Weierstrass-type representation for harmonic maps ... - Project Euclid

J. Math. Soc. Japan Vol. 57, No. 1, 2005

Weierstrass-type representation for harmonic maps into general symmetric spaces via loop groups By Vladimir BALAN and Josef D ORFMEISTER (Received Sept. 24, 2003)

Abstract. In [19] a method was presented, which constructs via loop group splittings all harmonic maps into a compact symmetric space. The present paper generalizes this method to all spaces G/K, where G is an arbitrary Lie group (semisimple or not) and K is the fixpoint group of some involution of G. The method is illustrated by a number of examples.

1. Introduction. In [19] a Weierstrass-type representation was introduced for harmonic maps ϕ from Riemannian surfaces M to compact symmetric spaces N = G/K, where K is a closed subgroup of G. The paper associates first with a harmonic map ϕ : M → N a frame F : M → G, such that the canonical projection π : G → N forms with ϕ and F a commutative diagram. Then, after choosing conformal coordinates on M one decomposes the Maurer Cartan form F −1 dF = α 0 + α 00 , where α 0 is of type (1,0) and α 00 of type (0,1). Similar to [4] and [41], [19] introduces a parameter λ from S1 ⊂ C . Then it turns out that the integrability for αλ is equivalent with the harmonicity of the map ϕ . This permits to integrate the Maurer Cartan equation F −1 dF = αλ , where F now depends on λ (“extended frame”). The most crucial feature of [19] then is that one can split F everywhere into F = H · F+ where H is holomorphic in z (“holomorphic extended frame”) and F+ is holomorphic in λ in the open unit disk. Moreover, the Maurer Cartan form η = H −1 dH of H is a holomorphic differential one-form. It is obvious that any η of this form trivially satisfies the integrability condition for the differential equation η = H −1 dH. We have thus replaced the Maurer-Cartan form of the extended frame F with its non-linear integrability condition with η , which satisfies its integrability condition trivially. It is crucial to observe, that from η one can reconstruct the (associated family of the) original harmonic map. Namely: starting, conversely, from an arbitary holomorphic one form η one can obtain a harmonic map ϕ : M → N as follows: first, one integrates the ordinary differential equation d H = H · η , H(z = 0) = I. Then one carries out an Iwasawa splitting H = F · V+ and sees that F is an extended frame for a harmonic map ϕ = F mod K. 1.1. Let 1 be a finite-dimensional real Lie algebra, satisfying the following conditions: • There exists an involutive automorphism of 1, σ ∈ End(1) \ {Id}, σ 2 = Id; • Ker(σ − Id) ∩ Z(1) = {0}, where Z(1) = {ξ ∈ 1 | [ξ , η ] = 0 for all η ∈ 1} is the center of 1. 2000 Mathematics Subject Classification. 58E20, 53C43, 22E67, 30F15. Key Words and Phrases. harmonic map, finite type harmonic map, symmetric space, Maurer-Cartan form, Birkhoff splitting, Iwasawa splitting, affine connection, loop group, tangent group, potential.

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V. BALAN and J. D ORFMEISTER

Then 1 admits the splitting 1 = k ⊕ p,

(1.1.1)

where k and p are the eigenspaces of σ k = Ker(σ − Id),

p = Ker(σ + Id).

(1.1.2)

In the following we consider two connected Lie groups K ⊂ G, K closed in G, such that Lie(K) = k ⊂ Lie(G) = 1. We assume as in [3] and [4] that G admits a faithful linear continuous representation. Actually, we assume that there exists some complex matrix group GC , the universal complexification of G in the sense of [30, Section 17.5], which has G as a real form. The vector space, on which GC acts linearly will be denoted by V . 1.2. In this paper we consider harmonic maps from non-compact simply connected, Riemann surfaces D to pseudo-Riemannian general symmetric spaces G/K. First we need to define the notion of general symmetric space (see, e.g., [35]): We consider a connected matrix group G, σ an automorphism of order two and denote Fix σ = {1 ∈ G | σ (1) = 1} and by (Fix σ )0 the connected component of Fix σ containing the identity I. Let K be a subgroup of G such that (Fix σ )0 ⊂ K ⊂ Fix σ , then G/K is called a general symmetric space. Starting in Section 1.3 we will assume in addition that we have a non-degenerate symmetric bilinear form invariant by 1. Then we have ([3]) the semidirect product G ≡ H.M, where H is a connected, reductive subgroup and M is a connected, simply connected solvable subgroup of G. For our purposes it will be important that the groups under consideration are invariant under σ . If we would only want a Levi factor to be invariant, then it would suffice to refer to [39] and for uniqueness questions also [40]. However, we also want a reductive subgroup to be invariant under σ . Therefore we prove: T HEOREM . There exists a choice of the subgroups H and M as above, such that σ H = H and σ M = M. P ROOF . We write G ≡ S.R, where S is semisimple and R is the radical of G. Since Lie(R) is a maximal solvable ideal, it is easy to see σ Lie(R) = Lie(R). Thus we obtain σ R = R. By [30, Theorem 18.4.3], and our assumptions we know R = LM, where L is maximal compact in R and abelian and M is simply connected. Analyzing the proof of loc.cit. it is easy to observe that M is preserved by σ . We will show next that one can, if necessary, replace L by some conjugate maximal compact subgroup L˜ = hLh−1 , such that L˜ is invariant under σ . For the proof we will use mostly Lie algebras. To simplify notation we abbreviate s = Lie(S) and m = Lie(M) etc. and recall the descending central series: r (0) = r and r ( j+1) = [rr ( j) , r ( j) ]. Moreover, since σ is semisimple and leaves every r ( j) invariant, there are σ -invariant subspaces W j , j = 0, 1, . . . , n such that W j +W j+1 + . . . +Wn = r ( j) and the sum is direct. First we note that

Weierstrass-type representation for harmonic maps

σ L = mLm−1

for some m ∈ M,

71

(1.2.1)

since maximal compact subgroups are conjugate to each other. Therefore for every l in L we have some l 0 ∈ L such that σ l = ml 0 m−1 . Applying σ again we obtain l = [σ (m)m]l 00 [σ (m)m]−1 . Multiplying from the left by (l 00 )−1 and observing that M is normal in R, we conclude that l = l 00 , and that σ (m)m = c commutes with every l ∈ L:

σ (m)m = c ∈ C,

(1.2.2)

C = {r ∈ M; rl = lr for all l ∈ L}.

(1.2.3)

where

Since the exponential map exp : m → M is a diffeomorphism, we can write m = exp(m) ˆ etc. Decomposing mˆ relative to the subspaces W j the idea of the proof is to replace iteratively L by some conjugate subgroup Lˆ such that the “m” associated with Lˆ and σ has fewer and fewer components. Altogether, we will finally obtain some h ∈ M such that L˜ = hLh−1 is invariant under σ , i.e. is so that for every l ∈ L there exists some l 0 ∈ L such that σ (hlh−1 ) = hl 0 h−1 holds. Applying (1.2.1) we obtain σ (h) · ml 00 m−1 · σ (h−1 ) = hl 0 h−1 . −1 Therefore [h−1 σ (h)m]l 00 [h−1 σ (h)m] = l 0 . As above one derives from this l 0 = l 00 and we obtain (σ (h))−1 hd = m,

(1.2.4)

for some d ∈ C. Taking logarithms and comparing the coefficients in W0 we obtain from (1.2.2) and (1.2.4) the equations

σ (mˆ 0 ) + mˆ 0 = cˆ0

(1.2.5)

−σ (hˆ 0 ) + hˆ 0 + dˆ0 = mˆ 0 .

(1.2.6)

and

Next we note that every subspace invariant under σ decomposes into its eigenspaces relative to 1 and −1. With the obvious notation we thus obtain from (1.2.5) and (1.2.6) the equations cˆ0 = 2mˆ 0 (+)

(1.2.7)

2hˆ 0 (−) + dˆ0 (−) = mˆ 0 (−) ,

(1.2.8)

1 dˆ0 (+) = mˆ 0 (+) = cˆ0 (+) = cˆ0 . 2

(1.2.9)

and

and

Note that m in equation (1.2.1) is only unique up to multiplication with elements of C. Thus, √ in view of (1.2.7) we would like to replace m with m0 = m/ c. To make this precise, we write √ c = exp(c) ˆ and note Ad(l)cˆ = c. ˆ Thus “ c”= exp(c/2) ˆ = d is still in C, and the W0 component of the logarithm of m0 = md −1 is in the −1−eigenspace of σ . This means, that we can assume (−) (−) w.r.g. mˆ 0 = mˆ 0 . Now we set s = exp(mˆ 0 /2) and L] = sLs−1 . Then σ (s) = s−1 and for

