Finite Type Lorentz Harmonic Maps and the Method of Symes

Josef Dorfmeister^{∗} Fakutät fr̈ Mathematik Technische Universität München Arcisstrasse 21, D-80290 München, Germany

Ivan Sterling^{†}

Department of Mathematics and Computer Science St. Mary’s College of Maryland St. Mary’s City, MD 20686, USA

Abstract

The purpose of this note is to define the notion of Symes type for Lorentz harmonic maps into S^{2}(1) and to prove that Lorentz harmonic maps into S^{2}(1) are finite-Symes type if and only if they are finite type. The analogous theorem for harmonic maps is a special case of a theorem due to Burstall and Pedit which states that harmonic maps into k-symmetric spaces are finite-Symes type if and only if they are finite type.

Keywords: Lorentz harmonic maps, finite type MS Classification: 53C43

∗dorfm@mathematik.tu-muenchen.de †isterling@smcm.edu

1 Introduction

The purpose of this note is to define the notion of Symes type for Lorentz harmonic maps into S^{2}(1) and to prove that Lorentz harmonic maps into S^{2}(1) are finite-Symes type if and only if they are finite type. The analogous theorem for harmonic maps is a special case of a theorem due to Burstall and Pedit [BP] which states that harmonic maps into k-symmetric spaces are finite-Symes type if and only if they are finite type. The term “finite-Symes type” is ours, but is very close to the terminology used in [BP].

In both cases (harmonic and Lorentz harmonic) the story line is related to the DPW representation of harmonic maps [DPW]. A DPW-like representation for (weakly regular) Lorentz harmonic maps into S^{2}(1) was given by Toda [T] in her study of (weakly regular) pseudo-spherical surfaces. In particular this includes all Lorentz harmonic maps into S^{2}(1) of finite type (or, as we will show, finite-Symes type).

It is important to note that Toda’s DPW-like representation produces all (weakly regular) Lorentz harmonic maps, including many which are not finite type. If one restricts to finite type maps, then an excellent reference is Terng-Uhlenbeck’s far-reaching work [TU]. They do not consider the question addressed in this paper. For another source of Burstall and Pedit’s result, and many related topics, see Guest’s book [G]. Guest introduces a notion of “generalized finite type”, it would be interesting to know if any of these are not finite type.

See [MS1] or [T] for the relationship between Lorentz harmonic maps in S^{2}(1), pseudo-spherical surfaces in R^{3} (K-surfaces) and the sine-Gordon equation. See [PS] for the historical and geometrical meaning of “finite type” in the context of soliton geometry and harmonic maps. In the physic literature the term “finite gap” is used.

For completeness and comparison we excised the harmonic case from [BP] and include it as the first half of the paper. Our proof follows the pattern of this paper. But due to the double loop group setting required for K-surfaces our proof has some new features. We would expect that at some time there will be a more general theory including most, if not all, present DPW-like theories. Then there also should be a general theorem relating finite type and finite-Symes type. We would be pleased if this note would be a steping stone towards such a general result. A preliminary attempt at this was already done in [MS2].

One central theme of this paper is to highlight the similarities and differences between the harmonic and the Lorentz harmonic cases. In order to better see the big picture. Particularly interesting is the delicate interplay between the Iwasawa and the Birkhoff factorization theorems. There has been recent work on this by I. McIntosh [S2].

In general in order to construct Lorentz harmonic maps one must first solve a (loop) system of ODE’s and then carry out a Birkhoff factorization. For finite type solutions the factorization step can be skipped. For finite-Symes type the solving of the ODE’s step can be skipped. It is the result of this paper that these apparently very different cheating tricks produce exactly the same set of solutions.

Finite type Lorentz harmonic maps enjoy many properties not shared by non-finite type Lorentz harmonic maps. Here we mention just a few.

• They can be constructed by solving a ODE’s [MS1].

• Alternatively, due to our main result (Theorem 3.2), they can be constructed by a Birkhoff factorization.

• They can be constructed by theta functions [D], [M]. (Even in the case of a singular spectral curve [E].)

• They can be classified via their spectral curves [KK].

• They can be deformed isospectrally via linear flows on the Jacobian of their spectral curves [EF].

• They include the Gauss maps of many of the classical K-surfaces, such as the pseudo-sphere, and the surfaces of Dini, Minding, Kuen, Enneper, and Dobriner. See [MS2] for a history of the subject and for several examples which are not finite type.

In another paper by Burstall and Pedit [BP2] they prove that every finite type harmonic map is in the dressing orbit of the trivial map. It is not known if this is also true for finite type Lorentz harmonic maps.

