Willmore surfaces in spheres via loop groups III: on minimal surfaces in space forms
aa r X i v : . [ m a t h . DG ] A ug WILLMORE SURFACES IN SPHERES VIA LOOP GROUPS III: ONMINIMAL SURFACES IN SPACE FORMS
PENG WANG
Abstract
Abstract.
The family of Willmore immersions from a Riemann surface into S n +2 canbe divided naturally into the subfamily of Willmore surfaces conformally equivalent toa minimal surface in R n +2 and those which are not conformally equivalent to a mini-mal surface in R n +2 . On the level of their conformal Gauss maps into Gr , ( R ,n +3 ) = SO + (1 , n + 3) /SO + (1 , × SO ( n ) these two classes of Willmore immersions into S n +2 correspond to conformally harmonic maps for which every image point, considered as a4-dimensional Lorentzian subspace of R ,n +3 , contains a fixed lightlike vector or whereit does not contain such a “constant lightlike vector”. Using the loop group formal-ism for the construction of Willmore immersions we characterize in this paper preciselythose normalized potentials which correspond to conformally harmonic maps contain-ing a lightlike vector. Since the special form of these potentials can easily be avoided,we also precisely characterize those potentials which produce Willmore immersions into S n +2 which are not conformal to a minimal surface in R n +2 . It turns out that our proofalso works analogously for minimal immersions into the other space forms. Keywords:
Willmore surfaces, normalized potential, minimal surfaces, Iwasawa de-compositions.
Mathematics Subject Classification . Primary 53A30; Secondary 58E20; 53C43; 53C351.
Introduction
This is the third paper of a series of papers concerning the global geometry of Willmoresurfaces in terms of loop group theory. Our aim is to derive a criterion characterizing thenormalized potentials of all strongly conformally harmonic maps which either correspondto minimal surfaces in R n +2 or to no Willmore surfaces at all. On the one hand, thisprovides a characterization of minimal surfaces in R n +2 , as well as this special class ofstrongly conformally harmonic maps. On the other hand, it also allows us to deriveWillmore surfaces different from minimal surfaces in R n +2 by excluding this special typeof normalized potentials. This is particularly important for the application of the mainresults of [14] to generic Willmore surfaces in S n +2 .It is well-known that minimal surfaces in Riemannian space forms provide standardexamples of Willmore surfaces [3], [4], [28]. To pick up all minimal surfaces in space formsamong Willmore surfaces, we provide a description of them via potentials. To be concrete, let F be a harmonic map from a Riemann surface M into Gr , ( R ,n +3 )(= SO + (1 , n + 3) /SO + (1 , × SO ( n )), with a lift F : M → SO + (1 , n + 3) and M-C form α = F − dF = (cid:18) A B − B t I , A (cid:19) dz + (cid:18) ¯ A ¯ B − ¯ B t I , ¯ A (cid:19) d ¯ z. Here A ∈ M at (4 × , C ) , A ∈ M at ( n × n, C ) , B ∈ M at (4 × n, C ) , I , = diag( − , , , . F is called strongly conformal if B satisfies B t I , B = 0. Note that this condition isindependent of the choice of F [14]. We also recall that the conformal Gauss map of aWillmore surface is a strongly conformally harmonic map [14]. Conversely, by Theorem3.10 of [14], there are two different kinds of strongly conformally harmonic maps: Those which contain a constant lightlike vector and those which do not contain a con-stant lightlike vector.
Moreover from Theorem 3.10 of [14], we see that if a strongly conformally harmonic map F does not contain a lightlike vector, f will always be the conformal Gauss map of someWillmore map. And, most importantly, these Willmore maps correspond exactly to allthe Willmore maps which are not M¨obius equivalent to any minimal surface in R n +2 ,since minimal surfaces in R n +2 can be characterized as those Willmore surfaces whoseconformal Gauss map contains a constant lightlike vector [19], [30], [21] (See Lemma 1.2below, see also [3], [17], [23], [22] and [5]). Since minimal surfaces in R n +2 can be con-structed by a straightforward way, one will be mainly interested in Willmore surfaces not M¨obius equivalent to minimal surfaces in R n +2 . It is therefore vital to derive a criteriondetermining whether a strongly conformally harmonic map f contains a lightlike vectoror not. Note that this will also yield an interesting description of minimal surfaces in R n +2 . Applying Wu’s formula, one will obtain the following description of the normalizedpotential of F when it contains a constant light-like vector. Theorem 1.1.