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V. BALAN and J. D ORFMEISTER −1

l ] ∈ L] we obtain σ (l ] ) = σ (sls−1 ) = s−1 ml 0 m−1 s = m] (sl 0 s−1 )m] , where m] = s−1 ms−1 . We note the crucial observation that the logarithm of m] does not have any W0 component anymore. Repeating the procedure for σ , L] and m] we obtain some new maximal compact subgroup and some new “m”, the logarithm of which does not contain any W1 -component nor any W0 ˜ which is component. Repeating this procedure finally yields a maximal compact subgroup L, invariant under σ . It now remains to prove that one can find not only some semisimple subgroup ˆ But this follows of G, which is invariant under σ , but one, which also commutes with A = L. precisely from the first part of the proof of [30, Theorem 18.4.2]. ¤ 1.3. Let F : D → G be a smooth map, D = D1 or C ; D1 = {z ∈ C | | z |< 1}. D, 1), deIn view of (1.1.1), the Maurer-Cartan form associated to F, α = F −1 dF ∈ Λ 1 (D composes canonically D, k ) ⊕ Λ 1 (D D, p ). α = αk + α p ∈ Λ 1 (D

(1.3.1)

Denoting by z, z¯ the complex coordinates in D ⊂ R 2 ≡ C , we have T D = T (1,0) D ⊕ T (0,1) D , and d = ∂ + ∂¯ . Then α p splits accordingly, D, pC ) ⊕ Λ (0,1) (D D, pC ). α p = α p0 + α p00 ∈ Λ (1,0) (D

(1.3.2)

D, 1C ) We consider the symmetric bracket on Λ 1 (D [α ∧ β ] = ([α 0 , β 00 ] − [α 00 , β 0 ])dz ∧ d¯z, D, 1C ) = Λ (1,0) (D D, 1C )⊕ Λ (0,1) (D D, 1C ), 1C ⊂ 1l(V ). for α = α 0 dz+ α 00 d¯z, β = β 0 dz+ β 00 d¯z ∈ Λ 1 (D Composing F with the projection π of the principal K-bundle (G, π , N = G/K), one obtains the mapping ϕ which closes the diagram G F¡ µ π ? ¡ D -N = G/K. ϕ We can characterize the harmonicity of the map ϕ (the quality of being minimizer of the energy map (1.5.3); [22], [23], [42], [19]), in terms of the Maurer-Cartan form α associated to its lift F, as follows T HEOREM . Assume G/K carries a non-degenerate symmetric bilinear form invariant by G. Then the following statements are equivalent: a) The map ϕ associated with F is harmonic. b) The form α satisfies the integrability and harmonicity equations:   dα + 1 [α ∧ α ] = −[α 0 ∧ α 00 ] k k p p 2 k (1.3.3) ¯ 0 ∂ α p + [αk ∧ α p0 ] = 0. D, k ) and λ −1 α p0 + λ α p00 ∈ Λ 1 (D D, p ) ∩ (Λ (1,0) (D D, pC ) ⊕ c) The forms αk ∈ Λ 1 (D D, pC )) satisfy the equations (1.3.3), for all λ ∈ S1 . Λ (0,1) (D


73

d) The “loopified form” D, 1) αλ = λ −1 α p0 + αk + λ α p00 ∈ Λ 1 (D

(1.3.4)

is integrable for all λ ∈ S1 , i.e., it satisfies for every λ ∈ S1 the integrability condition: 1 dαλ + [αλ ∧ αλ ] = 0. 2

(1.3.5)

P ROOF . b) ⇔ c). Straightforward, replacing in (1.3.3) α p0 with λ −1 α p0 and α p00 with λ α p00 , where λ ∈ S1 . c) ⇔ d) The relation (1.3.5) rewrites as

λ −1 {∂¯ α p0 + [α p0 ∧ αk ]} + λ {∂ α p00 + [α p00 ∧ αk ]} 1 +{dαk + [α p0 ∧ α p00 ] + [αk ∧ αk ]} = 0, 2

(1.3.6)

for all λ ∈ S1 . Then the three braces vanish separately, provided the last equation of (1.3.3), its conjugate and the first equation of (1.3.3) vanish, whence the whole system (1.3.3) vanishes. On the other hand, inserting λ −1 , λ and 1 respectively into the equations (1.3.3) and adding, we obtain (1.3.6), whence (1.3.5). a) ⇔ b). Since σ is an involution, one obtains by a direct computation that its eigenspaces satisfy the relations [kk , k ] ⊂ k , [kk , p] ⊂ p, [pp, p] ⊂ k .

(1.3.7)

Then the integrability condition for a 1-form α , 1 dα + [α ∧ α ] = 0 2 rewrites, using (1.3.7) and (1.3.1),   dα + 1 {[α ∧ α ] + [α ∧ α ]} = 0 p p k k k 2  dα p + [αk ∧ α p ] = 0.

(1.3.8)

(1.3.9)

Thus, if F and φ are as in the discussion preceding the theorem, then α = F −1 dF satisfies (1.3.8), whence (1.3.9). Conversely, since D is simply connected and open, any α satisfying (1.3.9), i.e. satisfying (1.3.8), integrates to an F via α = F −1 dF. The harmonicity of ϕ = F mod K is then equivalent to α satisfying the following equation (1.3.10), which we prove independently in Proposition 1.4 below:

∂¯ α p0 + [αk ∧ α p0 ] = 0.

(1.3.10)

Assuming this, it is easy to observe that the system (1.3.3) is equivalent to the system (1.3.8), (1.3.10). Indeed, using (1.3.2), the first equation in (1.3.9) becomes exactly the first one in (1.3.3). Also, using ∂ α p0 = ∂¯ α p00 = 0, the second equation in (1.3.9) rewrites (∂¯ α p0 + [αk ∧ α p0 ]) + (∂ α p00 + [αk ∧ α p00 ]) = 0,

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where the two parentheses are the complex conjugate of each other. Hence, this equation is just a consequence of (1.3.10), which is exactly the second equation of (1.3.3). ¤ 1.4. The basic result, used in the proof of Theorem 1.3, which characterizes the harmonicity of the mapping ϕ is provided by the following P ROPOSITION . Assume G/K carries a non-degenerate symmetric bilinear form invariant by G. The map ϕ : D → G/K induced by F : D → G is harmonic if and only if the associated Maurer-Cartan form α = F −1 dF satisfies (1.3.10). R EMARK . This result is restricted to harmonic maps from 2-dimensional domains to general symmetric spaces. We sketch its proof briefly throughout sections 1.4–1.7, following a procedure similar to [12], [7]. S KETCH OF P ROOF . We describe first the (right) Maurer-Cartan form of the homogeneous space N = G/K and characterize its tangent bundle. a) The tangent space T N is characterized by the property that there exist a canonical isomorphism of K-bundles

ψ0 : G ×K (1/kk ) → T (G/K),

(1.4.1)

given by

ψ0 (1, ξˆ ) =

d

¯ ¯ ¯

dt t=0

[exp t (Ad 1 ξ )].ˆ1,

(1.4.2)

where (1, ξˆ ) ∈ G ×K 1/kk , and 1ˆ ≡ 1 mod K. b) In the case when [kk , p ] ⊂ p , which is always satisfied under our assumptions (1.3.7), we have the isomorphism of K-bundles

ψ1 : G ×K (1/kk ) → [pp] ≡ G ×K p , given by ψ1 (1, ξˆ ) = (1, π p (ξ )), where ξˆ ≡ ξ mod k and π p : 1 → p is the projection associated to the decomposition (1.1.1). We consider the map

β 0 = ψ1 ◦ ψ0−1 : T N → [pp],

β 0 (X1ˆ ) = (1, π p (ξ )),

where X1ˆ ∈ T1ˆ N, and where ξ ∈ 1 is determined by the relation ¯ d ¯ X1ˆ = ¯ [exp t (Ad1 ξ )]. 1ˆ ∈ T1ˆ N. dt t=0 Then it is easy to verify that β 0 is an isomorphism of K-bundles, closing the diagram

β0 TN [pp] = G ×K p @ ¡ µ ψ ψ0−1 @ R ¡ 1 @ G ×K 1/kk

(1.4.3)

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Weierstrass-type representation for harmonic maps i

c) The bundle [pp] = G ×K p is a subbundle of [1] ≡ G ×K 1, via [pp],→[1], which is induced by the natural injection. d) [1] can be canonically identified with the trivial bundle N × 1, via the diffeomorphism

ψ2 : [1] → N × 1,

ψ2 (1, ξ ) = (ˆ1, Ad1 ξ ) for all (1, ξ ) ∈ [1]. ψ

2 e) As a consequence, T N can be identified with [pp] ⊂ [1] ∼ =N × 1, whence there exists the

canonical injective mapping

β˜ : T N → N × 1,

β˜ = ψ2 ◦ i ◦ β 0 ,

β˜ (X1ˆ ) = (1, Ad 1π p (ξ )) for all X1ˆ ∈ T1ˆ N, where ξ is determined by the relation (1.4.3). f ) The right Maurer-Cartan form of the homogeneous space N = G/K is the mapping [12, p. 6]

β : T N → 1, β = pr2 ◦ β˜ ,

β (X1ˆ ) = Ad1 π p ξ for all X1ˆ ∈ T1ˆ N

and represents a vector-valued 1-form β ∈ Λ 1 (N, 1), which is equivariant with respect to the action of G on T N (the left translation) and the action of G on 1 (the adjoint action), i.e., 1∗ β = Ad1 ◦ β for all 1 ∈ G. One can verify that its pullback to G satisfies (π ∗ β )1 = Ad1θ1 , where θ1 = π p ◦ θ , with θ ∈ Λ 1 (G, 1) being the (left) Maurer-Cartan 1-form of G,

θ (ζ1 ) = 1−1 ∗ (ζ1 ), for all ζ1 ∈ T1 G. 1) The map β 0 = β ◦ ∂ ϕ = ϕ ∗ β |T (1,0) D satisfies the relation

β 0 = AdF ◦ α p0 .

