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GROUP ACTIONS AND DEFORMATIONS FOR HARMONIC MAPS

arXiv:hep-th/9303037v1 5 Mar 1993GROUP ACTIONS AND DEFORMATIONS FOR HARMONIC MAPSMartin A. Guest and Yoshihiro OhnitaIntroductionFrom the theory of integrable systems it is known that harmonic maps from a Riemannsurface to a Lie group may be studied by infinite dimensional methods (cf.

[ZM],[ZS]). Thiswas clarified considerably by the papers [Uh],[Se], especially in the case of maps from theRiemann sphere S2 to a unitary group Un.

The basic connection with infinite dimensionalmethods is the correspondence between harmonic maps S2 −→G and “extended solutions”S2 −→ΩG, where G is any compact Lie group and ΩG is its (based) loop group. In [Uh]this was used in two ways (in the case G = Un):(1) to introduce a group action of matrix valued rational functions on harmonic maps, and(2) to prove a factorization theorem for harmonic maps, which unifies and extends many ofthe known results on the classification of harmonic maps from S2 into various homogeneousspaces.In [Se] it was shown that the factorization theorem can be proved very naturally by usingthe “Grassmannian model” of ΩG, which is an identification of ΩG with a certain infinitedimensional Grassmannian (see [PS]).

In this paper we shall show how the group actionmay be interpreted in terms of the Grassmannian model. The advantages of this point ofview are that the geometrical nature of the action is emphasized, and that calculationsbecome easier.

We shall illustrate this by giving some applications to deformations ofharmonic maps. By using some elementary ideas from Morse theory, we obtain new resultson the connectedness of spaces of harmonic maps, a subject which has been studied recentlyby various ad hoc methods (for example, in [Ve1],[Ve2],[Ve3],[Lo],[Kt]).The paper is arranged as follows.

In §1 we give the basic definitions, including that of a“generalized Birkhoffpseudo-action”. The latter is an action of k-tuples of loops γ on ex-tended solutions Φ, denoted by (γ, Φ) 7−→γ♯Φ.

This definition involves a Riemann-Hilbertfactorization (a generalization of the Birkhofffactorization for loops), and is an example ofa “dressing action” in the theory of integrable systems. Because the factorization cannotalways be carried out, the action is defined only for certain γ and Φ, so we call it a pseudo-action.

Nevertheless, it is possible to establish some general properties of the action byusing contour integral formulae, and we shall use these to show that the most importantcase of a generalized Birkhoffpseudo-action is precisely the one introduced by Uhlenbeck.In §2 we go on to show that the Uhlenbeck action on harmonic maps S2 −→Un of fixed

energy “collapses” to the pseudo-action of a finite dimensional group.This collapsingphenomenon has been described from a different point of view in [AJS],[AS1],[AS2],[JK].The Grassmannian model and its relevance for harmonic maps are reviewed in §3.From this point of view there is a natural action of the complex group ΛGC on extendedsolutions, where GC is the complexification of G and ΛGC is its (free) loop group. Thisaction is denoted by (γ, Φ) 7−→γ♮Φ.

Elementary properties of this action – which reallyis an action, not a pseudo-action – are given in §4. In particular, it is easy to see that thisaction, like the Uhlenbeck action, collapses to an action of a finite dimensional Lie group.Our first main result appears in §5, where we show that the actions ♯and ♮are essentiallythe same, despite their very different definitions.The essential point here is that theexplicit Riemann-Hilbert factorization needed for ♯is incorporated into the definition ofthe Grassmannian needed for ♮.

This result explains the similarities between the propertiesof the action ♯(described in §1 and §2) and the properties of the action ♮(described in §3and §4). In particular, it “explains” and extends Theorem 9.4 of [Uh].In §6, we discuss applications of the action ♮to deformations of harmonic maps.

A oneparameter subgroup {γt} of ΛGC gives rise to a deformation Φt = γt♮Φ of an extendedsolution Φ.This deformation has a simple geometrical interpretation: it is the resultof applying the gradient flow of a suitable Morse-Bott function on ΩG to the extendedsolution Φ. Hence, we obtain a new extended solution Φ∞= limt→∞Φt which takes values(almost everywhere) in a critical manifold of this Morse-Bott function.

In general, Φ∞has a finite number of (removable) singularities. This illustrates the well known fact (see[SU]) that a sequence of harmonic maps (of S2) has a convergent subsequence over thecomplement of a finite set, the latter being points at which “bubbling off” occurs.

Weshall give some examples where the singularities do not occur, so that Φ∞is joined to Φ bya continuous path in the space of extended solutions. The main example is the following.Let ϕ : S2 −→Un be a harmonic map, with corresponding normalized extended solutionΦ = Pmα=0 Tαλα (this notation will be explained later).

Then we have (see Theorem 6.2):(A) Assume that rank T0(z) ≥2 for all z. Then ϕ can be deformed continuously to aharmonic map ψ : S2 −→Un−1.It is well known that harmonic maps into an inner symmetric space G/K may be studiedas a special case of harmonic maps into G (by making use of a totally geodesic embeddingof G/K into G).So our method can be used to produce continuous deformations ofharmonic maps from S2 to G/K, for various G/K.

We shall give two examples, namelyG/K = CP n and G/K = Sn. In the first case we shall show:(B) The number of connected components of the space of harmonic maps S2 −→CP n isindependent of n, if n ≥2.This can be obtained as a consequence of the method for (A), but we shall also give adirect proof (Theorem 6.5).

We conjecture that the space of harmonic maps S2 →CP nof fixed energy and degree is connected. By (B), it would suffice to verify this conjecturein the case n = 2.

In the case G/K = Sn, for n ≥4, we shall use the same method to givea new proof of the following fact (Theorem 6.7; see also [Lo],[Ve3],[Kt]):2

(C) The space of harmonic maps S2 −→Sn of fixed energy is connected.The proof we give is quite elementary and does not depend on §1-§5 of this paper (thoughit was motivated by the method used for (A)).Most of our results in §6 generalize to the case of extended solutions M −→ΩG, whereM is any compact connected Riemann surface. In particular, the results on the connectedcomponents of harmonic maps from S2 into Sn or CP n generalize to the case of isotropicharmonic maps into Sn or complex isotropic harmonic maps into CP n. In fact, since ourmethod primarily involves the target space, one may go even further and obtain similarresults on pluriharmonic maps of compact connected complex manifolds (cf.

[OV]).Finally, we make some concluding remarks on the two main ingredients of this paper,i.e. group actions and deformations.

First, it should be emphasized that the group ac-tions discussed here do not represent a new idea. It is a well known principle in othercontexts to convert from real to complex geometry, in order to reveal a larger (complex)symmetry group.

(Here, one converts from harmonic maps into a Riemannian manifoldto “horizontal” holomorphic maps into a complex manifold.) Indeed, as mentioned above,the action ♯had its origins in the theory of integrable systems, while examples of theaction ♮have been treated explicitly in [Gu] and have been alluded to by other authors.Our contribution to this topic (in §5) is the unification of the two actions.

Second, theresults of §6 concerning deformations are essentially independent of §1-§5, although we feelthat the group action provides some motivation for these deformations. From a practicalpoint of view, the deformations have two main features.

One is the connection with Morsetheory which allows us to predict easily the end result of the deformations. The other isthat the horizontality condition, which is sometimes hard to deal with directly, is neverneeded explicitly in our calculations.Acknowledgements: Our results on the connectedness of spaces of harmonic maps wereinspired by work of N. Ejiri and M. Kotani (cf.

[EK]). The first author is indebted toW.

Richter for pointing out the importance of doing Morse theory on finite dimensionalsubvarieties of the loop group (cf. [Ri]).

He acknowledges financial support from the JapanSociety for Promotion of Science and the U.S. National Science Foundation.§1. Extended solutions and generalized Birkhoff pseudo-actionsLet M be a connected Riemann surface or, more generally, a connected complex mani-fold.

Let G be a compact connected Lie group equipped with a bi-invariant Riemannianmetric and let g denote its Lie algebra.If necessary, we choose a realization for thecomplexification GC of G as a subgroup of some general linear group GLn(C), withG = GC ∩U(n).Let µ denote the Maurer-Cartan form of GC.For a smooth mapϕ : M −→GC, set ϕ∗µ = α = α′ + α′′, where α′ and α′′ are the (1, 0)-component and(0, 1)-component of α, respectively.3

Definition. The map ϕ : M −→GC is said to be (pluri)harmonic if and only if ¯∂α′ = ∂α′′.If ϕ(M) ⊆G, then this definition coincides with the usual definition (see (8.5) of [EL], §2of [OV]).

We shall call such a map ϕ a real harmonic map.For each λ ∈C∗= C \ {0}, consider the 1-form on M with values in gC given byαλ = 12(1 −λ−1)α′ + 12(1 −λ)α′′,and consider the first order linear partial differential equation(∗)Φλ∗µ = αλ,for a map Φλ : M −→GC. Using an embedding GC −→GLn(C), this equation may bewritten as(∗∗) ∂Φλ = 12(1 −λ−1)Φλα′¯∂Φλ = 12(1 −λ)Φλα′′.Definition.

A family of solutions Φλ, λ ∈C∗, to (∗) or (∗∗) is called an extended solution([Uh]) or an extended (pluri)harmonic map ([OV]).The fundamental observation, proved in [Uh] for harmonic maps, and extended in [OV] topluriharmonic maps, is:Theorem 1.1. Assume that Hom(π1(M), G) = {e}.

Choose a base point z0 of M and amap σ : C∗−→GC. Let ϕ : M −→GC be a (pluri)harmonic map.

Then there exists aunique extended solution Φ : M × C∗−→GC such that Φλ(z0) = σ(λ). Conversely, if Φis an extended solution, then Φ−1 : M −→GC is a (pluri)harmonic map.□Moreover, the extended solution Φ (obtained from σ and ϕ) necessarily satisfies Φ−1 = aϕ,where a = σ(−1)ϕ(z0)−1.Let ϕ be a real harmonic map.

If we choose σ satisfying σ(1) = e and σ(S1) ⊆G, thenΦ1 ≡e and Φλ(M) ⊆G for any λ ∈S1 = {λ ∈C∗| |λ| = 1}. (For example, we maychoose σ ≡e.) In this case we call Φ a real extended solution.The smooth loop group of G is defined by:ΩG = {γ : S1 −→G | γ smooth, γ(1) = e}.Let π : ΩG −→G be the map π(γ) = γ(−1).

A real extended solution Φ can be consideredas a map into ΩG; conversely, if Φ : M −→ΩG satisfies (∗) or (∗∗) for λ ∈S1, thenthe same argument as for Theorem 1.1 shows that the map ϕ = π ◦Φ : M −→G is(pluri)harmonic. Because of this we shall (with abuse of notation) use the term “realextended solution” for any map Φ : M −→ΩG satisfying (∗) or (∗∗).4

It is known that ΩG has the structure of an infinite dimensional homogeneous K¨ahlermanifold (see [PS]). There is a left-invariant complex structure J such that the (+i)-eigenspace of J is the subspace spanned by the elements (λ−k −1)gC (k = 1, 2, .