Let D denote the Riemann surface S , C or the unit disk of C . Let F : D → SO + (1 , n + 3) /SO + (1 , × SO ( n ) be a strongly conformally harmonic mapwhich contains a constant light-like vector. Assume that f ( p ) = I n +4 · K w.r.t some basepoint p ∈ D and z is a local coordinate with z ( p ) = 0 . Then the normalized potential of f with reference point p is of the form (1.1) η = λ − (cid:18) B − ˆ B t I , (cid:19) dz, where ˆ B = ˆ f ˆ f . . . ˆ f n − ˆ f − ˆ f . . . − ˆ f n ˆ f ˆ f . . . ˆ f n i ˆ f i ˆ f . . . i ˆ f n . Here the functions f ij are meromorphic functions on D .Moreover, F is of finite uniton type with maximal uniton number ≤ . It is well-known that minimal surfaces in Riemannian space forms can be characterizedby the following lemma (The statements and proofs can be found in [19], see also [21] fora proof of Case (1)).
INIMAL SURFACES IN R n VIA POTENTIALS 3
Lemma 1.2. [19] , [30] Let y : M → S n +2 be a Willmore surface, with F as its conformalGauss map. We say that F contains a constant vector a ∈ R ,n +3 if for any p ∈ M , a isin the − dim Lorentzian subspace F ( p ) . Then (1) y is M¨obius equivalent to a minimal surface in R n +2 if and only if F contains anon-zero constant lightlike vector. (2) y is M¨obius equivalent to a minimal surface in some S n +2 ( c ) if and only if F contains a non-zero constant timelike vector. (3) y is M¨obius equivalent to a minimal surface in H n +2 ( c ) if and only if F containsa non-zero constant spacelike vector. Applying this lemma and Wu’s formula, one obtains the following descriptions of min-imal surfaces in space forms.
Theorem 1.3.
Let F : M → SO + (1 , n + 3) /SO (1 , × SO ( n ) be a strongly conformallyharmonic map. Let (1.2) η = λ − (cid:18) B − ˆ B t I , (cid:19) dz, ˆ B = (cid:0) v v . . . v n (cid:1) , be the normalized potential of F with respect to some base point z . Then, up to a conju-gation by some T ∈ O + (1 , × O ( n ) , (1) F contains a constant lightlike vector, if and only if every v j has the form (1.3) v j = f j (cid:0) f j − f j f j if j (cid:1) t , with f jl meromorphic . (2) F contains a constant timelike vector, if and only if every v j has the form (1.4) v j = g j (cid:0) g − g i (1 + g ) (cid:1) t , with g j , g meromorphic . (3) F contains a constant spacelike vector, if and only if every v j has the form (1.5) v j = h j (cid:0) ih − h i (1 + h ) (cid:1) t , with h j , h meromorphic . Potentials of above form will be called canonical potentials for the corresponding minimalsurfaces in space forms.
The proof of Case (1) of this is more elaborate than the proof for the other cases. It alsohas a more general background. The classical theorem of Bryant [3] tells us that everyWillmore two-sphere in S is M¨obius equivalent to some minimal surface with embeddedplaner ends in R . However, when the co-dimension increases, this does no longer hold trueand there are Willmore two-spheres in S different from minimal surfaces in R [17]. In[27], by using the loop group methods developed in [14], [15], we provide a classification ofWillmore two-spheres in S n +2 via the normalized potentials of their harmonic conformalGauss maps. The basic idea is to combine the work of Burstall and Guest [6], [16] withthe DPW method [15] and to characterize harmonic conformal Gauss maps of Willmoresurfaces by describing their normalized potentials. We would like to point out that thesepotentials take values in nilpotent Lie sub-algebras by the results of [6] and [15]. In [27]it was shown that there are m − n + 4 = 2 m . The possible forms of normalized potentials are also listed explicitly. PENG WANG
To derive concrete expressions for Willmore surfaces, and to understand their geometricproperties one needs to perform Iwasawa decompositions of the meromorphic frames ofthe given normalized potentials. Since the potentials take values in some nilpotent Liesub-algebra, the meromorphic frames, i.e., ODE solutions with the normalized potentialsas coefficient matrices, are Laurant polynomials in λ ∈ S [6], [16], [15]. A theoreticalprocedure for the Iwasawa decompositions of such algebraic elements in a loop grouphas been presented in [8]. However, for the first type of normalized potentials in theclassification theorem of [27], the Iwasawa decomposition can be obtained in an easierand more straightforward way.To be concrete, assume that the normalized potential is of the form (See Section 3 of[14], Section 2 of [27] for the definitions and notations)(1.6) η = λ − η − dz with(1.7) η − = (cid:18) B − ˆ B t I , (cid:19) , and ˆ B = f f . . . f m − , f m − , f f . . . f m − , f m − , f f . . . f m − , f m − , if if . . . if m − , if m − , , where f ij are meromorphic functions on the Riemann surface ˜ M . Note that this η isconjugate to the one in (1.1) by ˜ T = diag(1 , − , , . . . , Theorem 1.4.