(1.4.4)

1.5. Note that the generalization of the notion of harmonicity of a map ϕ : D → N to a pseudoRiemannian manifold N can be chosen in several ways. At one hand one can consider the variational problem [4, (2.1.2)] under compactness conditions. One obtains [4, (2.1.7), (2.2.1)]. Assuming that the metric on N is induced from a G-invariant form, then the equation obtained from the variational problem is equivalent to the condition τ (ϕ ) = 0, where τ is the tension of the map ϕ

τ (ϕ ) = Trγ (5dϕ ),

(1.5.1)

where γ is the pseudo-metric of N, and 5 is the connection on the bundle of forms D, ϕ −1 T N), induced from the (trivial) Levi-Civita connection of the Riemannian flat Λ 1 (D N manifold D and the Levi-Civita connection 5 of N. As the manifold D is of real dimension 2, the tension τ (ϕ ) is a multiple of the induced N covariant derivative (ϕ −1 5)(∂ /∂ z¯) ϕ∗ (∂ /∂ z), and its vanishing represents the holomorphicity condition for the section ϕ∗ (∂ /∂ z) of the induced bundle (ϕ −1 T N, pr1 , D ), ([34], [12], [7]).

76

V. BALAN and J. D ORFMEISTER N

Hence the harmonicity of the map ϕ can be rewritten in terms of 5 µ ¶ ∂ N = 0, ϕ 5 ϕ∗ (∂ /∂ z¯) ∗ ∂z

(1.5.2)

N

or briefly 5 00 ∂ ϕ = 0. N

R EMARK . In general, if 5 is not the Levi-Civita, but an arbitrarily given affine connection, then (1.5.1) characterizes affine harmonic maps ([28, Definition 2.1, p. 407]) and (1.5.2) N characterizes 5-harmonic maps. In our case, where the range of the harmonic map is a general symmetric space with two-dimensional contractive domain, the two definitions mentioned just N above are equivalent. Moreover, for the Levi Civita connection 5, the two definitions (1.5.1) and (1.5.2) both provide the classical harmonic maps ϕ : D → N, i.e. the extremals of the energy functional Z

E(ϕ ) =

N∗

|d ϕ |2 d vol1 ,

(1.5.3)

where N∗ ⊂ N is a compact subdomain of N. Moreover, τ (ϕ ) = 0 is exactly the Euler-Lagrange can N equation of the energy functional E(ϕ ). Since on general symmetric spaces 5 = 5 (see 1.6-f can below), the 5 -harmonic maps coincide with the (classical) harmonic maps, the minimizers of (1.5.3) ([28, Proposition 2.3, p. 408]). 1.6. N On the general symmetric space N = G/K, the connection 5 has a specific form, which permits the reformulation of (1.5.2) in terms of the Maurer-Cartan form α associated to F. N The following steps lead to an explicit expression for 5: a) The left translation on G provides by left shifts of p a horizontal distribution, which is right K-invariant and hence provides a connection on the K-principal bundle (G, π , N = G/K), which induces on the associated bundle (T N, π , N) ≡ ([pp], π˜ , N) a G-invariant canonical conneccan tion 5 of the general symmetric space N. K. Nomizu ([36]) has shown that any Ad(K)-invariant bilinear form γ : m × m → m induces via

γ (η , ξ ) = (5ξ¯ η¯ )eˆ , for all ξ , η ∈ m ,

(1.6.1)

where τ¯ = d/dt |t=0 (exp t [Ad 1 τ ]).ˆ1, for all τ ∈ p , a linear connection 5 on ([pp], π , N) ∼ = (T N, π , N). In general, the torsion of such a connection is ([28, p. 405]) T (ξ , η ) ≡ 5ξ η − 5η ξ − [ξ , η ] = γ (ξ , η ) − γ (η , ξ ) − π p ([ξ , η ]). can

(1.6.2)

In particular, for γ ≡ 0 in (1.6.1) one obtains exactly 5 ([28, Proposition 1.4, p. 404]), called can also the canonical affine connection of the second kind of N. The torsion of 5 is given by c

T eˆ (ξ , η ) = −π p ([ξ , η ]), for all ξ , η ∈ [pp]eˆ ≡ p ⊂ 1.

77


Note that T ≡ 0 ⇔ [pp, p ] ⊂ k . b) The connection associated via (1.6.1) to the Ad(K)-invariant bilinear form 1 γ (ξ , η ) = π p ([ξ , η ]), for all ξ , η ∈ p 2 is called the canonical affine connection of the first kind of N. Using (1.6.2), it is easy to see that N this connection is torsionless. Moreover, by [12] it coincides with the connection 5. can N c) If 5 is torsion-free, it coincides with 5 if and only if

γeˆ (ξ , π p [η , µ ]) = γeˆ (π p [ξ , η ], µ ) for all ξ , η , µ ∈ [pp]eˆ ≡ p ⊂ 1. This relation is obviously fulfilled if [pp, p ] ⊂ k . can N d) The two connections 5 and 5 are described by the relations can

β ( 5 X Y )= Xβ (Y ) − [β (X), β (Y )],

(1.6.3)

1 N β ( 5X Y )= Xβ (Y ) − [β (X), β (Y )] + π˜ ([β (X), β (Y )]), 2

(1.6.4)

for all X,Y ∈ Γ (T N), where π˜ is the projection π˜ : N × 1 → N ×k p . ∼ can d e) The canonical connection 5 coincides with the connection 5 = π[pp] ◦ 5 if and only if ∼ [pp, p ] ⊂ k , where 5 ≡ d is the trivial connection (≡ flat differentiation). N d f ) As a consequence, in view of part c), if the relations (1.3.7) are satisfied, then 5 ≡ 5. N N Hence, in this case, the Levi-Civita connection 5 on N is given by the relation β 0 ◦ 5 = π[pp] ◦ ∼ 5 ◦ β , which is expressed explicitly by the equation N

∼

β 0 ( 5X (Y )) = π[pp] ( 5X (β (Y ))) for all X,Y ∈ T N,

(1.6.5)

where π[pp] is the canonical projection along the fibers of [kk ] = G ×K k , induced on N × 1 via

π[pp] : N × 1 → [pp], π[pp] (1, ξ ) = (1, ξ p ) for all (1, ξ ) ∈ N × 1, ξ p = π p Ad1(ξ ).

1.7. Combining the previous results with (1.3.7), we insert X = ∂ /∂ z¯, and Y = ∂ /∂ z into (1.6.5), and obtain the harmonicity condition (1.5.2) rewritten in the form: Ã µ ¶ µ µ ¶¶! ∂ ∂ N ∼ 5 ϕ∗ (∂ /∂ z¯) ϕ∗ = 0 ⇔ π[pp] 5 (∂ /∂ z¯) β ϕ∗ = 0. ∂z ∂z Setting β 0 = β ◦ ∂ ϕ = β ϕ∗ (∂ /∂ z), this is equivalent with

π[pp] (∂¯ β 0 ) = 0. Then, using (1.4.4), the condition above becomes succesively

π[pp] (∂¯ AdF α p0 ) = 0 ⇔ π[pp] (∂¯ (F α p0 F −1 )) = 0 ⇔

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π[pp] (F ∂¯ α p0 F −1 + ∂¯ F ∧ α p0 F −1 − F α p0 ∧ F −1 .F −1 ∂¯ F.F −1 ) = 0 ⇔ π[pp] {F(∂¯ α p0 + [F −1 ∂¯ F ∧ α p0 ])F −1 } = 0 ⇔ π[pp] ◦ AdF(∂¯ α p0 + [F −1 ∂¯ F ∧ α p0 ]) = 0 ⇔ π p AdF −1 AdF(∂¯ α p0 + [(α p00 + αk00 ) ∧ α p0 ]) = 0 ⇔ ∂¯ α p0 + [αk ∧ α p0 ] = 0.

(1.7.1)

Thus we obtain exactly (1.3.10), whence the conclusion in Proposition 1.4. R EMARK . In (1.7.1) we used besides the splittings D, k ) = Λ (1,0) (D D, k ) ⊕ Λ (0,1) (D D, k ), αk = αk0 + αk00 ∈ Λ 1 (D D, 1) = Λ (0,1) (D D, k ) ⊕ Λ (0,1) (D D, p ), F −1 ∂¯ F = αk00 + α p00 ∈ Λ (0,1) (D D, Ker(π p )), which is a consequence of also the relation [αk0 ∧ α p0 ] = 0 and [α p00 ∧ α p0 ] ∈ Λ 2 (D (1.3.7). The result above was obtained in [28, Proposition 3.1, p. 409] and leads to N 00 can 00 ˜ 00 ∂ ϕ = 0 ⇔ ∂¯ α p0 + [αk ∧ α p0 ] = 0. 5 ∂ϕ = 0 ⇔ 5 ∂ϕ = 0 ⇔ 5

2.

Loop groups.

