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Definite affine spheres and loop groups
Journal of Geometry and Physics 60 (2010) 782–790
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Definite affine spheres and loop groups Mingheng Liang a , Qingchun Ji b,∗ a
Department of Mathematics, East China Normal University, Shanghai, 200062, China
b
School of Mathematics, Fudan University, Shanghai 200433, China
article
info
Article history: Received 24 February 2009 Received in revised form 16 January 2010 Accepted 25 January 2010 Available online 2 February 2010
abstract We obtain the Weierstrass-type representation and the dressing transformation for definite affine spheres in this paper. As an application of the Weierstrass-type representation, we construct the entire family of finite-Symes type affine spheres. © 2010 Elsevier B.V. All rights reserved.
MSC: primary 53A10 53A07 53C38 Keywords: Affine spheres Dressing transformation Finite type Loop groups
0. Introduction The technique of loop groups has played an increasingly important role in integrable systems. There have been many papers in this field; see [1–13], and the standard text on loop groups is [11]. In [5,7], J. Dorfmeister, F. Pedit and H. Wu (F. Helein) gave a construction of harmonic maps (Willmore immersions) through meromorphic or holomorphic maps and loop group factorization. It turns out that this method works in very general situations, and it has been known as the DPW method. The setup of this paper follows closely that in [7]. The appropriate loop group factorization is crucial in carrying out the DPW method. In this article, we will first establish the factorization theorem needed in our investigation of affine spheres. In the case of definite affine spheres, there is no global Iwasawa decomposition available, but we do have a local one just as in [7]. We will also discuss the Weierstrasstype representation, dressing transformation and finite type solutions of definite type affine spheres. The Weierstrass-type representation of indefinite affine sphere was obtained in [3]; in that case, the double loop group was required. In this article, we only use the single loop, but we have to deal with the different reality condition (2.1). In order to get the factorization theorem of the corresponding loop group (Theorem 2.2) we proceed as in [7] to prove the decomposition for corresponding loop algebra. This paper is organized as follows: In Section 1, we collect some necessary notions and results from affine differential geometry and derive the Maurer–Cartan equation with a spectral parameter for definite affine spheres. In Section 2, we deal with the factorization theorems of loop groups and as an application we obtain the dressing transformation for definite affine spheres. In Section 3, we deduce from factorization theorems established in Section 2 the Weierstrass-type representation formula for definite affine spheres. In Section 4, we prove the equivalence of finite type and finite-Symes.
∗
Corresponding author. E-mail addresses: [email protected] (M. Liang), [email protected] (Q. Ji).
0393-0440/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.geomphys.2010.01.010
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1. Preliminaries in affine geometry Let Σ be a Riemann surface, f : Σ → R3 an immersion. We will work with local equiaffine frame (e1 , e2 , e3 ) i.e., e1 , e2 are tangent to Σ , e3 is transversal to Σ , and det(e1 , e2 , e3 ) = 1.
(1.1)
We denote the dual coframe by (ω1 , ω2 )(ω3 = 0 on Σ ). We adopt the following ranges of indices: 1 ≤ i, j , k . . . ≤ 2 1 ≤ α, β, γ . . . ≤ 3. We define the 1-forms ωαβ by the following identities
X
deα =
ωαβ eβ
(1.2)
β
then we have the structure equations dω α =
X
ωβ ∧ ωβα
(1.3)
β
X
dωαβ =
γ
ωαγ ∧ ωγ β .
(1.4)
From Cartan’s lemma and (1.3), we can find hij such that
X
ωi3 =
hij ωj
hij = hji .
(1.5)
j
Let h=
1
X
|det(hij )|− 4 hij ωi ⊗ ωj
(1.6)
ij
then this is a global defined quadratic form, called affine metric. One can also prove that
Π
h=
(1.7)
1
|K | 4 where Π is the Euclidean second fundamental form, K is the Gaussian curvature of induced metric from the standard metric of R3 . To make h well defined, We should concentrate on non-degenerate affine surfaces, by definition, we know that in this case the quadratic form h is actually non-degenerate everywhere on Σ . From (1.7), we can see that it is equivalent to K 6= 0 holds on Σ . The case of K < 0 was discussed in [3]. In this paper, we will concentrate on the definite case, i.e. we assume K > 0 on Σ . Since the quadratic form h is non-degenerate, the next definition makes sense
ξ=
1
∆h ( f ) 2 where ∆h denotes the Beltrami–Laplace operator of h. The vector ξ will be called the affine normal. Exterior differentiating (1.5), it follows that
" X
(1.8)
# dhij − hij ω33 −
j
X
hik ωjk −
k
X
hkj ωik
∧ ωj = 0.
(1.9)
k
By using again Cartan’s lemma, we get hijk such that
X
hijk ωk = dhij − hij ω33 −
k
X k
hik ωjk −
X
hkj ωik .
(1.10)
k
It is easy to see that hijk is symmetric in i, j, k. Now we can define the following equiaffine invariant cubic form C =
X
hijk ωi ⊗ ωj ⊗ ωk .
(1.11)
ijk
For a non-degenerate affine surface, it is convenient to use the conformal coordinates (z , z¯ ) w.r.t. the definite affine metric h. Under such conformal coordinates, h, ξ , and C are of the following forms h = eω dz ⊗ dz¯ ,
ξ = e−ω fz z¯ ,
¯ z¯ ⊗3 C = Adz ⊗3 + Ad
where ω : Σ → R, A : Σ → C are functions defined on Σ .
(1.12)
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We have the evolution for moving frame Ω = (fz , fz¯ , ξ )t ,
ωz
Ae−ω 0 Az¯ e−2ω
Ωz = 0 −H
0 eω Ω , 0
0 ¯ −ω Ωz¯ = Ae A¯ z e−2ω
0
ωz¯ −H
eω 0 Ω 0
(1.13)
¯ −3ω , and in what follows H will be called affine mean curvature. The compatibility conditions where H = −e−ω ωz z¯ − AAe for (1.13) is Hz = e−3ω AA¯ z − e−ω (Az¯ e−ω )z¯ Hz¯ = e
−3ω
Az¯ A¯ − e
−ω
(A¯ z e
−ω
(1.14)
)z .