Let F : D → SO + (1 , m − /SO + (1 , × SO (2 m − be a stronglyconfomally harmonic map with its normalized potential being of the form in (1.7) . Then F contains a constant light-like vector.Moreover, if F is the conformal Gauss map of a strong Willmore map y : D → S m − ,then rank ( ˆ B ) ≤ and y is M¨obius equivalent to a minimal surface in R m − . It is easy to verify that such F is of finite uniton type. Moreover, F actually belongsto one of the simplest cases, called S − invariant (See [6], [11], [27]). For such harmonicmaps, by using a straightforward and lengthy computation, one can derive the harmonicmap explicitly and then read off all needed information, which will provide a proof ofTheorem 1.4. Corollary 1.5.
Let F : D → SO + (1 , n + 3) /SO + (1 , × SO ( n ) be a strongly confor-mally harmonic map with its normalized potential η of the form (1.1) and of maximal rank ( ˆ B ) = 2 . Then F can not be the conformal Gauss map of a Willmore surface. Inparticular, there exist conformally harmonic maps which are not related to any Willmoremap. As a consequence, at this point we have obtained a complete description of the stronglyconformally harmonic maps which produce either minimal surfaces in R n +2 or no Willmoresurfaces at all, which make our theory workable for the study of non-Euclidean-minimal INIMAL SURFACES IN R n VIA POTENTIALS 5
Willmore surfaces. This corollary shows that the characterization theorems here do makesense for the theory in [14] to deal with global Willmore surfaces different from minimalsurfaces in R n +2 .This paper is organized as follows. Section 2 provides the form of the potentials ofstrongly conformally harmonic maps containing a constant lightlike vector. The proofs ofCases (2) and (3) of Theorem 1.3 are also derived in this section. Section 3 contains thecharacterizations of minimal surfaces in R n +2 in terms of potentials and several technicallemmas providing a proof of our main theorem. The proofs of these technical lemmas arederived in Section 4.For simplicity we will, in this paper, always retain the notation of [14] and [27]. Formore details we also refer to [14] and [27].2. Potentials of strongly conformally harmonic maps containing aconstant lightlike vector
This section is to derive the forms of the normalized potentials of strongly confor-mally harmonic maps containing a non-zero constant real vector. The basic idea isto characterize the Maurer-Cartan form of such a strongly conformally harmonic map F : D → SO + (1 , n + 3) /SO + (1 , × SO ( n ).2.1. Proof of Theorem 1.1.
The proof of Theorem 1.1 relies on the following technicallemma.
Lemma 2.1.
Let A = a a − a − a a a a a a − a be a holomorphic matrix function on a contractible open Riemann surface U . Let F bea solution to the equation F − dF = A dz, F | z =0 = I . Then F = ( b + b ) ( b + b ) b b − ( b + b ) 1 − ( b + b ) − b − b b b b b ϕ sin ϕ − sin ϕ cos ϕ with ϕ = Z z a dw and b = Z z ( a cos ϕ + a sin ϕ ) dz, b = Z z ( − a sin ϕ + a cos ϕ ) dz. PENG WANG
Proof.
Set ˜ F = ϕ sin ϕ − sin ϕ cos ϕ , and ˆ F = ( b + b ) ( b + b ) b b − ( b + b ) 1 − ( b + b ) − b − b b b b b . Straightforward computations yield˜ F − d ˜ F = a − a dz, and ˆ F − d ˆ F = b ′ b ′ − b ′ − b ′ b ′ b ′ b ′ b ′ dz, with b ′ = a cos ϕ + a sin ϕ, b ′ = − a sin ϕ + a cos ϕ. Moreover, one obtains˜ F − (cid:16) ˆ F − d ˆ F (cid:17) ˜ F = a a − a − a a a a a . Since F = ˆ F ˜ F , one derives F − F z = ˜ F − (cid:16) ˆ F − d ˆ F (cid:17) ˜ F + ˜ F − ˜ F z = A . (cid:3) Proof of Theorem 1.1:
Let F ( z, ¯ z, λ ) = ( φ , φ , φ , φ , ψ , . . . , ψ n ) be a frame of f with the initial condition F (0 , , λ ) = I n +4 . W.l.g, we may assume that Y = φ − φ is the constant lightlike vector contained in f . As a consequence, we derive φ z = φ z = a φ + a φ + √ n X j =1 β j ψ j . INIMAL SURFACES IN R n VIA POTENTIALS 7
