A Weierstrass - Type Representation for Harmonic Maps from Riemann Surfaces to General Lie Groups

Vladimir Balan and Josef Dorfmeister

Dedicated to Prof.Dr. Constantin UDRIŞTE on the occasion of his sixtieth birthday

Abstract The first part of the paper describes the harmonicity equations for harmonic maps from Riemann surfaces to Lie groups which carry a left-invariant pseudo-Riemannian structure; §3 includes basic facts about loop groups and their factorizations; §4 presents the formalism of [10] and an extension of Wu’s formula to the case of generalized harmonic maps into arbitrary Lie groups. Finally, §5 includes three examples which outline the developed theory.

Mathematics Subject Classification: 58E20, 53C43, 22E67, 30F15 Key words: harmonic maps, loop groups, meromorphic potentials, bi-invariant metrics, Birkhoff decomposition, Iwasawa decomposition, extended frame.

1 Introduction

This paper continues the study of [2]. We consider an arbitrary real Lie group G admitting a faithful finite-dimensional representation, and discuss harmonic maps

φ : M → G,

from connected, simply-connected Riemann surfaces to G. This generalizes to some extent the work of K. Uhlenbeck [29] and provides Weierstrass data for all such harmonic maps in the spirit of [10]. The first part of the paper describes the harmonicity equations for harmonic maps from arbitrary Riemann surfaces to G, where G carries a left-invariant pseudo-Riemannian structure. If we specialize, as in the classical case [29], to bi-invariant pseudo-metrics, then the equations specialize to the Uhlenbeck equations [29]. It is remarkable that these equations do not depend on the bi-invariant structure chosen. In section 3 we recall the basic facts about loop groups and their factorizations, and specialize to the based loop group. In section 4 we recall the formalism of [10] and present an extension of Wu’s formula to the case of generalized harmonic maps into arbitrary Lie groups. It is here where [2] enters in an essential way. The final section illustrates with selected examples what types of ”potentials” one obtains and what type of solutions they provide to the equations listed in section 2. We illustrate in section 5 the equations obtained by three typical examples.

Balkan Journal of Geometry and Its Applications, Vol.5, No.1, 2000, pp. 7-37 \begin{matrix}c \\ ◯ \end{matrix}Balkan Society of Geometers, Geometry Balkan Press

2 The harmonicity equations

2.1 Let hereafter G be a real Lie group admitting a faithful finite-dimensional representation, and g =< ., . > a left-invariant pseudo-Riemannian metric on G. Moreover, let D be the unit disk in C or all of C and φ : D → G an immersion such that the pull-back metric φ^{∗}g on D is of the form

φ^{∗}g = (γ_{ij}) = λ(x, y)(δ_{ij}), (2.1.1)

where λ(x, y) > 0, for all (x, y) ∈ D. Then the energy of φ [12, 13, 30]

E(φ) = ∫_{D}ϵ(φ) dx dy, (2.1.2)

is obtained by integrating the energy-density

ϵ(φ) = \frac{1}{2} | dφ |^{2}= \frac{1}{2}h_{ij} \frac{dφ^{i}}{∂x^{a}} \frac{∂φ^{j}}{∂x^{b}} γ^{ab}, (x, y) ≡ (x^{1}, x^{2}) ∈ D, (2.1.3)

of φ, where (γ^{ab}) is the inverse of γ and the Einstein summation convention is used. Note that the integral (2.1.2) is considered over every open bounded set with compact closure in D and needs to be minimized for every variation with support in the open set D [12], [13]. It is known that E(φ) is conformally invariant (e.g., [30, 10.2, p. 42], [20, Lemma 1.32, p. 20]); moreover, [29, 3.3, p. 216] one can write the energy integral as

E(φ) = \frac{1}{2} ∫_{D}(| A_{(x)} |^{2} + | A_{(y)} |^{2}) dx dy, (2.1.4)

where A_{(x)} = φ^{−1}φ_{x}, A_{(y)} = φ^{−1}φ_{y} (2.1.5)

are g-valued 1-forms on D. Here we use g = Lie(G). Also note that subscripts x, y, xx, etc denote the partial differentiation with respect to the corresponding variable(s). The subscript (x) or (y), on the other hand, has no independent meaning. The map φ is called harmonic if φ is a critical point of E(φ), i.e., if for any smooth variation [30, 1.2, p124] with compact support φ_{t} ∈ C^{∞}( D, G), t ∈ (−ε, ε), φ_{0} = φ, we have

\frac{d}{dt} |_{t=0} E(φ_{t}) = 0. (2.1.6)

The harmonicity (Euler-Lagrange) equations for φ, are provided by

Theorem. Let G be a real Lie group admitting a left-invariant pseudo-Riemannian metric g. Then the map φ : D→ G is harmonic iff the associated g-valued 1-forms A_{(x)} and A_{(y)} of (2.1.5) satisfy the equation

(adA_{(x)})^{∗}A_{(x)} + (adA_{(y)})^{∗}A_{(y)} − (∂_{x}A_{(x)} + ∂_{y}A_{(y)}) = 0, (2.1.7)

where the star superscript indicates the adjoint w.r.t. the nondegenerate bilinear form ⟨ , ⟩ = g_{e} induced by g on the Lie algebra g.

Proof. Let φ̃ be a smooth variation of φ, and η = \frac{d}{dt} |_{t=0} φ^{−1}φ̃ the variation field; assume that η : D → g has compact support contained in D. Remark also that we have the relation

\frac{d}{dt} |_{t=0} φ^{−1} \frac{∂φ̃}{∂x} = η_{x} + [A_{(x)}, η]. (2.1.8)

By direct computation, using (2.1.8) and applying the Stokes formula, the derivative inside the harmonicity condition 2.1.6 for φ writes succesively

\frac{d}{dt} |_{t=0} E(φ̃) = ∫_{D}\{⟨ \frac{d}{dt} |_{t=0} φ^{−1} \frac{∂φ̃}{∂x}, A_{(x)}⟩ + ⟨ \frac{d}{dt} |_{t=0} φ^{−1} \frac{∂φ̃}{∂y}, A_{(y)}⟩\}dxdy =
(2.1.9)  = ∫_{D}{⟨η_{x}, A_{(x)}⟩ + ⟨[A_{(x)}, η], A_{(x)}⟩ + ⟨η_{y}, A_{(y)}⟩ + ⟨[A_{(y)}, η], A_{(y)}⟩}dxdy
= ∫_{D}{−⟨η, ∂_{x}A_{(x)}⟩ + ⟨adA_{(x)}(η), A_{(x)}⟩ −
−⟨η, ∂_{y}A_{(y)}⟩ + ⟨adA_{(y)}(η), A_{(y)}⟩}dxdy
= ∫_{D}{⟨η, (adA_{(x)})^{∗}A_{(x)} − ∂_{x}A_{(x)}⟩ + ⟨η, (adA_{(y)})^{∗}A_{(y)} − ∂_{y}A_{(y)}⟩}dxdy.

Since the equation (2.1.9) holds true for any variation field η with compact support, the claim follows. □

Remark. The proof above is provided for completeness. It is similar to the one in [23] and for the positive definite case it appears also in [7, Corollary 2.4, p. 145].

2.2 As consequence, we have the following result of Urakawa [30, Chap.6, Sec.3]

Corollary. If the pseudo-metric g is bi-invariant, then the harmonicity condition becomes:

∂_{x}A_{(x)} + ∂_{y}A_{(y)} = 0, (2.2.1)

with A_{(x)}, A_{(y)} defined in (2.1.5).

Proof. At the Lie algebra level, ad is antisymmetric with respect to the bilinear non-degenerate form ⟨ · , · ⟩ = g_{e} associated to g [16, p. 125], hence we have for every η ∈ g

⟨(adA_{(x)})^{∗}adA_{(x)}, η⟩ = ⟨adA_{(x)}(η), A_{(x)}⟩ = (2.2.2)

= −⟨η, adA_{(x)}(A_{(x)})⟩ = −⟨η, [A_{(x)}, A_{(x)}]⟩ = 0.

Therefore (adA_{(x)})^{∗}adA_{(x)} = 0. Similarly one establishes (adA_{(y)})^{∗}adA_{(y)} = 0. □

Remarks. 1. For complex coordinates z = x + iy, z̄ = x − iy on D, the equations (2.2.1) rewrite ∂z̄A_{(z)} + ∂_{z}A_{(z̄)} = 0, (2.2.3)

where ∂_{z} = (∂_{x} − i∂_{y})/2, ∂z̄ = (∂_{x} + i∂_{y})/2, and A_{(z)} = φ^{−1}φ_{z}, A_{(z̄)} = φ^{−1}φz̄.

2. Since (2.2.1) does not explicitely depend on the bi-invariant metric chosen, we can consider (2.2.1) also for Lie groups not carrying a bi-invariant metric. In this case the equation does not correspond to a known variational problem associated with a functional of type (2.1.2). It would be interesting to find geometric interpretations [30, Chapter 6, Section 3] for (2.2.1) also in this case.