In the last chapter, we have seen that introducing a parameter λ ∈ S1 reduces the number of equations to one and thus changes the discussion of harmonic maps to the investigation of PDE’s with parameter. In this chapter we will first discuss loop groups and then apply the results to harmonic maps. 2.1. As before we consider a connected real Lie group G. We assume that G is faithfully repreC ) we consider the norm sented by matrices in M (n, R ). On M (n,C Ã ! n

|A| = max j

∑ |Ai j |

.

(2.1.1)

i=1

Then |AB| ≤ |A| · |B| and |I| = 1. Next we consider the Wiener algebra ½ A = f : S1 → C

¯ ¯ ¯f= ¯

∑

fk λ k ,

Z k∈Z

This is a Banach algebra and so is ¯ ½ ¯ C ) ¯¯ A = M (n, A ) = A : S1 → M (n,C

(2.1.2)

¾ | f | < ∞ . ∑ k

(2.1.3)

Z k∈Z

∑ Ak λ k ,

Z k∈Z

¾ |A | < ∞ . ∑ k

Z k∈Z

(2.1.4)


79

Similarly C ) | Ai j ∈ A } Λ 1C = {A : S1 → 1C ⊂ M (n,C is a Banach Lie algebra with closed subalgebras ¯ ½ ¾ ¯ + C k C ¯ Λ 1 = A ∈ Λ 1 ¯ A = ∑ Ak λ , Ak = 0 if k < 0 ¯ ½ ¯ Λ − 1C = A ∈ Λ 1C ¯¯ A =

(2.1.5)

(2.1.6)

Z k∈Z

¾ k , A = 0 if k > 0 . A λ k ∑ k

(2.1.7)

Z k∈Z

Clearly,

Λ + 1C + Λ − 1C = Λ 1C

(2.1.8)

Λ + 1C ∩ Λ − 1C = 1C .

(2.1.9)

On the group level one can proceed similarly. For this we note, that according to [30, Chapter VII], one can always define for G a “universal complexification” GC . In this paper, we will assume that GC has a faithful linear representation, extending the faithful representation of G. Next, by a classical result for the Wiener algebra we have A ∈ M (n, A ) is invertible ⇔ det A 6= 0 or all λ ∈ S1 .

(2.1.10)

GL(n, A ) = {A ∈ M (n, A ) | det A 6= 0 for all λ ∈ S1 }.

(2.1.11)

GL(n, A ) is a Banach Lie group with Lie algebra M (n, A ).

(2.1.12)

We can thus define

Then we have

Analogously to the Lie algebra level we set

Λ GC = {A ∈ GL(n, A ) | A(λ ) ∈ GC for all λ ∈ S1 }, ¯ ½ ¾ ¯ + C k ¯ C ) ¯ A = ∑ λk A , Ak = 0 if k < 0 Λ G = A ∈ GL(n,C Z k∈Z ¯ ½ ¾ ¯ C ) ¯¯ A = ∑ λk Ak , Ak = 0 if k > 0 . Λ − GC = A ∈ GL(n,C

(2.1.13) (2.1.14) (2.1.15)

Z k∈Z

Then Λ GC is a Banach Lie subgroup of GL(n, A ) with Lie algebra Λ 1C . Similarly, Λ + GC and Λ − GC are Banach Lie subgroups of Λ GC with Lie algebras Λ + 1C and Λ − 1C respectively. In view of (1.3.4), we can expect that our discussion will require restrictions with regard to the distribution of powers of λ in the coefficients of the matrices occurring in our investigation. It seems that this restriction is exactly incorporated by using twisted algebras. For this we extend the automorphism σ of G to Λ GC : (σ A)(λ ) = σ (A(−λ )).

(2.1.16)

80


Then σ is an automorphism of Λ GC and we set

Λ GCσ = {A ∈ Λ GC | σ A = A}.

(2.1.17)

Then Λ GCσ is a Banach Lie group. On the Lie algebra level we obtain

Λ 1Cσ = {A ∈ Λ 1C | d σ A = A}.

(2.1.18)

Expanding here A = ∑k∈ZZ Ak λ k we see A ∈ Λ 1Cσ

⇔

d σ A2k = A2k and d σ A2k+1 = −A2k+1 .

(2.1.19)

Thus, setting k = {A ∈ 1; d σ A = A} and p = {A ∈ 1; d σ A = −A}, we see that for every A ∈ Λ 1Cσ we have A2k ∈ k and A2k+1 ∈ p . Analogously one defines the Banach Lie groups Λ ± GCσ with associated Lie algebras Λ ± 1Cσ . N OTE . An extended framing F : D ×S1 → G is analytic in λ ∈ C ∗ and therefore F(z, z¯, · ) ∈ Since F even takes values in G for all z, λ ∈ S1 , it is natural to consider also the Banach Lie group

Λ GCσ .

Λ Gσ = {A ∈ Λ GCσ | A(λ ) ∈ G for all λ ∈ S1 };

(2.1.20)

the corresponding Lie algebra will be denoted by Λ 1σ . For our purposes two results are of importance T HEOREM (Birkhoff decomposition). Every A ∈ Λ GCσ can be written in the form A = A−WA+ , where A− ∈ Λ − GCσ , A+ ∈ Λ + GCσ and W is a homomorphism of S1 into a maximal toral subgroup of GC . A proof of this result can be found in [3, Section 4.5]. The second important result is T HEOREM (Iwasawa decomposition, [3, Theorem 6.5, p. 604]). written in the form

Every A ∈ Λ GCσ can be

A = L ·W · B+ , where L ∈ Λ Gσ and B+ ∈ Λ + GCσ and W is as in [3]. 2.2. Factorization of generalized loop groups. In many instances it turns out that the choice of λ ∈ S1 is too narrow to facilitate discussion of geometric objects sufficiently well. One therefore uses sometimes generalized loop groups and the corresponding generalized Birkhoff and Iwasawa decompositions. The necessity for this was already recognized in [21] and later generalized in [10]. We present here briefly the main features of this generalization. For G and K as above we have the (classical, finite-dimensional) Iwasawa decompositions KC = K · B,

˜ GC = G · B,

81


where B is a Borel subgroup of KC and B˜ is a Borel subgroup of GC . We fix ε ∈ (0, 1) and set Cε = {z ∈ C | |z| = ε } ⊂ C ∪ {∞} = C¯ , Cε = Cε ∪C1/ε , ½ ¶¾ µ 1 ¯ ¯ E = λ ∈ C | |λ | ∈ ε , , I = C¯ \E. ε Then it is natural to consider the following loop groups and loop algebras (see, e.g., [10])   Λ ε G = {1 : Cε → GC | 1(λ ) = 1(1/λ¯ ), λ ∈ Cε }     ΛEε G = {1 ∈ Λ ε G| 1 extends holomorphically to E}      Λ ε ˜ G = {1 ∈ Λ ε G| 1 extends holomorphically to I and 1(0) ∈ B} ˜ I,B   Λ ε Gσ = {1 ∈ Λ ε G| σ 1(λ ) = 1(−λ )}     ΛEε Gσ = {1 ∈ Λ ε Gσ | 1 extends holomorphically to E}      Λ ε ˜ Gσ = {1 ∈ Λ ε Gσ | 1 extends holomorphically to I and 1(0) ∈ B} ˜ I,B    Lie(Λ ε G) = Λ ε 1 = {ξ : Cε → 1C | ξ (λ ) = ξ (1/λ ), for all λ ∈ Cε }       Lie(Λ ε G ) = Λ ε 1 = {ξ ∈ Λ ε 1| σ ξ (λ ) = ξ (−λ )}  σ

σ

  Lie(ΛEε Gσ ) = ΛEε 1σ = {ξ ∈ Λ ε 1σ | ξ extends holomorphically to E}        Lie(ΛI,ε B˜ Gσ ) = ΛI,ε B˜ 1σ = {ξ ∈ Λ ε 1σ | ξ extends holomorphically to I and ξ (0) ∈ b˜ }, ˜ With these notations we have the Iwasawa loop group decomposition ([5], where b˜ = Lie(B). ε ε [10]) Λ G ∼ ΛE G · ΛI,ε B˜ G and [28, Corollary 5.4, p. 415])

Λ ε Gσ ∼ ΛEε Gσ · ΛI,ε B˜ Gσ ,

(2.2.1)

where the intersection of the right factors is {e}. Moreover, one can show that in (2.2.1) one can replace the right side, up to a real analytic diffeomorphism, by the product of the corresponding groups. The symbol “∼” expresses the fact that the right side is not necessarily equal to the left side, but does contain at least an open neighbourhood of the identity element. A characterization of when the right side is open and dense would be of great interest. 2.3. Actions of loop groups. From (2.2.1) we obtain the following group actions [5], [13], [21], [28]: a) The group ΛI,ε B˜ Gσ acts on ΛEε Gσ via 1]h = (1 · h)E ,

82


where 1 ∈ ΛI,ε B˜ Gσ and h ∈ ΛEε Gσ and the right side is the first factor of 1 · h in its decomposition (2.2.1). Note that this group action is not globally defined, since in (2.2.1) there is (in general) no equality. b) If F˜ is an extended framing then 1 ∈ ΛI,ε B˜ Gσ acts on F˜ via ˜ ˜ (1]F)(p) = 1](F(p)), where p ∈ D .