. .

), underthe identification T Ce ΩG ∼= ΩgC. The condition (∗) or (∗∗) may be writtenΦ∗µ(T1,0M) = Φ−1dΦ(T1,0M) ⊆(λ−1 −1)gC.In particular, we see that any extended solution Φ : M −→ΩG is holomorphic relative toJ.Following [Uh], we say that a harmonic map ϕ has finite uniton number if there is anextended solution Φ such that π ◦Φ = aϕ for some a ∈GC and Φ(λ) = Pmα=0 Tαλα (forsome m).

The least such integer m is called the minimal uniton number of ϕ (or of Φ).The next fundamental result is that any harmonic map which admits a corresponding realextended solution has finite uniton number:Theorem 1.2 ([Uh]). Assume that M is compact.

Let Φ : M −→ΩUn be an extendedsolution. Then there exists a loop γ ∈ΩUn and a non-negative integer m ≤n −1 suchthat (i) γΦ(λ) = Pmα=0 Tαλα, (ii) Span{Im T0(z) | z ∈M} = Cn.

Here m is equal to theminimal uniton number of Φ−1.□We shall refer to property (ii) as the Uhlenbeck normalization.Now we discuss the group action studied by Uhlenbeck, and its generalizations. Theidea of a “dressing action” (see, for example, [ZM],[ZS],[Uh],[BG]) is as follows.

Let Gbe a group and G1, G2 two subgroups of G with G = G1G2 and G1 ∩G2 = {e}, where eis the identity element of G. For any g ∈G, we have a unique decomposition g = g1g2,g1 ∈G1, g2 ∈G2.For g, h ∈G, define g♯h by g♯h = gh(h−1gh)−12= h(h−1gh)1.Ifg, g′, h ∈G, then we have g♯(g′♯h) = (gg′)♯h, so this defines an action of G on itself.Let T C be the complexification of a maximal torus T of G. Let U+ = {λ ∈S2 | |λ| < 1}and U−= {λ ∈S2 | |λ| > 1} in the Riemann sphere S2 = C ∪{∞}. SetΛGC = {γ : S1 −→GC | γ smooth},Λ+GC = {γ ∈ΛGC | γ extends continuously to a holomorphic map U+ −→GC},Λ−GC = {γ ∈ΛGC | γ extends continuously to a holomorphic map U−−→GC},Λ∗−GC = {γ ∈Λ−GC | γ(1) = e},∆GC = {δ ∈ΛGC | δ : S1 −→T C ⊆GC is a homomorphism }.The following fact is known as the Birkhoffdecomposition ([PS]): the mapΛ−GC × ∆GC × Λ+GC −→ΛGC,(γ−, δ, γ+) −→γ−δγ+is surjective.

Moreover, Λ∗−GC × Λ+GC maps diffeomorphically to Λ−GCΛ+GC, which isan open dense subset of the identity component of ΛGC. We shall now take G = ΛGC,G1 = Λ∗−GC, G2 = Λ+GC in the definition of dressing action.

Since G1G2 is not quite equalto G here, we use the term “pseudo-action”:5

Definition. The Birkhoffpseudo-action of ΛGC on itself is defined by γ♯δ = γδ(δ−1γδ)−1+ =δ(δ−1γδ)−∈ΛGC, for γ, δ ∈ΛGC with δ−1γδ ∈Λ∗−GCΛ+GC.We can also consider “generalized Birkhoffpseudo-actions” ([BG]).

Let C1, . .

., Ck beoriented circles of radius r on the Riemann sphere S2 = C ∪{∞}. Let Ii and Ei denotethe interior and exterior of Ci for each i = 1, .

. ., k. Set C = C1 ∪· · ·∪Ck, I = I1 ∪· · ·∪Ikand E = E1 ∩· · · ∩Ek.

We assume in addition that ¯Ii ∩¯Ij = ∅for i ̸= j and 1 ∈E. LetΛ1,...,kGC = {γ : C −→GC | γ smooth },which is isomorphic to a direct product of k copies of ΛGC.

SetΛEGC = {γ ∈Λ1,...,kGC | γ extends continuously to a holomorphic map E −→GC},ΛIGC = {γ ∈Λ1,...,kGC | γ extends continuously to a holomorphic map I −→GC},Λ∗EGC = {γ ∈ΛEGC | γ(1) = e},∆1,...,kGC = {δ ∈Λ1,...,kGC | δ : C −→T C ⊆GC is a homomorphism}. (To say that the map δ is a homomorphism means that it can be written in the formδ(λ) = ({(λ −ci)/r}b1, .

. ., {(λ −ci)/r}bn) for λ ∈Ci = {λ ∈S2 | |λ −ci| = r}.

)There is an analogue of the Birkhoffdecomposition in this situation, namely (see [BG]):Λ1,...,kGC = ΛEGC∆GCΛIGC. Moreover, under the multiplication map, Λ∗EGC × ΛIGCis diffeomorphic to ΛEGCΛIGC, which is an open dense subset of the identity componentof Λ1,...,kGC.

If we take G = Λ1,...,kGC, G1 = Λ∗EGC, G2 = ΛIGC in the definition of adressing action, we obtain:Definition. The generalized Birkhoffpseudo-action of Λ1,...,kGC on itself is defined byγ♯δ = γδ(δ−1γδ)−1I= δ(δ−1γδ)E ∈Λ1,...,kGC, for γ, δ ∈Λ1,...,kGC with δ−1γδ ∈Λ∗EGCΛIGC.The main reason for studying such pseudo-actions is:Proposition 1.3 ([ZM],[ZS],[Uh],[BG]).

Let g ∈Λ1,...,kGC and let Φ be an extendedsolution. If Φ−1(z)gΦ(z) ∈Λ∗EGCΛIGC for each z ∈M, then the map g♯Φ is also anextended solution.□(We assume that Φλ is defined for all λ in some region which includes C. For example,this is the case if C does not contain the points 0, ∞and if we choose σ ≡e in Theorem1.1.) The pseudo-action of Λ1,...,kGC on extended solutions gives rise to a pseudo-action onharmonic maps, by means of the formula g♯(π◦Φ) = π◦g♯Φ.

This is not quite well-defined,as the extended solution Φ corresponding to a harmonic map M −→G is determined onlyup to left translation in ΩG. However, the non-uniqueness will be of no consequence inthis article.Let us impose now the following “reality conditions”: (1) the equator S1 is contained inE, (2) 0, ∞∈I, and (3) C = C1 ∪· · · ∪Ck is preserved by the transformation λ −→¯λ−1.We call an element g ∈Λ1,...,kGC real if g(¯λ−1)∗= g(λ)−1 for each λ ∈C.

It is easy tocheck that g♯Φ is a real extended solution if g and Φ are real. We denote by Λ1,...,kRGC6

the subgroup of real elements of Λ1,...,kGC, and by ΛE,RGC, Λ∗E,RGC, ΛI,RGC, ∆RGCthe subgroups of real elements of ΛEGC, Λ∗EGC,ΛIGC, ∆GC.We shall now give a contour integral expression for the generalized Birkhoffpseudo-action of Λ1,...,kGC on ΛEGC. Note that for δ ∈ΛEGC the formula for γ♯δ simplifies toγ♯δ = γδ(γδ)−1I= (γδ)E.Lemma 1.4.

Let g ∈Λ1,...,kGC and h ∈ΛEGC. Assume that h−1gh ∈Λ∗EGCΛIGC, sothat g♯h ∈Λ∗EGC is well-defined.

Then(g♯h)(λ) −h(λ) = λ −12πiZCh(λ)h−1(µ)(g−1(µ) −e)(g♯h)(µ)(µ −1)(µ −λ)dµfor each λ ∈E.Proof. By using Cauchy’s Integral Theorem, we obtain(h−1gh)E(λ) −e = λ −12πiZC((h−1gh)−1(µ) −e)(h−1gh)E(µ)(µ −1)(µ −λ)dµ.Multiplying by h(λ) on the left, we obtain the required formula.□Using this lemma, we derive a formula for the infinitesimal action of Λ1,...,kGC onΛEGC.

Let {gt}|t|<ε be a curve in Λ1,...,kGC with g0 = e and set V = ddtgt|t=0 ∈Λ1,...,kgC.Let h ∈ΛEGC. Note that for each t sufficiently close to 0, h−1gth ∈Λ∗EGCΛIGC andhence g♯th ∈Λ∗EGC is defined.

SetV ♯h = ddtg♯tht=0 ∈ThΛEGC.Proposition 1.5. For each λ ∈E, we havedL−1h (V ♯h)(λ) = −λ −12πiZCh−1(µ)V (µ)h(µ)(µ −1)(µ −λ) dµ.Here Lh denotes left translation by h in the group ΛEGC.Proof.

Replace g by gt in the formula of Lemma 1.4. By differentiating at t = 0, we obtainthe required formula.□Corollary 1.6.

Assume that 0 ∈I1, ∞∈I2. If g ∈ΛIGC satisfies g|Ii = e for i = 1, 2and h ∈ΛEGC extends to a holomorphic map C∗= S2 \ {0, ∞} −→GC, then g♯h existsand g♯h = h.□Thus, if Φ is a real extended solution, which without loss of generality we may assume isdefined for all λ ∈C∗, then it is only necessary to consider generalized Birkhoffpseudo-actions with C = C1 ∪C2, where C1, C2 are circles around 0, ∞respectively.7

§2. Properties of the Uhlenbeck pseudo-actionIn this section we shall study the pseudo-action introduced by Uhlenbeck in [Uh].

Itcan be regarded as the generalized Birkhoffpseudo-action given by the choice of circlesCε0 = {λ ∈S2 | |λ| = ε},Cε∞= {λ ∈S2 | |λ| = 1ε},where 0 < ε < 1.We shall call it the Uhlenbeck pseudo-action. This is the simplestchoice which is compatible with the reality conditions, and by Corollary 1.6 it containsthe essential features of all the other choices.We shall write ΛεGC for Λ1,2GC, where C1 = Cε0, C2 = Cε∞.

Using the notation of theprevious section, we have C = C1S C2, I = I1S I2 and E = S2 \ C S I1S I2, whereI1 = {λ ∈S2 | |λ| < ε},I2 = {λ ∈S2 | |λ| > 1ε}.We have subgroups ΛEGC, Λ∗EGC, ΛIGC of ΛεGC as in the previous section. We denoteby ΛεRGC the subgroup of all real elements γ of ΛεGC, namely elements satisfying thereality condition γ(¯λ−1)∗= γ(λ)−1 on C.LetG = {g : U −→GC | g holomorphic in some neighbourhood U of {0, ∞}},GR = {g ∈G | g(¯λ−1)∗= g(λ)−1 for all λ}.Note that G and GR are connected.

