Willmore Surfaces in Spheres via Loop Groups IV: On Totally Isotropic Willmore Two-Spheres in S6*

Peng WANG

Abstract In this paper the author derives a geometric characterization of totally isotropic Willmore two-spheres in S6， which also yields to a description of such surfaces in terms of the loop group language.Moreover， applying the loop group method， he also obtains an algorithm to construct totally isotropic Willmore two-spheres in S6.This allows him to derive new examples of geometric interests.He first obtains a new， totally isotropic Willmore two-sphere which is not S-Willmore (i.e.， has no dual surface) in S6.This gives a negative answer to an open problem of Ejiri in 1988.In this way he also derives many new totally isotropic， branched Willmore two-spheres which are not S-Willmore in S6.

Keywords Totally isotropic Willmore two-spheres， Normalized potential， Iwasawa decompositions

1 Introduction

Totally isotropic surfaces first appeared in the study of the global geometry of surfaces in the famous work of Calabi [11]， where twistor bundle theory was applied to describe the geometry of minimal two-spheres in Sn.This led later to much progress in geometry and the theory of integrable systems (see for example [4， 8， 10]).In the study of Willmore twospheres， totally isotropic surfaces play an important role as well.First we note that isotropic properties are conformally invariant.This indicates that they are of interest in the conformal geometry of surfaces.Moreover， the classical work of Ejiri [20] shows that isotropic surfaces in S4are automatically Willmore surfaces and furthermore they are Willmore surfaces with dual surfaces.He also showed that Willmore two-spheres in S4are either M¨obius equivalent to minimal surfaces with planer ends in R4，or isotropic two-spheres(see[20])(see also[7，28–29]).

In [20]， Ejiri also introduced the notion of S-Willmore surfaces.Roughly speaking， these surfaces can be viewed as Willmore surfaces admitting dual surfaces.Note that by Bryant’s classical work， every Willmore surface in S3has a dual Willmore surface (see [5–6]).But when the codimension is bigger than 1，a Willmore surface may not have a dual surface(see[7，20，27]).Using the duality properties of S-Willmore surfaces， Ejiri provided furthermore a classification of S-Willmore two-spheres in Sn+2by constructing the holomorphic forms for these surfaces(see [20]).Especially， for Willmore two-spheres in S4， a construction of a holomorphic 8-form indicates that these surfaces are automatically S-Willmore(see[7，20，26，28–29]).In the end of Ejiri’s paper， he asked whether all Willmore two-spheres in Sn+2are S-Willmore or not.If the answer is ‘no’， i.e.， if some Willmore， but not S-Willmore two-spheres would exist， how would one construct and characterize them?

In this paper， we will answer Ejiri’s open problem by a concrete construction of a totally isotropic Willmore two-sphere in S6which is not S-Willmore.Moreover， beyond the explicit construction of some new examples， the main goal of this paper is to characterize all totally isotropic Willmore two-spheres in S6via their geometric properties and their normalized potentials.This geometric description also supplies the basis for the work of [34]， where we provide a coarse classification of Willmore two-spheres in spheres by using the loop group method for the construction of harmonic maps (see [8， 15， 18–19]).

Different from the case in S4， where totally isotropic surfaces are automatically S-Willmore surfaces and of finite uniton type， totally isotropic surfaces in S6are not even Willmore in general.We refer to [8， 18， 33–34] for the definition of uniton.Moreover， even a totally isotropic Willmore surface in S6will， in general， not be of finite uniton type (see Remark 2.11).To this end， we first derive a geometric characterization of totally isotropic Willmore two-spheres in S6， which is similar to the description of minimal two-spheres in Sn(see[4，11]).Roughly speaking，the normal connection of a totally isotropic Willmore two-sphere has a special form and conversely， totally isotropic surfaces with such special normal connection are always Willmore and of finite uniton type(see Theorems 2.1–2.2).Application of this description yields a second description of such Willmore surfaces in terms of loop group language (see Theorems 2.8 and 3.3).The second description of totally isotropic Willmore two-spheres in S6contains also a concrete algorithm of constructions of explicit totally isotropic Willmore two-spheres.By this method， we derive many new examples of Willmore surfaces as follows， most of which have two branched points.

