G/K of a simply connected Riemann surface M may be obtained from Weierstrass-type data. More specifically, we will show that f can be reconstructed from a certain holomorphic 1-form with values in an infinite dimensional linear subspace of Λ𝔤ℂ_{σ}. Allowing certain singularities to occur we can do better and reconstruct f from a 𝔭ℂ-valued holomorphic 1-form. We also discuss how the finite type harmonic maps of [5, 7] arise from our point of view and finally apply our recipe to the construction of constant mean curvature surfaces with umbilic points.

Let M be either ℝ^{2} or the unit disk with global complex coordinate z. It will become clear below that our developments will not apply to M = S^{2} for which Uhlenbeck [29] already provided a complete description. It follows from Proposition 3.4, Definition 3.5 and Remark 3.6 that the space of harmonic maps f: M → G/K with f(0) = eK can be identified with the space of extended lifts F̃: M → ΛG_{σ} with F̃(0) = k ∈ K modulo gauge transformations H: M → K where f = π ∘ F̃_{1}.

Thus, if we denote the space of harmonic maps by

H = {f: M → G/K; f(0) = eK, f harmonic},  (4.1)

we have a bijective correspondence

H = {F̃: M → ΛG_{σ}; F̃(0) = k ∈ K, F̃ extended lift}/ ∼  (4.2)

where F̃_{1} ∼ F̃_{2} if and only if F̃_{2} = F̃_{1}H for a gauge H: M → K. This identification will be used in the remainder of this section.

We now turn to the construction of harmonic maps in terms of holomorphic data. Recall from section 2, Theorem 2.3, that there is a global decomposition of the complexified σtwisted loop group

ΛGℂ_{σ} = ΛG_{σ} · Λ_{B}^{+}Gℂ_{σ}  (4.3)

where Kℂ = KB is an Iwasawa decomposition for Kℂ. Correspondingly, the Lie algebra Λ𝔤ℂ_{σ} splits into a direct sum of Lie subalgebras

Λ𝔤ℂ_{σ} = Λ𝔤_{σ} ⊕ Λ_{𝔟}^{+}𝔤ℂ_{σ} .  (4.4)

We will need the following linear subspace of Λ𝔤ℂ_{σ}: for d ∈ ℕ we let

Λ_{d} = {ξ ∈ Λ𝔤_{σ} | ξ = ∑_{|k|≤d} λ^{k}ξ_{k}}  (4.5)

be the finite dimensional subspace of Laurant polynomial loops of degree not larger than d. Further we introduce the space of σ-twisted loops which extend holomorphically to the interior of the λ-disc and have a simple pole at the origin

Λ_{−1,∞}: = {ξ ∈ Λ𝔤ℂ_{σ} | λξ extends holomorphically to |λ| < 1} .  (4.6)

As a Fourier series, ξ ∈ Λ_{−1,∞} if and only if

ξ = ∑_{k≥−1} λ^{k}ξ_{k}, ξ_{even} ∈ 𝔨ℂ, ξ_{odd} ∈ 𝔭ℂ .

Notice that Λ_{−1,∞} is a closed subspace of Λ𝔤ℂ_{σ} which is stable under the adjoint action of Λ^{+}Gℂ_{σ} and so in particular under Λ_{B}^{+}Gℂ_{σ}.

Definition 4.1. The vector space P of holomorphic 1-forms on M with values in the Banach space Λ_{−1,∞} will be called the space of holomorphic potentials. Thus μ ∈ P is a holomorphic potential if μ = ∑_{k≥−1} λ^{k}μ_{k}, where μ_{even} resp. μ_{odd} are holomorphic 𝔨ℂ resp. 𝔭ℂ-valued 1-forms on M.

The construction outlined below shows why the space P serves as the space of “Weierstrassdata” for the construction of harmonic maps: if μ ∈ P is a holomorphic potential, then

μ = ξdz  (4.7)

for some holomorphic function ξ = ∑_{k≥−1} λ^{k}ξ_{k}: M → Λ_{−1,∞}. Thus

dμ + \frac{1}{2}[μ ∧ μ] = \overline{∂}μ = 0 ,

and we can integrate

g_{μ}^{−1}dg_{μ} = μ, g_{μ}(0) = e ,  (4.8)

to obtain a unique map

g_{μ}: M → ΛGℂ_{σ} .  (4.9)

Decomposing g_{μ} according to (4.3),

g_{μ} = Φ_{μ}b_{μ} ,  (4.10)

yields a map Φ_{μ}: M → ΛG_{σ} with Φ_{μ}(0) = e.