LetA = {g ∈G | g extends to a GC-valued rational function on S2 },AR = {g ∈A | g(¯λ−1)∗= g(λ)−1 for all λ}.For each ε with 0 < ε < 1, we consider ΛIGC and ΛI,RGC as subgroups of G and GR,respectively. We then have[0<ε<1ΛIGC = G,[0<ε<1ΛI,RGC = GR.Denote by Lie(G) and Lie(GR) the Lie algebras of G and GR, respectively.

For eachinteger k ≥0 or k = ∞, letLie(G)k = { V ∈Lie(G) | V (λ) =Xα≥kV (0)α λα around 0,V (λ) =Xα≥kV (∞)−α λ−α around ∞}.Then Lie(G)k is an ideal of Lie(G) and Lie(G)k ⊆Lie(G)k−1, Lie(G)0 = Lie(G). LetGk be the analytic subgroup of G generated by the Lie algebra Lie(G)k, which is a con-nected closed normal subgroup of G.(Thus, Lie(Gk) = Lie(G)k.)The quotient com-plex Lie algebra Lie(G)/Lie(G)k has complex dimension 2k dimC gC.We have a se-quence of surjective Lie group homomorphisms : G/Gk −→G/Gk−1 (k = 1, 2, .

. .

). Set8

Lie(G)k,R = Lie(G)R ∩Lie(G)k, which is a real Lie algebra. The Lie algebra Lie(G)k,Rgenerates an analytic subgroup Gk,R of GR, which is a connected closed normal subgroupof GR.

The quotient real Lie algebra Lie(GR)/Lie(G)k,R has real dimension 2k dim g.For each integer k ≥0 or k = ∞, we set Ak = A ∩Gk, Ak,R = AR ∩Gk. Note that Akis a closed normal subgroup of A.Proposition 2.1.

(i) For each k with 0 ≤k < ∞, the natural injective homomorphismof A into G induces a Lie group isomorphism of A/Ak onto G/Gk. (ii) For each k with0 ≤k < ∞, the natural injective homomorphism of AR into GR induces a Lie groupisomorphism of AR/Ak,R onto GR/Gk,R.Proof.

Denote by σ and dσ the Lie group homomorphism A/Ak −→G/Gk and itsderivative, respectively.We have only to show that σ is surjective.Let V be anyelement of Lie(G).We take the Taylor expansions of V around 0 and ∞: V (λ) =Pα≥0 V (0)α λα around 0, and V (λ) = Pα≥0 V (∞)−α λ−α around ∞. By the method of in-determinate coefficients, we can find U ∈Lie(A) such that U(λ) = Pk−1α=0 V (0)α λα +P∞α=k U (0)α λα around 0 and U(λ) = Pk−1α=0 V (∞)−α λ−α + P∞α=k U (∞)−α λ−α around ∞.

HenceU −V ∈Lie(Ak), namely U ≡V mod Lie(Ak). Thus dσ is surjective.

Since G/Gk isconnected, σ is also surjective. This proves (i).

The proof of (ii) is similar.□For each integer k ≥0 or k = ∞, letXk = {γ : C∗−→GC | γ holomorphic, γ(1) = e,and γ(λ) =X|α|≤kAαλα, γ−1(λ) =X|α|≤kBαλα}Xk,R = {γ ∈Xk | γ(¯λ−1)∗= γ(λ)−1 for all λ}.Similary, letX +k = {γ : C∗−→GC | γ holomorphic, γ(1) = e,and γ(λ) =kXα=0Aαλα, γ−1(λ) =kXα=0B−αλ−α}X +k,R = {γ ∈X +k | γ(¯λ−1)∗= γ(λ)−1 for all λ}.We can consider Xk,R and X +k,R as subspaces of ΩG. Set X = X∞and XR = X∞,R.

Thepoint of these definitions is that a harmonic map of finite uniton number gives rise to anextended solution with values in X +k,R, for some k.Uhlenbeck obtained the following theorem by showing that any element of AR decom-poses into a product of elements of “simplest type”, then by showing that the action isdefined for any element of simplest type. See also [Be].Theorem 2.2 ([Uh]).

For each g ∈AR and each γ ∈XR, g♯γ ∈XR is well-defined.□We call the action of AR on XR the Uhlenbeck action.9

Theorem 2.3. (i) If V ∈Lie(G)2k and γ ∈Xk, then V ♯γ = 0.

(ii) If g ∈G2k and γ ∈Xk,then g♯γ ∈Xk is defined and g♯γ = γ.Theorem 2.4. (i) If V ∈Lie(G)k and γ ∈X +k , then V ♯γ = 0.

(ii) If g ∈Gk and γ ∈X +k ,then g♯γ ∈Xk is defined and g♯γ = γ.Proof of Therem 2.3. (i) Let V ∈Lie(G)2k and γ ∈Xk.

Then we have γ(λ) = P|α|≤k Aαλαand γ−1(λ) = P|α|≤k Bαλα for λ ∈C∗. By Proposition 1.5 we have, for λ ∈S1,dL−1γ (V ♯γ )(λ) = −λ −12πi {ZC0γ−1(µ)V (µ)γ(µ)(µ −1)(µ −λ) dµ +ZC∞γ−1(µ)V (µ)γ(µ)(µ −1)(µ −λ) dµ}.Denote by (A) and (B) the first term and the second term on the right-hand side of thisformula.

By assumption we haveV (λ) =Xα≥2kV (0)α λα on ¯I1,V (λ) =Xα≥2kV (∞)−α λ−α on ¯I2.On the circle C0, we haveγ−1(µ)V (µ)γ(µ) =X|α|≤k,|α′|≤k,β≥2kBα′V (0)βAαµα′+β+α.Write1(µ −1)(µ −λ) =Xα′′≥0a(0)α′′µα′′around 0. Then the first integrand isXα′′≥0,|α|≤k,|α′|≤k,β≥2ka(0)α′′Bα′V (0)βAαµα′′+α′+β+α.Since α′′ + α′ + β + α ≥0, in particular α′′ + α′ + β + α ̸= −1, we obtain (A)= 0.

On thecircle C∞, we haveγ−1(µ)V (µ)γ(µ) =X|α|≤k,|α′|≤k,β≥2kBα′V (∞)−β Aαµα′−β+α.Write1(µ −1)(µ −λ) =Xα′′≥2a(∞)−α′′µ−α′′around ∞. Then the second integrand isXα′′≥2,|α|≤k,|α′|≤k,β≥2ka(∞)−α′′Bα′V (∞)−β Aαµ−α′′+α′−β+α.10

Since −α′′ + α′ −β + α ≤−2 + k −2k + k = −2, in particular −α′′ + α′ −β + α ̸= −1,we obtain (B)= 0. (ii) By (i), there is a neighbourhood U of e in G2k such that g♯γ exists and g♯γ = γ foreach γ ∈U.

Since the group G2k is connected, G2k is generated by elements of U. Hencewe obtain (ii).□Proof of Theorem 2.4. Let V ∈Lie(G)k and γ ∈X +k .

Then we have γ(λ) = Pkα=0 Aαλαand γ−1(λ) = Pkα=0 B−αλ−α for λ ∈C∗. By assumption, we haveV (λ) =Xα≥kV (0)α λα on ¯I1,V (λ) =Xα≥kV (∞)−α λ−α on ¯I2.As in the proof of Theorem 2.3, the first integrand in the expression for dL−1γ (V ♯γ ) isXα′′≥0,0≤α≤k,0≤α′≤k,β≥ka(0)α′′B−α′V (0)βAαµα′′−α′+β+α.Since α′′ −α′ + β + α ≥0 −k + k + 0 = 0, in particular α′′ −α′ + β + α ̸= −1, we obtain(A)= 0.

The second integrand isXα′′≥2,0≤α≤k,0≤α′≤k,β≥ka(∞)−α′′B−α′V (∞)−β Aαµ−α′′−α′−β+α.Since −α′′ −α′ −β + α ≤−2 + 0 −k + k = −2, in particular −α′′ −α′ −β + α ̸= −1,we obtain (B)= 0. This proves (i).

By the same argument as in the proof of Theorem 1.3,(ii) follows from (i).□Theorem 2.4 implies that, for each k with 0 ≤k < ∞, the pseudo-actions of the infinitedimensional Lie groups AR and GR on X +k,R collapse to the pseudo-actions of the finitedimensional Lie groups AR/Ak,R and GR/Gk,R, respectively. Moreover, by Theorem 2.2and Proposition 2.1, we see that these pseudo-actions are in fact actions.

In §5 we shallprove by a different argument that the pseudo-action of GR/Gk,R on X +k,R is an action, i.e.without using Theorem 2.2.§3. The natural actionIn this section we study a different group action on the space of extended solutionsM −→ΩG.

This approach depends on recognising explicitly the role of the loop group ΩG.It is well known that ΩG enjoys many of the properties of a finite dimensional generalizedflag manifold (or K¨ahler C-space); one reason for this is that ΩG arises as an orbit of the11

“adjoint action” for the Lie group S1 ˜×ΛG. The semi-direct product here is defined withrespect to the action of S1 on the free loop group ΛG = Map(S1, G) by rotation of the loopparameter.

(That is, (e2πiϕ, γ(e2πit)) · (e2πiψ, δ(e2πit)) = (e2πi(ϕ+ψ), γ(e2πit)δ(e2πi(t−ϕ))). )Indeed, the isotropy subgroup of the point (i, 0) ∈iR˜×Λg is the group S1 × G, soΩG ∼= ΛGG∼= S1 ˜×ΛGS1 × G .The analogy can be strengthened by introducing the “Grassmannian model” of ΩG (see[PS], Chapters 7,8).

This is a submanifold of an infinite dimensional Grassmannian onwhich S1 ˜×ΛG acts transitively, with isotropy subgroup S1 × G, and it provides a geomet-rical basis for the above identification. We shall review briefly this construction.Let e1, .

. ., en be an orthonormal basis of Cn.

Let H(n) be the Hilbert space L2(S1, Cn) =Span{λiej | i ∈Z, j = 1, . .

., n}, and let H+ be the subspace Span{λiej | i ≥0, j =1, . .

., n}.The group ΩUn acts naturally on H(n) by multiplication, and we have amap from ΩUn to the Grassmannian of all closed linear subspaces of H(n), given byγ 7−→γH+ = {γf | f ∈H+}. It is easy to see that this map is injective.