Example 1.1Let λ ∈S1and let (we refer to Section 2 for the definition of η)

with

Here p ∈Z+，p ≥2.The associated family of Willmore two-spheres xλ， corresponding to η， is

with

Moreover xλ: S2{0，∞} →S6is a Willmore immersion in S6， which is full， not S-Willmore，and totally isotropic.It is obvious that xλis S1-equivariant.Note that xλis also immersed at 0 and ∞when p = 2.When p ＞2， xλhas two branched points 0 and ∞， whose metrics tend to 0 with the same speed r2p-4.To be concrete， near the point z = 0， |?z(xλ)|2=2(p-1)2r2p-4+o(r2p-4).Near the point z =∞，setting

Recently there are several progresses on the discussions of branched points of Willmore surfaces(see for example[1，12，24–25]).We hope that these examples will help the understanding of branched points of Willmore surfaces.We only show the explicit computations in Appendix B for the case p=2， since the construction of xλis the same for the other ones.

This paper is organized as follow: In Section 2， we first recall basic results of Willmore surfaces and derive a new geometric description of isotropic Willmore two-spheres in S6.Moreover， we obtain a description of the normalized potentials of isotropic Willmore two-spheres in S6.The converse part， that generically such normalized potentials will produce special totally isotropic Willmore surfaces in S6， as well as new examples， makes up the main content of Section 3.The main idea is to perform a concrete Iwasawa decomposition for these normalized potentials to derive geometric properties of the corresponding Willmore surfaces， which also yields an algorithm to construct Willmore surfaces.We put the technical computations of Iwasawa decompositions and examples into two Appendixes for interested readers.

2 Isotropic Willmore Two-spheres in S6

In Subsection 2.1， we first recall the basic theory of Willmore surfaces and then focus on isotropic Willmore surfaces in S6.In Subsection 2.2 we will collect the basic DPW methods as well as Wu’s formula for harmonic maps and then derive the normalized potentials for isotropic Willmore two-spheres in S6.

2.1 Isotropic Willmore surfaces in S6 and related holomorphic differentials

2.1.1 Willmore surfaces in spheres

For completeness we first recall briefly the basic surface theory.For more details， we refer to [16， Section 2]， [17] and [34， Section 2] (see also [9， 26]).

Recall that y is a Willmore surface if and only if the Willmore equation holds (see [9])

Another equivalent condition of y being Willmore is the harmonicity of the conformal Gauss mapof y (see [5， 20， 26]) with Gr :=Y ∧Yu∧Yv∧N =-2i·Y ∧Yz∧Yz∧N.A local lift of Gr is chosen as

with its Maurer-Cartan formwhere

{ψj} is an orthonormal basis of V⊥on U and

2.1.2 Isotropic Willmore surfaces in S6

Recall that y is totally isotropic if and only if all the derivatives of y with respect to z are isotropic， that is，

Let y be a Willmore surface with an isotropic Hopf differential，i.e.，〈κ，κ〉≡0.Note that one derives straightforwardly that 〈κ，Dzκ〉 = 〈κ，Dzκ〉= 0 by differentiating 〈κ，κ〉= 0.Applying the Willmore equation (2.3)， we also have 〈Dzκ，Dzκ〉≡0.

For isotropic Willmore surfaces， Ma introduced several holomorphic differentials， see [26，Theorem 5.4].For our case， we only need that

is a globally defined holomorphic differential on M.The fact that Ωdz4is holomorphic can be derived from a direct computation using 〈κ，κ〉 = 0， Willmore equations and Ricci equations(see also [26]).Then， if M =S2， we will have 〈Dzκ，Dzκ〉≡0.