(2.3.1)

It is easy to see ([28, Proposition 6.1, p. 417]) that 1]F˜ is again an extended framing. However, due to the non-global nature of the group action used, the framing 1]F˜ may have singularities at points p, where F˜ does not have any singularity. c) The action (2.3.1) induces canonically an action of ΛI,ε B˜ Gσ on ΛEε Gσ /K ([28]). This extends via (2.3.1) to an action of ΛI,ε B˜ Gσ on all S1 -families of harmonic maps.

3. Generalized Weierstrass representation of general harmonic maps. 3.1. Following the procedure of [19] we want to construct a “holomorphic potential” and a “normalized potential” (called originally “meromorphic potential” in [19]) for each harmonic map. First we note that the proof of [19, Appendix] generalizes immediately to our setting and yields T HEOREM . Let F = F(z, z¯, λ ) be the extended frame of some harmonic map, z ∈ D , λ ∈ S1 . Then there exists some V+ : D → Λ + GCσ such that C(z, λ ) = F(z, z¯, λ )V+ (z, z¯, λ )

(3.1.1)

is holomorphic on D \ T1 , where T1 is the discrete subset of D , and the generalized Iwasawa decomposition needs some middle term. For the rest of this paper we normalize the extended frames of harmonic maps so that they satisfy F(z0 , z¯0 , λ ) = I at some fixed base point z0 ∈ D . D EFINITION . Any map C as in (3.1.1) will be called a holomorphic extended frame. Moreover, the Maurer-Cartan form of any holomorphic extended frame

η = C−1 dC

(3.1.2)

will be called a holomorphic potential for the harmonic map ϕ (or for the extended frame F). As usual one verifies L EMMA . Every holomorphic potential η is of the form

η = λ −1 η−1 + λ 0 η0 + λ 1 η1 + . . .

(3.1.3)

where η j is a holomorphic (1, 0)-form on D . We note that the integrability condition is trivially satisfied for any holomorphic (1, 0)-form on D . Thus, we have P ROPOSITION . Assume η is a holomorphic (1, 0)-form on D with values in Λ 1Cσ which is of the form (3.1.3). Then the equation


C−1 dC = η , C(z0 , λ ) = I

83

(3.1.4)

is globally solvable on D . Moreover, decomposing C via Iwasawa splitting C = FV+ ;

F ∈ Λ Gσ , V+ ∈ Λ + GCσ ,

(3.1.5)

we obtain an extended frame of some harmonic map ϕ : D \S → G/K, where S = {z ∈ D | C(z, · ) 6∈ Λ Gσ · Λ + GCσ }. R EMARK . Since C(z0 , λ ) = I for all λ ∈ S1 , we know that D \S does contain an open neighbourhood of z0 . 3.2. As in [19] the results of Section 3.1 are used to construct normalized potentials. First we note L EMMA . We retain the notations and the assumptions of Theorem 3.1. If C = C−C+ with C− ∈ Λ − GCσ , C+ ∈ Λ + GCσ and C− = I + O(λ −1 ) on some open subset U ⊂ D \S, then C− is holomorphic on U and −1 C− dC− = λ −1 ξ−1 dz

(3.2.1)

for some holomorphic map ξ−1 on U. P ROOF . By our assumptions C is a holomorphic map from U to the open set Λ − GCσ · Λ + GCσ . Since the splitting map

Λ − GCσ · Λ + GCσ → Λ − GCσ × Λ + GCσ −1 is also complex analytic, the maps C− and C+ are holomorphic. Therefore C− dC− is a holo−1 morphic 1-form. The usual argument shows that it is of the form λ ξ−1 dz. The main question is for which z ∈ D \S one can split C = C−C+ analytically, and how the (additional) singular set ˆ To address this issue we follow the argument of [19]. D\S)\S. Sˆ looks like; let U = (D ¤

T HEOREM . We retain the notations and the assumptions of Theorem 3.1. Then for every C holomorphic extended frame C : D \S → Λ Gσ there is a holomorphic function τD \S : D \S →C such C can be split C = C−C+ analytically exactly on the open set on which τD \S 6= 0. Thus Sˆ is discrete in D \S. Moreover, considered as functions on D\S, C− and C+ are meromorphic. In particular, the −1 dC− is a meromorphic differential (1, 0)- form on D \S. normalized potential ξ = C− P ROOF . Following the analogous argument of [19] we consider a representation πˇ of Λ GL(n, A ) in the group of automorphisms of an infinite dimensional Grassmannian like manifold Gr. Considering the dual determinant bundle and a holomorphic (highest weight) section τ we set τM (z) = τ (H(z)p0 ), where p0 denotes the canonical base point of Gr relative to τ . Then H splits analytically exactly at all points z, where τM (z) 6= 0. Pulling back the line bundle det∗ on Gr to M via M → Gr, z → H(z) p˙0 , we obtain a holomorphic line bundle L∗ on M with holomorphic section τL induced from τM as defined above. Since M is Stein, L∗ is trivial, and τM can be considered as a complex valued function on M . ¤

84


D EFINITION . The differential (1, 0)-form −1 ξ = C− dC−

(3.2.2)

is called the normalized potential for the harmonic map ϕ (or the extended frame associated to ϕ ). R EMARK . We reiterate that every normalized potential ξ is of the form ξ = λ −1 ξ−1 dz ˆ D\S)\S. where ξ−1 is meromorphic on some open subset U = (D 3.3. The main feature of the procedure developed in [19] is that every harmonic map can be constructed from some holomorphic or normalized potential. We recall that the integrability condition is trivially satisfied for the potentials under consideration. T HEOREM A. Let η = λ −1 η−1 + λ 0 η0 + λ 1 η1 + . . . be a holomorphic (1, 0)-form with values in pC defined on a simply connected subset L ⊂ C . Let C be the solution on L to the ODE dC = Cη , C(z0 , λ ) = I for all λ ∈ S1 . Then C is the extended holomorphic frame of some harmonic map ϕ : L \T → G/K. More precisely, splitting C = FV+ on some open subset L \T , we obtain the extended frame F of the harmonic map ϕ : L \T → G/K, given by ϕ = F mod K. P ROOF . It is easy to see that the Maurer-Cartan form of F has the form (1.3.4). The rest is straightforward. ¤ T HEOREM B. Let ξ = λ −1 ξ−1 dz be a meromorphic (1, 0)-form on Lˆ ⊂ C and assume that there exists a globally meromorphic solution C to the ODE dC = Cξ , C(z0 , λ ) = I for all ˆ Tˆ → G/K. λ ∈ S1 . Then C is the extended holomorphic frame of some harmonic map ϕ : L\ ˆ ˆ More precisely, splitting C = FV+ on some open subset L\T , we obtain the extended frame F of ˆ Tˆ → G/K, given by ϕ = F mod K. the harmonic map ϕ : L\ P ROOF . First we remove the points from Lˆ where C has a pole. Then proceed as in the proof of Theorem A. ¤ Finally, we would like to address the question to what extent the harmonic maps are uniquely determined by the associated analytical potentials. To avoid lengthy technical assumptions we state only a local result: T HEOREM C. Let ϕ : D → G/K be harmonic and fix a base point z0 ∈ D . Then the normalized potential ξ associated with ϕ is holomorphic in a neighborhood of z0 and it is uniquely determined by ϕ . Moreover, given a normalized potential ξ which is holomorphic near z0 , then Theorem B constructs a unique associated family of harmonic maps defined in some neighborhood of z0 . P ROOF . Similar to [19].

¤

R EMARK . Theorem C above shows that harmonic maps and normalized potentials are essentially in a 1-1 relation.

85


4. Finite type harmonic maps. 4.1. Among the harmonic maps investigated in the literature those of finite type play a particularly prominent role. We follow here primarily the approach of [11], [10]. For d ∈ 2 N + 1 we set ¯ ½ d ¾ ¯ n ¯ (4.1.1) Λd = ∑ ξn λ ∈ Λ 1σ ¯ ξd 6= 0 . n=−d

Note, if ξ ∈ Λd , then ξd−1 ∈ k C , since d is odd. Decomposing k C in the form k C = n + h + n¯ with b = h + n being a Borel subalgebra, we can project any τ ∈ kC onto b (see [11, (2.5)]) 1 r(τ ) = τn + τh . 2

(4.1.2)

Using this notation we obtain T HEOREM . For each d ∈ 2 N + 1 and ξ∗ ∈ Λd , there exists an open ball U of O ∈ R 2 where

∂ξ = [ξ , λ −1 ξd + r(ξd−1 )], ξ (z0 ) = ξ∗ ∂z

(4.1.3)

is integrable, for ξ = ∑dn=−d ξn λ n . Moreover, in this case, the Λ 1σ -valued 1-form given by

α = (λ −1 ξd + r(ξd−1 ))dz + (λ ξ−d + r(ξd−1 ))d¯z

(4.1.4)

satisfies the Maurer-Cartan equations (1.3.8). In addition, the extended frame F defined by F −1 dF = α , F(z0 , λ ) = I on U induces the harmonic map ϕ = F mod K on U. P ROOF . The proof can be taken almost verbatim from [11, Chapter 2, Theorem 2.5]. However, instead of the Killing form we take the non-degenerate invariant bilinear form κ which we have assumed to exist. At this point we also need to restrict to open balls U around O, since the argument for the completeness given in [11] doesn’t apply in our case. ¤ D EFINITION . Harmonic maps obtained by this construction outlined in the last Theorem will be called of finite type. We will see in the next section that harmonic maps of finite type have particularly simple holomorphic potentials. 4.2. Consider the potential

η = λ d−1 ξ∗ dz,

(4.2.1)

where ξ∗ ∈ Λd . Then the holomorphic extended frame C defined by dC = Cη , C(z0 , λ ) = I, is of the form C(z, λ ) = exp(zλ d−1 ξ∗ ).