That is ( φ z , φ z ) t = (cid:18) a a √ β . . . √ β n a a √ β . . . √ β n (cid:19) · F t . Comparing with F − F z = (cid:18) A B − B t I , A (cid:19) , we obtain A = a a − a − a a a a a a − a , and B = √ β . . . √ β n −√ β . . . −√ β n − k . . . − k n − ˆ k . . . − ˆ k n . Since B t I , B = 0,ˆ k = ik , . . . , ˆ k n = ik n , or ˆ k = − ik , . . . , ˆ k n = − ik n . Similar to the discussion in Lemma 3.8 of Section 3 of [14], without loss of generality, weassume that on ˜ M , ˆ k = ik , . . . , ˆ k n = ik n .For the computation of the normalized potential, we will apply Wu’s formula (Theorem4.23, Section 4.3 of [14], see also [29]). Let δ = (˜ a ij ) denote the “holomorphic part” of A with respect to the base point z = 0, i.e., the part of the Taylor expansion of A whichis independent of ¯ z . Let F be a solution to the equation(2.1) F − dF = δ dz, F | z =0 = I . By Lemma 2.1, F is equal to ( b + ˆ a ) ( b + ˆ a ) b cos ϕ − b sin ϕ b sin ϕ + b cos ϕ − ( b + b ) 1 − ( b + b ) − b cos ϕ + b sin ϕ − ( b sin ϕ + b cos ϕ ) b b cos ϕ sin ϕb b − sin ϕ cos ϕ with ϕ = Z z ˜ a dz and b = Z z (˜ a cos ϕ + ˜ a sin ϕ ) dz, b = Z z ( − ˜ a sin ϕ + ˜ a cos ϕ ) dz. Let δ denote the “holomorphic part” of A , with respect to the base point z = 0, and let F be a solution to the equation F − dF = δ dz, F | z =0 = I n . PENG WANG
Let ˜ B denote the holomorphic part of B . By Wu’s formula (Theorem 4.23 of [14]), thenormalized potential can be represented in the form η = λ − (cid:18) F F (cid:19) (cid:18) B − ˜ B t I , (cid:19) (cid:18) F F (cid:19) − dz = λ − (cid:18) B − ˆ B t I , (cid:19) dz, with ˆ B = F ˜ B F − = F · ˜ f ˜ f . . . ˜ f n − ˜ f − ˜ f . . . − ˜ f n − ˜ f − ˜ f . . . − ˜ f n − i ˜ f − i ˜ f . . . − i ˜ f n · F − = ˆ f ˆ f . . . ˆ f n − ˆ f − ˆ f . . . − ˆ f n − ˆ f − ˆ f . . . − ˆ f n − i ˆ f − i ˆ f . . . − i ˆ f n . The statement that F is of finite uniton type with maximal uniton number ≤ ✷ Proof of Theorem 1.3.
Proof of Theorem 1.3:
Case (1) comes from Theorem 1.1 and Theorem 1.4.Now we consider Case (2). Since F contains a constant timelike vector e . We canassume | e | = 1 and it is time forward. Then there exists a transformation T ∈ SO (1 , n +3) transforming e into (1 , , . . . , t and transforming F into T F . So without loss ofgenerality, we assume e = (1 , , . . . , t . Let F = ( e , ˆ e , e , e , ψ , . . . , ψ n ) be a lift of F .As a consequence, every entry of the first column and the first row of α = F − dF is zero.The same holds for F λ when introducing the loop parameter into F . Then using Wu’sformula [29] (see also Theorem 4.23, and Theorem 4.24 of [14] for the Willmore case), wesee that every entry of the first column and the first row of the normalized potential stayszero. Moreover, by Theorem 4.24, ˆ B also satisfies ˆ B t I , ˆ B = 0, which yields v tj I , v l = 0 , for all j, l = 1 , . . . , n. Formula (1.4) follows by a simple computation similar to the one in deriving the Weier-strass representation of minimal surfaces in R .The converse part is straightforward. In fact, integrating η , we see that all the entriesof the first column and the first row of F − are 0, except the (1 , F inherits the same property,that is, the harmonic map F contains e = (1 , , . . . , t at every point.Case (3) follows by a similar argument. INIMAL SURFACES IN R n VIA POTENTIALS 9 ✷ Minimal surfaces in R n as Willmore surfaces This and the next section aim to give by direct computations, a concrete description ofall Willmore surfaces corresponding to the first type of nilpotent Lie subalgebras in [27].To this end, since there are many lengthy and elementary computations, we will dividethe proof into several technical lemmas, which will be stated in this section. The proofsof these lemmas will be left to Section 4.The basic idea in our computations is to express the normalized potentials by somestrictly upper triangular matrix-valued 1-forms, since this will simply the computationssubstantially. For this purpose, we need to transform the original group into a differentone such that the matrix coefficients of the potentials in (1.7) will be upper triangularmatrices. So we will first recall the Lie group isometry in Section 3.1. Then we state fivetechnical lemmas in Section 3.2.3.1.
Preliminary.