Now we assume that y is not S-Willmore，then Dzκ is not parallel to κ(recall that y is called S-Willmore if y is Willmore with Dzκ‖κ，see[16–17，20]).So Dzκ and κ span a two-dimensional isotropic subspace SpanC{κ，Dzκ}.Since Dzκ is perpendicular to κ and Dzκ， Dzκ is contained in SpanC{κ，Dzκ}.As a consequence， we also have 〈Dzκ，Dzκ〉 = 0.Summing up， we obtain the following theorem.(This theorem can also be derived by the loop group theory.See the end of Subsection 2.2.)

Theorem 2.1Letybe a Willmore two-spheres inS6with isotropic Hopf differential， i.e.，〈κ，κ〉=0.Ifyis not S-Willmore， thenyis totally isotropic(and hence full)inS6.Moreover，locally there exists an isotropic frame{E1，E2}of the normal bundleV⊥Cofysuch that

That is， the normal connection is block diagonal under the frame{E1，E2，E1，E2}.

Note that (2.7) provides also sufficient conditions for y to be a Willmore surface.

Theorem 2.2Letybe a totally isotropic surface fromUintoS6， with complex coordinatez.If there exists an isotropic frame{E1，E2}of the normal bundleV⊥Cofysuch that(2.7)holds， thenyis a Willmore surface.

ProofBy (2.7)， we see thatis an isotropic vector.Sinceby (2.2)， we have=0.So y is Willmore.

2.2 Normalized potentials of totally isotropic Willmore two-spheres in S6

This subsection aims to derive the description of totally isotropic Willmore two-spheres in S6in terms of the loop group methods.To this end，we will first collect the basic theory concerning the DPW construction of harmonic maps and the applications to Willmore surfaces.Then， we will derive the construction of normalized potentials of totally isotropic Willmore two-spheres via Wu’s formula.For more details of the loop group method we refer to [17–19， 37].

2.2.1 Harmonic maps into a symmetric space

Let G/K be a symmetric space defined by the involution σ :G →G，with Gσ?K ?(Gσ)0，and Lie algebras g=Lie(G)， k=Lie(K).The Cartan decomposition induced by σ on g states that g=k ⊕p， [k，k]?k， [k，p]?p， [p，p]?k.

Let f be a conformal harmonic map from a Riemann surface M into G/K.Let U be an open connected subset of M with complex coordinate z.Then there exists a frame F :U →G of f with a Maurer-Cartan form F-1dF =α.The Maurer-Cartan equation reads dα+[α∧α] =0.Decomposing with respect to the Cartan decomposition， we obtain α = α0+α1with α0∈Γ(k ?T*M)， α1∈Γ(p ?T*M).And the Maurer-Cartan equation becomes

Decomposing α1further into the (1，0)-part α′1and the (0，1)-part α′′1and introducing λ ∈S1，we set

It is well known (see [15]) that f : M →G/K is harmonic if and only if0 for all λ ∈S1.

Definition 2.1LetF(z，λ)be a solution to the equationdF(z，λ) = F(z，λ)αλ， F(0，λ) =F(0).ThenF(z，λ)is called the extended frame of the harmonic mapf.Note thatF(z，1) =F(z).

2.2.2 Two decomposition theorems

To state the DPW constructions for harmonic maps， we need the Iwasawa and Birkhoffdecompositions for loop groups.For simplicity， from now on we consider the concrete case for Willmore surfaces (see [17]).In this case， G = SO+(1，n+3)， K = SO+(1，3)×SO(n) and g=so(1，n+3)={X ∈gl(n+4，R)|XtI1，n+3+I1，n+3X =0}.The involution is given by

Note that SO+(1，n+3)σ?SO+(1，3)×SO(n+2)=(SO+(1，n+3)σ)0.We also have g=k⊕p，with

Let GC=SO+(1，n+3，C):={X ∈SL(n+4，C)|XtI1，n+3X =I1，n+3}with so(1，n+3，C)its Lie algebra.We extend σ to an inner involution of SO+(1，n+3，C)with KC=S(O+(1，3，C)×O(n，C))its fixed point group.Let ΛGCσbe the group of loops in GC=SO+(1，n+3，C)twisted by σ.