The tools developed in this paper are currently being implemented numerically and it is hoped that soon a framework for studying K-surfaces can be created as Nick Schmidt [S1] has done for H-surfaces (surfaces with constant mean curvature).

Finally, similar results might hold for Hasimoto surfaces and the Nonlinear Schroedinger Equation. The loop group approach for these examples is less developed.

The authors are grateful to Uwe Abresch, Fran Burstall, and Magdalena Toda for helpful comments.

2 Harmonic Maps into a Symmetric Space

2.1 Loop Spaces

Let N = G/K be a compact Riemannian symmetric space with G a compact semisimple Lie group, π the coset projection, K the fixed set of an order two automorphism τ. Assume the corresponding Cartan decomposition is

𝔤 = 𝔨 ⊕ 𝔪.

An element v in 𝔤 decomposes as v_{𝔨} + v_{𝔪}. We consider the Iwasawa decomposition of 𝔨^{ℂ} in three forms. First

𝔨^{ℂ} = 𝔨 ⊕ 𝔞 ⊕ 𝔫

where 𝔞 := i𝔱 for some maximal toral subalgebra 𝔱 of 𝔨 and 𝔫 is the nilpotent subalgebra given by the positive root spaces. Secondly, if we let 𝔟 := 𝔞 ⊕ 𝔫 (a solvable subalgebra), then we have

𝔨^{ℂ} = 𝔨 ⊕ 𝔟.

Thirdly, if we let

𝔥 := 𝔱^{ℂ} = 𝔱 ⊕ 𝔞

then

𝔨^{ℂ} = 𝔫 ⊕ 𝔥 ⊕ 𝔫̄.

The third form allows us to define a map r : 𝔨^{ℂ} → 𝔨^{ℂ} by

r(η) := η𝔫̄ + \frac{1}{2}η_{𝔥},  (2.1)

where the subscript denotes the component of η in the corresponding subspace.

The second form is used in the definition of (twisted) loop algebras. Let τ : 𝔤 → 𝔤 be the involution determining N (and τ : 𝔤^{ℂ} → 𝔤^{ℂ} its extension).

Choose and fix s > \frac{1}{2} and in the following definitions consider only maps which are (restrictions of) maps in H^{s}.

Λ𝔤_{τ}^{ℂ} := {ξ : S^{1} → 𝔤^{ℂ} : ξ(−λ) = τξ(λ) ∀λ ∈ S^{1}};
Λ𝔤_{τ} := {ξ ∈ Λ𝔤_{τ}^{ℂ} : ξ : S^{1} → 𝔤};
Λ_{+}𝔤_{τ}^{ℂ} := {ξ ∈ Λ𝔤_{τ}^{ℂ} : ξ extends holomorphically to ξ : D → 𝔤^{ℂ} and ξ(0) ∈ 𝔟};

where D is the disc {|λ| < 1}.

By considering the Fourier expansions we have (bewaring of reverse indexing)

Λ𝔤_{τ}^{ℂ} = {ξ = ∑_{n∈ℤ}ξ_{−n}λ^{n} : ξ_{n} ∈ 𝔤^{ℂ} and τξ_{n} = (−1)^{n}ξ_{n} ∀n};
Λ𝔤_{τ} = \{ξ ∈ Λ𝔤_{τ}^{ℂ} : \overline{ξ_{n}} = ξ_{−n} ∀n\};
Λ_{+}𝔤_{τ}^{ℂ} = {ξ ∈ Λ𝔤_{τ}^{ℂ} : ξ_{n} = 0 for n > 0 and ξ_{0} ∈ 𝔟}.

To introduce the infinite-dimensional (loop) Lie groups that correspond to these (loop) Lie algebras we let K^{ℂ} =: KB be the Iwasawa decomposition of K^{ℂ} corresponding to that of 𝔨^{ℂ} (using the second form). Then we have:

ΛG_{τ}^{ℂ} = {γ : S^{1} → G^{ℂ} : γ(−λ) = τγ(λ) ∀ λ ∈ S^{1}};
ΛG_{τ} = {γ ∈ ΛG_{τ}^{ℂ} : γ : S^{1} → G};
Λ_{+}G_{τ}^{ℂ} = {γ ∈ ΛG_{τ}^{ℂ} : γ extends holomorphically to γ : D → G^{ℂ}, γ(0) ∈ B}.