(1.15)
The following theorem is well known in affine differential geometry. Theorem 1.1. Let ω : Σ → R, A : Σ → C be functions defined on simply connected Riemann surface Σ . If (1.14) and (1.15) hold then there exists an immersion f : Σ → R3 ,
¯ z¯ ⊗3 . Moreover, this immersion is unique up to an affine transformation with the affine cubic form h = eω dz ⊗ dz¯ , C = Adz ⊗3 + Ad of R3 . Definition 1.1. A non-degenerate affine immersion f : Σ → R3 is a proper affine sphere iff ξ = −H (f − x0 ), H 6= 0 holds on Σ , where ξ is the affine normal, H is the affine mean curvature, and x0 is a fixed point of R3 . Moreover, a proper affine sphere f is called definite, if K > 0 on Σ . A remark on Definition 1.1: Although K is not an affine invariant, the condition K > 0 is preserved by affine transformations. Without loss of generality, we assume that x0 is just the origin of R3 . It follows directly from ξ = −Hf that
ξz = −fHz + Hfz . Then by replacing Hz by (1.14), we get
ξz = −f [e−3ω AA¯ z − e−ω (Az¯ e−ω )z¯ ] − Hfz . From the first part of (1.13), we have
ξz = −Hfz + Az¯ e−2ω fz¯ . It is easy to see that (from the above identities) HAz¯ e−2ω fz¯ = ξ [e−3ω AA¯ z − e−ω (Az¯ e−ω )z¯ ]. Since fz , ξ are linearly independent, we have Az¯ = 0. Substituting Az¯ = 0, A¯ z = Az¯ = 0 into (1.14) and (1.15), we obtain H = constant 6= 0. Up to rescaling and replacing z by iz if necessary, we could assume that H = −1. Theorem 1.2. A non-degenerate affine surface f : Σ → R3 is a proper affine sphere if and only if Az¯ = 0 holds on Σ , where A is defined in (1.12) and (1.13). Proof. The previous argument above the statement of Theorem 1.2 shows that Az¯ = 0 holds on Σ for every proper affine sphere. Conversely, if Az¯ = 0 holds on Σ , then from (1.13) we know that
ξz = −Hfz
ξz¯ = −Hfz¯
and from (1.14) and (1.15) we have H = constant. Now it is easy to see that up to a fixed point x0 ∈ R3 , ξ = −H (f − x0 ).
In the case of affine sphere, (1.13)–(1.15) will take simpler forms. In [2], the modified frame was introduced
−1 − ω λ e 2 Φλ = 0
0 ω λe− 2 0
0
0 0 Ω . 1
(1.16)
The pulled-back Maurer–Cartan form under Φ λ is given by
ωz dz − ωdz¯ dz¯ dΦ λ (Φ λ )−1 = λ−1 ξ−1 dz¯ + ξ0 + λξ1 dz 2 where
0 ¯ −ω ξ−1 = Ae 0
0 0 ω e2
ω
e2 0 0
ξ0 =
1 0 0
0 −1 0
0 0 0
!
(1.17)
0 ξ1 = 0 ω e2
Ae−ω 0 0
0 ω e2 0.
(1.18)
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2. Twisted loop groups associated with affine spheres Let G be a Lie group, σ : G → G be an automorphism of order k, define the corresponding loop group as following
ΛGσ = {g : S 1 → G | g (θ λ) = σ g (λ)}. It is easy to see that the corresponding Lie algebra is
ΛGσ = {ξ : S 1 → G | ξ (θ λ) = σ ξ (λ)} 2π i
where G is the Lie algebra of G, θ = e k , and we have denoted the morphism on G induced by σ by the same symbol. Moreover we also need to use the following loop groups and loop algebras.
Λ+ Gcσ Λ− Gcσ c Λ+ ∗ Gσ − c Λ∗ Gσ Λ+ Gcσ Λ− Gcσ c Λ+ ∗ Gσ c Λ− ∗ Gσ
= {g = {g = {g = {g = {ξ = {ξ = {ξ = {ξ
∈ ΛGcσ ∈ ΛGcσ ∈ ΛGcσ ∈ ΛGcσ ∈ ΛGcσ ∈ ΛGcσ ∈ ΛGcσ ∈ ΛGcσ
| g extends holomorphically into |z | < 1} | g extends holomorphically into ∞ ≥ |z | > 1} | g extends holomorphically into |z | < 1, g (0) = 1} | g extends holomorphically into ∞ ≥ |z | > 1, g (∞) = 1} | ξ extends holomorphically into |z | < 1} | ξ extends holomorphically into ∞ ≥ |z | > 1} | ξ extends holomorphically into |z | < 1, ξ (0) = 0} | ξ extends holomorphically into ∞ ≥ |z | > 1, ξ (∞) = 0}.
In the above definitions, we have denoted the complexification of G, G by Gc , Gc respectively, we could define the corresponding loop groups and loop algebras of Gc and Gc in the standard way. We also introduce c c Λ− B Gσ = {g ∈ ΛGσ | g extends holomorphically into ∞ ≥ |z | > 1, g (∞) ∈ B} c c Λ− B Gσ = {ξ ∈ ΛGσ | ξ extends holomorphically into ∞ ≥ |z | > 1, ξ (∞) ∈ B }
where B is some subgroup of Gc , with Lie algebra B . The method of loop groups has played an increasingly important role in the study of integrable systems (see [1–13]). To establish the loop group formalism for definite affine spheres, algebraic characterization of the pulled-back Maurer–Cartan form (1.18) suggests that we should concentrate on the following Lie group G, Lie algebra G and the automorphism σ . G = {g ∈ SL(3, C ) | TgT = g¯ }
(2.1)
G = {X ∈ sl(3, C ) | TXT = X¯ }
(2.2)
where
T =
0 1 0
1 0 0
0 0 . 1
!
(2.3)
It is easy to see that G is a subgroup (in fact a real form) of SL(3, C ), and G is the corresponding Lie algebra. Set
0
E = θ 2 0
θ 0 0
0 0 , 1
θ =e
2π i 3
(2.4)
then we could define σ : G → G:
σ (g ) = E (g t )−1 E −1 .
(2.5)
This is an automorphism of order six, the induced automorphism on Lie algebra is given by:
σ (X ) = −EX t E −1 . It is easy to see that σ is well defined.
(2.6)
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Denote the e
π ij 3
-eigenspace by Gj ⊆ G, then Yj ∈ Gj will be of the following forms
a11 0 0
0 −a11 0
0 0 , 0
0 0 −a13
a13 0 0
!
Y6k+2 =
0 0 0
0 a23 0
!
Y6k+4 =
0 0 −a23
Y6k =
0 0 a23
!
0 0 0
Y6k+1 =
, ,
a12 0 0
0 a23 0
!
Y6k+3 =
a11 0 0
0 a11 0
0 0 −2a11
Y6k+5 =
0 a12 0
0 0 a13
a13 0 0
! (2.7)
! .
Now we can state the fundamental theorem for loop group method applying to affine sphere as [3]. Theorem 2.1. Let D ⊆ R2 be a simply connected domain containing (0,0). If the equation dΦ λ (Φ λ )−1 = α := (U0 + λU1 )dz + (V0 + λ−1 V−1 )dz¯
(2.8)
λ
with initial condition Φ (0, 0) = I is solvable, and (U1 )23 6= 0 on D then there exists a K -valued function C , unique up to sign, ˜ λ = C Φ λ C0−1 is the modified frame of the affine sphere satisfying Φ ˜ λ (0, 0) = I, where α : D → ΛGσ ⊗ T ∗ D, K is such that Φ the connected subgroup corresponding to G0 , and C0 = C (0, 0). Proof. It is easy to see that Eq. (2.8) and the initial condition ensure the resulting Φ lies in ΛGσ , and a gauge caused by C valued in K will preserve both the integrable condition and the loop group condition. So what we have to do is just to choose a K -valued function such that the resulting α has the form of (1.18). Let U0 =
u0 0 0
0 −u 0 0
0 0 0
0
0 0 0
−v0
U1 =
!
v0
0 0
V0 =
!
0
V−1 =
0 0 u3
u1 0 0
0
0 0
v1 0
v2
0 u2 0
!
v3
!