Regarding theimage, one has:Theorem 3.1 ([PS]). The image Gr(n)∞of the map γ 7−→γH+ consists of all closed linearsubspaces W of H(n) which satisfy(1) λW ⊆W,(2) the orthogonal projections W −→H+ and W −→(H+)⊥are respectively Fredholmand Hilbert-Schmidt, and(3) the images of the orthogonal projections W ⊥−→H+ and W −→(H+)⊥consistof smooth functions.Moreover, if γ ∈ΩUn and W = γH+, then deg(det γ) is minus the index of the orthogonalprojection operator W −→H+.□This is the Grassmannian model of ΩUn.Now suppose G is a compact connected Lie group with trivial centre.

Via the adjointrepresentation, we may consider G as a subgroup of Un (where n = dim G) and ΩG as asubgroup of ΩUn. The Hilbert space H(n) inherits the structure of a Lie algebra from gC,and its Hermitian inner product arises from the Killing form of g.Corollary 3.2 ([PS]).

The image of ΩG under the map γ 7−→γH+ consists of all closedlinear subspaces W of H(n) which satisfy(1) λW ⊆W,(2) the orthogonal projections W −→H+ and W −→(H+)⊥are respectively Fredholmand Hilbert-Schmidt, and(3) W sm is a subalgebra of the Lie algebra H(n), where W sm is the space of smoothfunctions in W, and(4) W⊥= λW.□This is the Grassmannian model of ΩG. If G′ is any locally isomorphic group, we canobtain a Grassmannian model for ΩG′, because it suffices to give a model for the identity12

component, and the identity components of ΩG and ΩG′ may be identified. In particular,this shows that one has a Grassmannian model for any compact semisimple Lie group.The complexified group ΛGC also acts transitively on the Grassmannian model, withisotropy subgroup Λ+GC at H+.

Hence one obtains the identificationΩG ∼= ΛGCΛ+GC .It follows that ΛGC = ΩG · Λ+GC. Since ΩG ∩Λ+GC = {e}, we have a factorizationtheorem: any γ ∈ΛGC can be written as γ = γuγ+, where γu, γ+ are uniquely de-fined elements of ΩG, Λ+GC respectively.

If γ ∈ΛGC, we shall write [γ] for the cosetγ(Λ+GC) ∈ΛGC/Λ+GC ∼= ΩG. Thus, the natural action of ΛGC on ΩG, denoted by thesymbol ♮, may be writtenγ♮δ = [γδ] = (γδ)u.Definition.

Let Φ : M −→ΩG be an extended solution. Let γ ∈ΛGC.

We define thenatural action of γ on Φ by γ♮Φ = [γΦ] = (γΦ)u.Let Φ : M −→ΩG be a smooth map.By the Grassmannian model, this may beidentified with a map W : M −→GrG∞, where W(z) = Φ(z)H+. The extended solutionequations for Φ are equivalent to the conditions∂∂¯z C∞W ⊆C∞W(1)∂∂z C∞W ⊆C∞λ−1W(2)where C∞W denotes the space of (locally defined) smooth maps f : M −→H(n) withf(z) ∈W(z) for all z.

The first condition is simply the condition that Φ be holomorphic.The second condition is a horizontality condition on the derivative of Φ (this terminologywill be explained in the next section).Proposition 3.3. Let Φ : M −→ΩG be an extended solution.

Let γ ∈ΛGC. Then γ♮Φis also an extended solution.Proof.

Let W : M −→GrG∞be the map corresponding to Φ; thus γW corresponds to γ♮Φ.If W satisfies equations (1) and (2), then so does γW, as multiplication by γ commuteswith the differentiation with respect to z or ¯z and with multiplication by λ−1.□To understand this action, it is helpful to consider the following concrete examples. Weshall show later that these examples represent special cases of the action.Example 3.4 Let ϕ : M −→Un be a harmonic map with minimal uniton number 1.Then ϕ = π ◦Φ, where Φ : M −→Grk(Cn) is a holomorphic map (for some k), and whereπ : Grk(Cn) −→Un is a totally geodesic embedding.

More explicitly, there exists somea ∈Un such that ϕ(z) = a(πΦ(z) −π⊥Φ(z)), where πΦ(z) denotes the orthogonal projection13

Cn −→Φ(z) with respect to the Hermitian inner product of Cn. The embedding π :Grk(Cn) −→Un is then given by V 7−→a(πV −π⊥V ).

Conversely, any map ϕ of thisform (with Φ non-constant) is a harmonic map with minimal uniton number 1. Since thestandard action of the complex group GLn(C) = Aut(Cn) on Grk(Cn) is holomorphic,we obtain an action of GLn(C) on holomorphic maps M −→Grk(Cn).

Thus, an elementA of GLn(C) gives rise to a new holomorphic map A♮ϕ = π(AΦ).Example 3.5 It is well known (see [EL]) that all harmonic maps ϕ : S2 −→CP n areof the form ϕ = π ◦Φ, where Φ : S2 −→Fr,r+1(Cn+1) is (a) holomorphic with respectto the natural complex structure of Fr,r+1(Cn+1), and (b) horizontal with respect tothe projection π : Fr,r+1(Cn+1) −→CP n. Here, Fr,r+1(Cn+1) is the space of flags ofthe form {0} ⊆Er ⊆Er+1 ⊆Cn+1. Conversely, given a holomorphic horizontal mapΦ, the map ϕ = π ◦Φ is harmonic.If the flag corresponding to Φ(z) is denoted by{0} ⊆Wr(z) ⊆Wr+1(z) ⊆Cn+1, then the holomorphicity and horizontality conditionsare∂∂¯z C∞Wi ⊆C∞Wi, i = r, r + 1(1)∂∂z C∞Wr ⊆C∞Wr+1.

(2)The standard action of GLn+1(C) on Fr,r+1(Cn+1) preserves both these conditions be-cause of the linearity of the derivative. Hence for any A ∈GLn+1(C), we obtain a newharmonic map A♮ϕ = π(AΦ).

This action of GLn+1(C) on harmonic maps S2 −→CP nwas studied in [Gu].More generally, if M is a Riemann surface, complex isotropic harmonic maps ϕ : M −→CP n correspond to holomorphic horizontal maps Φ : M −→Fr,r+1(Cn+1). Thus, weobtain an action of GLn+1(C) on complex isotropic harmonic maps.Example 3.6 There is a similar description of harmonic maps from S2 to Sn or RP n. Itsuffices to consider harmonic maps ϕ : S2 −→RP 2n, as the other cases can be deducedfrom this one.

Let Zn be the space of (complex) n-dimensional subspaces V of C2n+1 suchthat V and V are orthogonal with respect to the standard Hermitian inner product ofC2n+1, i.e. such that V is “isotropic”.

There is a projection map π : Zn −→RP 2n, whichassociates to V the (+1)-eigenspace of the operator x 7−→¯x on (V ⊕V )⊥. It is known (see[Ca1],[Ca2],[Ba]) that such harmonic maps are of the form ϕ = π ◦Φ where Φ : S2 −→Znis a holomorphic map which is horizontal with respect to π.The holomorphicity andhorizontality conditions are∂∂¯z C∞Φ ⊆C∞Φ(1)∂∂z C∞Φ ⊥C∞Φ.

(2)The standard action of SOC2n+1 on Zn preserves both these conditions, hence we obtainan action of SOC2n+1 on harmonic maps.More generally, if M is a Riemann surface, isotropic harmonic maps from M into Sn orRP n correspond to holomorphic horizontal maps Φ : M −→Zn, and we obtain an actionof SOC2n+1 on such maps.14

The harmonic maps arising in these three examples fit into a more general framework,described in [Br], [BR], which we shall recall briefly. Let G/H be a generalized flag mani-fold, i.e.

the orbit of a point P of g under the adjoint representation. It is well known thatthe complex group GC acts transitively on G/H.

If GP is the isotropy subgroup at P, thenwe have an identification G/H ∼= GC/GP . This endows G/H with a complex structure,and the holomorphic tangent bundle of G/H may be identified with the homogeneousbundle GC ×GP (gC/gP ).Without essential loss of generality (see [BR]) we may assume that the linear endomor-phism ad P on gC has eigenvalues in iZ.

If the (iℓ)-eigenspace is denoted by gℓ, then onehas g0 = hC, gP = Li≤0 gi, and [gi, gj] ⊆gi+j. Let kC = Li even gi.

Then (gC, kC)is a symmetric pair, and (up to local isomorphism) one obtains a symmetric space G/K,where K = {g ∈G | g(expπP) = (exp πP)g}.The natural map π : G/H −→G/K is a “twistor fibration”; it gives rise to a relationbetween harmonic maps M −→G/K and holomorphic maps M −→G/H. The simplestaspect of this relation may be expressed in terms of the super-horizontal distribution, whichis by definition the holomorphic subbundle GC ×GP (gP ⊕g1/gP ) of GC ×GP (gC/gP ) ∼=T1,0G/H.

A holomorphic map Φ : M −→G/H is said to be super-horizontal if it istangential to the super-horizontal distribution. It is shown in [Br],[BR] that:(†) If Φ is holomorphic and super-horizontal, then ϕ = π ◦Φ is harmonic.Clearly the action of GC preserves holomorphicity and super-horizontality.Hence weobtain an action of GC on the set of those harmonic maps M −→G/K which are of theabove form.

This is precisely the action described in Examples 3.5 and 3.6, since in thosecases it turns out that gi = 0 for |i| > 2, hence (for holomorphic maps) the conceptsof horizontality and super-horizontality coincide. (This is also, trivially, the action ofExample 3.4, where K = H.)Before leaving these examples, we make some brief comments on further generalizations.It is possible to weaken the hypothesis of super-horizontality in (†).

Indeed, in [BR], itis shown that holomorphicity and horizontality, or the even weaker condition of “J2-holomorphicity”, implies that ϕ is harmonic. In the case M = S2, one then has a converseto (†), namely that any harmonic map ϕ : S2 −→G/K is of the form ϕ = π ◦Φ forsome J2-holomorphic map Φ : S2 −→G/H, for a suitable twistor fibration π : G/H −→G/K.

These generalizations are not so useful from the point of view of the action of GC,because neither holomorphicity and horizontality nor J2-holomorphicity are preserved bythis action in general. On the other hand, there is a natural filtration of T1,0G/H by theholomorphic subbundles T (ℓ) = GC×GP (Li≤ℓgi)/gP .

Let us say that a holomorphic mapΦ : M −→G/H is ℓ-holomorphic if it is tangential to T (ℓ). Thus, a 1-holomorphic mapis a holomorphic super-horizontal map; an ∞-holomorphic map is simply a holomorphicmap.

Clearly the action of GC preserves ℓ-holomorphicity. However, the relevance of thisremark depends on the answer to the question: what is the geometrical significance of themaps ϕ = π ◦Φ, where Φ is ℓ-holomorphic?Finally, we shall explain why the actions in the above examples are special cases ofthe natural action of ΛGC on extended solutions.