3 Loop groups and factorization theorems

3.1 Let G be a connected Lie group, (or, as well, the connected component of a given Lie group), admitting a faithful finite dimensional representation. We assume that G and its Lie algebra g are realized by N × N matrices. As in [2] we consider various loop groups associated with G. To this end, let

A = A_{r} = { f(λ) = ∑_{n∈Z Z} λ^{n}f_{n} | f_{n} ∈ C, ∑_{n∈Z Z} w_{r}(n) · |f_{n}| < ∞}, (3.1.1)

where w_{r}(n) = (1 + |n|)^{r}, n ∈ ZZ, (3.1.2)

with r ∈ N\{0} fixed. Then A is organized as a complex Banach algebra with the norm

||f||_{r} = ||f|| = ∑_{n∈Z Z} w_{r}(n)|f_{n}|. (3.1.3)

On the set of mappings

ΛMat(N, A) = {A : S^{1} → M_{n×n}(C) | (A(λ))_{ij} ∈ A for all i, j = 1, …, N},

we introduce the norm

|||A|||_{r} = max_{j∈{1,...,N}} { Σ_{i=1}^{N}||A(λ)_{ij}||_{r}} . (3.1.4)

Then Mat(N, A) with the norm (3.1.4) becomes an associative Banach algebra. The complexification g^{C} = g + ig of the real Lie algebra g of G can also be regarded as a Lie algebra of complex matrices. Then (3.1.5) Λg^{C} = g^{C} ⊗ A = {f : S^{1} → g^{C} | (f(λ))_{ij} ∈ A, for all i, j ∈ {1, …, n}}, together with the norm (3.1.4) is a Banach Lie algebra, and the group (3.1.6) ΛG^{C} = {f : S^{1} → G^{C} | (f(λ))_{ij} ∈ A, for all i, j ∈ {1, …, n}}, becomes a Banach Lie group with the Lie algebra Λg^{C}. Moreover, the group (3.1.7) ΛG = {f ∈ ΛG^{C} | f(λ) ∈ G, for all λ ∈ S^{1}} is a real Banach subgroup of ΛG^{C}, and has the Lie algebra (3.1.8) Λg = {f ∈ Λg^{C} | f(λ) ∈ g, for all λ ∈ S^{1}}. In the following we shall consider based loops, i.e., the subset Λ_{1}G^{C} = {f ∈ ΛG^{C} | f(1) = I} ⊂ ΛG^{C}, (3.1.9) where we denoted by I the identity matrix of order N × N. Similarly, one can define Λ_{1}G. Then the corresponding Lie algebras Λ_{1}g^{C} and Λ_{1}g are defined by the condition f(1) = 0. The elements of Λ_{1}g^{C} have expansions of the form

f(λ) = ∑_{n∈Z Z\{0}}(λ^{n} − 1)f_{n}, f_{n} ∈ g^{C}. (3.1.10)

and each loop f ∈ Λ_{1}G^{C} has an expansion of the form

f(λ) = I + ∑_{n∈Z Z\{0}}(λ^{n} − 1)f_{n}, f_{n} ∈ M_{N×N}(C). (3.1.11)

3.2 Loop group splittings are essential tools in the DPW method [10]. Such splittings were used in the study of harmonic maps by Uhlenbeck [29], later by Pressley, Segal [27], and recently by Guest and Ohnita [14]. Also, in a slightly different differential geometric context, such splittings have been used in [11] and [15]. The splittings involve at the Lie algebra level the subalgebras Λ_{1}g, Λ_{1}^{+}g^{C} and Λ_{1}^{−}g^{C}, where (3.2.1) Λ_{1}^{+}g^{C} = {f ∈ Λ_{1}g^{C} | f_{n} = 0, for n < 0}. In particular, every element f ∈ Λ_{1}^{+}g^{C} is of the form

f(λ) = ∑_{n=1}^{∞}(λ^{n} − 1)f_{n}, (3.2.2)

and similarly one can describe the elements of Λ_{1}^{−}g^{C}. The following result is straightforward Lemma. The three algebras above have pairwise trivial intersections and (3.2.3) Λ_{1}g^{C} = Λ_{1}g + Λ_{1}^{−}g^{C} = Λ_{1}^{−}g^{C} + Λ_{1}^{+}g^{C}. We define the following subgroups of ΛG^{C}:

Λ^{+}G^{C} = {g ∈ ΛG^{C} | g and g^{−1} extend holomorphically to D},

Λ^{−}G^{C} = {g ∈ ΛG^{C} | g and g^{−1} extend holomorphically to C \ D},

where D = {z ∈ C | | z |< 1} and C = C∪{∞} denotes the Riemann sphere. Then the Lie subalgebras Λ_{1}^{+}g^{C} and Λ_{1}^{−}g^{C} above determine respectively the based subgroups

(3.2.4)  Λ_{1}^{+}G^{C} = Λ^{+}G^{C} ∩ Λ_{1}G^{C},  Λ_{1}^{−}G^{C} = Λ^{−}G^{C} ∩ Λ_{1}G^{C}.

Proposition. The groups Λ_{1}^{+}G^{C} and Λ_{1}^{−}G^{C} are connected Banach subgroups of Λ_{1}G^{C}. Proof. For g ∈ Λ_{1}^{+}G^{C} and r ∈ [0, 1], we have that also g_{r}(λ) = g(r)^{−1} · g(rλ) is in Λ_{1}^{+}G^{C} and the mapping r → g_{r} is continuous.

□

Similarly one verifies that Λ_{1}^{−}G^{C} is connected. From [2, Theorem 4.5] we recall Theorem. (Birkhoff Factorization Theorem).

Any element g ∈ ΛG^{C} can be written as g = g_{−}Dg_{+} (3.2.5) where g_{±} ∈ Λ^{±}G^{C}, D = sb_{−}^{+} ∈ Λ^{d}G^{C}, and

Λ^{d}G^{C} = ∪_{s∈Λ^{d}H^{C}}s(Λ^{−}B^{C})_{s}^{+} (3.2.6)

and (Λ^{−}B^{C})_{s}^{−} = {b ∈ Λ^{−}B^{C} | sbs^{−1} ∈ Λ^{−}B^{C}} (3.2.7) (Λ^{−}B^{C})_{s}^{+} = {b ∈ Λ^{−}B^{C} | sbs^{−1} ∈ Λ_{∗}^{+}B^{C}}. (3.2.8)

The expression (3.2.5) will be called a Birkhoff decomposition, of g.

Corollary. If g ∈ Λ_{1}G^{C} is contained in the big cell Λ^{−}G^{C} · Λ^{+}G^{C} of ΛG^{C}, then g = g_{−} g_{+} with uniquely determined g_{−} ∈ Λ_{1}^{−}G^{C} and g_{+} ∈ Λ_{1}^{+}G^{C}.

Moreover, the big cell Λ_{1}^{−}G^{C} · Λ_{1}^{+}G^{C} is open and dense in Λ_{1}^{+}G^{C} and the map

Λ_{1}^{−}G^{C} × Λ_{1}^{+}G^{C} → Λ_{1}^{−}G^{C} · Λ_{1}^{+}G^{C}, (g_{−}, g_{+}) → g_{−}g_{+}

is an analytic diffeomorphism.

Proof. By assumption g = g̃_{−}g̃_{+} with g̃_{−} ∈ Λ^{−}G^{C} and g̃_{+} ∈ Λ^{+}G^{C}. Then g_{−} = g̃_{−} · g̃_{−}(λ = 1)^{−1} ∈ Λ_{1}^{−}G^{C}, g_{+} = g̃_{−}(λ = 1)g̃_{+} ∈ Λ^{+}G^{C} and g = g_{+}g_{−}. Since g ∈ Λ_{1}G^{C}, also g_{+} ∈ Λ_{1}^{+}G^{C}. Assume now g_{−}g_{+} = g̃_{−}g̃_{+} with g_{−}, g̃_{−} ∈ Λ_{1}^{−}G^{C} and g_{+}, g̃_{+} ∈ Λ_{1}^{+}G^{C}. Then g_{−}^{−1}g̃_{−} = g_{+}(g̃_{+})^{−1} = A is independent of λ. But evaluating the left side at λ = 1 yields I, whence A = I and the decomposition is unique.

To show the last statements it suffices to prove that the big cell is dense in Λ_{1}G^{C}. Let g ∈ Λ_{1}G^{C}. Then, since the big cell of ΛG^{C} is dense, for every ε > 0 there exists some g̃ ∈ ΛG^{C} such that ||g − g̃|| < ε. Let h = g̃_{0}^{−1}g̃, where g̃_{0} = g_{(λ=1)}. Then h is still in the big cell of ΛG^{C} and we have

||g − h|| ≤ ||g − g̃|| + ||g̃ − g̃_{0}^{−1}g̃|| < ε + ||g̃|| · ||I − g̃_{0}^{−1}||.

Since evaluation at λ = 1 is a continuous map,

|| ∑_{i,j∈{1,...,N}}(g_{ij} − g̃_{ij})|| ≤ ∑_{i,j∈{1,...,N}} ||g_{ij} − g̃_{ij}|| = ||g − g̃|| < ε, (3.2.9)

we conclude that we can choose h arbitrarily close to g. □

3.3 The second splitting we are using in this paper is a generalized Iwasawa splitting. From [2, Theorem 6.5] we recall

Theorem. (Iwasawa Factorization Theorem).

Let G be a connected real Lie group, which admits a finite-dimensional faithful representation. Then ΛG^{C} = ΛG · Λ^{m}G^{C} · Λ^{+}G^{C}, (3.3.1)

is a disjoint union of double cosets indexed by the middle terms

Λ^{m}G^{C} = ∪_{s∈Λ^{m}H^{C}}(ΛB)_{s}^{#} · s, (3.3.2)

where (ΛB)_{s}^{#} = s.ΛB.s^{−1}.

More precisely, every g ∈ ΛG^{C} has a unique representation of the form g = hb.wsb_{+}h_{+} (3.3.3)

where g = h̃b̃ and h̃ = hsh_{+} is the unique representation of h̃ ∈ ΛH^{C} = ΛH · Λ^{d}H^{C} · Λ^{+}H^{C} as described in the Appendix of [2].

Corollary. If g ∈ Λ_{1}G^{C} is contained in ΛG · Λ^{+}G^{C}, then

g = hv_{+}, (3.3.4)

with uniquely determined h ∈ Λ_{1}G and v_{+} ∈ Λ_{1}G^{C}.

Moreover, if the big cell ΛG · Λ^{+}G^{C} is open and dense in ΛG^{C}, then the big cell Λ_{1}G · Λ_{1}^{+}G^{C} is open and dense in Λ_{1}G^{C} and the map

Λ_{1}G × Λ_{1}^{+}G^{C} → Λ_{1}G · Λ_{1}^{+}G^{C}, (h, v_{+}) → hv_{+}, (3.3.5)

is an analytic diffeomorphism.