Lemma 4.2. Φ_{μ}: M → ΛG_{σ} is an extended harmonic lift.

Proof. According to Proposition 3.3 we have to verify that

Φ_{μ}^{−1}dΦ_{μ} = λ^{−1}α′_{1} + α_{0} + λα′′_{1} .

Since Φ_{μ} = gb^{−1} (surpressing the subscript μ in g and b) we have

\begin{aligned}Φ_{μ}^{−1}dΦ_{μ} & = Ad b(g^{−1}dg) − dbb^{−1} \\ & = Ad b(μ) − dbb^{−1} . \end{aligned}  (4.11)

But b takes values in Λ_{B}^{+}Gℂ_{σ}, so that dbb^{−1} takes values in Λ_{𝔟}^{+}𝔤ℂ_{σ}, hence by (4.4)

Φ_{μ}^{−1}dΦ_{μ} = (Ad b(μ))_{Λ𝔤_{σ}} .  (4.12)

Since μ = ∑_{k≥−1} μ_{k}λ^{k} is Λ_{−1,∞}-valued and AdΛ_{B}^{+}Gℂ_{σ} acts on Λ_{−1,∞} we obtain, by putting b_{|_{λ=0}} = b_{0}: M → B,

(Ad b(μ))_{Λ𝔤_{σ}} = λ^{−1}Ad b_{0}(μ_{−1}) + α_{0} + λAd \overline{b_{0}}(\overline{μ_{−1}})  (4.13)

for some 𝔨-valued 1-form α_{0}: T M → 𝔨. Finally note that μ_{−1} = ξ_{−1}dz is a 𝔭ℂ-valued holomorphic differential and thus Ad b_{0}(μ_{−1}) is a 1-form of type (1, 0). □

If we recall the identification made in (4.2), the above proposition provides us with a map

Φ: P ⟶ H: μ ⟼ [Φ_{μ}]  (4.14)

which we think of as a Weierstrass-type representation for harmonic maps.

Before we investigate the properties of this map, let us pause and discuss how this construction relates to the finite type harmonic maps in [5]. It is shown in [7] that a harmonic map f: ℝ^{2} → G/K is of finite type if f is obtained from an extended lift Φ_{μ}: ℝ^{2} → ΛG_{σ} whose

holomorphic potential μ =ξ^{∘} dz is constant of the form

ξ^{∘}= λ^{d−1} η^{∘}, η^{∘}∈ Λ_{d} ,  (4.15)

for some d ∈ ℕ odd. Otherwise said, for d ∈ ℕ odd, we have natural subspaces

λ^{d−1}Λ_{d}dz ⊂ P ,  (4.16)

and the images of those subspaces under Φ comprise the finite type harmonic maps. The importance of finite type harmonic maps lies in the fact that they can be constructed by solving finite dimensional Hamiltonian ODE’s in Lax form. We briefly sketch these developments: let d ∈ ℕ be odd, η^{∘}∈ Λ_{d}, ξ^{∘}= λ^{d−1} η^{∘} and μ =ξ^{∘} dz. Then, by (4.8), (4.10),

g_{μ}(z) = exp(λ^{d−1} η^{∘} z) = Φ_{μ}(z)b_{μ}(z) .  (4.17)

Let

η: = Ad Φ_{μ}^{−1}(η^{∘}): M → Λ𝔤_{σ} .  (4.18)

Then, by (4.17),

η = Ad b_{μ}(η^{∘})  (4.19)

and thus

λ^{d}η = Adb_{μ}(λ^{d} η^{∘}) ,

which, since b_{μ} takes values in Λ_{B}^{+}Gℂ_{σ}, extends holomorphically to |λ| < 1. Hence η has a pole of order d at λ = 0 and is real due to (4.18), so

η: M → Λ_{d} ⊂ Λ𝔤_{σ}  (4.20)

is Λ_{d}-valued. Moreover,

dη = [η, Φ_{μ}^{−1}dΦ_{μ}]  (4.21)

and from (4.19), (4.12) we obtain

Φ_{μ}^{−1}dΦ_{μ} = (Ad b_{μ}(μ))_{Λ𝔤_{σ}} = (Ad b_{μ}(λ^{d−1} η^{∘} dz))_{Λ𝔤_{σ}} = (λ^{d−1}η)_{Λ𝔤_{σ}} .  (4.22)

Combining (4.21) and (4.22) we see that the Λ_{d}-valued map η satisfies the following system of ODE’s in Lax-form:

dη = [η, (λ^{d−1}ηdz)_{Λ𝔤_{σ}}], η(0) = η^{∘}∈ Λ_{d} .  (4.23)