Theorem 2.3(see[16，Theorem 4.5]，also see[15，17]Iwasawa decomposition)There exists a closed， connected solvable subgroupS ?KC，such that the multiplicationis a real analytic diffeomorphism onto the open subsetHere

Theorem 2.4(see [15–17] Birkhoffdecomposition)The multiplicationis an analytic diffeomorphism onto the open， dense subset(big Birkhoffcell).

2.2.3 The DPW construction

Let D ?C be a disk or C with complex coordinate z.

Theorem 2.5(see [15]) (1)Letf : D →G/Kdenote a harmonic map with an extended frameF(z，z，λ) ∈ΛGσandF(0，0，λ) = I.Then there exists a Birkhoffdecomposition ofF(z，z，λ)，

such thatF-(z，λ):is meromorphic and the Maurer-Cartan formηofF-is

withη-1independent ofλ.The meromorphic1-formηis called the normalized potential off.

(2)Letηbe aλ-1·p ?C-valued meromorphic1-form onD.LetF-(z，λ)be a solution toThen there exists an Iwasawa decomposition

on an open subsetDIofD.Moreover，(z，z，λ)is an extended frame of some harmonic map fromDItoG/Kwith(0，λ) = I.All harmonic maps can be obtained in this way， since the above two procedures are inverse to each other if the normalization at some based point is fixed.

Note that in this paper since we consider the case with Identity，initial condition the Birkhoffdecomposition(see Theorem 2.4)holds for our case(see[15，30]).Moreover，Theorem 2.6 holds only if the Iwasawa decomposition and Birkhoffdecomposition are satisfied， since the proof of the similar results in [15] replies only on these two decompositions.In this sense， [15] is sufficient for this paper， except the Iwasawa case， which is provided essentially in [16–17].We also refer to H′elein’s paper(see[22])for another Iwasawa decomposition for some non-compact symmetric space (i.e， SO+(1，4)/(SO+(1，1)×SO(3))) slightly differenting from the present one.We refer to[17]for more discussions on these two kinds of different harmonic maps related with Willmore surfaces.

The normalized potential can be determined from the Maurer-Cartan form of f (see [36]).Let f，F(xiàn)(z，λ)and αλdenote the stuffas above.Let δ1and δ0denote the sum of the holomorphic terms of z about z =0 in the Taylor expansion ofrespectively.

Theorem 2.6(see [36] Wu’s formula)We retain the notions in Theorem2.5.Then the normalized potential offwith respect to the based point0is given byη =λ-1F0(z)δ1F0(z)-1dz，whereF0(z):D →GCis the solution toF0(z)-1dF0(z)=δ0dz，F(xiàn)0(0)=I.

2.2.4 Normalized potentials of totally isotropic Willmore two-spheres in S6

Let F be a frame of a Willmore surface y withas above.Here

Let δ′1be the holomorphic part of α′1and δ′0be the holomorphic part of α′0.Letbe the holomorphic part of B1.Letbe the solution to

By Theorem 2.6， we have

Applying Wu’s formula， we obtain the following theorem.

Theorem 2.7Letybe a totally isotropic Willmore two-spheres inS6.Then the normal bundle ofysatisfies the properties(2.7)of Theorem2.1.The normalized potential ofyis of the form

with(hijare meromorphic functions)

Lemma 2.1Set

Then，is a Lie sub-algebra ofso(4，C).Moreover， letbe the subgroup ofSO(4，C)with Lie algebra.Then

ProofIt is direct to show that

Remark 2.1(2.13) shows that the subgroupis diffeomorphic to S3×S1.

Proof of Theorem 2.7If y is not S-Willmore， (2.7) comes from Theorem 2.1.If y is S-Willmore， first let E1be a basis of the bundle spanned by κ (this bundle is globally defined，since Dzκ ∈SpanC{κ}，see the proof of[17，Lemma 1.3]for a detailed proof).Next，we consider the sub-bundle V2of the normal bundle perpendicular toSince〈Dzκ，E1〉=0，we can chose an isotropic basisof V2，such thatand〈Dzκ，E2〉=0.Then it is straightforward to verify that (2.7) holds.