Proof. The proof is almost verbatim identical with the proof of Corollary 3.2. □

Remark. We would like to point out that one can show that ΛG × Λ^{+}G^{C} is dense in ΛG^{C} if the semisimple part of a maximal compact subgroup of G^{C} is simply connected [2, Theorem 7.2]. This applies when the semisimple part of ΛG^{C} is simply conneceted. Moreover, ΛG.Λ^{+}G^{C} = ΛG^{C} iff the reductive part of G (i.e., the group H in [2, Section 2]) is compact.

3.4 Using [27, 8.6,8.7], we characterize the group elements not in the big cell by the vanishing of some function. This will be of importance in Theorem 4.7.

We can also prove a different splitting by [27, 8.6,8.7], and give a more direct proof using [8] to a more particular result.

We consider the ”lexicographic” isomorphism [27, 6.6], [8, 3.1] of L^{n} with L = L^{2}(S^{1}). This induces an injection of ΛG^{C} into GLres(L). We first split L = L_{−} ⊕ L_{+}, where L_{−} is spanned by the functions λ^{n}, n ≤ 0 and L_{+} by the λ^{n}, n > 0. This way the bounded operators of L decompose naturally into 2 × 2 blocks. An algebra B of such blocks has been presented for a large class of weights in [8, 1.10]. In our case, we use the weight w_{r−1} = (1 + |n|)^{r−1}, for g^{C} defined by the weight w_{r}. From [8, Proposition 3.4] we know that the embedding of g^{C} into the bounded operators L is contained in B. Since the inclusion of B into the bounded operators with off-diagonal Hilbert-Schmidt operators is bounded, it follows that the map Λ^{0}G^{C} → GLres(L) is holomorphic, where Λ^{0}G^{C} denotes the connected component of ΛG^{C}. Since GLres(L) acts holomorphically on the Grassmannian Gr(L) as defined in [27, Chapter 7, Section 7], the map g ∈ Λ^{0}G^{C} → i(g).L_{−} ∈ Gr(L) (3.4.1) is holomorphic. Finally, since the τ-function defined in [28, Chapter 3, Paragraph 3], [27, Chapter 8, Paragraph 10] is holomorphic on Gr(L), we obtain altogether that the map (3.4.2) φ : Λ^{0}G^{C} → C, φ(g) = τ(i(g).L_{−}), for all g ∈ Λ^{0}G^{C}

is holomorphic. Hence we have

Theorem.

φ(g) ≠ 0 ⇔ g ∈ (Λ_{1}^{−}G^{C}).(Λ_{1}^{+}G^{C}). (3.4.3)

Proof. Since G^{C} ⊂ SL(n, C), one obtains the Birkhoff decomposition g = a_{−}da_{+} for all g ∈ Λ^{0}G^{C}. Then

φ(g) = τ (i(a_{−})i(d)i(a_{+})) .L_{−} = τ (i(a_{−})i(d).L_{−}) . (3.4.4)

Hence φ(g) ≠ 0 iff φ(d) ≠ 0 and it follows directly that φ(d) ≠ 0 iff d = e. □

4 Generalized Weierstrass representation

4.1 For a real Lie group G we can characterize the harmonicity of the map φ : D → G, in terms of its associated Maurer-Cartan form

α ≡ φ^{−1}dφ : D → g ≡ Lie(G).

We assume in the following that G is endowed with a bi-invariant pseudo-Riemannian metric. First we call [29, p. 5]:

Proposition. The following statements are equivalent:

a) The map φ is harmonic.

b) The form α satisfies the integrability and harmonicity equations:

\{ \begin{matrix}dα + \frac{1}{2}[α ∧ α] = 0 \\ ∂z̄α′ + ∂_{z}α′′ = 0, \end{matrix} (4.1.1)

where α′ and α′′ are the holomorphic and the antiholomorphic parts of the differential form α, respectively. K. Uhlenbeck’s proof of the Proposition above carries over without change to the present pseudo-metric situation.

4.2 Following a successful procedure in soliton theory and related geometric theories [29], [10, p. 647], [11, 2.5, p. 50], we introduce a parameter λ into our theory. We will use d = ∂_{z} + ∂z̄ and split forms as

α = α′ + α′′ ≡ A_{(z)}dz + A_{(z̄)}dz̄: T D^{1,0} ⊕ T D^{0,1} → g^{C}, for all λ ∈ S^{1}.

Then we introduce the g^{C}-valued ”loopified form” α_{λ} defined on D,

α_{λ} = \frac{1 − λ^{−1}}{2} α′ + \frac{1 − λ}{2}α′′, (4.2.1)

for all λ ∈ S^{1}. Then we see by a straightforward compuatation that the system of equations (4.1.1) can be replaced by one equation:

Proposition. φ is harmonic iff the loopified form α_{λ} is integrable for all λ ∈ S^{1}, i.e., it satisfies the integrability condition:

dα_{λ} + \frac{1}{2}[α_{λ} ∧ α_{λ}] = 0. (4.2.2)

We would like to point out that equation (4.2.2) implies that we can find some map (”extended harmonic map”)

φ : (x, y; λ) ∈ D × S^{1} → φ(x, y, λ) ∈ G,

such that

α_{λ} = φ^{−1}dφ : D × S^{1} → g,

for any λ ∈ S^{1}. This will be the basis for our group splitting method. We will call the family of extended harmonic maps definied above the ”associated family” for φ. We notice that though the parameter λ can be chosen arbitrary on C^{∗}, in certain cases, the condition α_{λ}(x, y; λ) ∈ g requires λ ∈ S^{1}.

4.3 We prove the following statement.

Theorem. Let u : D → Λ_{1}G be an extended harmonic map such that u(0, 0, 1) = I. Then there exists a map v_{+} : D → Λ_{1}^{+}G^{C} such that

g = uv_{+} : D → Λ_{1}G^{C} (4.3.1)

is holomorphic and g(0) ∈ Λ_{1}^{+}G^{C}. Also, for certain holomorphic maps Q, A_{n} : D → g we can write

g^{−1} d g = (\frac{1 − λ^{−1}}{2} Q(z) + ∑_{n=0}^{∞}(1 − λ^{n})A_{n}(z)) d z. (4.3.2)

Proof. Consider

u^{−1} du = \frac{1 − λ^{−1}}{2} A d z + \frac{1 − λ}{2}Ā d z̄. (4.3.3)

Then we have

(uv_{+})^{−1} d(uv_{+}) = v_{+}^{−1} · u^{−1} · du · v_{+} + v_{+}^{−1} dv_{+}. (4.3.4)

Hence the dz̄ part of this expression is

v_{+}^{−1} \frac{1 − λ}{2} Āv_{+} + v_{+}^{−1}∂z̄v_{+}. (4.3.5)

In order to make uv_{+} holomorphic we must annihilate this expression with v_{+} ∈ Λ_{1}^{+}G^{C}, i.e., we must satisfy the differential equation

∂z̄v_{+} = −\frac{1 − λ}{2}Āv_{+}, (4.3.6)

with v_{+} ∈ Λ_{1}^{+}G^{C}. Since Ā is real analytic, we can consider z and z̄ as independent variables and solve (4.3.6) for z̄ ∈ D. Since the coefficient matrix −\frac{1 − λ}{2}Ā ∈ Λ_{1}^{+}G^{C}, we can assume that the solution is in Λ_{1}^{+}G^{C}. Hence we have shown that exists some v_{+} : D → Λ_{1}^{+}G^{C}, such that g = uv_{+} is holomorphic. Since u(0, 0, λ) = I, we obtain g(0, λ) = v_{+}(0, 0, λ) ∈ Λ_{1}^{+}G^{C}. Finally, we have

g^{−1} dg = v_{+}^{−1} (\frac{1 − λ^{−1}}{2} A dz + \frac{1 − λ}{2}Ā dz̄) v_{+} + v_{+}^{−1} dv_{+} (4.3.7)

which shows that g^{−1} dg has the required form (4.3.2). □

4.4 Consider a holomorphic map R : D → Λ_{1}G^{C} of the form

R(z, λ) = \frac{1 − λ^{−1}}{2} Q(z) + ∑_{n=0}^{∞}(1 − λ^{n})A_{n}(z), (4.4.1)

i.e., Q and all A_{n} are holomorphic maps from D into g^{C}. Next we consider the differential equation g^{−1} dg = R dz, g(0) = I. (4.4.2)

This differential equation has a unique solution g ∈ Λ_{1}G^{C} once g(0) is given. We note that actually g ∈ Λ_{1}^{0}G^{C} holds since Λ_{1}^{+}G^{C} is connected. Since g ∈ Λ_{1}G^{C} and g(0, λ) = I, we can apply Corollary 3.3 and we split

g(z, λ) = u(z, z̄, λ)v_{+}(z, z̄, λ), (4.4.3)

where u ∈ Λ_{1}G and v_{+} ∈ Λ_{1}^{+}G^{C}.

Theorem. The mapping u is an extended harmonic map from D into G and u(0, 0, λ) = I.