Readers familiar with the Adler-Kostant-Symes integration scheme [1] will recognize (4.23) as a special case of commuting Hamiltonian flows on Λ𝔤_{σ} (cf.[7]). From this theory it follows that (4.23) has a unique, globally defined solution

η: M → Λ_{d}

to any choice of initial condition η_{0} ∈ Λ_{d}. (That η is globally defined follows from the fact that in our case the L^{2}-inner product on Λ𝔤_{σ} is positive definite and is preserved by the flows (4.23) which evolve on the finite dimensional subspace Λ_{d}.) Moreover, when d ∈ ℕ is odd,

A = (λ^{d−1}ηdz)_{Λ𝔤_{σ}}: T M → Λ𝔤_{σ}  (4.24)

is of the form

A = λ^{−1}α′_{1} + α_{0} + λα′′_{1} ,  (4.25)

satisfies the Maurer-Cartan equation dA + \frac{1}{2}[A, A] = 0 and can thus be integrated to a map

Φ: M → ΛG_{σ}, Φ(0) = e .

Clearly, Φ = Φ_{μ} with μ = λ^{d−1} η^{∘} dz, so that by solving (4.23) one finds all extended lifts Φ_{μ} of finite type harmonic maps f: M → G/K. For a more detailed description of this and also its relation to the r-matrix integration scheme see [7].

After this excursion into finite type harmonic maps, we return to the Weierstrass representation Φ: P → H (4.14). First we observe that Φ is equivariant with respect to natural group actions on P and H. The group decomposition (4.3)

ΛGℂ_{σ} = ΛG_{σ} · Λ_{B}^{+}Gℂ_{σ}

defines a “dressing action” [32] of ΛGℂ_{σ} on ΛG_{σ}: for h ∈ ΛGℂ_{σ} and g ∈ ΛG_{σ}

h#g: = (hg)_{ΛG_{σ}} .  (4.26)

In particular, Λ^{+}Gℂ_{σ} acts via dressing on ΛG_{σ}. Now let F: M → ΛG_{σ} be an extended lift and h ∈ Λ^{+}Gℂ_{σ}. Applying (4.26) pointwise, we obtain a new map

h#F: M → ΛG_{σ}, (h#F)(z) = h#F (z), z ∈ M .  (4.27)

Similar to the case of harmonic maps into Lie groups [14] this defines an action on H.

Proposition 4.3 (i) h#F is an extended lift.

(ii) Let H: M → K be a gauge transformation. Then there exists H̃: M → K with

h#(F H) = (h#F)H̃ ,

i.e., h#(F H) and h#F are extended lifts of the same harmonic map. Thus (4.27) defines a left action of Λ^{+}Gℂ_{σ} on H given by

h#f = π ∘ ((h#F)_{|_{λ=1}}) ,

where F: M → ΛG_{σ} is an extended lift of the harmonic map f: M → G/K.

Proof. (i) This computation is analogous to the one given in [14]. Since F is an extended lift there exists 𝔨 and 𝔭-valued 1-forms α_{0}, α_{1} on M with

F^{−1}dF = λ^{−1}α′_{1} + α_{0} + λα′′_{1} .

Now let h ∈ Λ^{+}Gℂ_{σ} and decompose

hF = (h#F)b

according to (4.3) where b: M → Λ_{B}^{+}Gℂ_{σ}. Then

\begin{aligned}(h#F)^{−1}d(h#F) & = bF^{−1}h^{−1}(hdF b^{−1} − hF b^{−1}dbb^{−1}) \\ & = Adb(F^{−1}dF) − dbb^{−1} \end{aligned}

and since dbb^{−1} takes values in Λ_{𝔟}^{+}𝔤ℂ_{σ} we obtain

(h#F)^{−1}d(h#F) = Adb(F^{−1}dF)_{Λ𝔤_{σ}} .

Recall that AdΛ_{B}^{+}Gℂ_{σ} preserves Λ_{−1,∞} so that, putting b_{|_{λ=0}} = b_{0} ∈ B ⊂ Kℂ,

(λAdb(F^{−1}dF))_{|_{λ=0}} = Adb_{0}(α′_{1})

and thus

(Adb(F^{−1}dF))_{Λ𝔤_{σ}} = λ^{−1}α̃′_{1} + α̃_{0} + λα̃′′_{1} ,

where the 𝔭-valued 1-form α̃_{1} = α̃′_{1} + α̃′′_{1} is given by

α̃′_{1} = Adb_{0}(α′_{1})

and α̃_{0} is some 𝔨-valued 1-form. Moreover, since F (0) = k ∈ K and ΛG_{σ} ∩ Λ^{+}Gℂ_{σ} = K, we get

(h#F)(0) = h#F (0) = h#k = (hk)_{ΛG_{σ}} ∈ K .