Now we apply (2.7).Set E1= ψ1+iψ2， E2= ψ3+iψ4.Then we have a frame F of the form (2.4).Under this frame， we have

Then the normalized potential of y is expressed by (2.9).The holomorphic partof B1has the same form as B1and since K1does not change the relations between the columns ofwe need only to consider the influence of K2on.Note that A2takes value in.So the holomorphic partof A2also takes value in.Therefore， the integrationofalso takes value in.By Lemma 2.1， K2takes value in.Summing up， we can assume that the following two equations hold:

Remark 2.2Different from the case in S4， where totally isotropic surfaces are all SWillmore surfaces of finite uniton type，totally isotropic surfaces in S6can be even not Willmore in general.Moreover， for a totally isotropic Willmore surface in S6， if the holomorphic 4-form Ωdz4/=0(hence not S-Willmore)，it is full in S6and is not of finite uniton type.Given the fact that such surfaces come from the twistor projection of holomorphic or anti-holomorphic curves of the twistor bundle TS6of S6， they can be expressed by rational functions on the Riemann surface.Such harmonic maps which are not of finite uniton type are somewhat unexpected since they correspond to holomorphic or anti-holomorphic curves in the twistor bundle of S6.And it will be an interesting topic to classify and/or to characterize such harmonic maps as well as the corresponding Willmore surfaces， especially when the Riemann surface is a torus.As a consequence， it will be an interesting topic to generalize the work of Bohle on Willmore tori (see [2]) to Willmore tori in S6.

We can use the DPW method to give another proof of Theorem 2.1.

Proof of Theorem 2.1If y is non S-Willmore with〈κ，κ〉=0，we claim that its normalized potential can only take the form of type 3.By Theorem 3.1 of Section 3， y is totally isotropic and its normal connection has the desired form.

Now let us prove the claim.[34， Theorem 2.8] and [16， Theorem 5.2] show that B1must be either of type 2 or of type 3 in[34，Theorem 2.8].On the other hand，as we have seen before，the isotropy condition and the Willmore equation showThis yields that the Maurer-Cartan form of y satisfiesthe holomorphic part of B1， also satisfiesAs a consequence， we haveIf the normalized potential η of y is of type 2 in [35， Theorem 2.8]， then

3 Construction of Totally Isotropic Willmore Two-spheres in S6

This section is to describe geometric properties of Willmore surfaces of type 3 of[34，Theorem 3.3] We will provide an algorithm to derive a concrete construction of such Willmore surfaces in S6from the normalized potentials of type 3 of [34， Theorem 3.3] by a concrete Iwasawa decomposition.The geometric properties of this kind of Willmore surfaces are also revealed naturally.During this procedure， we will see that Willmore surfaces of this type will be the special kind of totally isotropic Willmore surfaces in S6， which has been discussed in Section 2.This section has three parts.The main theorem and the new examples are stated first.The technical lemmas combining the proof of Theorem 3.1 are stated in the end.The concrete proofs and constructions of examples are postponed to two appendixes.

3.1 From potentials to surfaces

Theorem 3.1(Case of [34， Theorem 3.3])Letybe a Willmore surface inS6with its normalized potential being of the form(2.10).Thenyis totally isotropic inS6.Moreover，locally there exists an isotropic frame{E1，E2}of the normal bundleV⊥Cofysuch that(2.7)holds.

3.2 Examples of totally isotropic Willmore spheres in S6

We have two kinds of examples to illustrate the algorithm presented in the proof of Theorem 3.1.The isotropic minimal surfaces in R4are used to illustrate the algorithm with simpler computations.The new， totally isotropic， non S-Willmore， Willmore two-spheres in S6is constructed to answer Ejiri’s question explicitly.

Theorem 3.2Let

Heref2andf4are(non-constant)meromorphic functions onC.Thisis of both type1and type3in[34， Theorem 2.8].The corresponding associated family of Willmore surfaces is

Corollary 3.1The Willmore surface[Yλ]in Theorem3.2is conformal to the minimal surface

Note thatλis different from the usual parameter of the associated family of a minimal surface.