Proof. Differentiating (4.4.3), we obtain

R = g^{−1} dg = v_{+}^{−1} · u^{−1} · du · v_{+} + v_{+}^{−1} dv_{+}. (4.4.4)

We use (4.4.2) and conjugate by v_{+}. Then we have

v_{+}Rv_{+}^{−1} dz = u^{−1} du + dv_{+} · v_{+}^{−1}. (4.4.5)

From (4.4.5) we see that u^{−1} du is of the form

u^{−1} du = \frac{1 − λ^{−1}}{2} g_{1} + ∑_{k=1}^{∞}(1 − λ^{k})g_{k}. (4.4.6)

But u^{−1} du = \overline{u^{−1} du}, whence only 1−λ^{−1} and 1−λ can occur in u^{−1} du. Proposition 4.2 now shows that (4.1.1) holds. This shows that u is an extended harmonic map. Finally, since g(0, 0, λ) ∈ Λ_{1}^{+}G^{C}, from (4.4.3) and the uniqueness of the splitting we obtain u(0, 0, λ) = I. □

4.5 Considering the previous results we conclude that the set H of (extended) harmonic maps u : D → Λ_{1}G satisfying u(0, 0, λ) = I and the set R of ”potentials” R as in (4.4.1) are closely related. In the following we shall make this relation more precise. First we note that in (4.4.3) we can obtain a unique solution g associated with R by requiring g(0, λ) = I. In this case v_{+} defined in (4.4.2) then satisfies v_{+}(0, 0, λ) = I as well. This way we obtain a map ψ : R → H. Following the proof of Theorem 4.3, we see that the requirement v_{+}(0, 0, λ) = I determines v_{+}(z, z̄, λ) uniquely up to a holomorphic factor q_{+}(z, λ) satisfying q_{+}(0, λ) = I. Any choice of q_{+} will produce some R ∈ R and ψ(R) is the same given u ∈ H for all these R. This shows that ψ is surjective. To describe the fibers of ψ we consider the ”generalized gauge group”

(4.5.1) G^{+} = {q_{+} : D → Λ_{1}^{+}G^{C} | q_{+} holomorphic , q_{+}(0, λ) = I}.

We define the action of G^{+} on R as follows: let R ∈ R and define g as in (4.4.1) with g(0, λ) = I. Then split g = uv_{+} and form ĝ = uv_{+}q_{+} = gq_{+}. We consider ĝ^{−1} dĝ = R̂ dz and set R.q_{+} = R̂. It is straightforward to see that

R.q_{+} = q_{+}^{−1}Rq_{+} + q_{+}^{−1} dq_{+} (4.5.2)

holds, and also (R.q_{+}).p_{+} = R.(q_{+}p_{+}). Hence we have the following

Theorem. a) The group G^{+} acts on the right on R by gauge transformations.

b) The map ψ : R → H is surjective.

c) The fibers of ψ are orbits of G^{+} in R.

4.6 Further, we consider a group action on H by Λ_{1}^{+}G^{C}. Let w_{+} ∈ Λ_{1}^{+}G^{C}. Note that w_{+} = w_{+}(λ) does not depend on z or z̄. Let u ∈ H. Then we consider w_{+}(λ)u(z, z̄, λ)w_{+}(λ)^{−1} and split

w_{+}(λ)u(z, z̄, λ)w_{+}(λ)^{−1} = û(z, z̄, λ)v̂_{+}(z, z̄, λ). (4.6.1)

We set w_{+}.u = û. (4.6.2)

Then we have the following

Theorem. The operation (4.6.2) defines an action of Λ_{1}^{+}G^{C} on H.

Proof. Clearly, û(0, 0, λ) = I. To see that û is an extended harmonic map, it suffices to show that û^{−1} dû is of the form (4.2.1). For this it suffices to show that is of the form (4.4.1), since û ∈ Λ_{1}G. But

(w_{+}uw_{+}^{−1})^{−1} d(w_{+}uw_{+}^{−1}) = w_{+}(u^{−1} du)w_{+}^{−1} = (w_{+}Rw_{+}^{−1}) (4.6.3)

is of the form (4.4.1), whence û ∈ H. To see that the action (4.6.2) is a group action, we note that w_{+}.u = û is equivalent with w_{+}u = ûv_{+} for some v_{+} : D → Λ_{1}^{+}G^{C}. Hence p_{+}.(w_{+}.u) = p_{+}(w_{+}.u)a_{+} = p_{+}(w_{+}ub_{+})a_{+} = (p_{+}w_{+})u(b_{+}a_{+}), (4.6.4)

whence p_{+}.(w_{+}.u) = (p_{+}w_{+}).u. (4.6.5)

□

Comparing the two actions defined here and in section 4.5, we have

Proposition. For every R ∈ R and every w_{+} ∈ Λ_{1}^{+}G^{C} we have

ψ(w_{+}Rw_{+}^{−1}) = w_{+}.ψ(R). (4.6.6)

Proof. Let R ∈ R and solve g^{−1} dg = R dz, g(0, λ) = I. Then

w_{+}Rw_{+}^{−1} = ĝ^{−1}dĝ, ĝ(0, λ) = I (4.6.7)

has the solution ĝ = w_{+}gw_{+}^{−1}. Hence ψ(w_{+}Rw_{+}^{−1}) = û where ĝ = ûv̂_{+}. On the other hand ĝ = w_{+}gw_{+}^{−1} implies

ĝ = w_{+}uv_{+}w_{+}^{−1} = w_{+}uw_{+}^{−1}.w_{+}v_{+}w_{+}^{−1} = (w_{+}.u)q_{+}.w_{+}v_{+}w_{+}^{−1} (4.6.8)

for some v_{+}, q_{+} : D → Λ_{1}^{+}G^{C}. The uniqueness of the group splitting now shows w_{+}.u = û, whence the claim.

Remarks. a) To know all the (extended) harmonic maps it suffices to know a representative for each orbit of Λ_{1}^{+}G^{C} on H. It is an open question to provide geometric conditions which point out such a representative in each orbit.

b) The composition of the two actions looks like

w_{+}.(R.q_{+}) = w_{+}(q_{+}^{−1}Rq_{+} + q_{+}^{−1} dq_{+})w_{+}^{−1}, for w_{+} ∈ G_{0}^{+} ≡ Λ_{1}^{+}G^{C}, (4.6.9)

and respectively

(w_{+}.R).q_{+} = (w_{+}^{−1}q_{+})^{−1}R(w_{+}^{−1}q_{+}) + q_{+}^{−1} dq_{+}, (4.6.10)

for q_{+} : D → Λ_{1}^{+}G^{C} holomorphic, where

R : D → Λ_{1}G^{C}, R(z, λ) = \frac{1 − λ^{−1}}{2} Q(z) + ∑_{n≥0}(1 − λ^{n})A_{n}(z), (4.6.11)

and Q, A_{n} : D → C are holomorphic maps. We notice that the two (left resp. right) actions generally do not commute on the level of potentials. However, the induced actions on the level of harmonic maps trivially commute, since the gauge transformation acts trivially on the immersions.

4.7 We have seen in 4.5 that the group G^{+} acts on R by gauge transformations and that the G^{+} orbits correspond to the points of H. We shall examine if it is possible to find a natural representative in each G^{+} orbit. We guess that it might be a potential R without any component in Λ_{1}^{+}G^{C}. This can be achieved, if we drop the requirement that R stays holomorphic, as shows the following

Theorem. a) Let u ∈ H; then there exists a discrete subset S_{u} ⊂ D and a map

v_{+} : D\S_{u} → Λ_{1}^{+}G^{C}, v_{+}(0, 0, λ) = I,

such that g = uv_{+} is meromorphic on D and

g^{−1} dg = R dz (4.7.1)

is of the form R = \frac{1 − λ^{−1}}{2} Q(z) with Q : D → g^{C} meromorphic. Moreover, the poles of g and Q are contained in S_{u}.

b) On the other hand, let R = \frac{1 − λ^{−1}}{2} Q(z) be meromorphic on D with discrete pole set S. Assume that

g^{−1} dg = R dz, g(0, λ) = I

has a meromorphic solution in D with poles in S. Then we split g = uv_{+} for z ∈ D\S and obtain an extended harmonic map u : D\S → G.

Proof. a) Let u ∈ H. Then Theorem 4.3 shows that there exists some q_{+} : D → Λ_{1}^{+}G^{C} such that g̃ = uq_{+} is holomorphic and g̃(0, λ) = I. This shows that g̃(z, · ) ∈ Λ_{1}^{0}G^{C} for every z ∈ D. We consider the map σ : D → C, σ(z) = φ(g̃(z, · )), where φ has been defined in (3.4.2). Since φ and g̃ are holomorphic, σ is holomorphic as well. Moreover, σ(0) = φ(g̃(0, · )) = φ(I) = τ(H_{−}) ≠ 0. (4.7.2)

From Theorem 3.4 we now conclude

g̃(z, · ) ∈ Λ_{1}^{−}G^{C} · Λ_{1}^{+}G^{C} (4.7.3)

for all z ∈ D\S_{u}, where S_{u} = {z ∈ D | σ(z) = 0} is a discrete subset of D. In particular, u(z, z̄, λ) = a_{−}(z, z̄, λ)a_{+}(z, z̄, λ) (4.7.4)

for all z ∈ D\S_{u}. Moreover, from [10, Lemma 2.6] we obtain that the singularities of a_{−} and a_{+} are only poles. Hence g = a_{−} = u.q_{+}.a_{+}^{−1} for z ∈ D\S_{u}. Now it is easy to

see that g^{−1} dg is of the form \frac{1 − λ^{−1}}{2} Q(z) with Q holomorphic on D\S_{u} and has only poles in S_{u}.

b) can be proved as in 4.4. □

Remarks. a) The theorem above seems to suggest that one should consider ”singular harmonic maps” rather than holomorphic harmonic maps. However, in this case, similar to [9], one needs to investigate the question when a singular R gives a nonsingular extended harmonic map.

b) If u is an extended harmonic map and u = u_{−}u_{+} for z ∈ D\S then R associated with u_{−}, u_{−}^{−1} du_{−} = R dz makes the choice of R independent of gauge transformations (i.e., R is unique). This is justified by the following argument: for

R = \frac{1 − λ^{−1}}{2} Q(z),  R̂ = \frac{1 − λ^{−1}}{2} Q̂(z),

one obtains g, ĝ ∈ Λ_{1}^{−}G^{C} which split g = uv_{+}, ĝ = uv̂_{+} whence ĝ^{−1}g = v̂_{+}^{−1}v_{+} is holomorphic, and hence v_{+} = v̂_{+}^{−1}w_{+}(z, λ) implies g = ĝw_{+}. Since g, ĝ ∈ Λ_{1}^{−}G^{C}, w_{+} ∈ Λ_{1}^{+}G^{C}, from the uniqueness of the splitting, we get w_{+} = I, which infers R = R̂.

c) The action of Λ_{1}^{+}G^{C} on the general R was provided simply by the adjoint action. Fixing the form of R as \frac{1 − λ^{−1}}{2} Q(z), makes the Λ_{1}^{+}G^{C} action more complicated.