At the loop algebra level the (twisted) Iwasawa vector space decomposition is given by :

Λ𝔤_{τ}^{C} = Λ𝔤_{τ} ⊕ Λ_{+}𝔤_{τ}^{ℂ};

and at the loop group level the (twisted) Iwasawa decomposition is given by an onto diffeomorphism (via multiplication) [DPW]:

ΛG_{τ} × Λ_{+}G_{τ}^{ℂ} → ΛG_{τ}^{ℂ}.

We will denote the decomposition of an element γ ∈ ΛG_{τ}^{ℂ} via

γ = γ_{u}γ_{p}.

Finally, let

Λ_{d} := {ξ ∈ Λ𝔤_{τ} : ξ_{n} = 0 for all |n| > d}.

2.2 Maps

Let M be a Riemann surface and ϕ : M → N. A framing of ϕ is a map Φ : M → G such that π ∘ Φ = ϕ. Such lifts always exist locally. Let Φ be a framing of ϕ, α := Φ^{−1}dΦ and (λ ∈ S^{1})

α^{λ} := λ^{−1}α′_{𝔪} + α_{𝔨} + λα′′_{𝔪},  (2.2)

where α_{𝔪} decomposes into its (1, 0) and (0, 1)-parts α_{𝔪} = α′_{𝔪} + α′′_{𝔪}. A standard result is that if ϕ is harmonic then α^{λ} satisfies the Zero Curvature Equation (ZCE)

dα^{λ} + \frac{1}{2}[α^{λ} ∧ α^{λ}] = 0.

Conversely, if α^{λ} is a loop of 1-forms of the form (2.2), then locally dΦ^{λ} = Φ^{λ}α^{λ} can be (smoothly) integrated and projected to yield a loop of harmonic maps to N.

A map a : M → ΛG_{τ} is called an extended framing if

a^{−1}da = λ^{−1}α′_{𝔪} + α_{0} + λα′′_{𝔪}.

Note that any extended framing gives rise to a loop of harmonic maps (since α^{λ} := a^{−1}da satisfies the ZCE). Also note again that any harmonic map on a simply connected M defines an extended framing (since da = aα^{λ} can be smoothly integrated).

Let d ≡ 1 mod 2. If ξ : ℝ^{2} → Λ_{d} then α_{ξ}, a Λ𝔤_{τ}-valued 1-form on ℝ^{2}, is defined by

α_{ξ} := (λ^{−1}ξ_{d} + r(ξ_{d−1}))dz + (λξ_{−d} + \overline{r(ξ_{d−1})})dz̄,  (2.3)

where r is defined in (2.1). Note that d − 1 is even and hence ξ_{d−1} is in 𝔨^{ℂ}.

It is pivotal to the theory that such an α_{ξ} automatically satisfies the ZCE as soon as ξ satisfies dξ = [ξ, α_{ξ}] (see, for example, [BP]).

2.3 Main Result

A map Φ^{λ} : ℝ^{2} ⟶ ΛG_{τ} is called finite type if dΦ^{λ} = Φ^{λ}α_{ξ} , Φ^{λ}(0) = I where ξ : ℝ^{2} → Λ_{d} and dξ = [ξ, α_{ξ}] , ξ(0) = ξ_{o} for some ξ_{o} ∈ Λ_{d} where d ≡ 1 mod 2. The loop ξ is called the Killing field of Φ^{λ}. It is a well-known result that every ξ_{o} ∈ Λ_{d} with d ≡ 1 mod 2 yields such a map of finite type (again see, for example, [BP]). The function det(ξ(λ) − μI) is called the spectral curve of Φ^{λ} and does not depend on z. A harmonic map ϕ : ℝ^{2} ⟶ G/K is called finite type if ϕ = π ∘ ev_{1} ∘ Φ^{λ} where Φ^{λ} : ℝ^{2} ⟶ ΛG_{τ} is finite type. (Here ev_{1} means evaluation at λ = 1.)

A map a : ℝ^{2} ⟶ ΛG_{τ} is called finite-Symes type if a = exp(zλ^{d−1}ξ_{o})_{u} for some ξ_{o} ∈ Λ_{d} where d ≡ 1 mod 2. Note that every ξ_{o} ∈ Λ_{d} with d ≡ 1 mod 2 yields a map of finite-Symes type via the Iwasawa decomposition of exp(zλ^{d−1}ξ_{o}). A harmonic map ϕ : ℝ^{2} ⟶ G/K is called finite-Symes type if ϕ = π ∘ ev_{1} ∘ a where a : ℝ^{2} ⟶ ΛG_{τ} is finite-Symes type.