Because of the previous discussion, it15

suffices to do this for the action of GC on 1-holomorphic maps Φ : M −→G/H, whereG/H = Ad(G)P.First, let us define a loop γP ∈ΩG by γP (λ) = exp 2πtP, whereλ = e2πit. Then G/H may be realized as a submanifold of ΩG, namely as the orbit ofγP under conjugation by G. The associated symmetric space G/K may be realized as asubmanifold of G, namely as the conjugacy class of exp πP.

Thus, the twistor fibrationπ : G/H −→G/K is just a restriction of the map π : ΩG −→G (evaluation at −1):G/H −−−−→ΩGyyG/K −−−−→GRecall that we have the identifications T CP G/H = Li̸=0 gi, and T CγP ΩG ∼= T Ce ΩG ∼=Lℓ̸=0(λℓ−1)gC.Lemma 3.7. The derivative at P of the embedding G/H −→ΩG identifies gℓwith(λ−ℓ−1)gℓ.Proof.

Let U ∈gℓ. This corresponds to the initial tangent vector to the curve Ad(exp sU)Pthrough P in G/H = Ad(G)P, i.e.

to the curve (exp sU)γP (exp sU)−1 through γP inΩG. By left translation we obtain the curve γ−1P (exp sU)γP (exp sU)−1 through e in ΩG.Now,γ−1P (exp sU)γP(exp sU)−1 = exp Ad[exp(−2πtP)]sU (exp sU)−1= exp (e−2πt ad P sU) exp(−sU)= exp(e−2πtiℓsU) exp(−sU)= exp(s(λ−ℓ−1)U).The initial tangent vector of this curve is (λ−ℓ−1)U.□In particular, the super-horizontal distribution of T1,0G/H maps into the subbundle ofT1,0ΩG defined by (λ−1 −1)gC, so we obtain:Proposition 3.8.

Via the embedding G/H −→ΩG, a holomorphic super-horizontal mapinto G/H goes to an extended solution into ΩG. Moreover, the action of GC on G/Hcorresponds to the action of the subgroup GC of ΛGC on ΩG.□More generally, the concept of ℓ-holomorphicity for a map Φ : M −→G/H may beinterpreted in terms of the corresponding map M −→ΩG.

Let us say that a holomorphicmap M −→ΩG is ℓ-holomorphic if it is tangential to the (holomorphic) subbundle H(ℓ)of T1,0ΩG defined by L1≤i≤ℓ(λ−i −1)gC. Thus, ℓ-holomorphic maps interpolate betweenextended solutions (ℓ= 1) and general holomorphic maps (ℓ= ∞).

By Lemma 3.7, ℓ-holomorphic maps into G/H go (via the embedding G/H −→ΩG) to ℓ-holomorphic mapsinto ΩG. If Φ is ℓ-holomorphic, and γ ∈ΛGC, then γ♮Φ is clearly also ℓ-holomorphic.

As16

in the finite dimensional case, however, the geometrical significance of maps ϕ = π ◦Φ :M −→G, where Φ is ℓ- holomorphic, is not clear.In contrast to the actions described in §1 and §2, the natural action is very easy towork with. In particular, it has the advantage that it is always well defined (so we have anaction, rather than a pseudo-action).

In the next section we shall give some elementaryproperties of this action.§4. Properties of the natural actionIn this section we shall always take M to be a compact Riemann surface and G = Un.The version of the extended solution equations used in the last section is due to Segal(see [Se]), who used it to give a new proof of the factorization theorem of [Uh] for harmonicmaps S2 −→Un, and of the classification theorem (see [EL]) for harmonic maps S2 −→CP n. We shall review Segal’s approach here, before discussing further properties of thenatural action.

The main technical result is the following version of Theorem 1.2:Theorem 4.1 ([Se]). Let Φ : M −→ΩUn be an extended solution.

Then there exists aloop γ ∈ΩUn and a non-negative integer m such that the map W = γΦH+ satisfies(i) λmH+ ⊆W(z) ⊆H+, for all z ∈M, and (ii) Span{W(z) | z ∈M} = H+. Moreover,m ≤n −1.□The extended solution γΦ is said to be normalized.

For example, let f : M −→Grk(Cn)be a holomorphic map, so that ϕ = πf −π⊥f : M −→Un is a harmonic map (as in Example3.4). Then the corresponding extended solution Φ = πf + λπ⊥f is normalized if and only iff is “full”, i.e.

Span{f(z) | z ∈M} = Cn. It can be shown that condition (ii) of Theorem4.1 is equivalent to the Uhlenbeck normalization (condition (ii) of Theorem 1.2).Let us assume that W corresponds to a normalized extended solution, as in the theorem.Then there is a canonical flag associated to W, namelyλmH+ ⊆W = W(m) ⊆W(m−1) ⊆· · · ⊆W(0) = H+where W(i) : M −→Gr(n)∞is the holomorphic map defined by W(i) = λ−(m−i)W ∩H+.Strictly speaking, this formula defines a holomorphic map with a finite number of re-movable singularities, but we shall use the notation W(i) to mean the map obtained byremoving these.

The canonical flag satisfies the following conditions:λW(i) ⊆W(i+1)(0)∂∂¯z C∞W(i) ⊆C∞W(i)(1)∂∂z C∞W(i) ⊆C∞W(i−1). (2)17

In fact, these equations are equivalent to the extended solution equations for Φ, as theholomorphicity condition for W is given by (1), and the horizontality condition for Wfollows from (2) and the definition of Wm−1. It is an immediate consequence that eachmap W(i) satisfies the extended solution equations.

Hence, by the Grassmannian model,we have W(i) = Φ(i)H+ for some extended solution Φ(i).From Example 3.5, we see that condition (2) can be interpreted as saying that the(holomorphic) map (W(i), W(i−1)) is horizontal with respect to the map (E(i), E(i−1)) 7−→E⊥(i) ∩E(i−1). This is why the equation∂∂z C∞W ⊆C∞λ−1W is called the horizontalitycondition.

Each map W ⊥(i) ∩W(i−1) is a harmonic map into a Grassmannian.From condition (0) we have λW(i−1) ⊆W(i) ⊆W(i−1), so we can derive some furtherinformation. The map W(i−1)/λW(i−1) defines a holomorphic vector bundle on M, andmultiplication by Φ−1(i−1) defines a smooth isomorphism of this bundle with the trivialbundle M × H+/λH+ ∼= M × Cn.

Through this isomorphism, the map W(i)/λW(i−1)corresponds to a map Ψi, and we have Φ(i) = Φ(i−1)Ψi. Each map Ψi is necessarily ofthe form πfi + λπ⊥fi, where fi is a map from M to a Grassmannian.

By construction, fiis holomorphic with respect to a complex structure which is obtained by “twisting” thestandard complex structure by Φ(i−1). Hence we have the factorization theorem: Φ canbe written as Φ = Ψ1 .

. .

Ψm, where Ψi = πfi + λπ⊥fi, and each sub-product Ψ1 . .

. Ψi is anextended solution.This completes our review of [Se], to which the reader is referred for further details.

Asfor the generalization to a pluriharmonic maps, we can show that Theorem 4.1 holds alsofor a compact complex manifold M. Moreover, the above argument for the canonical flagand the factorization also works for the higher dimensional case, if we consider meromorpicmaps and coherent sheaves instead of holomorphic maps and holomorphic vector bundles(cf. [OV]).The finiteness properties of extended solutions described above may be expressed interms of a filtration of the “algebraic loop group” by certain finite dimensional varieties.The algebraic loop group is defined as follows:Definition.

ΩalgUn = {γ ∈ΩUn | γ(λ) is polynomial in λ, λ−1}.A Grassmannian model for ΩalgUn may be deduced from that of Gr(n)∞:Proposition 4.2 ([PS]). The image of ΩalgUn, under the map ΩUn −→Gr(n)∞, is thesubspace Gr(n)alg of Gr(n)∞consisting of linear subspaces W which satisfyλkH+ ⊆W ⊆λ−kH+ for some k.Moreover, if γ ∈ΩalgUn and W = γH+, then for such minimal k we have deg(det γ) =12(dim λ−kH+/W −dim W/λkH+).□18

If we defineΛalgGLn(C) = {γ ∈ΛGLn(C) | γ(λ), γ(λ)−1 are polynomial in λ, λ−1},then we obtain the identificationΩalgUn ∼= ΛalgGLn(C)Λ+algGLn(C),where ΛalgUn, Λ+algGLn(C) are defined in the obvious way. This is analogous to the identi-fication ΩUn ∼= ΛGLn(C)/Λ+GLn(C) described in the last section.

However, in the caseof algebraic loops, one can replace ΛalgGLn(C) by an even larger group, the semi-directproduct C∗˜×ΛalgGLn(C), where the action of C∗on ΛalgGLn(C) is given by “re-scaling”,i.e. (v · γ)(λ) = γ(v−1λ) for all v ∈C∗, γ ∈ΛalgGLn(C).

The group C∗also acts onΩalgUn, byv♮γ = [v · γ]where square brackets (as usual) denote cosets in ΛalgGLn(C)/Λ+algGLn(C) ∼= ΩalgUn.We use the “natural” notation for this action, because the formula(v, γ)♮δ = γ♮v♮δdefines an action of C∗˜×ΛalgGLn(C) on ΩalgUn, which extends the natural action ofΛalgGLn(C) on ΩalgUn. Thus we obtain finally the identificationsΩalgUn ∼= ΛalgGLn(C)Λ+algGLn(C)∼= C∗˜×ΛalgGLn(C)C∗˜×Λ+algGLn(C).Mitchell ([Mi]) introduced the following subspaces of Gr(n)alg (see also §1 of [Se]):Definition.

Fn,k = {W ⊆H(n) | λkH+ ⊆W ⊆H+, λW ⊆W, dim H+/W = k}.It can be shown that Fn,k is a connected complex algebraic subvariety of the Grass-mannian Grkn−k(Ckn).Explicitly, if we make the identification Ckn ∼= H+/λkH+ =Span{[λiej] | 0 ≤i ≤k −1, 1 ≤j ≤n}, thenFn,k ∼= {E ∈Grkn−k(Ckn) | NE ⊆E},where N is the nilpotent operator on Ckn given by the multiplication by λ. The spaceFn,k is preserved by the action of Λ+GLn(C), since (by definition) this group fixes H+.The action of Λ+GLn(C) on Fn,k collapses to the action of the finite dimensional groupGn,k = {X ∈GLkn(C) | XN = NX}.Indeed, the action of Λ+GLn(C) factors through the homomorphism Λ+GLn(C) −→GLn(C[λ]/(λk)) defined by Pi≥0 λiAi 7−→Pk−1i=0 λiAi, and we have GLn(C[λ]/(λk)) ∼=Gn,k.