To begin with, we first recall some basic notations and results. Thedetailed descriptions and proofs can be found in Section 3 of [27]. We will retain thenotation of [27].Recall that SO + (1 , n + 3) = SO (1 , n + 3) is the connected subgroup of SO (1 , n + 3) := { A ∈ M at ( n + 4 , R ) | A t I ,n +3 A = I ,n +3 , det A = 1 } , with I ,n +3 = diag {− , , . . . , } . The subgroup K = SO + (1 , × SO ( n ) is defined by the involution(3.1) σ : SO + (1 , n + 3) → SO + (1 , n + 3) A DAD − , where D = diag {− I , I n } . For simplicity, we assume that n is even and n + 4 = 2 m . Wealso have(3.2) G ( n + 4 , C ) := { A ∈ M at ( n + 4 , C ) | A t J n +4 A = J n +4 , det A = 1 } , with J n +4 = ( j k,l ) ( n +4) × ( n +4) , j k,l = δ k + l,n +5 for all 1 ≤ k, l ≤ n + 4 . By Lemma 3.1 of [27], one obtains a Lie group isometry from SO + (1 , m − , C ) into G (2 m, C ), defined by the following map(3.3) P : SO + (1 , m − , C ) → G (2 m, C ) A ˜ P − A ˜ P with(3.4) ˜ P = 1 √ −
11 1 − i i − i i . Under this isometry, we have that P ( SO + (1 , m − { F ∈ G (2 m, C ) | F = S − m ¯ F S m } containing I m . Here(3.5) S m = J m − . Moreover, this induces an involution of the loop group Λ G (2 m, C ):(3.6) ˆ τ : Λ G (2 m, C ) → Λ G (2 m, C ) F S − m ¯ F S m with P (Λ SO + (1 , m − { F ∈ Λ G (2 m, C ) | ˆ τ ( F ) = F } as its fixed point set. We alsohave that the image of SO + (1 , × SO (2 m −
4) under P is of the form(3.7) P (cid:0) ( SO + (1 , × SO (2 m − C (cid:1) = { F ∈ G (2 m, C ) | F = D − F D } with(3.8) D = ˜ P − D ˜ P = diag {− , − , I m − , − , − } = − − I m − − − . Technical Lemmas.
With the notations as above, we are able to state the followinglemmas:
Lemma 3.1.
Let η ∈ Λ − so (1 , n +3) σ be the normalized potential defined on D in Theorem1.4. Then P ( η ) = λ − f
00 0 − ˜ f ♯ dz, with ˜ f ∈ M at (2 × (2 m − , C ) , ˜ f ♯ := J m − ˜ f t J . Lemma 3.2.
Let η be as in Lemma 3.1. Then H = I m + λ − H + λ − H is the solutionto (3.9) H − dH = P ( η ) , H | z =0 = I m INIMAL SURFACES IN R n VIA POTENTIALS 11 with (3.10) H = f
00 0 − f ♯ , H = g , (3.11) f = Z z ˜ f dz, g = − Z z ( f ˜ f ♯ ) dz. Note that if η is derived from some strong Willmore map, then η is meromorphic andalso H , the integration of P ( η ), is meromorphic. If in Lemma 3.1 we start from somenormalized potential and want to construct a strong Willmore map defined on D , thenwe need to assume that η is meromorphic and also that H is meromorphic. Lemma 3.3.
Retaining the assumptions and the notation of the previous lemmas, assumethat P ( η ) is the normalized potential of some harmonic map, we obtain:The Iwasawa decomposition of H is H = ˜ F ˜ F + , with ˜ F ∈ P (Λ SO + (1 , m − σ ) ⊂ Λ G (2 m, C ) σ , ˜ F + ∈ Λ + G (2 m, C ) σ . And ˜ F is given by (see also (3.7) of [27] ) (3.12) ˜ F = H ˆ τ ( W ) L − . Here W , W and L are the solutions to the matrix equations ˆ τ ( H ) − H = W W ˆ τ ( W ) − , W = ˆ τ ( L ) − L with W = I m + λ − W + λ − W , W = u
00 0 − u ♯ , W = g , and W = a b q
00 0 d , L = l l l
00 0 l , with l , l upper triangular . Moreover, we have d = I + E ¯ f t♯ f ♯ + E ¯ g t E g, (3.13a) u ♯ d = f ♯ − ¯ f t E g, (3.13b) q + u ♯ dE ¯ u ♯t = I m − + ¯ f t E f, (3.13c) a + uq ¯ u t E + gE ¯ˆ d t ¯ g t E = I , (3.13d) b + uq ¯ u t E + gE ¯ˆ d t ¯ g t E = E ¯ f t♯ f ♯ + E ¯ g t E g, (3.13e) uq − gE ¯ u ♯t = f. (3.13f) Here E , E , E and E are defined as (3.14) E = (cid:18) (cid:19) , E = E t = (cid:18) (cid:19) , E = (cid:18) (cid:19) . Remark .
1. Since in Lemma 3.3 the matrices f and g , whence also f ♯ , are given, equation (3.13a)determines d , where d is invertible (certainly true for small z close to z = 0). Thenequation (3.13b) determines u ♯ , hence u . Inserting this into (3.13c) results in determining q . Inserting what we have so far into (3.13d) determines a and similarly from (3.13e) weobtain b . The last equation, (3.13f), is a consequence of the previous equations. Therefore,the only condition for the solvability of the system of equations is the invertibility of d .2. If f and g are rational functions of z , the invertibility of d is satisfied locally, whenceon an open dense subset due to the rational expression in z, ¯ z . Lemma 3.5.