We remark that more generally a map a : ℝ^{2} ⟶ ΛG_{τ} is called Symes type if a = exp(zη_{o})_{u} for some η_{o} ∈ Λ𝔤_{τ}^{ℂ} such that λη_{o} extends holomorphically to D.

In [BP] it is shown that if a is of Symes type than it is an extending framing. In particular it is shown there (p. 258) that

a^{−1}da = λ^{−1}η_{1}dz + (η_{0}dz)_{𝔨} + λη̄_{1}dz̄  (2.4)

where η := Ad a^{−1}η_{o} and η =: ∑_{n≥−1} λ^{n}η_{−n}.

Theorem 2.1. [BP] A harmonic map ϕ : ℝ^{2} ⟶ G/K is finite type if and only if it is finite-Symes type.

Proof: (⇐) First we will show that if ϕ is finite-Symes type, then it is finite type. If it is finite-Symes type, then, by definition, ϕ = π ∘ ev_{1} ∘ a, where a is finite-Symes type. That is, a = exp(zλ^{d−1}ξ_{o})_{u}, as above, for some ξ_{o} ∈ Λ_{d}. Let Φ^{λ} := a and let ξ := Φ^{λ^{−1}}ξ_{o}Φ^{λ}. We show Φ^{λ} is finite type (and hence ϕ is), by showing that ξ is the desired Killing field for Φ^{λ}. That is, dξ = [ξ, α_{ξ}], where α_{ξ} is given by 2.3. Furthermore, by definition, we require that ξ : ℝ^{2} → Λ_{d} and ξ(0) = ξ_{o}. In summary we need to show that

i) ξ : ℝ^{2} ⟶ Λ_{d},

ii) Φ^{λ^{−1}}dΦ^{λ} = (λ^{−1}ξ_{d} + r(ξ_{d−1}))dz + (λξ_{−d} + \overline{r(ξ_{d−1)})}dz̄,

iii) dξ = [ξ, Φ^{λ^{−1}}dΦ^{λ}] with ξ(0) = ξ_{o}.

Let b denote the “positive” factor of the Iwasawa decomposition of exp(zλ^{d−1}ξ_{o}). That is, b = Φ^{λ^{−1}} exp(zλ^{d−1}ξ_{o}). For (i) first note ξ = Ad(b)ξ_{o} since

bξ_{o}b^{−1} = Φ^{λ^{−1}}Ad(exp(zλ^{d−1}ξ_{o}))ξ_{o}Φ = ξ.

Then note ξ ∈ Λ_{d} if and only if

λ^{d}ξ ∈ Λ_{+}𝔤_{τ}^{ℂ} and ξ ∈ Λ𝔤_{τ}.

But b ∈ Λ_{+}G_{τ}^{ℂ}, so λ^{d}bξ_{o}b^{−1} ∈ Λ_{+}𝔤_{τ}^{ℂ}. Also Φ^{λ} ∈ ΛG_{τ}, so Φ^{λ^{−1}}ξ_{o}Φ^{λ} ∈ Λ𝔤_{τ}. Part (ii) follows easily after collecting the following facts (letting η_{o} := λ^{d−1}ξ_{o})

◯ 1 Φ^{−1}dΦ = λ^{−1}η_{1}dz + (η_{0}dz)_{𝔨} + λη̄_{1}dz̄ (this is 2.4).

◯ 2 ∀η ∈ 𝔨^{ℂ} (ηdz)_{𝔨} = (η𝔫̄ + \frac{1}{2}η_{𝔥})dz + (\overline{η𝔫̄ + \frac{1}{2}η_{𝔥}})dz̄ (see proof below, we apply this to η = η_{0}).

◯ 3 r(η) := η𝔫̄ + \frac{1}{2}η_{𝔥} (this is 2.1).

◯ 4 ξ = λ^{1−d}η (by definition, thus η_{1} = ξ_{d} & η_{0} = ξ_{d−1}).

For part (iii), we simply differentiate ξ and use Φ(0) = I.

(⇒) If Φ^{λ} is finite type, with ξ its Killing field (with ξ(0) = ξ_{o}), then let a := exp(zλ^{d−1}ξ_{o})_{u}. Note again that da = aα_{ξ} with a(0) = I. Thus, by the uniqueness theory of ODE’s, Φ^{λ} = a and Φ^{λ} is finite-Symes type. □

Proof of ◯ 2 . This is equation (2.5) p.236 [BP]. Note that (ηdz)_{𝔨} = Re(ηdz) and that η𝔫̄ = \overline{η_{𝔫}}. We calculate.