(4.2.2)

86


Consider the Iwasawa splitting (locally around z0 ) C = Fˆ Vˆ+ .

(4.2.3)

For the purposes of this section we will use the freedom in choosing the coefficient V0 in V+ at λ 0 and require V0 ∈ B, where KC = KB is the (classical) Iwasawa decomposition of KC . At any rate, by the general theory Fˆ defines an associate family of harmonic maps. D EFINITION . Maps obtained from potentials of the form (4.2.1) are called of Symes finite type. The main result of this section is T HEOREM . a) Every map of finite type is of Symes finite type. b) Every map of Symes finite type is of finite type. P ROOF . a) Let φ : D → G/K be of finite type and ξ∗ as in the last Theorem. Let F denote the extended framing of φ . On the other hand, by the arguments at the beginning of this section, ˆ Then and starting from η = λ d−1 ξ∗ dz, using (4.2.2) and (4.2.3) we obtain some frame F.

ξˆ = Fˆ −1 ξ∗ Fˆ ∈ Λd ,

(4.2.4)

which follows from ξˆ = Fˆ −1 ξ∗ Fˆ = Vˆ+ ξ∗Vˆ+−1 and the fact that ξ∗ is in Λd . It is straightforward to verify ˆ ∂z ξˆ = [ξˆ , Fˆ −1 ∂z F],

ξˆ (z0 , λ ) = ξ∗ .

(4.2.5)

Next we use the fact that under our assumptions we have

η∗ dz = λ d−1 ξ∗ dz = C−1 dC = dCC−1 .

(4.2.6)

Inserting the unique decomposition (4.2.3) into the right side yields ˆ Vˆ+Vˆ+−1 Fˆ −1 , η∗ dz = d Fˆ Fˆ −1 + Fd

(4.2.7)

ηˆ dz = λ d−1 ξˆ dz = Fˆ −1 d Fˆ − dVˆ+Vˆ+−1 .

(4.2.8)

from which we derive

Therefore, α = Fˆ −1 d Fˆ is the projection of ηˆ dz along Λ + 1σ C . In particular, we obtain λ −1 ηˆ −1 dz = α 0 p = λ −1 ξˆ−d and ηˆ 0 dz = ξˆ−d+1 dz = αk − (dVˆ+Vˆ+−1 )0 , whence αk = (ξˆ−d+1 dz)k , the projection of ξˆ−d+1 dz onto k along Lie B. But it is straightforward to verify (see e.g. [20, Section 2.3]) that (ξˆ−d+1 dz)k = r(ξˆ−d+1 )dz + r(ξˆ−d+1 )dz.

(4.2.9)

Fˆ −1 d Fˆ = (λ −1 ξˆ−d + r(ξˆ−d+1 ))dz + (λ ξˆd + r(ξ−d+1 ))d¯z

(4.2.10)

Hence

and we have ξˆ (0) = ξ∗ . The uniqueness of the solution with values in Λd to the differential ˆ Thus F = AFˆ with some matrix A equation (4.1.3) now shows ξ = ξˆ and F −1 dF = Fˆ −1 d F.


87

independent of z. Evaluating at the base point z0 yields A = I. This shows that the harmonic map (of Symes finite type) derived from η = λ d−1 ξ∗ coincides with the given associated harmonic map of finite type derived in 4.1. b) This was actually part of the argument in the proof of a). ¤ 5. The Lie group case. 5.1. As an application of the theory presented in the paper we would like to consider harmonic maps into Lie groups, which were discussed in a somewhat different setting via loop groups in [41], [4]. In the context of this paper we consider a real Lie group G as a symmetric space G ∼ (G × G)/∆ ,

(5.1.1)

∆ = {(1, 1) | 1 ∈ G}.

(5.1.2)

where

The canonical projection π is given by

π : G × G → G, (1, h) → 1−1 h.

(5.1.3)

Thus our approach requires to consider the loop group with values in (G × G)C = GC × GC . Similar to [21] we set H = G×G

(5.1.4)

Λ H C = Λ GC × Λ GC .

(5.1.5)

Λ H = Λ G × Λ G.

(5.1.6)

and

Analogously we set

For our approach we also need some group Λ + H C . It is natural to consider pairs (1(λ ), h(λ )) of functions λ ∈ S1 , which have holomorphic extensions to the interior of the unit disk

Λ + H C = Λ + GC × Λ + GC ,

(5.1.7)

Λ − H C = Λ − GC × Λ − GC .

(5.1.8)

and analogously

Finally, we mention that the involution σ ( f , 1) = (1, f ) of Λ H has the fixpoint group ∆ and thus defines the symmetric space G. Therefore, by our theory, we need to twist Λ H C by σ and obtain the twisted groups Λ H Cσ , Λ ± H Cσ and Λ Hσ . Note that for ( f , 1) ∈ Λ H C the twisting condition is f (λ ) = 1(−λ ).

(5.1.9)

While this makes the second component in our group basically superfluous, we will nevertheless continue to use Λ H Cσ , since we want to illustrate the general theory with this example.

88


5.2. The rest of this example follows closely [14, Section 9]. We consider a harmonic map ϕ : D → G and lift it to an extended framing F˜ : D → G × G, F˜ = (e, ϕ ).

(5.2.1)

For the Maurer-Cartan form we obtain

α˜ = F˜ −1 d F˜ = (0, ϕ −1 d ϕ ).

(5.2.2)

Abbreviating α = ϕ −1 d ϕ we need to decompose α˜ = (0, α ) in the form α˜ = α˜ k + α˜ p , where k , p ⊂ h = 1 × 1 are defined by σ : k = Lie(∆ ) = {(A, A) | A ∈ 1},

(5.2.3)

p = {(A, −A) | A ∈ 1}.

(5.2.4)

1 1 α˜ k = (α , α ), α˜ p = (−α , α ). 2 2

(5.2.5)

Hence

Next we need to introduce the loop parameter λ . Decomposing α p0 + α p00 = α p into the (1, 0)-part α p0 and the (0, 1)-part α p00 we define

α˜ λ = λ −1 α˜ p0 + α˜ k + λ α˜ p00 , which yields by a straightforward computation ¶ µ 1 1 1 1 α˜ λ = (1 − λ −1 )α p0 + (1 − λ )α p00 , (1 + λ −1 )α p0 + (1 + λ )α p00 . 2 2 2 2

(5.2.6)

(5.2.7)

At this point it is useful to recall ([4, Proposition 4.2]) T HEOREM . A smooth map ϕ : D → G is harmonic if and only if the 1-form 1 1 αλ = (1 + λ −1 )α p0 + (1 + λ )α p00 2 2

(5.2.8)

is integrable for all λ ∈ S1 . R EMARK . 1. For G = U(n) this is a classical result (see e.g. [41]). 2. The Theorem also holds, of course, if one replaces λ by −λ . Thus (5.2.8) is equivalent with α˜ λ = (α−λ , αλ ) being integrable. Since αλ is integrable, we can solve F −1 dF = αλ .

(5.2.9)

For G = U(n) this is Uhlenbeck’s “extended framing”. In our setting, an extended framing associated with α˜ λ is given by F (z, z¯, λ ) = (F(z, z¯, −λ ), F(z, z¯, λ )).

(5.2.10)


89

R EMARK . Note that we usually normalize framings in a way, such that at some base point z∗ ∈ C we have ϕ (z∗ ) = I and F(z∗ , z¯∗ , λ ) = I, for all λ ∈ S1 . Then also F(z∗ , z¯∗ , −λ ) = I, for all λ ∈ S1 . But a glance at (5.2.7) shows that F(z, z¯, −1) =const., whence F(z, z¯, −1) = I. Since ϕ (z) and F(z, z¯, 1) satisfy the same differential equation with same initial condition, ϕ (z) = F(z, z¯, 1) follows. Incidentally we have shown that the framing F is “based at λ = −1”, i.e., F(z, z¯, −1) = I. Since our extended framing F is not based at any λ ∈ S1 , we do not use “based loop groups”, opposite to [41] or [4]. As a consequence, in the discussion above, the associated family ϕλ of harmonic maps containing ϕ±1 is given by

ϕλ (z, z¯) = F (z, z¯, λ ) mod ∆ = F(z, z¯, −λ )−1 F(z, z¯, λ ).