Hence h#F is an extended lift.

(ii) Let H: M → K and h ∈ Λ^{+}Gℂ_{σ}. Then

\begin{aligned}hF & = (h#F)b \\ hF H & = (h#(F H))b̃ = (h#F)bH \end{aligned}

for b, b̃: M → Λ_{B}^{+}Gℂ_{σ}. Thus

h#(F H) = (h#F)bHb̃^{−1}

and we conclude that bHb̃^{−1} takes values in ΛG_{σ}. But bHb̃^{−1} also takes values in Λ^{+}Gℂ_{σ} so that

bHb̃^{−1} =: H̃

takes values in K. □

The space P of holomorphic potentials admits a natural left action of the holomorphic gauge group

G = \{h: M → Λ^{+}Gℂ_{σ}; h\overline{z} = 0\}  (4.28)

by gauge transformations, i.e., for μ ∈ P and h ∈ G

h · μ = Ad h(μ) − dh h^{−1} .  (4.29)

Lemma 4.4. The map Φ: P → H: μ → [Φ_{μ}] is equivariant with respect to the gauge action of G on P and the dressing action of Λ^{+}Gℂ_{σ} on H. More specifically, for h ∈ G

[Φ_{h·μ}] = h(0)#[Φ_{μ}] .

Proof. We have to show that for some H: M → K

Φ_{h·μ} = (h(0)#Φ_{μ})H .

From (4.8) and (4.10) we see that Φ_{μ} = (g_{μ})_{ΛG_{σ}} where g_{μ}: M → ΛGℂ_{σ} solves g^{−1}dg = μ subject to g(0) = e. Since h(0)g_{μ}h^{−1} solves g^{−1}dg = h · μ subject to g(0) = e we conclude

g_{h·μ} = h(0)g_{μ}h^{−1} .  (4.30)

Let b_{μ}: M → Λ_{B}^{+}Gℂ_{σ} such that g_{μ} = Φ_{μ}b_{μ}. Since Λ^{+}Gℂ_{σ} = K · Λ_{B}^{+}Gℂ_{σ} and h takes values in

Λ^{+}Gℂ_{σ} we have

Φ_{h·μ} = (g_{h·μ})_{ΛG_{σ}} = (h(0)g_{μ}h^{−1})_{ΛG_{σ}}
= (h(0)Φ_{μ}b_{μ}h^{−1})_{ΛG_{σ}} = (h(0)Φ_{μ}H̃)_{ΛG_{σ}}

where b_{μ}h^{−1} = H̃b̃ with H̃: M → K and b̃: M → Λ_{B}^{+}Gℂ_{σ}. But by (4.27), the latter is h(0)#(Φ_{μ}H̃) and so proposition 4.3 (ii) gives

Φ_{h·μ} = h(0)#(Φ_{μ}H̃) = (h(0)#Φ_{μ})H

for some H: M → K. □

A crucial property of our construction is the fact that every harmonic map can be obtained from some holomorphic potential. This reduces the harmonic map equation to a linear ODE with holomorphic data similar to the classical Weierstrass representation for minimal surfaces.

Lemma 4.5. The map Φ: P → H: μ → [Φ_{μ}] is surjective and its fibres are the orbits of the based holomorphic gauge group G_{0} = {h ∈ G; ⟨(′) = ⌉}.

Proof. Let f: M → G/K be harmonic and let F: M → ΛG_{σ} be an extended lift of f which we may assume has F (0) = e. We have to find a holomorphic potential μ ∈ P with [Φ_{μ}] = [F]. The idea is to construct h: M → Λ^{+}Gℂ_{σ} with h(0) = e so that g = F h has g\overline{z} = 0, or equivalently, that μ = g^{−1}dg is a 1-form of type (1,0) which, due to the Maurer-Cartan equation, is then holomorphic. Since F^{−1}dF = λ^{−1}α′_{1} + α_{0} + λα′′_{1} the holomorphic 1-form μ takes values in Λ_{−1,∞}, i.e., μ ∈ P. Moreover, due to Λ^{+}Gℂ_{σ} = K · Λ_{B}^{+}Gℂ_{σ}, we have h = Hb with H: M → K and b: M → Λ_{B}^{+}Gℂ_{σ} so that

g = (F H)b .