ηdz = (η_{𝔫} + η_{𝔥} + η𝔫̄)dz = (η_{𝔫} + η_{𝔥} + η𝔫̄)(dx + idy).

Thus,

LHS = (ηdz)_{𝔨} = Re(ηdz)
= \frac{1}{2}((η_{𝔫} + η_{𝔥} + η𝔫̄) + (\overline{η_{𝔫}} + \overline{η_{𝔥}} + \overline{η𝔫̄}))dx + \frac{i}{2}((η_{𝔫} + η_{𝔥} + η𝔫̄) − (\overline{η_{𝔫}} + \overline{η_{𝔥}} + \overline{η𝔫̄}))dy
= (\frac{1}{2}(η_{𝔫} + \overline{η_{𝔫}}) + \frac{1}{2}(η𝔫̄ + \overline{η𝔫̄}) + \frac{1}{2}(η_{𝔥} + \overline{η_{𝔥}}))dx

+(\frac{i}{2}(η_{𝔫} − \overline{η_{𝔫}}) + \frac{i}{2}(η𝔫̄ − \overline{η𝔫̄}) + \frac{i}{2}(η_{𝔥} − \overline{η_{𝔥}}))dy.

Meanwhile,

RHS = (η𝔫̄ + \frac{1}{2}η_{𝔥})dz + (\overline{η𝔫̄ + \frac{1}{2}η_{𝔥}})dz̄
= (\frac{1}{2}(η𝔫̄ + \overline{η𝔫̄}) + \frac{1}{4}(η_{𝔥} + \overline{η_{𝔥}}))(dz + dz̄)

+(\frac{1}{2}(η𝔫̄ − \overline{η𝔫̄}) + \frac{1}{4}(η_{𝔥} − \overline{η_{𝔥}}))(dz − dz̄)

= ((η𝔫̄ + \overline{η𝔫̄}) + \frac{1}{2}(η_{𝔥} + \overline{η_{𝔥}}))dx

+i((η𝔫̄ − \overline{η𝔫̄}) + \frac{1}{2}(η_{𝔥} − \overline{η_{𝔥}}))dy.

Since η𝔫̄ = \overline{η_{𝔫}} we have LHS = RHS.

3 Lorentz Harmonic Maps into S^{2}(1)

We believe that the results of this section hold with little modification for Lorentz harmonic maps into a symmetric space (or for that matter into a k-symmetric space). However several of the tools we apply have been proved only for the case of S^{2}(1) and we shall therefore restrict to this case. We also mention that all the results remain true for weakly regular Lorentz harmonic maps. In that setting it is possible to study global questions. Thus all the results should otherwise be considered as local. It is unknown what happens in the non-weakly regular cases.

3.1 Loop Spaces

We therefore let N = S^{2}(1). We consider G = S^{3}(1) = SU(2) = unit quaternions,

and K = S^{1}. The map τ is given by Ad ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ). (See [MS1] for de-

tails.) Alternatively ([T]), G = SO(3), K = SO(2) and τ is given by

Ad ( \begin{matrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & −1 \end{matrix} ) .

When studying Lorentz harmonic maps ϕ : ℝ^{1,1} → S^{2}(1) we need “loop” spaces where the variable λ ∈ ℝ_{∗}. The H^{s} norm is replaced by a norm introduced in [T] and we consider the completion of the following spaces with respect to that norm.

Λ𝔤_{τ} = {ξ = ∑_{n∈ℤ}ξ_{n}λ^{n} : ξ_{n} ∈ 𝔤 and τξ_{n} = (−1)^{n}ξ_{n} ∀n};
Λ_{+}𝔤_{τ} = {ξ ∈ Λ𝔤_{τ} : ξ_{n} = 0 for n < 0};
Λ_{−}𝔤_{τ} = {ξ ∈ Λ𝔤_{τ} : ξ_{n} = 0 for n > 0};
Λ_{+_{∗}}𝔤_{τ} = {ξ ∈ Λ_{+}𝔤_{τ} : ξ(0) = 0};
Λ_{−_{∗}}𝔤_{τ} = {ξ ∈ Λ_{−}𝔤 : ξ(∞) = 0}.

The corresponding “loop” groups are denoted by ΛG_{τ}, Λ_{+}G_{τ}, Λ_{−}G_{τ}, Λ_{+_{∗}}G_{τ} and Λ_{−_{∗}}G_{τ}.