4.8 In general it is fairly difficult to carry out the required group splittings. However, if one knows the harmonic map then it is not difficult to find the meromorphic potential. The result below extends Wu’s formula to the case of generalized harmonic maps into arbitrary Lie groups.

Theorem. Let u(z, z̄, λ) : D → G be an extended harmonic map. Then u has a meromorphic extension to D × \overline{D} and the meromorphic potential R associated with u satisfies R(z, λ) = u(z, 0, λ)^{−1}∂_{z}u(z, 0, λ). (4.8.1)

Proof. We consider the based double loop group X = Λ_{1}G^{C} × Λ_{1}G^{C} and its

subgroups

(4.8.2)  X^{+} = Λ_{1}^{+}G^{C} × Λ_{1}^{−}G^{C}
X^{−} = Λ_{1}^{−}G^{C} × Λ_{1}^{+}G^{C}

and X^{Δ} = {(g, g) | g ∈ Λ_{1}G}. Let σ denote complex conjugation in Λ_{1}G^{C} relative to Λ_{1}G. Then for every h ∈ Λ_{1}G^{C} we set i(h) = (h, σ(h)). Splitting

i(h) = (p, p)(h_{+}, h_{−}) p ∈ Λ_{1}G, h_{ε} ∈ Λ^{ε}G^{C}, (4.8.3)

we obtain h = ph_{+}, σ(h) = ph_{−}. (Note, if G = SU(N), then this is the classical Iwasawa decomposition). Splitting

i(h) = (q_{+}, q_{−})(l_{+}, l_{−}) (4.8.4)

we obtain h = q_{+}l_{−} and σ(h) = q_{−}l_{+}. If h = u is an extended harmonic map, then σ(h) = h and q_{−} leads to the holomorphic potential R, while q_{+} = σ(q_{−}) leads to the ”complex conjugate” meromorphic potential σ(R). Consider now the pair of potentials R = ( R(z) dz, σ(R)(w̄) dw̄ ). Solving the ODE B^{−1} dB = R yields a function B : D × \overline{D} → X which is meromorphic in (z, w̄). Clearly B = (g(z), σ(g(w))). Splitting B = (p, p)(h_{+}, h_{−}) (4.8.5)

yields g(z) = p(z, w)h_{+}(z, w) and σ(g(w)) = p(z, w)h_{+}(z, w). Because of

g^{−1}(z)σ(g(w)) = h_{+}^{−1}(z, w)h_{−}(z, w), (4.8.6)

we have that h_{+} and h_{−} are meromorphic in (z, w). Therefore also p is meromorphic in (z, w). Finally setting w = z̄ we obtain g(z) = p(z, z̄)h_{+}(z), whence p(z, z̄) = u(z, z̄) and p(z, w) is a meromorphic extension of u. Let now v_{+}(z, z̄, λ) be chosen as in Theorem 4.7. Then g(z, λ) = u(z, z̄, λ)v_{+}(z, z̄, λ). Setting z̄ = 0 this yields

g(z, λ) = u(z, 0, λ)v_{+}(z, 0, λ). (4.8.7)

But the equation u^{−1}∂_{z}u = \frac{1−λ^{−1}}{2} α′ can be evaluated at z̄ = 0 and we obtain

u(z, 0, λ)^{−1}∂_{z}u(z, 0, λ) = \frac{1 − λ^{−1}}{2} α′. (4.8.8)

Since the right side is in Λ_{1}^{−}G^{C}, this ODE has a unique solution h ∈ Λ_{1}^{−}G^{C} satisfying the initial condition h(z = 0) = I. Moreover, h(z, λ) = c(λ)u(z, 0, λ). But I = u(0, 0, λ) implies c(λ) = I and h(z, λ) = u(z, 0, λ) ∈ Λ_{1}^{−}G^{C}. Therefore g(z, λ) = u(z, 0, λ) and the claim follows. □

Corollary. Let φ : D → G be a harmonic map and u its harmonic extension. Then u(z, z̄, λ = −1) = φ(z, z̄) (4.8.9)

and φ has a meromorphic extension to D × \overline{D} and the meromorphic potential R associated with u is of the form

R(z, λ) = u(z, 0, λ)^{−1}∂_{z}u(z, 0, λ) = \frac{1 − λ^{−1}}{2} · φ(z, 0)^{−1}∂_{z}φ(z, 0). (4.8.10)

Proof. This follows from (4.8.8) if one sets λ = −1.

5 Applications

Below we list several Lie groups and determine their bi-invariant metrics. Moreover, we spell out the harmonic map equations (2.2.1) for these groups and illustrate these equations with some examples. We would like to recall, that in the cases, where there does exist a (non-degenerate) bi-invariant metric, these equations follow from a variational principle. In the other cases we can write down the equations, but do not know of any variational and/or geometric interpretation. We note that throughout the section, the loop group decompositions, as stated in the theorems, are in the based loop groups. Moreover, due to our conventions (see Section 4), for every harmonic map φ we assume φ(0, 0) = I.

5.1 Let G be the nilpotent group

G = \{( \begin{matrix}1 & a \\ 0 & 1 \end{matrix} )| a ∈ R\} . (5.1.1)

In this case Lie(G) is one dimensional; therefore every nondegenerate symmetric bilinear form is determined by a non-zero scalar. Since G is abelian, the left-invariant metrics are also right-invariant. Consider now a map φ : D → G,

φ(x, y) = ( \begin{matrix}1 & a(x, y) \\ 0 & 1 \end{matrix} ) , a ∈ C^{2}( D, R), (5.1.2)

for all (x, y) in D ⊂ R. Applying (2.2.1) we obtain that φ is harmonic if and only if

Δφ ≡ (φ_{x})_{x} + (φ_{y})_{y} = 0, (5.1.3)

which is the canonical harmonicity equation for maps with values in ( R, +). Let φ : D → G be a harmonic map (5.1.2). Then a = Re (a_{h}), with a_{h} a holomorphic function. The Maurer Cartan form associated to φ is then

ω = φ^{−1}φ = ( \begin{matrix}0 & da \\ 0 & 0 \end{matrix} ) = ( \begin{matrix}0 & ∂_{z}a \\ 0 & 0 \end{matrix} ) dz + ( \begin{matrix}0 & ∂z̄a \\ 0 & 0 \end{matrix} ) dz̄, (5.1.4)

and, denoting f = \frac{1−λ^{−1}}{2} , the loopified form is,

ω_{λ} = f ( \begin{matrix}0 & ∂_{z}a \\ 0 & 0 \end{matrix} ) dz + f̄( \begin{matrix}0 & ∂z̄a \\ 0 & 0 \end{matrix} ) dz̄. (5.1.5)

We note that ∂z̄a = \overline{∂_{z}a} and see that one can integrate the equation g_{λ}^{−1} dg_{λ} = ω_{λ}, with initial condition g_{λ}(z, z̄, λ)|_{z=z̄=0} = I_{2} and obtains

g_{λ}(z, z̄, λ) = ( \begin{matrix}1 & (fa_{h}(z) + f̄\overline{a_{h}(z)})/2 \\ 0 & 1 \end{matrix} ) . (5.1.6)

The Birkhoff decomposition g_{λ} = g_{λ}_{−} · g_{λ}_{+} provides then

g_{λ}_{−} = ( \begin{matrix}1 & fa_{h}(z)/2 \\ 0 & 1 \end{matrix} ) , (5.1.7)

whence the meromorphic potential is

R = g_{λ}_{−}^{−1}∂_{z}g_{λ}_{−} = ( \begin{matrix}0 & f∂_{z}a_{h}(z)/2 \\ 0 & 0 \end{matrix} ) = f ( \begin{matrix}0 & ∂_{z}a \\ 0 & 0 \end{matrix} ) . (5.1.8)

We note that Wu’s extended formula (4.8.10) in Corollary 4.8 produces directly

ξ = φ^{−1}∂_{z}φ |_{z̄=0}= ( \begin{matrix}0 & ∂_{z}a \\ 0 & 0 \end{matrix} ) , (5.1.9)

which yields (5.1.8). We have traced above explicitely all the steps of our general theory, starting with a harmonic map finally arriving at its potential. Conversely, it is easy to follow the splitting procedure outlined in Theorem 4.7 in this example: let a be some meromor-

phic function and R = \frac{1 − λ^{−1}}{2} ( \begin{matrix}0 & a \\ 0 & 0 \end{matrix} ) the corresponding meromorphic potential.

We solve the differential equation g^{−1} dg = R dz (4.7.1) with the initial condition g(0, λ) = I. We see that in our case

g = ( \begin{matrix}1 & \frac{1−λ^{−1}}{2} ∫_{0}^{z} a(w) dw \\ 0 & 1 \end{matrix} ) (5.1.10)

and g is meromorphic if all residues of a vanish. The Iwasawa splitting g = u·v_{+} then is

g = ( \begin{matrix}1 & Re ((1 − λ^{−1}) ∫_{0}^{z} a(w) dw) \\ 0 & 1 \end{matrix} ) · ( \begin{matrix}1 & \frac{λ−1}{2} ∫_{0}^{z} ā(w) dw \\ 0 & 1 \end{matrix} ) . (5.1.11)

This shows that in this case we recover the well known fact that every harmonic function is the real part of a holomorphic function.