Theorem 3.3(The case p=2 in (1.2))Let

The associated family of unbranched Willmore two-spheresxλ，λ ∈S1， corresponding toη， is

Moreoverxλ: S2→S6is a Willmore immersion inS6， which is full， not S-Willmore， and totally isotropic.Note that for allλ ∈S1，xλis isometric to each other inS6.

3.3 Technical lemmas

3.3.1 The basic ideas

To begin with， we first explain our basic ideas， since the computations are very technical.We will divide the proof of Theorem 3.1 into two steps:

1.To derive the harmonic maps from the given normalized potentials.

2.To derive the geometric properties of the corresponding Willmore surfaces.

The main method in Step 1 is a concrete performing of Iwasawa decompositions.The main idea in Step 2 is to read offthe Maurer-Cartan forms of the corresponding Willmore surfaces.

For Step 1， we first transform SO+(1，7，C) into G(8，C) (see (3.6)) so that the normalized potentials in Theorem 3.1 are strictly upper-triangular in g(8，C) = Lie(G(8，C)) (see Lemma 3.1).Then Lemma 3.2 provides the concrete expressions of the normalized potential and its meromorphic frame.Lemma 3.3 gives the Iwasawa decompositions of the meromorphic frame by the method of undetermined coefficients.This finishes Step 1.For Step 2，we first derive the forms of the Maurer-Cartan forms of the extended frame derived in Step 1.Then translating into the computations of moving frames，one will obtain the isotropic properties of the corresponding Willmore surfaces.

3.3.2 Step 1: Iwasawa decompositions

Set

with Jn=(jk，l)n×n， jk，l=δk+l，n+1for all 1 ≤k，l ≤n.

Lemma 3.1Let

with

Thenis a Lie group isomorphism.

We also have thatwith

This induces an involution ofΛG(8，C):

withas its fixed point set.

The image of the subgroup(SO+(1，3)×SO(4))Cis

with

Set

For anyF ∈G(8，C)， we have

Lemma 3.2Letηbe the normalized potential of Theorem3.1.Then

with

Moreover，H =I8+λ-1H1+λ-2H2is a solution to

Here

Lemma 3.3Retaining the assumptions and the notations of the previous lemmas， assume that(η)is the normalized potential of some harmonic map， we obtain:

Assume that the Iwasawa decomposition ofHisH =，withΛG(8，C)σand∈Λ+G(8，C)σ.Then

HereW，W0andL0are the solutions to the matrix equationswithW =I8+λ-1W1+λ-2W2and

Here the sub-matricesa，q，danduare determined by the following equations:

Moreover，can be expressed by these sub-matrices as below

Remark 3.11.Since in Lemma 3.2， the matrices f and g are given， (3.14a)determines d，where d is invertible(true for z close to z =0).Then (3.14b)determines u?， hence u.Inserting this into(3.14c)results in determining q.Inserting what we have so far into(3.14d)determines a.The last equation (3.14e) is a consequence of the previous equations.Therefore， the only condition for the solvability of (3.14) is the invertibility of d.If f and g are rational functions of z， the invertibility of d is satisfied on an open dense subset as a rational expression in z，z.

2.For a general procedure for the computations of Iwasawa decompositions for algebraic loops， or more generally for rational loops， see [13， Section I.2].

3.In [14， 21]， a different method is used to produce all harmonic maps of finite uniton type into U(n)， the complex Grassmannian U(n+m)/(U(n)×U(m)) and G2.The treatment of these papers basically follows the spirit of Wood[35]， Uhlenbeck[33]and Segal[31]， using some special unitons.In [32]， the converse part of this procedure is also used for the computations of the Iwasawa decompositions of elements of the algebraic loop group λalgU(n)C.It will be an interesting and very hard question to apply they results to detect the geometry of harmonic maps.

3.3.3 Step 2: Maurer-Cartan forms

Lemma 3.4Retaining the assumptions and the notations of the previous lemmas， the Maurer-Cartan form ofin(3.15)is of the form

with

Note that these three equations for a1，a0and a4actually should be read as ordinary differential equations for l1， l0and l4， as initial conditions we may use lj(0)=I，j =0，1，4.