0 0
then it follows from the reality condition that ui = v¯ i ,
i = 0, 1, 2, 3. v¯
By choosing a = |v2 | , after the gauge transformation caused by C = diag{a, a¯ , 1}, the resulting U˜ = Cz C −1 + CUC −1 , V˜ = 2 Cz¯ C −1 + CVC −1 have the following property
v˜ 2 = u˜ 2 = |v2 | where u˜ i , v˜ i are defined in the same way as ui , vi . ω Let A = u˜ 1 u˜ 22 , e 2 = |v2 | then U˜1 and V˜ 1 are of the form of (1.18). From the Zero-Curvature condition (U˜1 )z¯ = [V˜ 0 , U˜ 1 ],
(V˜ −1 )z = [U˜ 0 , V˜ −1 ], it follows that ωz ωz¯ , v˜ 0 = u˜ 0 = 2
2
which concludes our proof.
By this theorem, to find a modified frame of affine sphere is equivalent to find some Φ λ : D → ΛGσ with certain nonvanishing condition such that dΦ λ (Φ λ )−1 only involves λ, λ−1 , λ0 and dz¯ does not appear in the coefficient of λ, dz does not appear in the coefficient of λ−1 . Remark. For our convenience, we will from now on call it a modified frame even if the non-vanishing condition does not hold. Now we introduce the decomposition theorems of loop groups. We will focus on the following subgroup
r B = 0 0
0 1 r 0
0
0 | r is a positive number
1
.
(2.9)
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c c Theorem 2.2. The multiplication operator from Λ− B Gσ × ΛGσ into ΛGσ defines a local diffeomorphism.
Proof. We proceed in two steps. Step 1. The addition operator c c Λ− B Gσ × ΛGσ → ΛGσ
(ξ , η) 7→ ξ + η
(2.10)
defines a diffeomorphism. We write down the Lie group K in Theorem 2.1 explicitly
( K =
a 0 0
0 a¯ 0
0 0 1
!
) |a∈S
1
(2.11)
it is easy to see that its complexification is given by
a c K = 0
0 1
0
0 | a is a nonzero complex number
a 0
0
1
.
Then the multiplication operator K × B → Kc
(k, b) 7→ kb
(2.12)
is a diffeomorphism, where B is defined in (2.9). So we have the direct sum of corresponding Lie algebras K ⊕ B = K c , and we will denote the K -part, B -part of ξ0 , an element of K c , by (ξ0 )K , (ξ0 )B respectively. It is easy to see, in our case, (ξ0 )K means taking the purely imaginary part of the entries of ξ0 and (ξ0 )B means taking the real part. P+∞ i c For arbitrary ξ = i=−∞ ξi λ ∈ ΛGσ denote
ξ− =
X
ξi λi
ξ+ =
i <0
X
ξi λi .
i>0
Now we have the following factorization of loop algebra c ΛGcσ = Λ− B Gσ ⊕ ΛGσ
ξ 7→ (ξ − −
T ξ +T
(2.13)
+ (ξ0 )B , ξ + +
T ξ +T
+ (ξ0 )K )
and in what follows by (ξ )Λ− Gcσ , (ξ )ΛGσ we mean taking the above factorization. B Hence we have proved the addition operator in (2.10) is a diffeomorphism, and what we have done above provides exactly the inverse of (2.10). Step2. The multiplication operator c c Λ− B Gσ × ΛGσ → ΛGσ
is a local diffeomorphism. c c First we note that there are some open neighborhoods of I in Λ− B Gσ , ΛGσ and ΛGσ respectively on which the exponential maps are diffeomorphic. Then the multiplication in Theorem 2.2 is given by the following formula in these small neighborhoods
(eξ , eη ) 7→ eξ eη . By the Baker–Campbell–Hausdorff formula log(eξ eη ) = ξ + η +
1 2
[ξ , η] +
1 12
[ξ , [ξ , η]] −
1 12
[η, [ξ , η]] + · · ·
and the diffeomorphism (2.10), we know that the differentiation at (I , I ) of the multiplication in theorem equals to the c c identity map. So there are open neighborhoods of I in Λ− B Gσ , ΛGσ and ΛGσ respectively such that the multiplication from U1 × U2 , to U3 is a diffeomorphism. c For arbitrary (g0 , h0 ) ∈ Λ− b Gσ × ΛGσ we have the following commutative diagram
U1 × U2
Rh Lg0 y 0
multiplication
−−−−−−→
U3
Rh Lg0 0 multiplication
y
g0 U1 × U2 h0 −−−−−−→ g0 U3 h0
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where Rh0 Lg0 (g , h) = (g0 g , hh0 ) on the left side and Rh0 Lg0 (g ) = g0 gh0 on the right side. Because the multiplication from left or right is a diffeomorphic map on loop groups, It follows that the multiplication in Theorem 2.2 is a diffeomorphism from g0 U1 × U2 h0 to g0 U3 h0 , which concludes our proof. Our next theorem follows directly from [5]. Theorem 2.3. The multiplication operator c c Λ− Gcσ × Λ+ ∗ Gσ → ΛGσ
is a diffeomorphism onto the open and dense subset V (called big cell) of ΛGcσ . As an easy application of Theorem 2.2, we obtain the dressing transformation for modified frame. Theorem 2.4. If Φ λ is a modified frame, R ∈ ΛGcσ (independent of z , z¯ ), and Φ λ R is decomposable by Theorem 2.2. Then we get another modified frame as the second factor in the factorization indicated by Theorem 2.2. Proof. Assume that − c 1 λ Φ λ R = L− − Ψ ∈ ΛB Gσ × ΛGσ .
From Ψ λ = L− Φ λ R we have 1 λ λ −1 −1 dΨ λ (Ψ λ )−1 = dL− L− − + L− dΦ (Φ ) L−
(2.14)
which shows that the powers of λ appeared in dΨ λ (Ψ λ )−1 are less than 1. From the reality condition of G, we know that the powers of λ appeared in dΨ λ (Ψ λ )−1 are not less than −1. It follows from (2.14) directly that the coefficient of λ in dΨ λ (Ψ λ )−1 only contains dz. Thus we have finished the proof. Remark. The proof of the theorem is due to the following observation. After the transformation
Φ λ 7→ RΦ λ L, where L is independent of z , z¯ dΨ λ (Ψ λ )−1 is changed by gauge transformation with R. 3. The Weierstrass-type representation of definite affine spheres We will show in this section that from each potential we can get a corresponding modified frame and conversely any modified frame comes from a potential. Following [5], we introduce
( Λ−∞,1 =
)
ξ : D → ΛGσ | ξ = c
X
ξi (z )λ
i
.
(3.1)
i≤1
In the above notation, ξi (z ) (i = 1, 0, −1, . . .) are holomorphic. Each ξ ∈ Λ−∞,1 will be called a potential. Theorem 3.1. Let ξ be a potential, then one can find a modified frame by the following algorithm. (1) Integrating
Ψz = ξ Ψ with initial value I at z = 0, to get a holomorphic Ψ (2) Factorizing, in some neighborhood of z = 0, Ψ (z ) by Theorem 2.2 to get Φ λ as the second factor then this Φ λ is a modified frame.
(3.2)
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Proof. We should justify that dΦ λ (Φ λ )−1 only involves the powers of 1, 0, −1 and the coefficient of λ only contains dz. (Note that we already know Φ λ takes value in ΛGσ .) By Theorem 2.2, we have c Ψ = Ψ−−1 Φ λ ∈ Λ− B Gσ × ΛGσ .