5.2 Let G be the (Heisenberg) group of upper triangular unipotent matrices

G = \{( \begin{matrix}1 & a & c \\ 0 & 1 & b \\ 0 & 0 & 1 \end{matrix} )| a, b, c ∈ R\} . (5.2.1)

From [24] it follows that the Heisenberg group does not have a bi-invariant metric. This can also easily be verified directly. Let now φ : D → G,

φ(x, y) = ( \begin{matrix}1 & a(x, y) & c(x, y) \\ 0 & 1 & b(x, y) \\ 0 & 0 & 1 \end{matrix} ), a, b, c ∈ C^{2}( D, R) (5.2.2)

for all (x, y) in D ⊂ R^{2}. Then, denoting

z = x + iy, a_{z} = (a_{x} − ia_{y})/2, az̄ = (a_{x} + ia_{y})/2,

the (formal) harmonicity equation (2.2.3) reads

\{ \begin{matrix}Δa ≡ 4a_{zz̄} = 0, b_{zz̄} = 0 \\ c_{zz̄} = (a_{z}bz̄ + az̄b_{z})/2. \end{matrix} (5.2.3)

It is easy to see that (5.2.3) is equivalent with a, b and c− \frac{1}{2}ab are harmonic functions. From Wu’s extended formula (4.8.10) we obtain that the meromorphic potential associated with any harmonic map φ of the form (5.2.2) is given by R = \frac{1−λ^{−1}}{2} ·V^{−1}·∂_{z}V dz, where V = φ(z/2, −iz/2). (5.2.4)

A straightforward computation yields

R = \frac{1 − λ^{−1}}{2} ( \begin{matrix}0 & ∂_{z}a & ∂_{z}c − a∂_{z}b \\ 0 & 0 & ∂_{z}b \\ 0 & 0 & 0 \end{matrix} ), (5.2.5)

where a = a(z/2, −iz/2), b = b(z/2, −iz/2), c = c(z/2, −iz/2). As in section 5.1, also for the Heisenberg group every step of the general theory leading from a harmonic map to the potential can be made explicit.

For a given harmonic map φ = ( \begin{matrix}1 & a & c \\ 0 & 1 & b \\ 0 & 0 & 1 \end{matrix} ) as in (5.2.2), its Maurer-Cartan form

is

φ^{−1} dφ = ( \begin{matrix}0 & da & dc − a db \\ 0 & 0 & db \\ 0 & 0 & 0 \end{matrix} ) (5.2.6)

whence by splitting da = ∂_{z}adz +∂z̄adz̄, and similar for db, dc, one gets the loopified Maurer-Cartan form

(5.2.7)  ω_{λ} = \frac{1 − λ^{−1}}{2} ( \begin{matrix}0 & ∂_{z}a & ∂_{z}c − a∂_{z}b \\ 0 & 0 & ∂_{z}b \\ 0 & 0 & 0 \end{matrix} ) dz +

+\frac{1 − λ}{2}( \begin{matrix}0 & ∂z̄a & ∂z̄c − a∂z̄b \\ 0 & 0 & ∂z̄b \\ 0 & 0 & 0 \end{matrix} ) dz̄.

Then an extended frame φ_{λ} = ( \begin{matrix}1 & A & C \\ 0 & 1 & B \\ 0 & 0 & 1 \end{matrix} ) has the Maurer-Cartan form

φ_{λ}^{−1} dφ_{λ} = ( \begin{matrix}0 & dA & dC − A dB \\ 0 & 0 & dB \\ 0 & 0 & 0 \end{matrix} ). (5.2.8)

To find its coefficients A, B, C requires to solve the system

dA = \frac{1−λ^{−1}}{2} ∂_{z}a dz + \frac{1−λ}{2}∂z̄a dz̄
dB = \frac{1−λ^{−1}}{2} ∂_{z}b dz + \frac{1−λ}{2}∂z̄b dz̄
dC = A dB + \frac{1−λ^{−1}}{2} (∂_{z}c − a∂_{z}b) dz + \frac{1−λ}{2}(∂z̄c − a∂z̄b) dz̄.

(5.2.9)

Remark that a, b and k = c − \frac{ab}{2} are real parts of holomorphic functions, and hence

\{ \begin{matrix}a(z) = (a_{h}(z) + \overline{a_{h}(z)})/2 \\ b(z) = (b_{h}(z) + \overline{b_{h}(z)})/2, \\ k(z) = (k_{h}(z) + \overline{k_{h}(z)})/2, \end{matrix} (5.2.10)

where a_{h}, b_{h}, k_{h} are holomorphic functions which vanish at z = 0. As a consequence, we have the relations \overline{∂_{z}a} = ∂z̄a, \overline{∂_{z}b} = ∂z̄b, \overline{∂_{z}k} = ∂z̄k, (5.2.11) Setting f = \frac{1−λ^{−1}}{2} , the first two relations yield ∂_{z}A = f∂_{z}a, ∂_{z}B = f∂_{z}b, (5.2.12) and the equations can be rewritten as

\{ \begin{matrix}dA = f∂_{z}a dz + \overline{f∂_{z}a} dz̄, A(0) = 0 \\ dB = f∂_{z}b dz + \overline{f∂_{z}b} dz̄, B(0) = 0, \end{matrix} (5.2.13)

whence, using an argument similar to that of case 1, we integrate and obtain A(z) = Re (fa_{h}(z)), B(z) = Re (fb_{h}(z)). (5.2.14) The last equation of (5.2.9) writes dC = A dB + f(∂_{z}c − a∂_{z}b) dz + f̄(∂z̄c − a∂z̄b) dz̄. (5.2.15) We assume C(0) = 0. Since dC = ∂_{z}C dz + ∂z̄C dz̄, ∂_{z}B = f∂_{z}b, and A = Re(fa_{h}) we have

(5.2.16)  ∂_{z}C = A∂_{z}B + f(∂_{z}c − a∂_{z}b) =
= f · [ ∂_{z}c + (Re (fa_{h}) − a)∂_{z}b ],

whence

(5.2.17)  C(z, z̄) = f(c(z) + u(z)) + c_{p}(z̄) =
= f · (k(z) + v(z)) + c_{p}(z̄),

where we denoted

(5.2.18) u = ∫_{0}^{z} [Re(fa_{h}(w)) − a(w)] ∂_{z}b(w) dw
v = \frac{ab}{2} + u = ∫_{0}^{z} \frac{fa_{h} + f̄\overline{a_{h}}}{2} ∂_{z}b + (\frac{ab}{2} − ∫_{0}^{z} a(w)∂_{z}b(w) dw) .

Since k = k̄ and C = C̄, we have 0 = C − C̄ = (f − f̄)k + (fv − f̄v̄) + c_{p} − \overline{c_{p}}, (5.2.19) whence c_{p} − \overline{c_{p}} = (f̄− f)k + (f̄v̄ − fv). (5.2.20) Note that in view of (5.2.10), the factor k is the sum of a holomorphic and an antiholomorphic function. Therefore also the last term on the right splits this way, (f̄v̄ − fv) = w_{h} + w_{a}. (5.2.21)

Actually, the expansion of (5.2.21) in terms of holomorphic and antiholomorphic parts yields the given terms in the right, since the terms which are not purely holomorphic or antiholomorphic drop out by the harmonicity equation. Then explicit expressions of the two outcoming terms are

(5.2.22)  w_{h} = \frac{f − f^{2}}{4} ∫_{0}^{z} a_{h}(w)∂_{z}b_{h}(w) dw + \frac{f̄− f}{8} a_{h}b_{h}
(5.2.23)  w_{a} = \frac{f̄^{2} − f̄}{4} · \overline{∫_{0}^{z} a_{h}(w)∂_{z}b_{h}(w) dw} + \frac{f̄− f}{8} \overline{a_{h}} · \overline{b_{h}},

and separate the holomorphic/antiholomorphic parts in (5.2.20), which implies

c_{p} − w_{a} − (f̄− f)\overline{k_{h}}/2 = \overline{c_{p}} + (f̄− f)k_{h}/2 + w_{h}. (5.2.24)

Now, taking into account that v(0) = k(0) = C(0) = c_{p}(0) = 0, this yields that both terms in (5.2.24) vanish, and

c_{p} = w_{a} + (f − f̄)\overline{k_{h}}/2, (5.2.25)

whence C = Re (fk_{h}) + fv + w_{a}. (5.2.26)

Then the extended frame is

φ_{λ} = ( \begin{matrix}1 & Re (fa_{h}) & C \\ 0 & 1 & Re (fb_{h}) \\ 0 & 0 & 1 \end{matrix} ). (5.2.27)

Denoting φ_{λ} = ( \begin{matrix}1 & A & C \\ 0 & 1 & B \\ 0 & 0 & 1 \end{matrix} ), we perform its Birkhoff splitting

( \begin{matrix}1 & A & C \\ 0 & 1 & B \\ 0 & 0 & 1 \end{matrix} ) = φ_{λ}_{−}·φ_{λ}_{+} = ( \begin{matrix}1 & A_{−} & C_{−} \\ 0 & 1 & B_{−} \\ 0 & 0 & 1 \end{matrix} )·( \begin{matrix}1 & A_{+} & C_{+} \\ 0 & 1 & B_{+} \\ 0 & 0 & 1 \end{matrix} ) ∈ Λ_{1}^{−}G^{C}·Λ_{1}^{+}G^{C},

(5.2.28) which leads to the relations

(5.2.29)  A = A_{−} + A_{+} ∈ Λ^{−}C + Λ^{+}C
(5.2.30)  B = B_{−} + B_{+} ∈ Λ^{−}C + Λ^{+}C
(5.2.31)  C = C_{−} + C_{+} + A_{−}B_{+}

where A_{±}, B_{±} ∈ Λ^{±}C are uniquely determined by the Birkhoff decomposition of the entries A and B,

\{ \begin{matrix}A_{−} = fa_{h}/2, A_{+} = \overline{A_{−}} \\ B_{−} = fb_{h}/2, B_{+} = \overline{B_{−}} \end{matrix} . (5.2.32)

The entry C rewrites as Birkhoff sum C = 2 Re (C_{m}) = C_{m} + C_{p} ∈ Λ^{−}C + Λ^{+}C. (5.2.33) Also, in view of the equality f · f̄ = (f + f̄)/2, (5.2.34) it follows that the term A_{−}B_{+} in the third relation of (5.2.29) splits