At the loop algebra level the (twisted) Birkoff vector space decomposition is given by :

Λ𝔤_{τ} = Λ_{+}𝔤_{τ} ⊕ Λ_{−_{∗}}𝔤_{τ} (resp. Λ_{τ} = Λ_{−}𝔤_{τ} ⊕ Λ_{+_{∗}}𝔤_{τ});

and at the loop group level the (twisted) Birkoff decomposition is given (via multiplication) by a diffeomorphism onto an open dense set of products [T]:

Λ_{+}G_{τ} × Λ_{−_{∗}}G_{τ} → Λ_{+}G_{τ} · Λ_{−_{∗}}G_{τ} ⊂ ΛG_{τ}
(resp. Λ_{−}G_{τ} × Λ_{+_{∗}}G_{τ} → Λ_{−}G_{τ} · Λ_{+_{∗}}G_{τ} ⊂ ΛG_{τ}).

We will denote the decomposition of an element g ∈ ΛG_{τ}^{ℂ} via

g = g_{+_{ℓ}}g_{−_{r}}(resp. g = g_{−_{ℓ}}g_{+_{r}}).

Finally, again let

Λ_{d} := {ξ ∈ Λ𝔤_{τ} : ξ_{n} = 0 for all |n| > d}.

3.2 Maps

M, which is a Riemann surface in the harmonic case, is now a simply connected domain in ℝ^{1,1} and we consider maps ϕ : M ⊂ ℝ^{1,1} → N = S^{2}(1). Again a framing of ϕ is a map Φ : M ⊂ ℝ^{1,1} → G = S^{3}(1) such that π∘Φ = ϕ. Such lifts always exist globally. Let Φ be a framing of ϕ, α := Φ^{−1}dΦ and (λ ∈ ℝ − {0})

α^{λ} := λα′_{𝔪} + α_{𝔨} + λ^{−1}α′′_{𝔪} (= λα̃′_{𝔪}dx + α_{𝔨} + λ^{−1}α̃′′_{𝔪}dy),  (3.1)

where α_{𝔪} decomposes into its dx and dy-parts α_{𝔪} = α′_{𝔪} + α′′_{𝔪}. A standard result is that if ϕ is Lorentz harmonic then α^{λ} satisfies the Zero Curvature Equation (ZCE)

dα^{λ} + \frac{1}{2}[α^{λ} ∧ α^{λ}] = 0.

Conversely, if α^{λ} is a loop of 1-forms of the form (3.1), then dΦ^{λ} = Φ^{λ}α^{λ} can be (smoothly) integrated and projected to yield a loop of Lorentz harmonic maps to N.

A map a : M → ΛG_{τ} is called an extended framing if

a^{−1}da = λα′_{𝔪} + α_{0} + λ^{−1}α′′_{𝔪}.

Note that any extended framing gives rise to a loop of Lorentz harmonic maps (since α^{λ} := a^{−1}da satisfies the ZCE). Also note again that any Lorentz harmonic map on M defines an extended framing (since da = aα^{λ} can be smoothly integrated).

Let d ≡ 1 mod 2. If ξ : M → Λ_{d} then α_{ξ}^{+} (resp. α_{ξ}^{−}), a Λ𝔤_{τ}-valued 1-from on M, is defined by

α_{ξ}^{+} = λξ_{d}dx + ξ_{d−1}dx + λ^{−1}ξ_{−d}dy,

(resp. α_{ξ}^{−} = λξ_{d}dx − ξ_{1−d}dy + λ^{−1}ξ_{−d}dy). It is pivotal to the theory that such an α_{ξ}^{+} (resp. α_{ξ}^{−}) automatically satisfies the ZCE as soon as ξ satisfies dξ = [ξ, α_{ξ}^{+}] (resp. dξ = [ξ, α_{ξ}^{−}]). See for example p.82-83 of [MS1].

Alternatively one could use the normalization in [MS1]:

α_{ξ} = (λξ_{d} + \frac{1}{2}ξ_{d−1})dx − (λ^{−1}ξ_{−d} + \frac{1}{2}ξ_{1−d})dy.

However this choice is not (as naturally) compatible with the normalization suggested by [DPW] and [T].