(3.3)
From Φ λ = Ψ− Ψ , we have dΦ λ (Φ λ )−1 = dΨ− Ψ−−1 − Ψ− dΨ Ψ −1 Ψ−−1
(3.4)
which implies that the powers of λ appeared in dΦ λ (Φ λ )−1 are not more than 1, and dz¯ does not appear in the coefficient of λ. By the reality condition on G, we see that the powers of λ appeared in dΦ λ (Φ λ )−1 are not less than −1, which concludes the proof. Next we should reverse the above procedure. Theorem 3.2. Let Φ λ be a modified frame satisfying Φ λ (0, 0) = I, by Theorem 2.3 we can factorize Φ λ , in some neighborhood of z = 0, to get Ψ as the second factor. Then Ψ is a potential, moreover it only involves power of 1, and each potential of this type is called a normalized potential. Proof. By Theorem 2.3 we can take
Φ λ = Φ−−1 Ψ ∈ Λ− Gcσ × Λ+ ∗ Gσ so it follows that −1 dΨ Ψ −1 = dΦ− Φ− + Φ− dΦ λ (Φ λ )−1 Φ−−1 .
It is easy to see that dz¯ does not appear in dΨ Ψ −1 , that is, Ψ is holomorphic in z. Ψ ∈ Λ+ ∗ Gσ also implies that non-positive powers does not appear in dΨ+ Ψ+−1 . So the loop Ψ constructed in this way indeed provides a potential. Remark. If Φ λ is a modified frame, from the following compatibility condition of (1.17)
¯ −2ω , ωz z¯ = eω − AAe
Az¯ = 0
and the Cauchy–Kowalewski theorem, we know that Φ λ is analytic in z , z¯ . Then by the method in [5], we can factorize Φ λ to get the normalized potential apart from a discrete subset of D. 4. Finite type solutions Let d ≡1 mod 6, we define
( Λd =
ξ ∈ ΛGσ | ξ =
d X
) ξi λ
i
i=−d
Definition 4.1. A Modified frame Φ λ is said to be of finite type (w.r.t. ξ ) iff there exists a ξ : D → Λd (for some d ≡1 mod 6), such that the following Eqs. (4.1) and (4.2) hold. dξ = [(λ1−d ξ dz )ΛGσ , ξ ]
(4.1)
dΦ λ (Φ λ )−1 = (λ1−d ξ dz )ΛGσ .
(4.2)
If ξ˚ is taken to be a loop in Λd (independent of z and z¯ ), then we have a simple potential λ could be solved explicitly
Ψ = ez λ
1−d ξ˚
.
1−d
ξ˚ . In this case, Eq. (3.2) (4.3)
λ
Using Theorem 3.1 to find a modified frame Φ locally, we get
Ψ = Φ−−1 Φ λ .
(4.4)
Following [6], we introduce the definition of finite-Symes type. Definition 4.2. A modified frame Φ λ obtained from (4.3) and (4.4) up to multiplying a loop in ΛGσ from right side is said to be of finite-Symes type.
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FromΦ λ = Φ− Ψ we have −1 dΦ λ (Φ λ )−1 = dΦ− Φ− + Φ− dΨ Ψ −1 Φ−−1
= dΦ− Φ−−1 + λ1−d Φ− ξ˚ Φ−−1 dz = dΦ− Φ−−1 + λ1−d Φ− Ψ ξ˚ Ψ −1 Φ−−1 dz = dΦ− Φ−−1 + λ1−d Φ λ ξ˚ (Φ λ )−1 dz .
(4.5)
Let
ξ = Φ λ ξ˚ (Φ λ )−1
(4.6)
then from the above computation we have
ξ = Φ− ξ˚ Φ−−1 .
(4.7)
From definition (4.6) we know that ξ ∈ ΛGσ , and from (4.7) we know the powers of λ appeared in ξ will exceed d. By using again the reality condition, we see that ξ is valued in Λd . Moreover, (4.6) implies dΦ λ (Φ λ )−1 = (λ1−d ξ dz )ΛGσ . The condition (4.1) follows directly from definition (4.6), so we have indeed constructed modified frames of finite type by DPW method. The above argument shows that finite-Symes type implies finite type, and it remains to prove the reverse direction. Let Φ λ be of finite type w.r.t. ξ . We choose ξ˚ = ξ (0), and Ψ λ to be the frame of finite-Symes type w.r.t. ξ˚ . Then we know from what has been proved that Ψ λ is of finite type w.r.t. η = Ψ λ ξ˚ (Ψ λ )−1 . From Eq. (4.1) and the initial condition ξ (0) = η(0), it follows that ξ ≡ η and then we know from Eq. (4.2) that Φ λ is of finite type. In this way, we proved that finite type implies finite-Symes type. Put the above conclusion more formally, we have the following theorem Theorem 4.1. Finite type is equivalent to finite-Symes type. Acknowledgements We are grateful to the referees for their valuable comments and extremely careful reading. The second author would like to express his special gratitude to Professor J. Dorfmeister for his stimulating suggestions and enlightening comments. This research was partially supported by NSFC 10701025/A010501. References [1] M.J. Bergvelt, M. Guest, Actions of loop groups on harmonic maps, Trans. Amer. Math. Soc. 326 (1991) 861–886. [2] David Brander, Loop group decompositions in almost split real forms and applications to soliton theory and geometry, J. Geom. Phys. 58 (12) (2008) 1792–1800. [3] J. Dorfmeister, U. Eitner, Weierstrass-type representation of affine spheres, Abh. Math. Sem. Univ. Hamburg 71 (2001) 225–250. [4] J. Dorfmeister, G. Haak, Meromorphic potentials and smooth CMC surfaces, Math. Z 224 (1997) 603–640. [5] J. Dorfmeister, F. Pedit, H. Wu, Weierstrass type representation of harmonic maps into symmetyic spaces, Comm. Anal. Geom. 6 (1998) 633–668. [6] J. Dorfmeister, I. Sterling, Finite type Lorentz harmonic maps and the method of Symes, Differential Geom. Appl. 17 (1) (2002) 43–53. [7] F. Helein, Willmore immersioms and loop groups, J. Differential Geom. 50 (1998) 331–385. [8] J. Inoguchi, Timelike surfaces of constant mean curvature in Minkowski 3-space, Tokyo J. Math 21 (1998) 141–152. [9] Q. Ji, Darboux Transformation for MZM-I,II equations, Phys. Lett. A 311 (2003) 384–388. [10] M. Kilian, S.-p. Kobayashi, W. Rossman, N. Schmitt, Unitarization of monodromy representations and constant mean curvature trinoids in 3-dimensional space forms, J. Lond. Math. Soc. (2) 75 (3) (2007) 563–581. [11] A. Pressley, G. Segal, Loop Groups, in: Oxford Math. Monographs, 1986. [12] C.L. Terng, K. Uhlenbeck, Backlund transformations and loop group actions, Comm. Pure Appl. Math 53 (2000) 1–75. [13] M. Toda, Pseudo spherical surfaces via moving frames and loop groups, Ph.D Thesis, Univ.of Kansas, 2000.