Comparing with (4.10) and noting that g(0) = e we see that

Φ_{μ} = F H

and thus [Φ_{μ}] = [F]. As outlined above, we want to find h: M → Λ^{+}Gℂ_{σ} such that the (0, 1)-part of μ vanishes, i.e., we need to solve the equation

μ^{(0,1)} = 0 .  (4.31)

But

g^{−1}dg = Adh^{−1}(F^{−1}dF) + h^{−1}dh
= Adh^{−1}(λ^{−1}α′_{1} + α_{0} + λα′′_{1}) + h^{−1}dh

so that (4.31) becomes

\overline{∂}hh^{−1} = −(α′′_{0} + λα′′_{1})  (4.32)

subject to h(0) = e. This is a \overline{∂}-problem with right hand side in the Banach Lie algebra Λ^{+}𝔤ℂ_{σ} which can be solved over the simply connected domain M (see Appendix). Note that solutions of (4.32) subject to h(0) = e are determined only up to right multiplication by elements in G_{′}. So let h: M → Λ^{+}Gℂ_{σ} be a solution of (4.32) with h(0) = e. Then, by construction, g = F h has

μ = g^{−1}dg = Adh^{−1}(λ^{−1}α′_{1} + α′_{0}) + h^{−1}∂h

which is of type (1,0) and thus holomorphic. Moreover, since h takes values in Λ^{+}Gℂ_{σ} and AdΛ^{+}Gℂ_{σ} leaves Λ_{−1,∞} invariant, μ takes values in Λ_{−1,∞} which is to say μ ∈ P. Thus Φ: P → H is surjective. To determine the fibres of Φ, let μ, η ∈ P such that

[Φ_{μ}] = [Φ_{η}] .  (4.33)

Then Φ_{μ} = Φ_{η}H with H: M → K and since by (4.10) g_{μ} = Φ_{μ}b_{μ}, g_{η} = Φ_{η}b_{η} we see that (4.33) is equivalent with

g_{μ} = g_{η}(b_{η}^{−1}Hb_{μ}) = g_{η}h^{−1}  (4.34)

where h: M → Λ^{+}Gℂ_{σ} with h(0) = e. Thus

μ = g_{μ}^{−1}dg_{μ} = Adh(g_{η}^{−1}dg_{η}) − dhh^{−1} = h · η

and from (4.34) we see that h ∈ G_{0}. Thus μ and η are gauge equivalent under G_{0}. The converse is obvious from Lemma 4.4. □

Corollary 4.6. Every harmonic map f: M → G/K is obtained from a holomorphic potential μ ∈ P such that μ = ∑_{k odd, k≥−1} λ^{k}μ_{k}, i.e., μ has only odd terms in λ, in particular, μ has no constant term.

Proof. This follows at once from the previous lemma and the fact that every η ∈ P is G_{0}-equivalent to a μ ∈ P with no even terms: let η ∈ P and decompose

η = η_{e} + η_{o}

into even and odd terms, i.e., η_{e}(−λ) = η_{e}(λ) and η_{o}(−λ) = −η_{o}(λ). In particular, η_{e} takes values in Λ^{+}Gℂ_{σ}, so that there is a unique solution h ∈ G_{0} solving

h^{−1}∂h = η_{e}, h(0) = e

and, as a consequence,

h(−λ) = h(λ) .

Thus

μ: = h · η = Adh(η) − dhh^{−1}
= Adh(η_{e}) − dhh^{−1} + Adh(η_{0}) = Adh(η_{0}) ,

where

μ(−λ) = Adh(−λ)η_{o}(−λ) = −μ(λ) ,

so that

μ = ∑_{k odd, k≥−1} λ^{k}μ_{k} .

□

We summarize the above results in the following:

Theorem 4.7. Let Φ: P → H be the Weierstrass representation of harmonic maps from an open simply connected Riemann surface into a symmetric space. Then

(i) Φ is equivariant, i.e., if h ∈ G then

[Φ_{h·μ}] = h(0)#[Φ_{μ}] ,

(ii) Φ is surjective and

(iii) the fibres of Φ are the G_{0}-gauge orbits in P.