This is a complex Lie group of dimension kn2. The action of C∗also preserves Fn,k,19

for the action of an element u ∈C∗induces the linear transformation Tu[λiej] = [uiλiej],and so TuN = uNTu.In these terms, we see that a normalized extended solution is a “horizontal” holomor-phic map Φ : M −→Fn,k, where the integer k = deg det Φ(z) represents the connectedcomponent of ΩUn which contains the image of Φ. The minimal uniton number m satisfiesthe conditions m ≤n −1, m ≤k.

It is known (see [Mi]) that H2(Fn,k; Z) ∼= Z, so Φ has atopological “degree” d, which (with appropriate choice of orientations) is a non-negativeinteger. The geometrical significance of d is that it represents the energy of the corre-sponding harmonic map ϕ : M −→Un (see [EL],[Va],[OV]).

From the discussion above wehave:Proposition 4.3. The natural action of Λ+GLn(C) on normalized extended solutions Φpreserves(1) the connected component k of ΩUn containing the image of Φ,(2) the minimal uniton number m of Φ, and(3) the degree d of Φ (i.e.

the energy of the corresponding harmonic map).Moreover, for a fixed choice of k, the action of Λ+GLn(C) on normalized extended solu-tions collapses to the action of the finite dimensional (complex) Lie group Gn,k.□In this proposition, Λ+GLn(C) could be replaced by the group C∗˜×Λ+algGLn(C), and Gn,kby C∗˜×Gn,k; we leave the verification of this to the reader. The existence of an action ofC∗on extended solutions was first noticed by Terng (see §7 of [Uh]).§5.

Relation between the Uhlenbeckpseudo-action and the natural actionIn this section we shall show that the Uhlenbeck pseudo-action discussed in §2 and thenatural action defined in §3 coincide on harmonic maps of finite uniton number. We beginby considering a special case.For any ε with 0 < ε < 1, we have an injective homomorphism as real Lie groupsΛ+GLn(C) −→ΛI,RGLn(C) ⊆GR,γ 7−→ˆγdefined byˆγ(λ) = γ(λ)for |λ| ≤ε,γ(¯λ−1)−1∗for |λ| ≥1/εfor γ ∈Λ+GLn(C).Theorem 5.1.

If γ ∈Λ+GLn(C) and δ ∈Xk,R ⊆ΩUn for 0 ≤k ≤∞, then ˆγ♯δ ∈Xk,Ris well-defined andγ♮δ = ˆγ♯δ.20

Proof. By the decomposition ΛGLn(C) ∼= ΩUn · Λ+GLn(C), we have γδ = (γδ)u(γδ)+,where (γδ)u ∈ΩUn, (γδ)+ ∈Λ+GLn(C).

Note that (γδ)u(λ) = γ(λ)δ(λ)(γδ)−1+ (λ) extendsholomorphically to {λ ∈C | 0 < |λ| < 1}. Define(ˆγδ)I(λ) = (γδ)+(λ)for |λ| ≤ε,{(γδ)+(¯λ−1)}−1∗for |λ| ≥1/ε,namely, (ˆγδ)I = (γδ)ˆ+ ∈ΛI,RGLn(C).

Define(ˆγδ)E(λ) = (γδ)u(λ)for 0 < |λ| ≤1,{(γδ)u(¯λ−1)}−1∗for1 ≤|λ| < ∞.By Painlev´e’s Theorem we have (ˆγδ)E ∈ΛE,RGLn(C), and moreover (ˆγδ)E ∈Xk,R,because (γδ)u = γδ(γδ)−1+ , δ ∈Xk,R.For 0 < |λ| ≤ε, we have(ˆγδ(ˆγδ)−1I )(λ) = γ(λ)δ(λ)(γδ)−1+ (λ)= (γδ)u(λ) = (ˆγδ)E(λ).For 1/ε ≤|λ| < ∞, we have(ˆγδ(ˆγδ)−1I )(λ) = γ(¯λ−1)−1∗δ(¯λ−1)−1∗{(γδ)+(¯λ−1)}∗= {γ(¯λ−1)δ(¯λ−1)(γδ)+(¯λ−1)}−1∗= {(γδ)u(¯λ−1)}−1∗= (ˆγδ)E(λ).Hence ˆγδ(ˆγδ)−1I= (ˆγδ)E = ˆγ♯δ. Thus we obtain ˆγ♯δ = (γδ)u = γ♮δ.□Corollary 5.2.

If γ ∈Λ+GLn(C) and Φ : M −→ΩUn is an extended solution such thatΦλ is holomorphic in λ ∈C∗, then we haveγ♮Φ = ˆγ♯Φ.Proof. By assumption the image of Φ is contained in XR = X∞,R.

Hence the corollaryfollows from Theorem 5.1.□In §2, we saw that the Uhlenbeck pseudo-action of GR on Xk,R collapses to the pseudo-action of the finite dimensional Lie group GR/Gk,R ∼= AR/Ak,R, and in §4 that the naturalaction of Λ+GLn(C) on Fn,k collapses to the action of the Lie group Gn,k. Evidently, wehave GR/Gk,R ∼= Gn,k as real Lie groups.

From Theorem 5.1, and by using the sameargument as was used at the end of §2, we see that the pseudo-action of GR (or GR/Gk,R)on extended solutions with finite uniton number is an action, and coincides with the actionof Λ+GLn(C) (or Gn,k). Hence:Corollary 5.3.

The Uhlenbeck pseudo-action of GR on extended solutions (or harmonicmaps) with finite uniton number coincides with the natural action of Λ+GLn(C).□21

§6. Deformations of harmonic mapsLet {gt} be a curve in ΛGC, i.e.

a continuous map t 7−→gt from an open interval of Rto ΛGC, with g0 = e. Let Φ : M −→ΩG be an extended solution. Then the formulaΦt = g♮tΦdefines a continuous family of extended solutions passing through Φ (a “deformation” ofΦ).

For example, we can take {gt} to be a one parameter subgroup {exp tβ}, for β ∈ΛgC.The same observation applies to a curve in C∗˜×ΛalgGLn(C), providing that the extendedsolution Φ takes values in ΩalgG.Now, it may happen that limt→∞Φt exists, even if limt→∞gt does not exist, and inthis case we obtain an extended solution Φ∞= limt→∞Φt which is not, a priori, of theform g♮Φ. Some examples of this “completion” process were studied in [BG] for the caseof the Uhlenbeck action ♯.

By using the action ♮, however, we can obtain more detailedinformation. The reason for this is that, for certain β, the curve γ 7−→(exp tβ)♮γ has asimple geometrical interpretation: it is a flow line of the gradient vector field of a naturalMorse-Bott function on ΩG.The basic example of a Morse-Bott function on ΩG is the “perturbed energy functional”E + cKQ, where E is the energy functionalE(γ) = 12ZS1 ||γ−1γ′||2,and KQ is the momentum functionalKQ(γ) =ZS1 ⟨⟨γ−1γ′, Q⟩⟩,for some fixed Q ∈g, and where c is a non-zero constant.

The critical points of E + cKQare simply the homomorphisms S1 −→C(TQ), where C(TQ) is the centralizer in G ofthe torus TQ generated by Q.It is classical that this is a Morse-Bott function.Theflow of −∇E with respect to the K¨ahler metric is given by the re-scaling action of the oneparameter semi-group {e−t | t ≥0}, and the flow of −∇KQ is given by the (natural) actionof {exp itQ}. Hence the flow of E + cKQ is given by the action of {exp it(i, cQ) | t ≥0}(which is contained in C∗× GC, and hence in C∗˜×ΛalgGC).As a first application, let us consider the case where Q is a regular point of g. SinceQ generates (by definition) a maximal torus T, which is equal to its own centralizer, thecritical points are the homomorphisms S1 −→T; in particular, they are isolated.

Thestable manifold of a critical point is a cell in ΩG of finite codimension, the so calledBirkhoffcell (see [PS]). If Φ : M −→ΩG is a holomorphic map, then Φ(z) must lie in asingle Birkhoffmanifold for all but a finite number of points z ∈M, so we obtain:Proposition 6.1.

Let Φ : M −→ΩalgG be an extended solution. Then there exists acurve {gt} in C∗˜×ΛalgGC such thatΦ∞(z) = limt→∞gt♮Φ(z)22

defines a constant (extended solution) Φ∞: M \ S −→ΩG, where the set S consists of afinite number of removable singularities of Φ∞.□This is an example of the “bubbling off” phenomenon for harmonic maps ([SU]). Propo-sition 6.1 answers positively the question posed at the end of §7 of [BG], namely whetherany extended solution can be reduced to a constant map by applying the “modified com-pletion” procedure.

However, it is perhaps of more interest to find deformations where thesingularities do not occur, and this we shall do next.For our second application, we shall consider the function KQ.The set of criticalpoints is ΩC(TQ), which is infinite dimensional. However, for the application to extendedsolutions, we are primarily interested in the restriction of KQ to the finite dimensionalsubvariety Fn,k (with G = Un).

Let us now consider the flow of −∇KQ, which is givenby the natural action of {exp itQ} on ΩUn.We may consider {exp itQ} to be a oneparameter subgroup of Λ+GLn(C), so it preserves Fn,k. (Indeed, {exp itQ} is a oneparameter subgroup of GLn(C) = Gn,1 ⊆Gn,k, in the notation of §4.) We shall use thisflow, with a suitable choice of Q, to prove:Theorem 6.2.

Let Φ = Pmα=0 Tαλα : M −→Fn,k be a normalized extended solution.If rank T0(z) ≥2 for all z ∈M, then Φ can be deformed continuously through extendedsolutions to an extended solution Ψ : M −→ΩUn−1.Proof. Let Q = iπL where πL : Cn −→Cn denotes orthogonal projection onto a complexline L in Cn.

The homomorphism GLn(C) −→Gn,1 −→Gn,k ⊆GLkn(C) will be denotedX 7−→X′. Thus, if Ckn is identified with H+/λkH+ as usual, we have X′(λiv) = λiXvfor any v ∈Cn.

Observe that (πL)′ = πL′, where L′ is the k-plane L ⊕λL ⊕· · · ⊕λk−1L.Consider the flow on the Grassmannian Grkn−k(Ckn) which is given by the action ofthe one parameter subgroup {(exp itQ)′} (= {exp itQ′}) of GLkn(C). It is well knownthat this is the downwards gradient flow of a Morse-Bott function on Grkn−k(Ckn), suchthat(1) the set of absolute minima is GL = {W | L′ ⊆W}, and(2) the stable manifold of GL (i.e.

the union of the flow lines which terminate on GL) isSL = {W | W ⊥∩L′ = {0}}. (These assertions represent a mild generalization of the standard Schubert cell decompo-sition of a Grassmannian.

They are explained in more detail in the Appendix. )Observe that GL ∩Fn,k = Fn−1,k if we take L = Span{en}.