Contents lists available at ScienceDirect
Journal of Geometry and Physics journal homepage: www.elsevier.com/locate/jgp
Definite affine spheres and loop groups Mingheng Liang a , Qingchun Ji b,∗ a
Department of Mathematics, East China Normal University, Shanghai, 200062, China
b
School of Mathematics, Fudan University, Shanghai 200433, China
article
info
Article history: Received 24 February 2009 Received in revised form 16 January 2010 Accepted 25 January 2010 Available online 2 February 2010
abstract We obtain the Weierstrass-type representation and the dressing transformation for definite affine spheres in this paper. As an application of the Weierstrass-type representation, we construct the entire family of finite-Symes type affine spheres. © 2010 Elsevier B.V. All rights reserved.
MSC: primary 53A10 53A07 53C38 Keywords: Affine spheres Dressing transformation Finite type Loop groups
0. Introduction The technique of loop groups has played an increasingly important role in integrable systems. There have been many papers in this field; see [1–13], and the standard text on loop groups is [11]. In [5,7], J. Dorfmeister, F. Pedit and H. Wu (F. Helein) gave a construction of harmonic maps (Willmore immersions) through meromorphic or holomorphic maps and loop group factorization. It turns out that this method works in very general situations, and it has been known as the DPW method. The setup of this paper follows closely that in [7]. The appropriate loop group factorization is crucial in carrying out the DPW method. In this article, we will first establish the factorization theorem needed in our investigation of affine spheres. In the case of definite affine spheres, there is no global Iwasawa decomposition available, but we do have a local one just as in [7]. We will also discuss the Weierstrasstype representation, dressing transformation and finite type solutions of definite type affine spheres. The Weierstrass-type representation of indefinite affine sphere was obtained in [3]; in that case, the double loop group was required. In this article, we only use the single loop, but we have to deal with the different reality condition (2.1). In order to get the factorization theorem of the corresponding loop group (Theorem 2.2) we proceed as in [7] to prove the decomposition for corresponding loop algebra. This paper is organized as follows: In Section 1, we collect some necessary notions and results from affine differential geometry and derive the Maurer–Cartan equation with a spectral parameter for definite affine spheres. In Section 2, we deal with the factorization theorems of loop groups and as an application we obtain the dressing transformation for definite affine spheres. In Section 3, we deduce from factorization theorems established in Section 2 the Weierstrass-type representation formula for definite affine spheres. In Section 4, we prove the equivalence of finite type and finite-Symes.
∗
Corresponding author. E-mail addresses: [email protected] (M. Liang), [email protected] (Q. Ji).
0393-0440/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.geomphys.2010.01.010
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1. Preliminaries in affine geometry Let Σ be a Riemann surface, f : Σ → R3 an immersion. We will work with local equiaffine frame (e1 , e2 , e3 ) i.e., e1 , e2 are tangent to Σ , e3 is transversal to Σ , and det(e1 , e2 , e3 ) = 1.
(1.1)
We denote the dual coframe by (ω1 , ω2 )(ω3 = 0 on Σ ). We adopt the following ranges of indices: 1 ≤ i, j , k . . . ≤ 2 1 ≤ α, β, γ . . . ≤ 3. We define the 1-forms ωαβ by the following identities
X
deα =
ωαβ eβ
(1.2)
β
then we have the structure equations dω α =
X
ωβ ∧ ωβα
(1.3)
β
X
dωαβ =
γ
ωαγ ∧ ωγ β .
(1.4)
From Cartan’s lemma and (1.3), we can find hij such that
X
ωi3 =
hij ωj
hij = hji .
(1.5)
j
Let h=
1
X
|det(hij )|− 4 hij ωi ⊗ ωj
(1.6)
ij
then this is a global defined quadratic form, called affine metric. One can also prove that
Π
h=
(1.7)
1
|K | 4 where Π is the Euclidean second fundamental form, K is the Gaussian curvature of induced metric from the standard metric of R3 . To make h well defined, We should concentrate on non-degenerate affine surfaces, by definition, we know that in this case the quadratic form h is actually non-degenerate everywhere on Σ . From (1.7), we can see that it is equivalent to K 6= 0 holds on Σ . The case of K < 0 was discussed in [3]. In this paper, we will concentrate on the definite case, i.e. we assume K > 0 on Σ . Since the quadratic form h is non-degenerate, the next definition makes sense
ξ=
1
∆h ( f ) 2 where ∆h denotes the Beltrami–Laplace operator of h. The vector ξ will be called the affine normal. Exterior differentiating (1.5), it follows that
" X
(1.8)
# dhij − hij ω33 −
j
X
hik ωjk −
k
X
hkj ωik
∧ ωj = 0.
(1.9)
k
By using again Cartan’s lemma, we get hijk such that
X
hijk ωk = dhij − hij ω33 −
k
X k
hik ωjk −
X
hkj ωik .
(1.10)
k
It is easy to see that hijk is symmetric in i, j, k. Now we can define the following equiaffine invariant cubic form C =
X
hijk ωi ⊗ ωj ⊗ ωk .
(1.11)
ijk
For a non-degenerate affine surface, it is convenient to use the conformal coordinates (z , z¯ ) w.r.t. the definite affine metric h. Under such conformal coordinates, h, ξ , and C are of the following forms h = eω dz ⊗ dz¯ ,
ξ = e−ω fz z¯ ,
¯ z¯ ⊗3 C = Adz ⊗3 + Ad
where ω : Σ → R, A : Σ → C are functions defined on Σ .
(1.12)
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We have the evolution for moving frame Ω = (fz , fz¯ , ξ )t ,
ωz
Ae−ω 0 Az¯ e−2ω
Ωz = 0 −H
0 eω Ω , 0
0 ¯ −ω Ωz¯ = Ae A¯ z e−2ω
0
ωz¯ −H
eω 0 Ω 0
(1.13)
¯ −3ω , and in what follows H will be called affine mean curvature. The compatibility conditions where H = −e−ω ωz z¯ − AAe for (1.13) is Hz = e−3ω AA¯ z − e−ω (Az¯ e−ω )z¯ Hz¯ = e
−3ω
Az¯ A¯ − e
−ω
(A¯ z e
−ω
(1.14)
)z .