3.3 Main Result

A map Φ^{λ} : M ⟶ ΛG_{τ} is called finite type if dΦ^{λ} = Φ^{λ}α_{ξ}^{+} , Φ^{λ}(0) = I where ξ : M → Λ_{d} and dξ = [ξ, α_{ξ}^{+}] , ξ(0) = ξ_{o} for some ξ_{o} ∈ Λ_{d} where d ≡ 1 mod 2. The loop ξ is called the Killing field of Φ^{λ}. It is a well-known result that every ξ_{o} ∈ Λ_{d} with d ≡ 1 mod 2 yields such a map of finite type (again see, for example, [MS1]). The function det(ξ(λ) − μI) is called the spectral curve of Φ^{λ} and does not depend on (x, y). A harmonic map ϕ : M ⟶ S^{2}(1) is called finite type if ϕ = π∘ev_{1}∘Φ^{λ} where Φ^{λ} : M ⟶ ΛG_{τ} is finite type. (Again ev_{1} is the evaluation map.)

A map a : M ⟶ ΛG_{τ} is called finite-Symes type if a = g^{y}g_{+_{ℓ}} = g^{x}g_{−_{r}}^{−1} where g^{y} := exp(yλ^{d−1}ξ_{o}), g^{x} := exp(xλ^{−d+1}ξ_{o}), and g := exp(xλ^{−d+1}ξ_{o} − yλ^{d−1}ξ_{o}) = g^{y^{−1}}g^{x} for some ξ_{o} ∈ Λ_{d} where d ≡ 1 mod 2. Note g(0, 0) = I and recall g = g_{+_{ℓ}}g_{−_{r}}. Thus g_{+_{ℓ}}(0, 0) = g_{−_{r}}(0, 0) = I by the uniqueness of the Birkhoff factorization. Note also that g^{y}(0, 0) = g^{x}(0, 0) = I and hence a(0, 0) = I. We note that every ξ_{o} ∈ Λ_{d} with d ≡ 1 mod 2 yields a finite-Symes type map: we define g^{y}, g^{x} and g from ξ_{o} as above and we consider the Birkhoff splitting of g. Then a = g^{y}g_{+_{ℓ}} = g^{x}g_{−_{r}}^{−1} is a map of finite-Symes type. A Lorentz harmonic map ϕ : ℝ^{2} ⟶ G/K is called finite-Symes type if ϕ = π ∘ ev_{1} ∘ a where a : M ⟶ ΛG_{τ} is finite-Symes type.

Lemma 3.1. If a is finite-Symes type, then a is an extended framing and a^{−1}da is of the form

a^{−1}da = λA_{1}dx + (A_{0}dx + B_{0}dy) + B_{−1}λ^{−1}dy.  (3.2)

Proof: In this proof, let g_{+} := g_{+_{ℓ}} and g_{−} := g_{−_{r}}^{−1}. We follow [T].

a^{−1}da = (g_{+}^{−1}g^{y^{−1}})(g^{y}dg_{+} + dg^{y}g_{+}),  (3.3)
= g_{+}^{−1}dg_{+} + g_{+}^{−1}λ^{d−1}ξ_{0}dy g_{+} .  (3.4)

λa^{−1}da contains only non-negative terms and the constant term has no dxpart. Similarly (using a = g^{x}g_{−}) we see λ^{−1}a^{−1}da contains only non-positive terms and the constant term has no dy-part and we have a formula similar to 3.4 but with g_{−}:

a^{−1}da = (g_{−}^{−1}g^{x^{−1}})(g^{x}dg_{−} + dg^{x}g_{−}),  (3.5)
= g_{−}^{−1}dg_{−} + g_{−}^{−1}λ^{−d+1}ξ_{0}dx g_{−} .  (3.6)

The lemma follows directly from this. □

Theorem 3.2. A Lorentz harmonic map ϕ : M ⟶ G/K is finite type if and only if it is finite-Symes type.

Proof: (⇐) Let ξ = Φ^{λ^{−1}}ξ_{o}Φ^{λ}. We need to show that

i) ξ : M ⟶ Λ_{d}, ii) Φ^{λ^{−1}}dΦ^{λ} = λξ_{d}dx + ξ_{d−1}dx + λ^{−1}ξ_{−d}dy,

iii) dξ = [ξ, Φ^{λ^{−1}}dΦ^{λ}] with ξ(0) = ξ_{o}.

For part (i), for shorthand, we again let g_{+} := g_{+_{ℓ}} and g_{−} := g_{−r}^{−1}. Thus

Φ^{λ^{−1}}ξ_{o}Φ^{λ} = g_{−}^{−1}g_{x}^{−1}ξ_{o}g_{x}g_{−}
= g_{−}^{−1}ξ_{o}g_{−}

Similarly Φ^{λ^{−1}}ξ_{o}Φ^{λ} = g_{+}^{−1}ξ_{o}g_{+}. Since g_{+}, g_{+}^{−1} (resp. g_{−}, g_{−}^{−1}) have only nonnegative (resp. nonpositive) terms, we conclude ξ ∈ Λ_{d} .