Remark 4.8. The above equivariance statement is the starting point for an investigation of the harmonic map equation along the lines of Segal and Wilson’s [26] approach to the KdV -equation. The basic idea is that a sufficiently interesting class of solutions to the equations at hand is comprised by the dressing orbit through a “vacuum” solution. In the case of harmonic maps f: M → G/K into a symmetric space “vacuum” solutions are orbits of 2-parameter subgroups

f_{C}(z) = exp(Cz + \overline{C}\overline{z}) · o

where C ∈ 𝔭ℂ with [C, \overline{C}] = 0 and o = eK. An extended lift is easily seen to be

F_{C}(z) = exp(λ^{−1}Cz + λ\overline{C}\overline{z}) .

Since

exp(λ^{−1}Cz) = F_{C} · exp(−λ\overline{C}\overline{z})

and exp(−λ\overline{C}\overline{z}) takes values in Λ_{B}^{+}Gℂ_{σ} we obtain by comparison to (4.10) that

F_{C} = Φ_{μ} with the holomorphic potential μ = λ^{−1}Cdz. Recall that a harmonic map is of finite type (4.15) if it is of the form [Φ_{μ}] with μ = λ^{d−1} η^{∘}, η^{∘}∈ Λ_{d}, d ∈ ℕ odd. It has been shown in [5], [7] that every harmonic 2-torus f: T^{2} → G/K whose derivative \frac{∂f}{∂z}: T^{2} → T (G/K)ℂ is regular semisimple at one point is of finite type. It is interesting to note that all such harmonic 2-tori are in fact contained in the dressing orbit of some f_{C}. Without going too much into details (one has to enlargen the loop groups involved slightly) we have the following more general result which is proven in [6].

Theorem 4.9. Let μ = ∑_{k≥−1} ξ_{k}λ^{k}dz ∈ Λ_{−1,∞} be a constant (holomorphic) potential, i.e., ξ_{even} ∈ 𝔨ℂ, ξ_{odd} ∈ 𝔭ℂ, such that ξ_{−1} ∈ 𝔭ℂ is semisimple. Then there exists h ∈ Λ^{+}Gℂ_{σ} and C ∈ AdKℂ · ξ_{−1} with [C, \overline{C}] = 0 so that

h#f_{C} = [Φ_{μ}] .

Also in the general case (of not necessarily finite type harmonic maps) a natural class are those harmonic maps [Φ_{μ}], which arise from holomorphic potentials

μ = λ^{−1}η ,  (4.35)

where η is a 𝔭ℂ-valued holomorphic 1-form on M. In particular, such harmonic maps [Φ_{μ}] are given by finitely many holomorphic differentials. Interestingly enough, any harmonic map f: M → G/K may be obtained from a potential of the form (4.35) if one allows η to be meromorphic.

To see this let f: M → G/K be harmonic with holomorphic potential μ ∈ P. Then (4.10) f = [Φ_{μ}] where g = Φ_{μ}b and g: M → ΛGℂ_{σ} integrates

g^{−1}∂g = μ, g(0) = e .

Since g: M → ΛGℂ_{σ} is holomorphic and τ: ΛGℂ_{σ} → Det^{∗} is holomorphic we know from Corollary 2.5 that the zero locus of τ ∘ g,

S = {p ∈ M; g(p) not contained in the big cell } ⊂ M ,

is discrete. But g = Φ_{μ}b with b taking values in Λ_{B}^{+}Gℂ_{σ}, so that g(p) is in the big cell if and only if Φ_{μ}(p) is, in particular

\begin{matrix}g_{−} = (Φ_{μ})_{−}, \\ S = (τ ∘ Φ_{μ})^{−1}(0) , \end{matrix}  (4.36)

and Lemma 2.6 implies that g_{−} extends meromorphically across S. Notice that if F: M → ΛG_{σ} is another extended lift for f then (4.2) F̃ = kΦ_{μ}H for some k ∈ K and H: M → K. Thus the set S ⊂ M depends only on the harmonic map f: M → G/K.

Theorem 4.10. Let f: M → G/K be harmonic with f(0) = eK and S ⊂ M as defined above. Then there exists a 𝔭ℂ-valued meromorphic 1-form η on M with poles contained in S so that

f = [Φ_{λ^{−1}η}]

on M \ S.