(1.15)
The following theorem is well known in affine differential geometry. Theorem 1.1. Let ω : Σ → R, A : Σ → C be functions defined on simply connected Riemann surface Σ . If (1.14) and (1.15) hold then there exists an immersion f : Σ → R3 ,
¯ z¯ ⊗3 . Moreover, this immersion is unique up to an affine transformation with the affine cubic form h = eω dz ⊗ dz¯ , C = Adz ⊗3 + Ad of R3 . Definition 1.1. A non-degenerate affine immersion f : Σ → R3 is a proper affine sphere iff ξ = −H (f − x0 ), H 6= 0 holds on Σ , where ξ is the affine normal, H is the affine mean curvature, and x0 is a fixed point of R3 . Moreover, a proper affine sphere f is called definite, if K > 0 on Σ . A remark on Definition 1.1: Although K is not an affine invariant, the condition K > 0 is preserved by affine transformations. Without loss of generality, we assume that x0 is just the origin of R3 . It follows directly from ξ = −Hf that
ξz = −fHz + Hfz . Then by replacing Hz by (1.14), we get
ξz = −f [e−3ω AA¯ z − e−ω (Az¯ e−ω )z¯ ] − Hfz . From the first part of (1.13), we have
ξz = −Hfz + Az¯ e−2ω fz¯ . It is easy to see that (from the above identities) HAz¯ e−2ω fz¯ = ξ [e−3ω AA¯ z − e−ω (Az¯ e−ω )z¯ ]. Since fz , ξ are linearly independent, we have Az¯ = 0. Substituting Az¯ = 0, A¯ z = Az¯ = 0 into (1.14) and (1.15), we obtain H = constant 6= 0. Up to rescaling and replacing z by iz if necessary, we could assume that H = −1. Theorem 1.2. A non-degenerate affine surface f : Σ → R3 is a proper affine sphere if and only if Az¯ = 0 holds on Σ , where A is defined in (1.12) and (1.13). Proof. The previous argument above the statement of Theorem 1.2 shows that Az¯ = 0 holds on Σ for every proper affine sphere. Conversely, if Az¯ = 0 holds on Σ , then from (1.13) we know that
ξz = −Hfz
ξz¯ = −Hfz¯
and from (1.14) and (1.15) we have H = constant. Now it is easy to see that up to a fixed point x0 ∈ R3 , ξ = −H (f − x0 ).
In the case of affine sphere, (1.13)–(1.15) will take simpler forms. In [2], the modified frame was introduced
−1 − ω λ e 2 Φλ = 0
0 ω λe− 2 0
0
0 0 Ω . 1
(1.16)
The pulled-back Maurer–Cartan form under Φ λ is given by
ωz dz − ωdz¯ dz¯ dΦ λ (Φ λ )−1 = λ−1 ξ−1 dz¯ + ξ0 + λξ1 dz 2 where
0 ¯ −ω ξ−1 = Ae 0
0 0 ω e2
ω
e2 0 0
ξ0 =
1 0 0
0 −1 0
0 0 0
!
(1.17)
0 ξ1 = 0 ω e2
Ae−ω 0 0
0 ω e2 0.
(1.18)
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2. Twisted loop groups associated with affine spheres Let G be a Lie group, σ : G → G be an automorphism of order k, define the corresponding loop group as following
ΛGσ = {g : S 1 → G | g (θ λ) = σ g (λ)}. It is easy to see that the corresponding Lie algebra is
ΛGσ = {ξ : S 1 → G | ξ (θ λ) = σ ξ (λ)} 2π i
where G is the Lie algebra of G, θ = e k , and we have denoted the morphism on G induced by σ by the same symbol. Moreover we also need to use the following loop groups and loop algebras.
Λ+ Gcσ Λ− Gcσ c Λ+ ∗ Gσ − c Λ∗ Gσ Λ+ Gcσ Λ− Gcσ c Λ+ ∗ Gσ c Λ− ∗ Gσ
= {g = {g = {g = {g = {ξ = {ξ = {ξ = {ξ
∈ ΛGcσ ∈ ΛGcσ ∈ ΛGcσ ∈ ΛGcσ ∈ ΛGcσ ∈ ΛGcσ ∈ ΛGcσ ∈ ΛGcσ
| g extends holomorphically into |z | < 1} | g extends holomorphically into ∞ ≥ |z | > 1} | g extends holomorphically into |z | < 1, g (0) = 1} | g extends holomorphically into ∞ ≥ |z | > 1, g (∞) = 1} | ξ extends holomorphically into |z | < 1} | ξ extends holomorphically into ∞ ≥ |z | > 1} | ξ extends holomorphically into |z | < 1, ξ (0) = 0} | ξ extends holomorphically into ∞ ≥ |z | > 1, ξ (∞) = 0}.
In the above definitions, we have denoted the complexification of G, G by Gc , Gc respectively, we could define the corresponding loop groups and loop algebras of Gc and Gc in the standard way. We also introduce c c Λ− B Gσ = {g ∈ ΛGσ | g extends holomorphically into ∞ ≥ |z | > 1, g (∞) ∈ B} c c Λ− B Gσ = {ξ ∈ ΛGσ | ξ extends holomorphically into ∞ ≥ |z | > 1, ξ (∞) ∈ B }
where B is some subgroup of Gc , with Lie algebra B . The method of loop groups has played an increasingly important role in the study of integrable systems (see [1–13]). To establish the loop group formalism for definite affine spheres, algebraic characterization of the pulled-back Maurer–Cartan form (1.18) suggests that we should concentrate on the following Lie group G, Lie algebra G and the automorphism σ . G = {g ∈ SL(3, C ) | TgT = g¯ }
(2.1)
G = {X ∈ sl(3, C ) | TXT = X¯ }
(2.2)
where
T =
0 1 0
1 0 0
0 0 . 1
!
(2.3)
It is easy to see that G is a subgroup (in fact a real form) of SL(3, C ), and G is the corresponding Lie algebra. Set
0
E = θ 2 0
θ 0 0
0 0 , 1
θ =e
2π i 3
(2.4)
then we could define σ : G → G:
σ (g ) = E (g t )−1 E −1 .
(2.5)
This is an automorphism of order six, the induced automorphism on Lie algebra is given by:
σ (X ) = −EX t E −1 . It is easy to see that σ is well defined.
(2.6)
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Denote the e
π ij 3
-eigenspace by Gj ⊆ G, then Yj ∈ Gj will be of the following forms
a11 0 0
0 −a11 0
0 0 , 0
0 0 −a13
a13 0 0
!
Y6k+2 =
0 0 0
0 a23 0
!
Y6k+4 =
0 0 −a23
Y6k =
0 0 a23
!
0 0 0
Y6k+1 =
, ,
a12 0 0
0 a23 0
!
Y6k+3 =
a11 0 0
0 a11 0
0 0 −2a11
Y6k+5 =
0 a12 0
0 0 a13
a13 0 0
! (2.7)
! .
Now we can state the fundamental theorem for loop group method applying to affine sphere as [3]. Theorem 2.1. Let D ⊆ R2 be a simply connected domain containing (0,0). If the equation dΦ λ (Φ λ )−1 = α := (U0 + λU1 )dz + (V0 + λ−1 V−1 )dz¯
(2.8)
λ
with initial condition Φ (0, 0) = I is solvable, and (U1 )23 6= 0 on D then there exists a K -valued function C , unique up to sign, ˜ λ = C Φ λ C0−1 is the modified frame of the affine sphere satisfying Φ ˜ λ (0, 0) = I, where α : D → ΛGσ ⊗ T ∗ D, K is such that Φ the connected subgroup corresponding to G0 , and C0 = C (0, 0). Proof. It is easy to see that Eq. (2.8) and the initial condition ensure the resulting Φ lies in ΛGσ , and a gauge caused by C valued in K will preserve both the integrable condition and the loop group condition. So what we have to do is just to choose a K -valued function such that the resulting α has the form of (1.18). Let U0 =
u0 0 0
0 −u 0 0
0 0 0
0
0 0 0
−v0
U1 =
!
v0
0 0
V0 =
!
0
V−1 =
0 0 u3
u1 0 0
0
0 0
v1 0
v2
0 u2 0
!
v3
!