Thus, if the image of theextended solution Φ is contained entirely in SL∩Fn,k, the formula Φt = (exp itQ′)♮Φ givesa continuous deformation of Φ into Fn−1,k. To prove the theorem, therefore, it suffices toshow that any extended solution satisfying the hypotheses lies in SL ∩Fn,k, for some lineL.23

Let Φ be an extended solution satisfying the hypotheses. LetY Φ = {L | Φ(z) /∈SL for some z ∈M}.Thus, Y Φ is the set of “bad” lines in Cn.

We shall show that dimC Y Φ < n −1, whichimplies immediately that not all lines are “bad”.To do this, note thatΦ(z) /∈SL ⇐⇒Φ(z)⊥∩L′ ̸= {0}⇐⇒Φ(z)⊥∩L ̸= {0}⇐⇒Φ(z) ⊆L⊥(the middle step follows from the fact that both Φ(z)⊥and L′ are preserved by theadjoint of multiplication by λ, i.e. by the linear transformation λ∗of H+/λkH+ givenby λ∗(λiej) = λi−1ej, 1 ≤i ≤k −1, and λ∗(ej) = 0).

Let X = {(L, W) ∈CP n−1 ×Fn,k | W ⊆L⊥}. Let p1 : X −→CP n−1, p2 : X −→Fn,k be the projection maps.Then we have Y Φ = p1(p−12 (Φ(M))), so dimC Y Φ ≤dimC p−12 (Φ(M)).

We claim thatdimC p−12 (Φ(z)) ≤n −3 for all z ∈M. Since dimC M = 1, we may then conclude thatdimC Y Φ < n −1, as required.

From the expression Φ = Pmα=0 Tαλα we see thatp−12 (Φ(z)) = {L | Φ(z) ⊆L⊥} = P(Ker T ∗0 (z)),so the claim follows from the hypothesis.□It is appropriate at this point to make some comments on the use of Morse theory inthe proof of Theorem 6.2. The fact that Fn,k is in general a singular variety (to whichordinary Morse theory does not apply) is irrelevant for our purposes, as we are concernedonly with the given flow.

However, to study this flow in practice, it is useful to regardit as the restriction of a flow on the Grassmannian Grkn−k(Ckn), where it is indeed thedownwards gradient flow of a Morse-Bott function.This type of Morse-Bott functionis well understood: it is an example of a “height function” on an orbit of the adjointrepresentation of a compact Lie group. In the Appendix to this paper, we summarize thebasic facts concerning such height functions.

Briefly, the situation is as follows. Considera finite dimensional generalized flag manifold of G, i.e.

an orbit Ad(G)P of a point P ofg under the adjoint representation. Let Q be any element of g. Then one may define theheight function hQ : Ad(G)P −→R by hQ(X) = ⟨⟨X, Q⟩⟩.

This is a Morse-Bott functionand its non-degenerate critical manifolds can be described explicitly in Lie theoretic terms.Let ∇hQ be the gradient of hQ with respect to the natural K¨ahler metric on Ad(G)P.Then the flow line of −∇hQ which passes through a point X of Ad(G)P is given byt 7−→(exp itQ)♮X.This can be used to obtain results analogous to Theorem 6.2 for harmonic mapsM −→G/K, for various inner symmetric spaces G/K, because the total space of thecorresponding twistor fibration is a generalized flag manifold. Although this is simply aspecial case of the discussion above, it is instructive to give a direct argument (avoiding theparaphernalia of extended solutions), and this we shall do for each of the three examples24

considered in §3. This will, incidentally, provide some examples of extended solutions Φwhich satisfy the hypotheses of Theorem 6.2.Example 6.3 (cf.

Example 3.4). Let Hold(S2, Grk(Cn)) denote the space of holomor-phic maps Φ : S2 −→Grk(Cn) which have degree d. It is well known that this space isconnected.

However, we shall give a proof of this fact as an illustration of the techniqueintroduced above.We identify Grk(Cn) with the orbit Ad(Un)P in un, where P = iπV for some k-planeV . Let Q = iπ1, where π1 : Cn −→C denotes orthogonal projection onto the line spannedby the first standard basis vector.

The action of the one parameter subgroup {exp itQ}gives the downwards gradient flow of a Morse-Bott function Grk(Cn) −→R. (See theAppendix.) The critical points are those k-planes W ∈Grk(Cn) for which [iπ1, iπW ] = 0,i.e.

for which C ⊆W or W ⊆C⊥. Thus there are two connected critical manifolds:G+ = {W | C ⊆W} ∼= Grk−1(Cn−1)G−= {W | W ⊆C⊥} ∼= Grk(Cn−1).The corresponding stable manifolds are:SQ(G+) = G+SQ(G−) = {W | W ∩C = {0}}.We claim that the inclusionsHold(S2, Grk(Cn−1)) ∼= Hold(S2, G−) −→Hold(S2, SQ(G−)) −→Hold(S2, Grk(Cn))induce bijections on the sets of connected components.

In the case of the first inclusion,this is so because, if Φ(S2) ⊆SQ(G−), then {(exp itQ)♮Φ}0≤t≤∞provides a continuousdeformation of Φ into G−. For the second inclusion, it is because Hold(S2, SQ(G−)) isobtained from the manifold Hold(S2, Grk(Cn)) by removing a closed subvariety of com-plex codimension 1.

By induction it follows that Hold(S2, Grk(Cn)) has the same numberof connected components as Hold(S2, CP k). However, from the usual description of holo-morphic maps S2 −→CP k in terms of polynomials, it follows that this space is connected.By modifying this argument slightly (see the proof of Theorem 6.5 below), it can beshown that Hold(M, Grk(Cn)) is connected for any compact Riemann surface M, providingthat d ≥2g, where g is the genus of M. The last restriction ensures that Hold(M, S2) isconnected (see Corollary 1.3.13 of [Na]).

In fact, these conditions may be weakened ; forexample in [To] it is shown that Hold(M, S2) is connected when d ≥g, and it follows from[FL] that Hold(M, S2) is connected for “generic ” M when d ≥(g + 3)/2.Example 6.4 (cf. Example 3.5).

Let Harmd(S2, CP n) denote the space of harmonicmaps ϕ : S2 −→CP n which have degree d. If Φ : S2 −→Fr,r+1(Cn+1) corresponds to aharmonic map ϕ as in Example 3.5, and if deg Φ = (deg Wr, deg Wr+1) = (k, l), then wehaved = l −k,E = l + kwhere E denotes the (suitably normalized) energy.If n > 1, it is easy to constructexamples of harmonic maps ϕ1, ϕ2 with deg ϕ1 = deg ϕ2 but E(ϕ1) ̸= E(ϕ2), so the spaceof harmonic maps of fixed degree cannot be connected. However, we can prove:25

Theorem 6.5. (i) The inclusion Harmd(S2, CP 2) −→Harmd(S2, CP n) induces a bi-jection on the sets of connected components, if n ≥2.

(ii) More generally, the same istrue if harmonic maps from S2 are replaced by complex isotropic harmonic maps from anycompact Riemann surface M.Proof. It suffices to give the proof of (ii).

Let Φ = (Wr, Wr+1) : M −→Fr,r+1(Cn+1)be a holomorphic horizontal map associated to ϕ. We shall use the method of Example6.3 to show that Φ may be deformed into Fr,r+1(Cn), if r < n −1.

Hence, by induction,we obtain a map (also denoted by Φ) whose image lies in Fr,r+1(Cr+2). By repeatingthis argument with Φ∗= (W ⊥r+1, W ⊥r ), we can similarly deform Φ into {(Er, Er+1) ∈Fr,r+1(Cr+2) | Cr−1 ⊆Er}.

Thus we obtain a deformation of ϕ into P(Cr+2/Cr−1), andhence (by applying a projective transformation) into CP 2.We identify Fr,r+1(Cn+1) with the orbit Ad(Un+1)(iπVr +iπVr+1), where (Vr, Vr+1) is afixed element of Fr,r+1(Cn+1). Let π⊥n denote orthogonal projection onto the line (Cn)⊥inCn+1 spanned by the last standard basis vector, and set Q = iπ⊥n .

We shall use the Morse-Bott function on Fr,r+1(Cn+1) whose downwards gradient flow is given by the action of{exp itQ}. A point (Er, Er+1) is a critical point if and only if [iπ⊥n , iπEr + iπEr+1] = 0,i.e.

the line (Cn)⊥is contained in Er, E⊥r ∩Er+1, or E⊥r+1. The three connected criticalmanifolds are:F + = {(Er, Er+1) | (Cn)⊥⊆Er} = Fr−1,r(Cn)F 0 = {(Er, Er+1) | (Cn)⊥= E⊥r ∩Er+1} ∼= Grr(Cn)F −= {(Er, Er+1) | (Cn)⊥⊆E⊥r+1} = Fr,r+1(Cn).The corresponding stable manifolds are:SQ(F +) = F +SQ(F 0) = {(Er, Er+1) | (Cn)⊥⊆Er+1, (Cn)⊥∩Er = {0}}SQ(F −) = {(Er, Er+1) | (Cn)⊥∩Er+1 = {0}}.If Φ(S2) ⊆SQ(F −), then {(exp itQ)♮Φ}0≤t≤∞provides a continuous deformation of Φinto F −= Fr,r+1(Cn).So it suffices to show that Φ can be deformed into SQ(F −).In Example 6.3, the corresponding fact was true for dimensional reasons, but a differentargument is necessary in the present situation as the space of holomorphic horizontal mapsis not in general a manifold.

(The argument we are about to give is also needed in Example6.3, in the case of a Riemann surface. )We claim that there exists some A ∈Un+1 such that A♮Φ(M) ⊆SQ(F −), i.e.

AWr+1(z) ̸⊇(Cn)⊥for all z ∈M; from this one can construct the required deformation, as Un+1 isconnected. It suffices to find some line L such that Wr+1(z) ̸⊇L for all z ∈M.

LetY Φ = {L ∈CP n | L ⊆Wr+1(z) for some z ∈M}.Then our claim is that Y Φ ̸= CP n. Let X = {(L, Er, Er+1) ∈CP n × Fr,r+1(Cn+1) | L ⊆Er+1}. Let p1 and p2 be the projections to CP n and Fr,r+1(Cn+1).

Then Y Φ = p1(p−12 (Φ(M))).26

We have dimC Y Φ ≤dimC p−12 (Φ(M)) ≤r+dimC Φ(M) (as the fibre of p2 is CP r) ≤r+1.Hence Y Φ cannot be equal to CP n if r < n −1. This completes the proof.□Remark: We have extended solutions of the form πf + λπ⊥f in Example 6.3, and (πfr +λπ⊥fr)(πfr+1 + λπ⊥fr+1) in Example 6.4.It follows that the deformations used in theseexamples could have been obtained by applying Theorem 6.2, because the deformation ofTheorem 6.2 preserves the relevant Grassmannian or flag manifold and the hypotheses ofthat theorem are satisfied.Example 6.6 (cf.