Lemma 3.5LetF : M →SO+(1，7)/SO+(1，3)×SO(4)be the conformal Gauss map of a Willmore surfacey， with an extended frameF.If the Maurer-Cartan form of=(F)has the form(3.16)， thenyis totally isotropic inS6.Moreover， locally there exists an isotropic frame{E1，E2}of the normal bundleV⊥Cofysuch that(2.7)holds.

A combination of the above lemmas provides a complete proof of Theorem 3.1.Lemmas 3.1–3.2 can be verified by straightforward matrix computations since the concrete formulas are provided(compare also[34]).So we leave these computations to the readers.The proofs of the other lemmas will be contained in the following section.

4 Appendix A: Iwasawa Decompositions

4.1 Proof of Lemma 3.3

Firstly one computes

Hence we obtain

Since

from the second matrix equation of (4.1)， one derives easily that vq =0， -v?a-u?c=0， uq-Since q is invertible， v =0.Therefore we have

Next we consider the third matrix equation in (4.1).Since

comparing with the λ-independent part of ˇτ-1(H)H， we derive directly that c=b=c=b=0.Substituting these results into the matrix equations in (4.1)， a straightforward computation yields (3.14).

In the end， let L0be of the form as in Lemma 3.3， it is easy to compute

4.2 The Maurer-Cartan form of ˇF and the geometry of Willmore surfaces

Proof of Lemma 3.4We haveSince

we obtain

Proof of Lemma 3.5By (3.16) in Lemma 3.4， there exists a framesuch that the(1，0)-part ˇα′of the Maurer-Cartan form of ˇF has the form

with

and

Therefore， one obtains

Now assume that Y is a canonical lift of the Willmore surface y.Note that SpanC{Y，Yz，Yz，N}=SpanC{φ1，φ2，φ3，φ4}.So Yzis a linear combination of {φ1，φ2，φ3，φ4}.Then we compute the Hopf differential κ=Yzzmod {Y，Yz，Yz，N}:

Hence 〈κ，κ〉≡0， i.e.， κ is isotropic.To show that Y is totally isotropic， we need only to verify that Dzκ is isotropic.From the Maurer-Cartan form of F， we derive that

So Dz(ψ1+iψ2)=(ψ1+iψ2)+(ψ3+iψ4)， Dz(ψ3+iψ4)=(ψ1+iψ2)+(ψ3+iψ4).As a consequence， we obtain that Dzκ = λ-1(δ1(ψ1+iψ2)+δ2(ψ3+iψ4)) for some complex valued function δ1and δ2.This indicates that Dzκ is also isotropic，i.e.，Y as well as y is totally isotropic.

4.3 An Algorithm to derive Willmore surfaces from frames

This subsection is to derive an algorithm permitting to read offy from the frame F.Although the harmonic maps have been constructed in the above subsections， to obtain the Willmore surfaces from the harmonic maps needs more computations.We retain the notation， in the proof of Lemma 3.5.

Set B1= (h1，ih1，h3，ih3) with hj= (h1j，h2j，h3j，h4j)t， j = 1， 3.Since B1satisfiesI1，3B1= 0， we haveI1，3hl= 0， j， l = 1， 3.Therefore h1and h2are contained in one of the following two subspaces (see also [34]):

Let Y be a canonical lift of y.Hence Y ∈SpanR{φ1，φ2，φ3，φ4}.Since Y is real and lightlike，we may assume that

Hence， to ensure that Yz∈SpanC{φ1，φ2，φ3，φ4}，needs to satisfy

Without loss of generality， we assume that h1=ρ0(1+ρ1ρ2，1-ρ1ρ2，ρ1+ρ2，-i(ρ1-ρ2))t.If the maximal rank of B1is 1， then h3=ρ01(1+ρ1ρ2，1-ρ1ρ2，ρ1+ρ2，-i(ρ1-ρ2))t.So (4.5)is equivalent toHenceThese two solutions provide a pair of dual Willmore (therefore S-Willmore) surfaces y andwith the same conformal Gauss map.