0 0
then it follows from the reality condition that ui = v¯ i ,
i = 0, 1, 2, 3. v¯
By choosing a = |v2 | , after the gauge transformation caused by C = diag{a, a¯ , 1}, the resulting U˜ = Cz C −1 + CUC −1 , V˜ = 2 Cz¯ C −1 + CVC −1 have the following property
v˜ 2 = u˜ 2 = |v2 | where u˜ i , v˜ i are defined in the same way as ui , vi . ω Let A = u˜ 1 u˜ 22 , e 2 = |v2 | then U˜1 and V˜ 1 are of the form of (1.18). From the Zero-Curvature condition (U˜1 )z¯ = [V˜ 0 , U˜ 1 ],
(V˜ −1 )z = [U˜ 0 , V˜ −1 ], it follows that ωz ωz¯ , v˜ 0 = u˜ 0 = 2
2
which concludes our proof.
By this theorem, to find a modified frame of affine sphere is equivalent to find some Φ λ : D → ΛGσ with certain nonvanishing condition such that dΦ λ (Φ λ )−1 only involves λ, λ−1 , λ0 and dz¯ does not appear in the coefficient of λ, dz does not appear in the coefficient of λ−1 . Remark. For our convenience, we will from now on call it a modified frame even if the non-vanishing condition does not hold. Now we introduce the decomposition theorems of loop groups. We will focus on the following subgroup
r B = 0 0
0 1 r 0
0
0 | r is a positive number
1
.
(2.9)
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c c Theorem 2.2. The multiplication operator from Λ− B Gσ × ΛGσ into ΛGσ defines a local diffeomorphism.
Proof. We proceed in two steps. Step 1. The addition operator c c Λ− B Gσ × ΛGσ → ΛGσ
(ξ , η) 7→ ξ + η
(2.10)
defines a diffeomorphism. We write down the Lie group K in Theorem 2.1 explicitly
( K =
a 0 0
0 a¯ 0
0 0 1
!
) |a∈S
1
(2.11)
it is easy to see that its complexification is given by
a c K = 0
0 1
0
0 | a is a nonzero complex number
a 0
0
1
.
Then the multiplication operator K × B → Kc
(k, b) 7→ kb
(2.12)
is a diffeomorphism, where B is defined in (2.9). So we have the direct sum of corresponding Lie algebras K ⊕ B = K c , and we will denote the K -part, B -part of ξ0 , an element of K c , by (ξ0 )K , (ξ0 )B respectively. It is easy to see, in our case, (ξ0 )K means taking the purely imaginary part of the entries of ξ0 and (ξ0 )B means taking the real part. P+∞ i c For arbitrary ξ = i=−∞ ξi λ ∈ ΛGσ denote
ξ− =
X
ξi λi
ξ+ =
i <0
X
ξi λi .
i>0
Now we have the following factorization of loop algebra c ΛGcσ = Λ− B Gσ ⊕ ΛGσ
ξ 7→ (ξ − −
T ξ +T
(2.13)
+ (ξ0 )B , ξ + +
T ξ +T
+ (ξ0 )K )
and in what follows by (ξ )Λ− Gcσ , (ξ )ΛGσ we mean taking the above factorization. B Hence we have proved the addition operator in (2.10) is a diffeomorphism, and what we have done above provides exactly the inverse of (2.10). Step2. The multiplication operator c c Λ− B Gσ × ΛGσ → ΛGσ
is a local diffeomorphism. c c First we note that there are some open neighborhoods of I in Λ− B Gσ , ΛGσ and ΛGσ respectively on which the exponential maps are diffeomorphic. Then the multiplication in Theorem 2.2 is given by the following formula in these small neighborhoods
(eξ , eη ) 7→ eξ eη . By the Baker–Campbell–Hausdorff formula log(eξ eη ) = ξ + η +
1 2
[ξ , η] +
1 12
[ξ , [ξ , η]] −
1 12
[η, [ξ , η]] + · · ·
and the diffeomorphism (2.10), we know that the differentiation at (I , I ) of the multiplication in theorem equals to the c c identity map. So there are open neighborhoods of I in Λ− B Gσ , ΛGσ and ΛGσ respectively such that the multiplication from U1 × U2 , to U3 is a diffeomorphism. c For arbitrary (g0 , h0 ) ∈ Λ− b Gσ × ΛGσ we have the following commutative diagram
U1 × U2
Rh Lg0 y 0
multiplication
−−−−−−→
U3
Rh Lg0 0 multiplication
y
g0 U1 × U2 h0 −−−−−−→ g0 U3 h0
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where Rh0 Lg0 (g , h) = (g0 g , hh0 ) on the left side and Rh0 Lg0 (g ) = g0 gh0 on the right side. Because the multiplication from left or right is a diffeomorphic map on loop groups, It follows that the multiplication in Theorem 2.2 is a diffeomorphism from g0 U1 × U2 h0 to g0 U3 h0 , which concludes our proof. Our next theorem follows directly from [5]. Theorem 2.3. The multiplication operator c c Λ− Gcσ × Λ+ ∗ Gσ → ΛGσ
is a diffeomorphism onto the open and dense subset V (called big cell) of ΛGcσ . As an easy application of Theorem 2.2, we obtain the dressing transformation for modified frame. Theorem 2.4. If Φ λ is a modified frame, R ∈ ΛGcσ (independent of z , z¯ ), and Φ λ R is decomposable by Theorem 2.2. Then we get another modified frame as the second factor in the factorization indicated by Theorem 2.2. Proof. Assume that − c 1 λ Φ λ R = L− − Ψ ∈ ΛB Gσ × ΛGσ .
From Ψ λ = L− Φ λ R we have 1 λ λ −1 −1 dΨ λ (Ψ λ )−1 = dL− L− − + L− dΦ (Φ ) L−
(2.14)
which shows that the powers of λ appeared in dΨ λ (Ψ λ )−1 are less than 1. From the reality condition of G, we know that the powers of λ appeared in dΨ λ (Ψ λ )−1 are not less than −1. It follows from (2.14) directly that the coefficient of λ in dΨ λ (Ψ λ )−1 only contains dz. Thus we have finished the proof. Remark. The proof of the theorem is due to the following observation. After the transformation
Φ λ 7→ RΦ λ L, where L is independent of z , z¯ dΨ λ (Ψ λ )−1 is changed by gauge transformation with R. 3. The Weierstrass-type representation of definite affine spheres We will show in this section that from each potential we can get a corresponding modified frame and conversely any modified frame comes from a potential. Following [5], we introduce
( Λ−∞,1 =
)
ξ : D → ΛGσ | ξ = c
X
ξi (z )λ
i
.
(3.1)
i≤1
In the above notation, ξi (z ) (i = 1, 0, −1, . . .) are holomorphic. Each ξ ∈ Λ−∞,1 will be called a potential. Theorem 3.1. Let ξ be a potential, then one can find a modified frame by the following algorithm. (1) Integrating
Ψz = ξ Ψ with initial value I at z = 0, to get a holomorphic Ψ (2) Factorizing, in some neighborhood of z = 0, Ψ (z ) by Theorem 2.2 to get Φ λ as the second factor then this Φ λ is a modified frame.
(3.2)
M. Liang, Q. Ji / Journal of Geometry and Physics 60 (2010) 782–790
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Proof. We should justify that dΦ λ (Φ λ )−1 only involves the powers of 1, 0, −1 and the coefficient of λ only contains dz. (Note that we already know Φ λ takes value in ΛGσ .) By Theorem 2.2, we have c Ψ = Ψ−−1 Φ λ ∈ Λ− B Gσ × ΛGσ .