Retaining the assumptions and the notation of the previous lemmas, theMaurer-Cartan form of ˜ F in (3.12) is of the form (3.15) ˜ α ′ p = λ − b
00 0 − ˜ b ♯ dz, ˜ α ′ k = a a a
00 0 a dz. Lemma 3.6.
Let F : M → SO + (1 , m − /SO + (1 , × SO (2 m − be a strongly confor-mally harmonic map with an extended frame F . If the Maurer-Cartan form of ˜ F = P ( F ) is of the form (3.15) in g (2 m, C ) , then F contains a constant light-like vector. Therefore,if F is the conformal Gauss map of some Willmore map y , y is M¨obius equivalent to aminimal surface in R m . Lemma 3.1 and Lemma 3.2 can be verified by straightforward computations. And theother lemmas will be proven in the following section.
Proof of Theorem 1.4:
Combination of the above five lemmas provides the proof of Theorem 1.4. ✷ Note that Corollary 1.5 already follows from the above three lemmas. The fact that η − takes values in a nilpotent Lie subalgebra of rank 2 comes from Theorem 2.6 and Lemma3.5 of [27]. Remark .
1. For a general procedure for the computation of Iwasawa decompositions for algebraicloops, or more generally for rational loops, see § I. SU (1 ,
1) can be found in [2].3. Recently there have been several publications concerning harmonic maps into com-pact Lie groups and compact symmetric spaces which have used methods which are differ-ent from ours, see [18], [9], [25] and reference therein. Most of their work basically follows
INIMAL SURFACES IN R n VIA POTENTIALS 13 the techniques developed by Uhlenbeck [26] and Segal [24]. We also note that in [25], aconverse procedure is used for the computations for the Iwasawa decompositions of loopsin λ alg U ( n ) C , which provides another way to do the concrete Iwasawa decomposition foralgebraic loops. 4. Iwasawa decompositions
In this section we first provide the proof of Lemma 3.3, which corresponds to the Iwa-sawa decompositions. Then, we derive the M-C forms by the information from the explicitIwasawa decompositions, which yields the geometric descriptions of the correspondingharmonic maps.4.1.
Iwasawa decompositions and Lemma 3.3.
Proof of Lemma 3.3:
The first question is the existence of the Iwasawa decomposition for H . This is guar-anteed by the existence of an Iwasawa decomposition on an open subset containing theidentity and H | z =0 = I (see Theorem 4.1 of [14], Theorem 2.3 of [15], also [20]). So cer-tainly near z = 0 we have H = ˜ F ˜ F + . Since ˆ τ ( ˜ F ) = ˜ F , the maximal and minimal powersof λ of ˜ F are λ and λ − respectively. Hence in ˜ F + the maximal power of λ is at most 2.Moreover, from the definition of G ( n + 4 , C ) we infer that also ( ˜ F + ) − only contains thepowers of λ from − I n +4 = ˆ τ ( ˜ F ) − ˜ F = (cid:16) ˆ τ ( ˜ F − ) (cid:17) − ˆ τ ( H ) − H ˜ F + − implies that ˆ τ ( H ) − H = ˆ τ ( ˜ F − ) ˜ F + . Let’s write ˆ τ ( ˜ F − ) = W ˆ τ ( L ) − with W = I m + λ − W + λ − W , where W = u − v ♯ − u ♯ v . Set W = ˆ τ ( L ) − L with W = a b q c d , W − = ˆ a b q − c d , and L = l l l
00 0 l . With these notations, we obtainˆ τ ( H ) − H = W W ˆ τ ( W ) − . For explicit computations we recall from (3.6) (See also (3.7) of [27]) that for any F ∈ G (2 m, C ) we have(4.1) ˆ τ ( F ) − = ˆ J m ¯ F t ˆ J m , where(4.2) ˆ J m = S m J m = E E I m − E E . Recall that (3.14) E = (cid:18) (cid:19) , E = E t = (cid:18) (cid:19) , E = (cid:18) (cid:19) . As a consequence, we derive(4.3) W W = H ,W W + W W ( ˆ J m ¯ W t ˆ J m ) = H + ( ˆ J m ¯ H t ˆ J m ) H , c W = I m + ( ˆ J m ¯ H t ˆ J m ) H + ( ˆ J m ¯ H t ˆ J m ) H . Here c W = W + W W ( ˆ J m ¯ W t ˆ J m ) + W W ( ˆ J m ¯ W t ˆ J m ) . The first equation of (4.3) yields W = H W − = g ˆ c g ˆ d . Next we evaluate the second matrix equation of (4.3). We compute H + ( ˆ J m ¯ H t ˆ J m ) H = f