A_{−}B_{+} = (fa_{h} · f̄\overline{b_{h}})/4 = \frac{1}{8}fa_{h}\overline{b_{h}} + \frac{1}{8}f̄a_{h}\overline{b_{h}} ∈ Λ^{−}C + Λ^{+}C, (5.2.35)

which is exactly the Birkhoff splitting of A_{−}B_{+}. Hence (5.2.31) becomes

C = (C_{−} + \frac{1}{8}fa_{h}\overline{b_{h}}) + (C_{+} + \frac{1}{8}f̄a_{h}\overline{b_{h}}). (5.2.36)

and, using the uniqueness of the Birkhoff decomposition (5.2.33), one gets

\{ \begin{matrix}C_{−} = C_{m} − fa_{h}\overline{b_{h}}/8 \\ C_{+} = C_{p} − f̄a_{h}\overline{b_{h}}/8. \end{matrix} (5.2.37)

So we have

φ_{λ}_{−} = ( \begin{matrix}1 & fa_{h}/2 & C_{m} − fa_{h}\overline{b_{h}}/8 \\ 0 & 1 & fb_{h}/2 \\ 0 & 0 & 1 \end{matrix} ), (5.2.38)

We use that

a = \frac{fa_{h} + f̄\overline{a_{h}}}{2} (5.2.39)

whence ∂_{z}a = f∂_{z}a_{h}/2 (5.2.40) and ∂_{z}A_{−} = f∂_{z}a, (5.2.41) and obtain the meromorphic potential R = φ_{λ}_{−}^{−1}∂_{z}φ_{λ}_{−},

(5.2.42)  R = ( \begin{matrix}1 & A_{−} & C_{−} \\ 0 & 1 & B_{−} \\ 0 & 0 & 1 \end{matrix} )^{−1} · ( \begin{matrix}0 & ∂_{z}A_{−} & ∂_{z}C_{−} \\ 0 & 0 & ∂_{z}B_{−} \\ 0 & 0 & 0 \end{matrix} ) =
(5.2.43)  = ( \begin{matrix}0 & ∂_{z}a & ∂_{z}C_{−} − A_{−}∂_{z}B_{−} \\ 0 & 0 & ∂_{z}b \\ 0 & 0 & 0 \end{matrix} ).

The computation of the right-upper corner yields

∂_{z}C_{−} − A_{−}∂_{z}B_{−} = ∂_{z}C_{m} − ∂_{z} (f \frac{a_{h}\overline{b_{h}}}{8} ) − f^{2} \frac{a_{h}∂_{z}b}{2}. (5.2.44)

Comparing (5.2.43) to (5.2.5) we see that it suffices to show that the functions in the right upper corner of both matrices coincide. To this end we first note

(5.2.45) ∂_{z}C = ∂_{z}(f(c + u) + c_{p}) = f∂_{z}c + f (\frac{fa_{h} + f̄\overline{a_{h}}}{2} − a) ∂_{z}b =
(5.2.46)  = f∂_{z}c + f^{2} \frac{a_{h}∂_{z}b}{2} + (f + f̄)\frac{\overline{a_{h}}∂_{z}b}{4} − fa∂_{z}b =
(5.2.47)  = f^{2} \frac{a_{h}∂_{z}b}{2} + f (∂_{z}c − a∂_{z}b + \frac{\overline{a_{h}}∂_{z}b}{4}) + f̄\frac{\overline{a_{h}}∂_{z}b}{4}.

This implies

∂_{z}C_{m} = f^{2} \frac{a_{h}∂_{z}b}{2} + f (∂_{z}c − a∂_{z}b + \frac{\overline{a_{h}}∂_{z}b}{4}) . (5.2.48)

Therefore

∂_{z}C_{−} − A_{−}∂_{z}B_{−} = f (∂_{z}c − a∂_{z}b + \frac{\overline{a_{h}}∂_{z}b}{4} − ∂_{z} (\frac{a_{h}\overline{b_{h}}}{8} )) . (5.2.49)

From this we infer that ∂z̄ annihilates this expression which hence does not depend on z̄, and is equal to the one obtained when setting z̄ = 0. But a_{h}(0) = b_{h}(0) = 0, and hence, for z̄ = 0 we obtain

∂_{z}C_{−} − A_{−}∂_{z}B_{−} = f (∂_{z}c − a∂_{z}b) . (5.2.50)

Thus (5.2.5) and (5.2.43) coincide. As shown in [7, Section 6, p. 158] maps of the form

φ : R^{2} → G, φ(x, y) = e^{xX} · e^{yY} , (5.2.51)

where X, Y ∈ g, satisfy the harmonicity equations (5.2.3). In particular one can choose for the Heisenberg group X, Y from the generators of g,

A_{1} = ( \begin{matrix}0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix} ), A_{2} = ( \begin{matrix}0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{matrix} ), A_{3} = ( \begin{matrix}0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix} ). (5.2.52)

More precisely, starting from a harmonic map of type (5.2.51)

(5.2.53) φ = exp \{ x( \begin{matrix}0 & a & c \\ 0 & 0 & b \\ 0 & 0 & 0 \end{matrix} )\} · exp \{ y( \begin{matrix}0 & α & γ \\ 0 & 0 & β \\ 0 & 0 & 0 \end{matrix} )\} =

= ( \begin{matrix}1 & ax + αy & cx + γy + aβxy + \frac{abx^{2}+αβy^{2}}{2} \\ 0 & 1 & bx + βy \\ 0 & 0 & 1 \end{matrix} ), a, b, c, α, β, γ ∈ R,

we apply Wu’s formula and derive its meromorphic potential of the form

(5.2.54)  ξ = \frac{1}{4}( \begin{matrix}0 & 2(a − iα) & iz(αb − aβ) + 2(c − iγ) \\ 0 & 0 & 2(b − iβ) \\ 0 & 0 & 0 \end{matrix} ).

As in 5.1 we also want to make explicit the converse chain of steps. From a given potential it is easy to follow the construction principle given in Theorem 4.7.

Let a, b, c provide the meromorphic potential R = \frac{1−λ^{−1}}{2} ( \begin{matrix}0 & a & c \\ 0 & 0 & b \\ 0 & 0 & 0 \end{matrix} ). Then solving

the differential equation g^{−1} dg = R dz (4.7.1) with the initial condition g(0, λ) = I, we obtain

g = ( \begin{matrix}1 & \frac{1−λ^{−1}}{2} ∫_{0}^{z} a(w) dw & \frac{1−λ^{−1}}{2} ∫_{0}^{z} (c(w) + b(w) ∫_{0}^{w} a(v) dv) dw \\ 0 & 1 & \frac{1−λ^{−1}}{2} ∫_{0}^{z} b(w) dw \\ 0 & 0 & 1 \end{matrix} ). (5.2.55)

In order to make this meromorphic we need to assume that the residues of a, b and c(z) + b(z) ∫_{0}^{z} a(w) dw all vanish.

The Iwasawa splitting of g = ( \begin{matrix}1 & A & C \\ 0 & 1 & B \\ 0 & 0 & 1 \end{matrix} ) can be carried out explicitely in this

case:

g = u·v_{+} = ( \begin{matrix}1 & A + Ā & 2Re(C + Ā · B) \\ 0 & 1 & B + B̄ \\ 0 & 0 & 1 \end{matrix} )·( \begin{matrix}1 & −Ā & ĀB̄ − C̄ − BĀ \\ 0 & 1 & −B̄ \\ 0 & 0 & 1 \end{matrix} ).

(5.2.56) Thus every extended harmonic map from D to G is of the form

u = ( \begin{matrix}1 & A + Ā & 2Re(C + Ā · B) \\ 0 & 1 & B + B̄ \\ 0 & 0 & 1 \end{matrix} ), (5.2.57)

where

\{ \begin{matrix}A = \frac{1−λ^{−1}}{2} ∫_{0}^{z} a(w) dw, B = \frac{1−λ^{−1}}{2} ∫_{0}^{z} b(w) dw \\ C = \frac{1−λ^{−1}}{2} ∫_{0}^{z} (c(w) + b(w) ∫_{0}^{w} a(v) dv) dw. \end{matrix} (5.2.58)

This implies that in terms of the coefficient functions of the potential every harmonic map from D to G is of the form

φ = ( \begin{matrix}1 & 2 Re ∫_{0}^{z} a(w) dw & C_{∗} \\ 0 & 1 & 2 Re ∫_{0}^{z} b(w) dw \\ 0 & 0 & 1 \end{matrix} ), (5.2.59)

where

(5.2.60)  C_{∗} = 2 Re (∫_{0}^{z} (c(w) + b(w) ∫_{0}^{w} a(v) dv) dw+
+\overline{∫_{0}^{z} a(w) dw} · ∫_{0}^{z} b(w) dw) .

It is straightforward to verify that, indeed, starting from ξ of the form (5.2.54) formula (5.2.59) yields exactly (5.2.53).