Part (ii). Since Φ^{λ} is finite-Symes type we can apply equations 3.4 and 3.6 with a replaced with Φ^{λ}. Thus we have:

Φ^{λ^{−1}}dΦ = g_{+}^{−1}dg_{+} + g_{+}^{−1}λ^{d−1}ξ_{o}dy g_{+}  (3.7)
Φ^{λ^{−1}}dΦ = g_{−}^{−1}dg_{−} + g_{−}^{−1}λ^{1−d}ξ_{o}dx g_{−}.  (3.8)

Next we note that this yields

Φ^{λ^{−1}}dΦ^{λ} = AdΦ^{λ^{−1}}λ^{−d+1}ξ_{o}dx + g_{−}^{−1}g_{−_{x}}dx + g_{−}^{−1}g_{−_{y}}dy,  (3.9)
Φ^{λ^{−1}}dΦ^{λ} = AdΦ^{λ^{−1}}λ^{d−1}ξ_{o}dy + g_{+}^{−1}g_{+_{x}}dx + g_{+}^{−1}g_{+_{y}}dy.  (3.10)

To derive 3.9 and 3.10 we use 3.7 and 3.8 and the following easy facts:

◯ 1 dg_{−} = g_{−_{x}}dx + g_{−_{y}}dy,

◯ 2 dg_{+} = g_{+_{x}}dx + g_{+_{y}}dy,

◯ 3 g^{x^{−1}}λ^{−d+1}ξ_{o}g^{x} = λ^{−d+1}ξ_{o},

◯ 4 g^{y^{−1}}λ^{d−1}ξ_{o}g^{y} = λ^{d−1}ξ_{o}.

By Lemma 3.1 we need only consider the terms (Φ^{λ^{−1}}dΦ^{λ})_{1}, (Φ^{λ^{−1}}dΦ^{λ})_{0}, and (Φ^{λ^{−1}}dΦ^{λ})_{−1}. This yields six equations:

(Φ^{λ^{−1}}dΦ^{λ})_{1} = ξ_{d}dx + 0dy,  (3.11)
(Φ^{λ^{−1}}dΦ^{λ})_{1} = ξ_{2−d}dy + (g_{+}^{−1}g_{+_{x}})_{1}dx + (g_{+}^{−1}g_{+_{y}})_{1}dy,  (3.12)
(Φ^{λ^{−1}}dΦ^{λ})_{0} = ξ_{d−1}dx + (g_{−}^{−1}g_{−_{x}})_{0}dx + (g_{−}^{−1}g_{−_{y}})_{0}dy,  (3.13)
(Φ^{λ^{−1}}dΦ^{λ})_{0} = ξ_{1−d}dy + (g_{+}^{−1}g_{+_{x}})_{0}dx + (g_{+}^{−1}g_{+_{y}})_{0}dy,  (3.14)
(Φ^{λ^{−1}}dΦ^{λ})_{−1} = ξ_{d−2}dx + (g_{−}^{−1}g_{−_{x}})_{−1}dx + (g_{−}^{−1}g_{−_{y}})_{−1}dy, (3.15)
(Φ^{λ^{−1}}dΦ^{λ})_{−1} = 0dx + ξ_{−d}dy  (3.16)

From 3.11 we have (Φ^{λ^{−1}}dΦ^{λ})_{1} = ξ_{d}dx. From 3.16 we have (Φ^{λ^{−1}}dΦ^{λ})_{−1} = ξ_{−d}dy. Recalling that g_{−_{0x}} = 0 and g_{−_{0y}} = 0, from 3.13 we have (Φ^{λ^{−1}}dΦ^{λ})_{0} = ξ_{d−1}dx.

The proof of iii) is the same as the corresponding part in the harmonic case.

(⇒) If Φ^{λ} is finite type with ξ its Killing field and ξ(0) = ξ_{o}, then ξ_{o} is in Λ_{d} and we define a := g^{y}g_{+ℓ} as in front of Lemma 3.1. Then the first part of the proof is applicable, replacing Φ^{λ} by a and we see that both, ξ̃ = a^{−1}ξ_{o}a and ξ = Φ^{λ^{−1}}ξ_{o}Φ^{λ} satisfy (iii). Hence ξ = ξ̃ and α_{ξ} = αξ̃ follows. Therefore, (ii) has the unique solution Φ^{λ} = a, since both functions satisfy the same differential equation and have the same initial condition. Therefore Φ^{λ} is of finite-Symes type.□

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