00 0 − f ♯ + ¯ f t E g , ˆ J m ¯ W t ˆ J m = − E ¯ v ♯t − E ¯ u ♯t u t E + ¯ v t E u t E + ¯ v t E − E ¯ v ♯t − E ¯ u ♯t ,H ( ˆ J m ¯ W t ˆ J m ) = − gE ¯ v ♯t − gE ¯ u ♯t
00 0 00 0 0 ,W W = uq − v ♯ a − u ♯ c − v ♯ b − u ♯ d vq . As a consequence we read off the equations vq = 0 , − v ♯ a − u ♯ c = 0 , uq − gE ¯ v ♯t − gE ¯ u ♯t = f, − v ♯ b − u ♯ d = − f ♯ + ¯ f t E g. INIMAL SURFACES IN R n VIA POTENTIALS 15
Since q is invertible, v = 0. Therefore these equations reduce to the following ones: v = 0 , u ♯ c = 0 , uq − gE ¯ u ♯t = f, u ♯ d = f ♯ − ¯ f t E g. Similarly, for the third equation of (4.3), we first compute matrix expressions for thesummands: ( ˆ J ¯ H t ˆ J ) H = E ¯ f t♯ f ♯ f t E f
00 0 E ¯ f t♯ f ♯ , ( ˆ J ¯ H t ˆ J ) H = E ¯ g t E g E ¯ g t E g ,W W ( ˆ J ¯ W t ˆ J ) = uq ¯ u t E uq ¯ u t E u ♯ dE ¯ u ♯t
00 0 0 , ˆ J ¯ W t ˆ J = E ¯ˆ c t ¯ g t E + E ¯ˆ d t ¯ g t E E ¯ˆ c t ¯ g t E + E ¯ˆ d t ¯ g t E E ¯ˆ c t ¯ g t E + E ¯ˆ d t ¯ g t E E ¯ˆ c t ¯ g t E + E ¯ˆ d t ¯ g t E ,W W ( ˆ J m ¯ W t ˆ J ) = gE ¯ˆ c t ¯ g t E + gE ¯ˆ d t ¯ g t E gE ¯ˆ c t ¯ g t E + gE ¯ˆ d t ¯ g t E . Substituting these expressions into the third matrix equation of (4.3), we derive that c = 0 . Therefore we have c = 0 , a + uq ¯ u t E + gE ¯ˆ d t ¯ g t E = I ,b + uq ¯ u t E + gE ¯ˆ d t ¯ g t E = E ¯ f t♯ f ♯ + E ¯ g t E g,q + u ♯ dE ¯ u ♯t = I m − + ¯ f t E f, d = I + E ¯ f t♯ f ♯ + E ¯ g t E g. Summing up, we obtain (3.13).The only thing left to prove Lemma 3.3 is the statement of the form of L . To derivethis, we first consider W . Recall that W = W − ˆ τ ( H ) − H ˆ τ ( W ) = ˆ τ ( W ) − = ˆ J m ¯ W t ˆ J m , and W ∈ G (2 m, C ). Hence, in particular, we also have W t J m W = J m . A direct computation using these equations shows˜ W = (cid:18) a bc d (cid:19) = a a a a a a a ¯ a a , with | a | = 1 , a a = 1 , ¯ a = − a ¯ a a , ¯ a = | a | a . Set ˜ l = (cid:18) l l l (cid:19) = l l l l l l l l l , with l ij satisfying a = l , a = l l + ¯ l l , a = | l | , and l = − l l l , l = − l l l , l = 1 l , l = − l l l , l = 1 l . Moreover, let l be a solution to q = ¯ l l . Applying (4.1), it is a straightforward computation to verify thatˆ τ ( L ) − L = ˆ J m ¯ L t ˆ J m L = W . This finishes the proof of Lemma 3.3. ✷ Remark . In the proof above the splitting q = ¯ l l implies a strong restriction on q .This condition will always be satisfied near the identity. While in general it may happenthat one needs some middle term in the splitting q = ¯ l l due to the non-globality ofthe Iwasawa splitting in our case. Actually the situation is much more complicated (fora similar situation, see [2]). Since we have two open Iwasawa cells by Section 6 [14],Theorem 6.7, it may happen that q starts at I in the first open Iwasawa cell I for some z and moves forward to the boundary between the two open Iwasawa cells I and I . Itcould touch the boundary and return to I or it could cross into I . What this meansgeometrically has been investigated only to a very small extent so far, but would seem tobe a highly interesting project. But there are certainly cases where everything works justfine. See the example below. Example 4.2.