5.3 Consider the special linear Lie group

G = SL(2, R) = \{( \begin{matrix}a & b \\ c & d \end{matrix} )| a, b, c, d ∈ R, ad − bc = 1\} . (5.3.1)

For G, as for any simple real Lie group, there exists only one bi-invariant metric, up to scalar multiples, the one produced by the Killing form. In this case the harmonicity equations (2.2.1) follow from applying the variational principle and rewrite

aΔc − cΔa = 0
bΔd − dΔb = 0
aΔd − dΔa = cΔb − bΔc,

(5.3.2)

or, at points where all four functions do not vanish, these have the equivalent form,

\frac{Δa}{a} = \frac{Δb}{b} = \frac{Δc}{c} = \frac{Δd}{d}. (5.3.3)

In the previous sections we have shown what meromorphic potentials one obtains from a given harmonic map, both by applying Wu’s Formula and by direct computation of a Birkhoff splitting. We also followed explicitely the construction of the harmonic map from a potential, in particular carrying out an Iwasawa splitting. In the case under consideration in this section the group splittings seem to be very difficult if not impossible to carry out ”by hand” explicitely. In a sense this underlines the strength of the theory presented. We will show in several examples what harmonicity means in the case of SL(2, R) and compute the corresponding potentials via Wu’s Formula. As a first class of examples we consider harmonic maps for which the coefficient functions a, b, c, d all are of the form p(x) + q(y) and satisfy

\frac{Δa}{a} = \frac{Δb}{b} = \frac{Δc}{c} = \frac{Δd}{d} = 1. (5.3.4)

It follows that the solutions of (5.3.4) are of the form

φ(x, y) = ( \begin{matrix}a_{1}e^{x} + a_{2}e^{−x} + a_{3}e^{y} + a_{4}e^{−y} & b_{1}e^{x} + b_{2}e^{−x} + b_{3}e^{y} + b_{4}e^{−y} \\ c_{1}e^{x} + c_{2}e^{−x} + c_{3}e^{y} + c_{4}e^{−y} & d_{1}e^{x} + d_{2}e^{−x} + d_{3}e^{y} + d_{4}e^{−y} \end{matrix} ) ,

(5.3.5) where the condition det φ = 1 is equivalent to the condition that the coefficients a_{i}, b_{i}, c_{i}, d_{i}, i ∈ {1, 2, 3, 4} satisfy the system

a_{i}d_{i} = b_{i}c_{i}, for all i ∈ {1, 2, 3, 4}
a_{i}d_{j} + a_{j}d_{i} = b_{i}c_{j} + b_{j}c_{i}, for all (i, j) ∈ {1, 2} × {3, 4}
a_{1}d_{2} + a_{2}d_{1} + a_{3}d_{4} + a_{4}d_{3} − (b_{1}c_{2} + b_{2}c_{1} + b_{3}c_{4} + b_{4}c_{3}) = 1.

(5.3.6)

Also, the condition φ(0, 0) = I implies

\{ \begin{matrix}a_{1} + a_{2} + a_{3} + a_{4} = d_{1} + d_{2} + d_{3} + d_{4} = 1 \\ b_{1} + b_{2} + b_{3} + b_{4} = c_{1} + c_{2} + c_{3} + c_{4} = 0. \end{matrix} (5.3.7)

The family of harmonic maps (5.3.5) provides via Wu’s extended formula potentials ξ = φ^{−1}∂_{z}φ|_{z̄=0} of the form

(5.3.8) ξ = (1 − i) · e^{(1+i)z/2} ( \begin{matrix}b_{1}c_{4} − d_{1}a_{4} & b_{1}d_{4} − d_{1}b_{4} \\ c_{1}a_{4} − c_{4}a_{1} & −(b_{1}c_{4} − d_{1}a_{4}) \end{matrix} ) +
+ (1 + i) · e^{(1−i)z/2} ( \begin{matrix}b_{1}c_{3} − a_{3}d_{1} & b_{1}d_{3} − d_{1}b_{3} \\ c_{1}a_{3} − c_{3}a_{1} & −(b_{1}c_{3} − a_{3}d_{1}) \end{matrix} ) +
+ (1 + i) · e^{(−1+i)z/2} ( \begin{matrix}a_{4}d_{2} − b_{2}c_{4} & b_{4}d_{2} − d_{4}b_{2} \\ c_{4}a_{2} − c_{2}a_{4} & −(a_{4}d_{2} − b_{2}c_{4}) \end{matrix} ) +
+ (1 − i) · e^{(−1−i)z/2} ( \begin{matrix}a_{3}d_{2} − b_{2}c_{3} & b_{3}d_{2} − d_{2}b_{3} \\ a_{2}c_{3} − a_{3}c_{2} & −(a_{3}d_{2} − b_{2}c_{3}) \end{matrix} ) + ( \begin{matrix}a_{∗} & b_{∗} \\ c_{∗} & d_{∗} \end{matrix} ) ,

where

\{ \begin{matrix}a_{∗} = −d_{∗} = (a_{1}d_{2} − a_{2}d_{1}) + (b_{1}c_{2} − b_{2}c_{1}) + i[(a_{4}d_{3} − a_{3}d_{4}) + (b_{4}c_{3} − b_{3}c_{4})] \\ b_{∗} = 2[(b_{1}d_{2} − b_{2}d_{1}) + i(b_{4}d_{3} − b_{3}d_{4})] \\ c_{∗} = 2[(c_{1}a_{2} − c_{2}a_{1}) + i(c_{4}a_{3} − c_{3}a_{4})]. \end{matrix}

(5.3.9) Note that for b_{1}b_{2}b_{3}b_{4}d_{1}d_{2}d_{3}d_{4} ≠ 0, the first equations in (5.3.6) rewrite

\frac{a_{i}}{b_{i}} = \frac{c_{i}}{d_{i}} = k_{i}, i = \overline{1, 4}, (5.3.10)

and hence the second set in (5.3.6) yields

(k_{i} − k_{j}) · (b_{i}d_{j} − b_{j}d_{i}) = 0, for all (i, j) ∈ {1, 2} × {3, 4}. (5.3.11)

In this case (5.3.8) rewrites

ξ = (1 − i) · (b_{1}d_{4} − d_{1}b_{4})e^{(1+i)z/2} ( \begin{matrix}k_{4} & 1 \\ −k_{1}k_{4} & −k_{4} \end{matrix} ) +
+ (1 + i) · (b_{1}d_{3} − d_{1}b_{3})e^{(1−i)z/2} ( \begin{matrix}k_{3} & 1 \\ −k_{1}k_{3} & −k_{3} \end{matrix} ) +
+ (1 + i) · (b_{4}d_{2} − d_{4}b_{2})e^{(−1+i)z/2} ( \begin{matrix}k_{4} & 1 \\ −k_{2}k_{4} & −k_{4} \end{matrix} ) +
+ (1 − i) · (b_{3}d_{2} − d_{2}b_{3})e^{(−1−i)z/2} ( \begin{matrix}k_{3} & 1 \\ −k_{2}k_{3} & −k_{3} \end{matrix} ) +
+ (b_{1}d_{2} − b_{2}d_{1}) ( \begin{matrix}k_{1} + k_{2} & 2 \\ −2k_{1}k_{2} & −(k_{1} + k_{2}) \end{matrix} ) +
+ i(b_{4}d_{3} − b_{3}d_{4}) ( \begin{matrix}k_{3} + k_{4} & 2 \\ −2k_{3}k_{4} & −(k_{3} + k_{4}) \end{matrix} ) .

In particular, for a_{2} = b_{1} = c_{3} = −d_{4} = \frac{1}{\sqrt{2}} and the rest of coefficients zero in (5.3.5), we obtain the following special example of a harmonic map φ_{∗} : R^{2} → SL(2, R)

φ_{∗}(x, y) = \frac{1}{\sqrt{2}} ( \begin{matrix}e^{−x} & −e^{−y} \\ e^{y} & e^{x} \end{matrix} ) . (5.3.12)

When multiplying φ_{∗} by the constant matrix φ_{∗}|_{x=y=0}^{−1} = \frac{1}{\sqrt{2}}( \begin{matrix}1 & 1 \\ −1 & 1 \end{matrix} ) on the left, which does not change harmonicity, we obtain the harmonic map φ which satisfies the condition φ(0, 0) = I,

φ(x, y) = \frac{1}{2} ( \begin{matrix}e^{−x} + e^{y} & e^{x} − e^{−y} \\ −e^{−x} + e^{y} & e^{x} + e^{−y} \end{matrix} ) . (5.3.13)

This is a harmonic map given by J.Inoguchi, which can be obtained from (5.3.5) for a_{2} = a_{3} = c_{3} = b_{1} = d_{1} = d_{4} = 1/2 and c_{2} = b_{4} = −1/2 and the rest of coefficients zero. For z̄ = 0 this writes

φ|_{z̄=0} = \frac{1}{2} ( \begin{matrix}e^{−z/2} + e^{−iz/2} & e^{z/2} − e^{iz/2} \\ −e^{−z/2} + e^{−iz/2} & e^{z/2} + e^{iz/2} \end{matrix} ) (5.3.14)

and this leads to the holomorphic potential - a particular form of (5.3.8),

(5.3.15) φ^{−1}∂_{z}φ|_{z̄=0} = \frac{1}{2} ( \begin{matrix}−(1 + i) & (1 − i)e^{z(1+i)/2} \\ (1 − i)e^{−z(1+i)/2} & 1 + i \end{matrix} ) .

If a map φ : D → SL(2, R) has all its values in the subgroup H_{0} = SO(2) ⊂ SL(2, R), then

(5.3.16) φ = ( \begin{matrix}cos t(x, y) & − sin t(x, y) \\ sin t(x, y) & cos t(x, y) \end{matrix} )

and φ is harmonic, i.e., it satisfies (5.3.2), iff t satisfies the Laplace equation Δt = 0. We remark that for the subgroup

H_{1} = \{( \begin{matrix}a & b \\ 0 & d \end{matrix} )| ad = 1, a, b, d ∈ R\} ⊂ SL(2, R), (5.3.17)

the equations (5.3.2) lead to

\{ \begin{matrix}bΔd − dΔb = 0 \\ d_{x}^{2} + d_{y}^{2} − dΔd = 0. \end{matrix} (5.3.18)

Moreover, for the subgroup

H_{2} = \{( \begin{matrix}1 & b \\ 0 & 1 \end{matrix} )| b ∈ R\} ⊂ H_{1} ⊂ SL(2, R), (5.3.19)

the system (5.3.18) reduces to Δb = 0.

Acknowledgement. The present work was partially supported by the Fulbright Grant #21945-5/13/97 and respectively by the NSF Grant DMS - 9705479 and Sonderforschungsbereich 288, Differentialgeometrie und Quantenphysik, TU - Berlin.

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V. Balan Department Mathematics I Politehnica University of Bucharest Splaiul Independenţei 313, RO-77206 Bucharest, Romania

J. Dorfmeister Department of Mathematics TU-Muenchen, Arcistrasse 123, D-80333 Muenchen, Germany