If the maximal rank of B1is 2， then

or

For the first case， (4.5) is equivalent toFor the second case， (4.5) is equivalent toIn both cases， we obtain a unique non-S-Willmore surface.

From the above discussions， clearly it is necessary to obtain the first four columns of F.By (3.15)，can be derived from the Iwasawa decompositions.SetandWriting F = (φ1，φ2，φ3，φ4，ψ1，ψ2，ψ3，ψ4)， and setting(φ1+φ2，φ1-φ2，φ3-iφ4，φ3+iφ4)， one obtains straightforwardly from (3.7) that

5 Appendix B: Construction of Examples

5.1 Proof of Theorem 3.2

By the procedures in Subsection 4.3， to derive the expression of y， one needs to figure out B1of the Maurer-Cartan form and the first four columns of the frame F.Applying Lemmas 3.2–3.3 to(η)，F(xiàn) and the Maurer-Cartan form can be derived by solving(3.14)for the Iwasawa decompositions.Therefore we have three steps to derive y:

1.Computation of the first four columns of F.

2.Computation of the Maurer-Cartan form of F.

3.Computation of Y.

Step 1: Computation of the first four columns of F.By (3.11)， it is straightforward to derive that

Since W0∈G(8，C)， we derive at=Jd-1J ==a.By (3.14b) and (3.14c)， we have

It is straightforward to verify thatwith l0∈G(4，C):

Moreover，by (3.15)， we have

Step 2: Computation of the Maurer-Cartan form of F.Applying Lemma 3.4， thepart of the Maurer-Cartan form of ˇF is of the form

Step 3: Computation of Y.Here we follow the discussions in Subsection 4.3.First from，the Maurer-Cartan form we have

Set E1=φ1-φ2，=φ1+φ2， E2=φ3-iφ4.Assume thatfor some μ.We have that

So DzY =0 if and only ifThis yields (3.2).

Remark 5.1Note that the above Iwasawa decomposition only blows up at the poles of f2and f4， showing that the above decomposition does not cross the boundary of an Iwasawa big cell.

5.2 Proof of Theorem 3.3

Here we have four steps:

1.Computation of the first four columns of F.

2.Computation of the Maurer-Cartan form of F.

3.Computation of Y.

4.Computation of metric of Y (to check the immersion properties of y).

Step 1: Computation of the first four columns of F.Since η is of the form stated in (3.4)，by (3.11)， it is easy to derive that

with

Moreover，by (3.14b)， one computes

Substituting these into (3.14c)， we obtain q =(qij) with

It is straightforward to check thatwith

with 1 ≤j ≤8，3 ≤l ≤6， and

Step 2: Computation of the Maurer-Cartan form of F.By (3.16)–(3.17) of Lemma 3.4， the Maurer-Cartan form of ˇF has the expression

with

Transforming back to so(1，7，C)， we derive

Step 3: Computation of Y.Here we follow the discussions in Subsection 4.3.It is easy to verify that h1and h3can be expressed as a(functional)linear combination of(1，1，-iρ，ρ)tand (ρ，-ρ，i，1)t.Therefore one obtains easily that ˇρ1=-iρ is the unique solution to(4.5).Substitutinginto (4.4)， we obtain

Then by (5.1) and (4.6) we have that

Step 4: Let [Y] be a global immersion.Let xλbe of the form (3.5).Then xλ: S2→S6is well-defined on S2with Y as its lift.Since

x has no branch point at z ∈C.As to ∞， setwe derive thatat the point=0.

Remark 5.2Note that in the above Iwasawa decomposition， there exists a circle 1 += 0 such that the frame (5.1) obtained from the Iwasawa decomposition blows up.However， this blowing up can be avoided by a change of frames and hence the corresponding harmonic map is in fact globally well defined.This also means that the decomposition of the corresponding harmonic map does not cross the boundary of an Iwasawa big cell (compare [4，24]).

AcknowledgementThe author is thankful to Professor Josef Dorfmeister， Professor Changping Wang and Professor Xiang Ma for their suggestions and encouragement.