Proof. Let F̃: M → ΛG_{σ} be an extended lift of f with F̃(0) = e. Then, on M \ S, we can decompose

F̃ = F_{−}F_{+}

where F_{−} takes values in Λ_{∗}^{−}Gℂ_{σ}, F_{+} takes values in Λ^{+}Gℂ_{σ} and F_{−} extends meromorphically across S (4.36). But

\begin{aligned}F_{−}^{−1}dF_{−} & = AdF_{+}(F̃^{−1}dF̃) − dF_{+}F_{+}^{−1} \\ & = AdF_{+}(λ^{−1}α′_{1} + α_{0} + λα′′_{1}) − dF_{+}F_{+}^{−1} \\ & = λ^{−1}Ad(F_{+})_{0}(α′_{1}) =: λ^{−1}η , \end{aligned}  (4.37)

due to the direct sum decomposition Λ𝔤ℂ_{σ} = Λ_{∗}^{−}𝔤ℂ_{σ} ⊕ Λ^{+}𝔤ℂ_{σ}, where η: M → 𝔭ℂ is meromorphic with poles contained in S. Moreover, F_{−}(0) = e so that, together with (4.37), (4.8)

F_{−} = g_{λ^{−1}η}

on M \ S. If we let

F_{+} = b^{−1}H^{−1} ,

with H: M \ S → K and b: M \ S → Λ_{B}^{+}Gℂ_{σ}, we obtain

g_{λ^{−1}η} = F_{−} = F̃F_{+}^{−1} = (F̃H)b ,  (4.38)

so that by (4.10)

Φ_{λ^{−1}η} = F̃H

and thus

f = [Φ_{λ^{−1}η}] .
□

We conclude this section with a discussion of how to apply the above theory to the construction of constant mean curvature surfaces with prescribed umbilic points. Let ϕ: M → ℝ^{3} be a (simply connected) constant mean curvature surface and let f: M → S^{2} be its Gauss map. Then [25] f is harmonic and the quadratic differential given by the (2, 0)-part of the induced metric

Q = (df, df)^{(2,0)}  (4.39)

is holomorphic. The zeros of Q are the umbilic points of the constant mean curvature surface ϕ: M → ℝ^{3}. Unless M is the Riemann sphere, in which case Q ≡ 0 and ϕ(M) ⊂ ℝ^{3} is a round sphere, f: M → S^{2} is nonconformal. We view S^{2} as the symmetric space SU(2)/S^{1}

with symmetric involution σ(g) = sgs^{−1}, s = ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ), and Cartan decomposition 𝔤 =

𝔨 ⊕ 𝔭 where

𝔨 = \{( \begin{matrix}ia & 0 \\ 0 & −ia \end{matrix} ); a ∈ ℝ\} ,

𝔭 = \{( \begin{matrix}0 & b \\ −\overline{b} & 0 \end{matrix} ); b ∈ ℂ\} .

If F: M → SU(2) is a frame of f then, (3.5),

F^{−1}dF = α_{0} + α_{1}

and by Definition 3.4, F̃: M → ΛSU(2)_{σ} solving

F̃^{−1}dF̃ = λ^{−1}α′_{1} + α_{0} + λα′′_{1} ,  (4.40)

is an extended lift. Notice that under the identification (3.2), (3.4)

df = AdF (α_{1})

so that

Q = (α′_{1}, α′_{1}) ,

where ( , ) denotes the Killing form on 𝔭ℂ. From Lemma 4.5 we know that every harmonic map f: M → S^{2} is obtained from an extended lift F̃ arising from a holomorphic potential μ = ∑_{k≥−1} λ^{k}μ_{k} via the decomposition (4.10)

g_{μ} = F̃b ,

where g_{μ} integrates g^{−1}dg = μ, g(0) = e. Moreover, (4.40) and (4.13) imply that

α′_{1} = Ad b_{0}(μ_{−1}) ,

where b_{0} = b_{|_{λ=0}}: M → B. Thus, the holomorphic differential Q is given by the product of the leading term μ_{−1} of the holomorphic potential μ,

Q = (μ_{−1}, μ_{−1}) .  (4.41)

Note that by Theorem 4.10 one may assume that, away from a discrete set of points in M, μ = λ^{−1}η for a holomorphic 𝔭ℂ-valued 1-form η. This suggests the following recipe to construct special constant mean curvature surfaces with prescribed umbilics: choose two holomorphic functions a_{1}, a_{2}: M → ℂ and consider the holomorphic potential

μ = λ^{−1} ( \begin{matrix}0 & a_{1} \\ a_{2} & 0 \end{matrix} ) dz .  (4.42)

Then F̃: M → ΛSU(2)_{σ}, where g_{μ} = F̃b, is an extended lift of the harmonic map f = π ∘ F̃_{1}: M → SU(2)/S^{1}. From Sym’s formula [2] we know that f is the Gauss map of the constant mean curvature surface