Let 2 m = 6 and assume f = (cid:18) f f f f (cid:19) , g = (cid:18) g g g g (cid:19) , with f j , g j , j = 1 , . . . ,
4, meromorphic functions. It is easy to derive that g + g = − f f − f f , g = − f f , g = − f f holds. From the last equation in (3.13a), and from¯ f t♯ f ♯ = J ¯ f f t J = (cid:18) | f | + | f | ¯ f f + ¯ f f ¯ f f + ¯ f f | f | + | f | (cid:19) , and ¯ g t E g = (cid:18) | g | ¯ g g ¯ g g | g | (cid:19) , INIMAL SURFACES IN R n VIA POTENTIALS 17 we obtain d = (cid:18) (1 + | f | )(1 + | f | ) ¯ f f + ¯ f f (cid:19) = (cid:18) d d (cid:19) . Therefore a = 1 , a − = (1 + | f | )(1 + | f | ) , a = − f ¯ f + f ¯ f + ¯ g g | f | + | f | + | g | . Moreover, we have d − = (cid:18) d − − d − d (cid:19) . From (3.13b) we obtain u ♯ = (cid:18) f − ¯ f g f − ¯ f g f − ¯ f g f − ¯ f g (cid:19) · d − = (cid:18) u u u u (cid:19) with u = f − f f ¯ f − ¯ f g | f | , u = f − f ¯ f f − ¯ f g | f | , u = f | f | , u = f | f | . Moreover, since ¯ f t E f = (cid:18) ¯ f f ¯ f f ¯ f f ¯ f f (cid:19) ,u ♯ dE ¯ u ♯t = (cid:18) | u | d u ¯ u d u ¯ u d | u | d (cid:19) = | f | (1+ | f | )1+ | f | ¯ f f ¯ f f | f | (1+ | f | )1+ | f | ! , from (3.13c), we have q = | f | | f | | f | | f | ! . Clearly, such expression can be written in the form q = ¯ l t l , and the above computationsprovides a global solution of equation (3.13). This shows that for the case 2 m = 6 thereexists a global Iwasawa splitting for harmonic maps with normalized potential of the formin Theorem 1.4.4.2. The Maurer-Cartan form of ˜ F . Proof of Lemma 3.5:
To take a look at the Maurer-Cartan form of ˜ F , we first recall that H − H z = λ − η − = λ − f
00 0 − ˜ f ♯ . Since ˜ F − ˜ F z = λ − L ˆ τ ( W ) − η − ˆ τ ( W ) L − + L ˆ τ ( W ) − ˆ τ ( W ) z L − + L ( L − ) z , and ˆ τ ( W ) = I m + λ ˆ τ ( W ) + λ ˆ τ ( W ) , we may assume that˜ F − ˜ F z dz = λ − ˜ α − + ˜ α + λ ˜ α + λ ˜ α + λ ˜ α + λ ˜ α , ˜ α − = L η − L − dz. On the other hand, the reality condition yieldsˆ τ ( ˜ F − d ˜ F ) = ˜ F − d ˜ F , and ˜ F − ˜ F z dz = λ − ˜ α − + ˜ α + λ ˜ α with ˜ α = ˆ τ ( ˜ α − ) , and ˜ α = ˆ τ ( ˜ α ) . Moreover, a straightforward computation yields˜ α ′ = L [ η − , ˆ τ ( W )] L − + L ( L − ) z dz. Since L is an upper triangular block matrix, ˜ α ′ is of the desired form stated in (3.15). ✷ Proof of Lemma 3.6:
By Lemma 3.5, there exists a frame ˜ F such that ˜ α ′ is of the form stated in (3.15). By(3.3), (3.5) and (3.7), we obtain α ′ = P − ( ˜ α ′ ) = (cid:18) A λ − B − λ − B t I , A (cid:19) dz, with A = a a a a a a a − a a a − a − a , and B = h ˆ h h ˆ h . . . h ,m − ˆ h ,m − h ˆ h h ˆ h . . . h ,m − ˆ h ,m − h ˆ h h ˆ h . . . h ,m − ˆ h ,m − ih i ˆ h ih i ˆ h . . . ih ,m − i ˆ h ,m − . Set F = P − ( ˜ F ) = ( φ , φ , φ , φ , ψ , . . . , ψ m − ) . We have now (cid:26) φ z = a φ + a φ + a φ + h ψ + . . . + ˆ h ,m − ψ m − ,φ z = a φ − a φ − a φ − h ψ − . . . − ˆ h ,m − ψ m − . This indicates ( φ + φ ) z = a ( φ + φ ) . Since φ + φ is a non-zero real vector-valued function, let φ be the first coordinate of φ + φ , it is straightforward to verify that φ ( φ + φ ) is a non-zero constant lightlike INIMAL SURFACES IN R n VIA POTENTIALS 19 vector. As a consequence, a well-known fact states that if F is the conformal Gauss mapof a Willmore surface y , y is M¨obius equivalent to some minimal surface in R m − [19]. ✷ Acknowledgements : The author is thankful to Professor Josef. Dorfmeister, Profes-sor Changping Wang and Professor Xiang Ma for their suggestions and encouragement.This work is supported by the Project 11201340 of NSFC.
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