(5.2.11)

We would like to point out that Uhlenbeck’s extended framing F(z, z¯, λ ) yields a harmonic map into G only for λ = ±1, while our setting produces naturally an S1 -family of harmonic maps into G. 5.3. In Section 3.1 we have introduced holomorphic potentials and normalized potentials. The general theory states that the extended framing F ∈ Λ Hσ can be multiplied by some V+ ∈ Λ + H Cσ such that C (z, λ ) = F (z, z¯, λ )V+ (z, z¯, λ )

(5.3.1)

is “holomorphic” in z. For the present discussion we thus obtain C1 (z, λ ) = F (z, z¯, λ )V+ (z, z¯, λ ),

(5.3.2)

where C1 denotes the first component of C . The second component is determined by (5.1.8). Hence, all holomorphic potentials are of the form

η˜ = (η (z, λ )dz, η (z, −λ )dz),

(5.3.3)

η = λ −1 η−1 + λ 0 η0 + λ η1 + . . .

(5.3.4)

where

where all the matrix functions ηk are holomorphic in z. Analogously, normalized potentials are of the form

ξ˜ = (ξ , −ξ ),

(5.3.5)

ξ = λ −1 ξ−1 , ξ−1 meromorphic.

(5.3.6)

5.4. For the discussion of maps of finite type, we consider “Λd ”, i.e., those elements in Lie(Λ Hσ ), which only involve finitely many powers of λ . Then these elements need to be shifted by λ d−1 . In our setting we thus obtain T HEOREM .

Harmonic maps ϕ : D → G of finite type are exactly those maps which are

90


obtained from potentials of the type

η˜ = (λ d−1 µ (λ )dz, (−λ )d−1 µ (−λ )dz) = λ d−1 (µ (λ ), µ (−λ ))dz,

(5.4.1)

where µ ∈ Λd , i.e., µ = ∑−d≤ j≤d µ j λ j , d odd, µd 6= 0. 5.5. Finally, we would like to mention some explicit examples. Comparing [4, 4.2.1] to (5.2.8) we see that our framing F is of the form (5.2.10), where F is the framing considered in [4]. In view of (5.3.3) and (5.3.5) it thus suffices to consider one component. Therefore the potentials used in this section can be read off directly from the potentials used in [4]. In particular 1. For the nilpotent group (Ã !¯ ) 1 a ¯¯ G = (5.5.1) ¯ a∈R , 0 1 ¯ Ã ! 0 a −1 one gets the potential ξ = (A(λ ), A(−λ )), where A(λ ) = ((1 − λ )/2) dz and a is 0 0 a meromorphic function. 2. For the Heisenberg group of upper triangular unipotent matrices   ¯ ¯    0 a b ¯   ¯ (5.5.2) G =  0 0 c ¯ a, b, c ∈ R ,    0 0 0 ¯¯    0 ∂z a ∂z c − a∂z b   the potential is ξ = (A(λ ), A(−λ )), where A(λ ) = ((1 − λ −1 )/2)  0 0  dz ∂z b and a, b, c : D → R are meromorphic functions. 0 0 0 3. For the special linear Lie group (Ã !¯ ) a b ¯¯ G = SL(2, R ) = (5.5.3) ¯ a, b, c, d ∈ R , ad − bc = 1 , c d ¯ the potential is ξ˜ = ((1 − λ −1 )/2, (1 + λ −1 )/2)ξ dz, for ξ given in [4, (5.3.8)]. In all three cases one obtains the normalized potential in the sense of Section 3.2 by gauging away the coefficient at λ 0 . 6. More examples. 6.1. CMC surfaces. The immersions of constant mean curvature (CMC) surfaces can be characterized [38] by the harmonicity of the Gauss map ϕ : D → G/K = SO(3)/SO(2) ≡ SU(2)/SU(1). Here GC = C ) and K = SU(1) is the subgroup of diagonal elements in G = SU(2). The normalized SL(2,C potentials have the form [15, (3.2.1); Theorem 3.13] Ã ! 0 f −1 dz, (6.1.1) ξ =λ E/ f 0


91

where f has only poles of even order ([15, Corollary 2.3]) and does not vanish identically. The solution to 1−1 − d1− = ξ can be decomposed in the form (Iwasawa splitting) 1− (z, λ ) = F(z, λ )1−1 + (z, λ ) C ) are smooth in z. Then the Symand we can assume that F ∈ Λσ SU(2) and 1+ ∈ Λσ+ SL(2,C Bobenko formula ([15, (1.1.4)], [16, (2.2.11)]) provides the associated family of immersions, which consists of CMC surfaces without branch points, ϕ : S1 × D → su(2) ≡ so(3) ≡ R 3 . The holomorphic potentials of the form ([19, Section 4]) Ã ! 0 f −1 dz, (6.1.2) η =λ 1 0 with f and 1 holomorphic, provide CMC surfaces with umbilic points at the zeros of 1 and branch points at the zeros of f . E.g., we have: a) for f = 1, 1 = 0, the punctured sphere S2 \{one point}; b) for f = 1, 1 = 1, the right circular cylinder; c) for f = 1, 1 = czm (c ∈ C ∗ , m ≥ 0), B. Smyth’s CMC surface with an umbilic of order m at the origin z = 0 [17, Proposition 4.1]. This is nondegenerate only for m = 0 and |c| = 6 1 ([17, Proposition 4.4]). d) for f = 1, 1 = (z − z1 ) · . . . · (z − zn ), B. Smyth’s CMC surface with n umbilic points at zk , k = 1, . . . , n. Similarly, one can obtain surfaces with branch points; e.g, the potential Ã ! 0 z − z 0 ξ = λ −1 dz 1 0 yields a CMC surface with one branch point at z = z0 . 6.2. Willmore surfaces. The DPW method was applied for a description of the Willmore surfaces in terms of potentials [26] by F. Helein. The Willmore surfaces S are the minimizers of the Willmore functional Z

W (S) = S

H 2dσ =

1 4

Z S

(k1 − k2 )2d σ + 4π (1 − 1),

where the variation is made within the set S of surfaces S immersed in R 3 , which are oriented and without boundary. Moreover, H denotes the mean curvature, k1 and k2 the principal curvatures, 1 the genus and d σ the area element of S ∈ S . Willmore surfaces are characterized by the equation ∆ H + 2H(H 2 − K) = 0. The only CMC Willmore surfaces are S2 and the minimal surfaces. In the DPW approach one associates to a conformal Willmore immersion a frame in R 4,1 which encodes the tangent plane of the surface and the conformal Gauss map ([26], [1]). Outside the umbilic set, this frame F : U ⊂ D → G = SO(4, 1) incorporates the conformal transform of the surface. The immersion is Willmore if and only if the loopified Maurer-Cartan form (1.3.4) of F is integrable; DPW works for (1) the noncompact subgroup K = SO(3, 1) ⊂ SO(4, 1), and (2) for K = SO(3) × SO(1, 1). Thus S is Willmore if (1) the induced map Fˆ : U → G/K is harmonic,

92


or respectively (2) if Fˆ is “roughly harmonic”, i.e. it provides a harmonic map by a K-right ˆ The second case has the peculiarity that α p0 is not necessarily holomorphic; gauge shift of F. still, this alternative is constructive and the meromorphic potentials can be explicitly described [26, Section 4.2, Theorem 9, p. 38]. 6.3. Minimal surfaces. These, regarded as special cases of CMC surfaces, are characterized as well by the holomorphy of the Gauss map. The associated meromorphic potentials have the form [18, Thorem 3.1, p. 5] Ã ! 0 0 −1 dz, (6.3.1) ξ =λ 1 0 a particular form of (6.1.2). Then the classical Weierstrass representation produces directly from 1 and the coefficients of the extended frame F obtained from C = FV+ the minimal surface [18, Section 4]. 6.4. The tangent group case. Given a connected real Lie group G with an involution σ , let K = (Fix σ )0 ⊂ G. We set ˜ ˜ K˜ = T G/T K ∼ T (G/K) [6]. We note G = T G, K˜ = T K and consider the homogeneous space G/ ˜ ˜ that for σ˜ = (σ , σ∗,e ), we have K = (Fix σ )0 ⊂ 1˜ . Then G = T G ∼ G n 1, where 1 = Lie(G). Let k = Lie(K) and set k˜ = {(A, a) | A, a ∈ k } ∼ k × k . Let 1 = k ⊕ p be the Cartan decomposition relative to σ and p˜ = {(A, a) | A, a ∈ k } ∼ p × p . Then 1˜ = k˜ ⊕ p˜ is the induced decomposition on the tangent group level. Consider the group operation on G˜ given by (1, X) ◦ (h,Y ) = (1h, X + Ad1Y ), where (1, X), (h,Y ) ∈ G n 1 ∼ T G. Then the Lie bracket on T G ∼ 1 n 1 reads "Ã ! Ã !# Ã ! A a B b [A, B] [A, b] − [B, a] [(a, A), (b, B)] ≡ , = . 0 0 0 0 0 0

(6.4.1)

(6.4.2)

˜ K, ˜ the corresponding lift F : D → G˜ induces a normalized For a harmonic function ϕ : D → G/ −1 potential of the form ξ = λ ξ−1 dz, with ξ−1 : D → p˜ C = pC n pC . Conversely, let any ξ = λ −1 ξ−1 dz, with ξ−1 : D → p˜ C = pC n pC be given. We can assume that for this potential the differential equation dC = Cξ has a meromorphic solution C : D → G˜ C where ξ is of the form ξ = λ −1 (ξ (1) , ξ (2) )dz and C = (1, X), whence C−1 dC = (1, X)−1 d(1, X) = (1−1 d1, (−Ad1 X + Ad1−1 X 0 )dz). For the solution C = (1, X) to this differential equation we perform an Iwasawa splitting (1, X) = (1r , Xr ) ◦ (1+ , X+ ) ∈ Λ G˜ · Λ + G˜ C , which is equivalent to (

1 = 1r 1+ X = Xr + Ad1r X+ .