ϕ = (\frac{d}{dθ}_{|θ=0}F̃) F̃^{−1} + f: M → su(2) ≅ ℝ^{3}

where λ = e^{iθ}. Moreover, in view of (4.39), (4.41) and (4.42) one can show by computing the differential of ϕ that the umbilic points of ϕ are given by the zeros of a_{2} and ϕ has branch points at the zeros of a_{1}. (The parallel constant mean curvature surface to ϕ is given by ϕ−2f which has umbilics at the zeroes of a_{1} and branch points at the zeros of a_{2}.) Hence, by selecting a_{1}, a_{2}: M → ℂ accordingly we can arrange for any prescribed umbilic configuration. A detailed study of constant mean curvature surfaces so obtained will be done elsewhere. We content ourselves here with some examples:

(i) a_{1} ≡ 1, a_{2} ≡ 0, M = ℂ, gives the round sphere minus a point;

(ii) a_{1} = −a_{2} ≡ 1, M = ℂ, gives the cylinder;

(iii) a_{1} = 1, a_{2} ≡ z^{m}, M = ℂ, gives properly immersed constant mean curvature planes with one umbilic of order m at the origin and rotational induced metric initially constructed by B. Smyth [27], [13], [2] (see Picture 1);

(iv) a_{1} = 1, a_{2} = Π_{k=1}^{n}(z − p_{k}), p_{k} ∈ ℂ, M = ℂ, gives the so called generalized Smyth surfaces with umbilics at p_{k} ∈ ℂ (see Picture 2).

Picture 1: a_{2} = z^{5}.

Picture 2: a_{2} = Π_{k=1}^{5}(z − e^{2πik/5}).

For the construction of compact constant mean curvature surfaces one expects that in general the holomorphic potential will have higher order terms in λ. Except for the leading term μ_{−1} we have no geometrical interpretation of those higher order terms.

APPENDIX: The \overline{∂}-problem.

Since we could not find a reference in the literature we give a brief outline how to solve the \overline{∂}-problem in the setting of Lemma 4.5. Let M be either ℂ or the unit disc and A + λB: M → Λ^{+}Gℂ_{σ} so that A: M → 𝔨ℂ and B: M → 𝔭ℂ. Since A and B are the (0, 1)- components of the derivative of a harmonic map they are real analytic on M. Consider the \overline{∂}-problem

\begin{matrix}g\overline{z} = (A + λB)g \\ g(0) = 1 \end{matrix}  (4.43)

where z is a global holomorphic coordinate on M.

Lemma 4.11. (4.43) has a (real analytic) solution g: M → Λ^{+}Gℂ_{σ}.

Proof. First we show that we have local solutions around every point p_{0} ∈ M. Since A, B are real analytic there exist holomorphic maps

\begin{matrix}Ã: D_{ϵ}(p_{0}) × D_{ϵ}(\overline{p}_{0}) ⟶ 𝔨ℂ \\ B̃: D_{ϵ}(p_{0}) × D_{ϵ}(\overline{p}_{0}) ⟶ 𝔭ℂ \end{matrix}

where D_{ϵ}(p) is the ϵ-disc around p ∈ M, so that

Ã(z, \overline{z}) = A(z, \overline{z}), B̃(z, \overline{z}) = B(z, \overline{z}) .

Consider the equation

\begin{matrix}g̃_{w} = (Ã + λB̃)g̃ \\ g̃(z, \overline{p}_{0}) = 1 \end{matrix}

which has a unique (holomorphic) solution

g̃: D_{ϵ}(p_{0}) × D_{ϵ}(\overline{p}_{0}) → Λ^{+}Gℂ_{σ} .

Then g: D_{ϵ}(p_{0}) → Λ^{+}Gℂ_{σ} defined by

g(z, \overline{z}): = g̃(z, \overline{z})

solves

g\overline{z} = (A + λB)g .

Thus we have an open cover (U_{α}) of M with (real analytic) solutions g_{α}: U_{α} → Λ^{+}Gℂ_{σ} of

g\overline{z} = (A + λB)g .

On the overlap U_{α} ∩ U_{β} the maps

h_{αβ} = g_{α}^{−1}g_{β}: U_{α} ∩ U_{β} → Λ^{+}Gℂ_{σ}

are holomorphic and satisfy the co-cycle condition

h_{αβ}h_{βγ} = h_{αγ} .

But those are the data for a holomorphic Λ^{+}Gℂ_{σ}-principal fiber bundle P → M whose (local) holomorphic sections are described by the \overline{∂}-equation (4.43). Since M is contractible P admits a continuous global section. By a generalization of Grauert’s Theorem [3] there exists a global holomorphic section g: M → Λ^{+}Gℂ_{σ}. To satisfy the initial condition take g g(0)^{−1}. □

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