Example 3.6). Let Harmd(S2, Sn) be the space of harmonic mapsϕ : S2 −→Sn of energy d, with a similar definition for Harmd(S2, RP n).Theorem 6.7.

(i) Harmd(S2, Sn) and Harmd(S2, RP n) are connected, if n ≥3. (ii) More generally, the space of isotropic harmonic maps of energy d of any compactRiemann surface M into Sn (or RP n) is connected, if n ≥3 and if d ≥2g, where g isthe genus of M.Remark: This result is elementary if n = 3.

Part (i) was proved by Loo ([Lo]) and byVerdier ([Ve3]) for n = 4, and extended to n ≥4 by Kotani ([Kt]).Proof. It suffices to give the proof of (ii).

The result for Sn follows from that for RP n, asthe natural map Sn −→RP n induces a non-trivial double covering Harmisod (M, Sn) −→Harmisod (M, RP n), where Harmisoddenotes isotropic harmonic maps of energy d.By[Ca1],[Ca2] it suffices to take n even, say n = 2m, and it suffices to show that the spaceHHd(S2, Zm) of holomorphic horizontal maps Φ : M −→Zm of degree d is connected, asthe map π : Zm −→RP 2m induces a surjection HHd(M, Zm) −→Harmisod (M, RP 2m). (The degree of Φ is equal to the energy of ϕ = π ◦Φ, if the energy is normalized suitably.

)We shall prove that HHd(M, Zm) is connected by induction on m. For m = 1, thehorizontality condition is vacuous, so HHd(M, Zm) may be identified with the spaceHold(M, S2).This is known to be connected if d ≥2g (see Corollary 1.3.13 of [Na]and also the comments in Example 6.3),and so the induction begins.For the inductive step, we shall identify Zm with the orbit Ad(SO2m+1)(iπV −iπ ¯V ),where V is a fixed element of Zm. Let L be an isotropic line in C2m+1, and set Q =iπL −iπ¯L.

The critical points W ∈Zm of the Morse-Bott function whose downwardsgradient flow is given by the action of {exp itQ} are given by [iπW −iπ ¯W , iπL −iπ¯L] = 0,i.e. W = W1 ⊕W2 ⊕W3 with W1 ⊆L, W2 ⊆¯L, W3 ⊆(L ⊕¯L)⊥.

There are two connectedcritical manifolds, namely:Z+ = {W | L ⊆W ⊆¯L⊥} ∼= Zm−1Z−= {W | ¯L ⊆W ⊆L⊥} ∼= Zm−1.The corresponding stable manifolds areSQ(Z+) = Z+SQ(Z−) = {W | W ∩L = {0}}.27

The embeddings I± : Zm−1 −→Zm defined by the inclusions of Z± in Zm are holomorphic.They also respect the horizontality condition∂∂zC∞Φ ⊥C∞¯Φ, in the sense that a mapΦ : M −→Zm−1 is horizontal if and only if either of the maps I± ◦Φ : M −→Zmare horizontal. We shall accomplish the inductive step by showing that any element Φ ofHHd(M, Zm) may be deformed into Z−.If Φ(M) ⊆SQ(Z−), then {(exp itQ)♮Φ}0≤t≤∞provides a continuous deformation ofΦ into Z−.

So it suffices to show that Φ can be deformed into SQ(Z−). We claim thatthere exists some A ∈SO2m+1 such that A♮Φ(M) ⊆SQ(Z−), i.e.

AΦ(z) ̸⊇L for allz ∈M; this will give the required deformation, as SO2m+1 is connected. Since SO2m+1acts transitively on the space Ym of all isotropic lines in C2m+1, it suffices to find someisotropic line L′ such that Φ(z) ̸⊇L′ for all z ∈M.

LetY Φm = {L′ ∈Ym | L′ ⊆Φ(z) for some z ∈M}.Then our claim is that Y Φm ̸= Ym.Let Xm = {(L′, W) ∈Ym × Zm | L′ ⊆W}. Letp1, p2 be the projections to Ym, Zm.

Then Y Φm = p1(p−12 (Φ(M))). We have dimC Y Φm ≤dimC p−12 (Φ(M)) ≤m −1 + dimC Φ(M) (as the fibre of p2 is CP m−1) ≤m.SincedimC Ym = 2m −1, Y Φm cannot be equal to Ym if m ≥2.

This completes the proof.□It should be clear from these examples that a similar method applies to those harmonicmaps ϕ : M −→G/K which are of the form ϕ = π ◦Φ, where Φ is holomorphic and super-horizontal with respect to a twistor fibration π : G/H −→G/K. That is, for a heightfunction hQ : G/H −→R (where G/H = Ad(G)P), we obtain deformations Φt of Φ suchthat Φ∞takes values (generically) in a critical manifold C(TQ)/C(TQ)X = Ad C(TQ)X ofhQ.

To obtain a continuous deformation, one must ensure that the image of Φ lies entirelyin the stable manifold of this critical manifold.Without loss of generality we may assume that X = P. A calculation similar to thatof Lemma 3.7 then shows that the bundle C(TQ)/C(TQ)X −→C(TQ)/C(TQ)X ∩K is a“twistor sub-fibration” of G/H −→G/K. Lemma 3.7 provides an infinite dimensionalversion of this phenomena ; namely, that the bundle G/H −→G/H may be regarded asa twistor sub-fibration of the fibration ΩG −→G.

As explained in §3, ΩG can be realisedas the orbit of the point α = (i, 0) ∈iR˜×Λg, under the action of S1 ˜×ΛG. The theorydescribed in the Appendix for a finite dimensional adjoint orbit extends almost entirely toΩG (cf.

[AP], §8.9 of [PS], and [Ko]), although there are some new features. For example,the inner product ⟨, ⟩is not bi-invariant with respect to the action of S1 ˜×ΛG.

Fromour point of view, the main difference is that it is not in general possible to integrate thegradient vector field on the infinite dimensional manifold ΩG. In fact (Theorem 8.9.9 of[PS]), every point γ ∈ΩG admits a “downwards” flow line, but only points of ΩalgG admit“upwards” flow lines.

The asymmetrical nature of the flow reflects the fact that the actionof C∗on ΩalgG extends to an action of C∗1 = {λ ∈C∗| |λ| < 1} on ΩG, but not to anaction of C∗.28

Appendix: Height functions on generalized flag manifoldsLet G be a compact connected Lie group. The orbit MP = Ad(G)P of a point P ∈gunder the adjoint representation is called a generalized flag manifold.

It is known that theisotropy subgroup of P is the centralizer, C(TP ), of that torus TP which is the closure ofthe one parameter subgroup {exp tP}. The complex group GC also acts transitively onMP , and the isotropy subgroup of P is a parabolic subgroup GP of GC.

Thus, we havenatural diffeomorphismsMP ∼= G/C(TP ) ∼= GC/GP .We denote the natural action of GC on MP by (g, X) 7−→g♮X. (If g ∈G, then g♮X =Ad(g)X.) The standard example of this is given by G = Un and P = iπV ∈un, where Vis a k-dimensional subspace of Cn and πV denotes orthogonal projection from Cn to Vwith respect to the Hermitian inner product of Cn.

ThenMP ∼= Un/Uk × Un−k ∼= GLn(C)/GP ,where GP = {A ∈GLn(C) | AV ⊆V }. This can be identified with the GrassmanianGrk(Cn), by identifying Ad(A)P with the k-plane AV .

The action of GLn(C) on Grk(Cn)is then given by the formula A♮V = AV . The homogeneous space MP has a natural K¨ahlerstructure, which is determined by the choice of P and a choice of an Ad(G)- invariant innerproduct ⟨⟨, ⟩⟩on g.For any Q ∈g, we define the “height function” hQ : MP −→R byhQ(X) = ⟨⟨X, Q⟩⟩.A point X ∈MP is a critical point of hQ if and only if [Q, X] = 0.

It follows from thisthat the critical points of hQ form a finite number of orbits of the group C(TQ), sayN1 = Ad(C(TQ))X1,. .

., Nr = Ad(C(TQ))Xr.These critical manifolds are non-degenerate; in other words, hQ is a “Morse-Bott function”.In the standard example, where MP ∼= Grk(Cn), let us choose Q = iπl where πl : Cn −→Cl is orthogonal projection onto the span of the first l standard basis vectors. A pointAd(A)P = iπW is a critical point of hQ if and only if [πl, πW ] = 0, i.e.

W = W0 ⊕W1where W0 ⊆Cl, W1 ⊆(Cl)⊥. The critical manifold N containing W = W0 ⊕W1 is the setof k-planes U such that U = U0 ⊕U1, where U0 ⊆Cl, U1 ⊆(Cl)⊥, and dim Ui = dim Wifor i = 0, 1.

It is the orbit of iπW under the group C(TQ) = Ul × Un−l, and hence is acopy of Grw0(Cl) × Grw1(Cn−l), where wi = dim Wi. The index of a critical manifoldmay be computed using the Stiefel diagram of G. This theory is due to Bott ([Bo]).Let ∇hQ be the gradient of hQ with respect to the K¨ahler metric.

The integral curvesof ∇hQ may be calculated explicitly, since−∇hQ = JQ∗where Q∗is the vector field on MP associated to the one parameter subgroup {exp tQ}.This observation is due to Frankel ([Fr]). It follows that the flow line of −∇hQ whichpasses through a non-critical point X ist 7−→(exp itQ)♮X.29

In the standard example, the flow line of −∇hQ passing through a non-critical point iπWis given byt 7−→πWt,Wt = e−tπlWwhere e−tπl is the n×n diagonal matrix with diagonal terms e−t, . .

., e−t (l times) 1, . .

., 1(n −l times).The stable (or unstable) manifold SQ(X) (or U Q(X)) of a critical point X is by defini-tion the union of the flow lines of −∇hQ which converge to X as t →∞(or as t →−∞).The stable manifold of the critical manifold N is defined by SQ(N) = SY ∈N SQ(Y ), witha similar definition of the unstable manifold U Q(N). Using the above description of theflow lines, it can be shown thatSQ(N) = (GQ)♮Xi.e.

the orbit of X under the (complex) group GQ. Similarly,U Q(N) = (GoppQ )♮Xwhere GoppQis the “opposite” parabolic subgroup to GQ.

In the standard example, thestable manifold of the critical manifold N is the set of k-planes U such that dim U ∩Cl =w0. This is the orbit of W under the group GQ = {A ∈GLn(C) | A(Cl) ⊆Cl}.

Theunstable manifold is the set of k-planes U such that dim U ∩(Cl)⊥= w1, i.e. the orbit ofW under the group GoppQ= {A ∈GLn(C) | A(Cl)⊥⊆(Cl)⊥}.30

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