(3.3)
From Φ λ = Ψ− Ψ , we have dΦ λ (Φ λ )−1 = dΨ− Ψ−−1 − Ψ− dΨ Ψ −1 Ψ−−1
(3.4)
which implies that the powers of λ appeared in dΦ λ (Φ λ )−1 are not more than 1, and dz¯ does not appear in the coefficient of λ. By the reality condition on G, we see that the powers of λ appeared in dΦ λ (Φ λ )−1 are not less than −1, which concludes the proof. Next we should reverse the above procedure. Theorem 3.2. Let Φ λ be a modified frame satisfying Φ λ (0, 0) = I, by Theorem 2.3 we can factorize Φ λ , in some neighborhood of z = 0, to get Ψ as the second factor. Then Ψ is a potential, moreover it only involves power of 1, and each potential of this type is called a normalized potential. Proof. By Theorem 2.3 we can take
Φ λ = Φ−−1 Ψ ∈ Λ− Gcσ × Λ+ ∗ Gσ so it follows that −1 dΨ Ψ −1 = dΦ− Φ− + Φ− dΦ λ (Φ λ )−1 Φ−−1 .
It is easy to see that dz¯ does not appear in dΨ Ψ −1 , that is, Ψ is holomorphic in z. Ψ ∈ Λ+ ∗ Gσ also implies that non-positive powers does not appear in dΨ+ Ψ+−1 . So the loop Ψ constructed in this way indeed provides a potential. Remark. If Φ λ is a modified frame, from the following compatibility condition of (1.17)
¯ −2ω , ωz z¯ = eω − AAe
Az¯ = 0
and the Cauchy–Kowalewski theorem, we know that Φ λ is analytic in z , z¯ . Then by the method in [5], we can factorize Φ λ to get the normalized potential apart from a discrete subset of D. 4. Finite type solutions Let d ≡1 mod 6, we define
( Λd =
ξ ∈ ΛGσ | ξ =
d X
) ξi λ
i
i=−d
Definition 4.1. A Modified frame Φ λ is said to be of finite type (w.r.t. ξ ) iff there exists a ξ : D → Λd (for some d ≡1 mod 6), such that the following Eqs. (4.1) and (4.2) hold. dξ = [(λ1−d ξ dz )ΛGσ , ξ ]
(4.1)
dΦ λ (Φ λ )−1 = (λ1−d ξ dz )ΛGσ .
(4.2)
If ξ˚ is taken to be a loop in Λd (independent of z and z¯ ), then we have a simple potential λ could be solved explicitly
Ψ = ez λ
1−d ξ˚
.
1−d
ξ˚ . In this case, Eq. (3.2) (4.3)
λ
Using Theorem 3.1 to find a modified frame Φ locally, we get
Ψ = Φ−−1 Φ λ .
(4.4)
Following [6], we introduce the definition of finite-Symes type. Definition 4.2. A modified frame Φ λ obtained from (4.3) and (4.4) up to multiplying a loop in ΛGσ from right side is said to be of finite-Symes type.
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FromΦ λ = Φ− Ψ we have −1 dΦ λ (Φ λ )−1 = dΦ− Φ− + Φ− dΨ Ψ −1 Φ−−1
= dΦ− Φ−−1 + λ1−d Φ− ξ˚ Φ−−1 dz = dΦ− Φ−−1 + λ1−d Φ− Ψ ξ˚ Ψ −1 Φ−−1 dz = dΦ− Φ−−1 + λ1−d Φ λ ξ˚ (Φ λ )−1 dz .
(4.5)
Let
ξ = Φ λ ξ˚ (Φ λ )−1
(4.6)
then from the above computation we have
ξ = Φ− ξ˚ Φ−−1 .
(4.7)
From definition (4.6) we know that ξ ∈ ΛGσ , and from (4.7) we know the powers of λ appeared in ξ will exceed d. By using again the reality condition, we see that ξ is valued in Λd . Moreover, (4.6) implies dΦ λ (Φ λ )−1 = (λ1−d ξ dz )ΛGσ . The condition (4.1) follows directly from definition (4.6), so we have indeed constructed modified frames of finite type by DPW method. The above argument shows that finite-Symes type implies finite type, and it remains to prove the reverse direction. Let Φ λ be of finite type w.r.t. ξ . We choose ξ˚ = ξ (0), and Ψ λ to be the frame of finite-Symes type w.r.t. ξ˚ . Then we know from what has been proved that Ψ λ is of finite type w.r.t. η = Ψ λ ξ˚ (Ψ λ )−1 . From Eq. (4.1) and the initial condition ξ (0) = η(0), it follows that ξ ≡ η and then we know from Eq. (4.2) that Φ λ is of finite type. In this way, we proved that finite type implies finite-Symes type. Put the above conclusion more formally, we have the following theorem Theorem 4.1. Finite type is equivalent to finite-Symes type. Acknowledgements We are grateful to the referees for their valuable comments and extremely careful reading. The second author would like to express his special gratitude to Professor J. Dorfmeister for his stimulating suggestions and enlightening comments. This research was partially supported by NSFC 10701025/A010501. References [1] M.J. Bergvelt, M. Guest, Actions of loop groups on harmonic maps, Trans. Amer. Math. Soc. 326 (1991) 861–886. [2] David Brander, Loop group decompositions in almost split real forms and applications to soliton theory and geometry, J. Geom. Phys. 58 (12) (2008) 1792–1800. [3] J. Dorfmeister, U. Eitner, Weierstrass-type representation of affine spheres, Abh. Math. Sem. Univ. Hamburg 71 (2001) 225–250. [4] J. Dorfmeister, G. Haak, Meromorphic potentials and smooth CMC surfaces, Math. Z 224 (1997) 603–640. [5] J. Dorfmeister, F. Pedit, H. Wu, Weierstrass type representation of harmonic maps into symmetyic spaces, Comm. Anal. Geom. 6 (1998) 633–668. [6] J. Dorfmeister, I. Sterling, Finite type Lorentz harmonic maps and the method of Symes, Differential Geom. Appl. 17 (1) (2002) 43–53. [7] F. Helein, Willmore immersioms and loop groups, J. Differential Geom. 50 (1998) 331–385. [8] J. Inoguchi, Timelike surfaces of constant mean curvature in Minkowski 3-space, Tokyo J. Math 21 (1998) 141–152. [9] Q. Ji, Darboux Transformation for MZM-I,II equations, Phys. Lett. A 311 (2003) 384–388. [10] M. Kilian, S.-p. Kobayashi, W. Rossman, N. Schmitt, Unitarization of monodromy representations and constant mean curvature trinoids in 3-dimensional space forms, J. Lond. Math. Soc. (2) 75 (3) (2007) 563–581. [11] A. Pressley, G. Segal, Loop Groups, in: Oxford Math. Monographs, 1986. [12] C.L. Terng, K. Uhlenbeck, Backlund transformations and loop group actions, Comm. Pure Appl. Math 53 (2000) 1–75. [13] M. Toda, Pseudo spherical surfaces via moving frames and loop groups, Ph.D Thesis, Univ.of Kansas, 2000.