93

Note that the first equation is the usual Iwasawa splitting equation for 1 ∈ Λ GC and yields 1r and 1+ . Rewriting the second equation we obtain (Ad1r )−1 X = (Ad1r )−1 Xr + X+ . Since the first summand is real, we can find X+ and Xr as follows: first, decompose (Ad1r )−1 X = Yr +Y+ ; second, set X+ = Y+ and Xr = X − Ad1r X+ ; then 1r , 1+ , Xr , X+ are the components of the Iwasawa splitting of (1, X). By the general theory (Proposition 3.1), (1r , Xr ) ∈ Λ G˜ is the extended framing of some harmonic map. In particular, the Maurer-Cartan form αλ = λ α p0˜ + αk˜ + λ α p00˜ of (1r , Xr ) is integrable. Moreover, ϕ = (1r , Xr ) mod K˜ is an associate framing of a harmonic map from D to T (G/K). For) G = SU(2), σ = Ad σ3 we obtain (see [15]): K = SU(1) = ( Ã E XAMPLE !¯. ¯ ¯ z 0 ¯ z ∈ C ∗ = (Fix σ )0 . Then T G = SU(2) n su(2), T K = SU(1) n su(1) and 0 z−1 ¯ T G/T K = T (SU(2)/SU(1)) = T S2 . Our costruction then produces the harmonic maps ϕ : D → T S2 [2]. ACKNOWLEDGMENT . The present work was partially supported by NSF Grant DMS - 9705479 and Sonderforschungsbereich 288, Differentialgeometrie und Quantenphysik, TU Berlin, and by Fulbright Grant #21945-5/13/97 and Grant CNCSIS MEN 1478/2004.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]

V. Balan, Willmore surfaces and loop groups, Mat. Bilten, 25 (2001), 17–24. V. Balan, On the DPW Method for the Tangent Group, In: Proc. Conf. Diff. Geom. Thessaloniki, 1998, Proceedings, 4, Geometry Balkan Press, Bucharest, 2000, 1–9. V. Balan and J. Dorfmeister, Birkhoff Decompositions and Iwasawa Decompositions for Loop Groups, Tohoku Math. J., 53 (2001), 4, 593–615. V. Balan and J. Dorfmeister, A Weierstrass—Type Representation for harmonic maps from Riemannian surfaces to general Lie groups, Balkan J. Geom. Appl., 5 (2000), 1, 8–37. M. I. Bergvelt and M. A. Guest, Actions of loop groups on harmonic maps, Trans. Amer. Math. Soc., 326 (1991), 2, 861–886. R. W. Brockett and H. J. Sussmann, Tangent bundles of homogeneous spaces are homogeneous spaces, Proc. Amer. Math. Soc., 35 (1972), 550–551. M. Black, Harmonic Maps into Homogeneous Spaces, Longman Scientific and Technical, 1991. F. Burstall, Twistor methods for harmonic maps, In: Differential Geometry, (ed. V. L. Hansen), Lecture Notes in Math., vol. 1263, Springer, Berlin, 1987, pp. 158–176. F. Burstall, Recent developments in twistor methods for harmonic maps, In: Harmonic mappings, twistors and σ -models, (ed. P. Gauduchon), World Scientific, Singapore, 1988, pp. 158–176. F. Burstall and F. Pedit, Dressing orbits of harmonic maps, Duke Math. J., 80 (1995), no. 2, 353–382. F. Burstall and F. Pedit, Harmonic maps via Adler-Kostant-Symes theory, In: Harmonic Maps and Integrable Systems, Vieweg, 1994, pp. 221–272. F. E. Burstall and J. H. Rawnsley, Twistor Theory for Riemannian Symmetric Spaces, Lecture Notes in Math., vol. 1424, Springer, Berlin, 1990. I. McIntosh, Global solutions of the elliptic 2D Toda lattice, Nonlinearity, 7 (1994), 1, 85–108. J. Dorfmeister and J. H. Eschenburg, Pluriharmonic maps, loop groups and twistor theory, Ann. Global Anal. Geom., 24 (2003), 4, 301–321. J. Dorfmeister and G. Haak, Meromorphic potentials and smooth surfaces of constant mean curvature, Math. Z., 224 (1997), 4, 603–640.

94 [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32]

[33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43]

V. BALAN and J. D ORFMEISTER J. Dorfmeister and G. Haak, On constant mean curvature surfaces with periodic metric, Pacific J. Math., 182 (1998), 2, 229–287. J. Dorfmeister and G. Haak, On symmetries of constant mean curvature surfaces. Part II: Symmetries in a Weierstrass-type representation, Int. J. Math. Game Theory Algebra, 10 (2000), 2, 121–146. J. Dorfmeister, F. Pedit and M. Toda, Minimal surfaces via loop groups, Balkan J. Geom. Appl., 2 (1997), 1, 25–40. J. Dorfmeister, F. Pedit and H. Wu, Weierstrass type representations of harmonic maps into symmetric spaces, Comm. Anal. Geom., 6 (1998), 633–668. J. Dorfmeister and I. Sterling, Finite type Lorentz harmonic maps and the method of Symes, Differential Geom. Appl., 17 (2002), 43–53. J. Dorfmeister and H. Wu, Constant Mean Curvature Surfaces and Loop Groups, J. Reine Angew. Math., 440 (1993), 43–76. J. Eells and L. Lemaire, A Report on Harmonic Maps, Bull. London Math. Soc., 20 (1988), 385–524. J. Eells and L. Lemaire, Another Report on Harmonic Maps, Bull. London Math. Soc., 10 (1978), 1–68. M. A. Guest, Harmonic maps, loop groups, and integrable systems, London Math. Soc. Stud. Texts, 38, Cambridge Univ. Press, Cambridge, 1997. M. A. Guest and Y. Ohnita, Group actions and deformations for harmonic maps, J. Math. Soc. Japan, 45 (1993), 671–704. F. Helein, Willmore immersions and loop groups, Centre de Mathématiques et Leurs Applications, no. 9618, CACHAN Cedex, France; J. Differential Geom., 50 (1998), 331–385. S. Helgason, Differential Geometry, Lie Groups and Symmetric Spaces, Academic Press, New York, San Francisco, London, 1978. M. Higaki, Actions of loop groups on the space of harmonic maps into reductive homogeneous spaces, J. Math. Sci. Univ. Tokyo, 5 (1998), 401–421. J. Hilgert and K.-H. Neeb, Lie Semigroups and their Applications, Lecture Notes in Math., vol. 1552, Springer, Berlin, 1993. G. Hochschild, The Structure of Lie Groups, Holden-Day San Francisco, 1965. J. Jost, Two-dimensional geometric variational Problems, John Wiley & Sons, Ltd., Chichester, 1991. P. Kellersch, The Iwasawa decomposition for the untwisted group of loops in semisimple Lie groups, [Eine Verallgemeinerung der Iwasawa Zerlegung in Loop Gruppen] (in German), Ph.D. Thesis, TU-Munich, 1999; DGDS Monographs #4, Geometry Balkan Press, Bucharest, 2004, http://www.mathem.pub.ro/dgds/mono/kep.zip (electronic). S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Vol. I: Vol. II, Interscience Publishers, 1963, 1969. J.-L. Koszul and B. Malgrange, Sur certaines structures fibrées complexes, Arch. Math. (Basel), 9, Birkhäuser, 1958, 102–109. O. Loos, Symmetric spaces. I: General theory; Symmetric spaces. II: Compact spaces and classification, W. A. Benjamin, Inc., New York-Amsterdam, 1969. K. Nomizu, Invariant affine conections on homogeneous spaces, Amer. J. Math., 76 (1954), 33–65. A. N. Pressley and G. Segal, Loop Groups, Oxford Math. Monogr., VIII, Clarendon Press, Oxford, 1986. E. A. Ruh and J. Vilms, The torsion field of the Gauss map, Trans. Amer. Math. Soc., 149 (1970), 569–573. E. J. Taft, Invariant Wedderburn factors, Illinois J. Math., 1 (1957), 565–573. E. J. Taft, Invariant Levi factors, Michigan Math. J., 9 (1962), 65–68. K. Uhlenbeck, Harmonic Maps into Lie Groups (classical solutions of the chiral model), J. Differential Geom., 30 (1989), 1–50. H. Urakawa, Calculus of Variations and Harmonic Maps, Trans. Amer. Math. Soc., vol. 132, 1990. N. R. Wallach, Harmonic Analysis on Homogeneous Spaces, Pure Appl. Math., 19, Dekker, New York, 1973.

Vladimir BALAN

Josef D ORFMEISTER

Department Mathematics I Politehnica University of Bucharest Splaiul Independenctei 313 RO-060042 Bucharest Romania

Zentrum Mathematik M8 Munich University of Technology Boltzmannstrasse 3 D-85747 Garching Germany