Harmonic tori in symmetric spaces and commuting Hamiltonian systems on loop algebras

F. E. Burstall, D. Ferus, F. Pedit, and U. Pinkall

Introduction

Background

The study of harmonic maps of compact Riemann surfaces into symmetric spaces can be motivated by considerations from several areas of Geometry and Physics:

For example, minimal surfaces are (conformal) harmonic maps while constant mean curvature surfaces, constant Gauss curvature surfaces and Willmore surfaces all have Gauss maps which are harmonic maps into various symmetric spaces.

Again, the harmonic map equations on 2-dimensional domains enjoy a number of properties (conformal invariance, instanton solutions) analogous to those of the Yang–Mills equations in 4 dimensions. This observation has encouraged Physicists to study the (simpler) harmonic map equations in the hope of gaining insight into the Yang–Mills equations.

Much of the qualitative behaviour of the problem can be seen in embryo form in the simplest possible case: that of harmonic maps of a compact Riemann surface M into S^{2}.

If M is the Riemann sphere, all harmonic maps into S^{2} are ±-holomorphic and so are given by rational functions. A similar picture obtains when the target is a higher dimensional symmetric space [6,7,11,34,35]: in this setting, harmonic maps are obtained from holomorphic curves in some auxiliary complex manifold (a twistor space) such as a flag manifold [6,11] or a loop group [34].

A model for such results are the theorems of Calabi [7] and Eells–Wood [11] concerning S^{2n} and ℂ P^{n} respectively which show that all harmonic 2-spheres in such spaces may be obtained by explicit algebraic procedures from holomorphic maps (in this regard, see also the results of Wood [36] concerning explicit construction of harmonic 2-spheres in complex Grassmannians). In these results, a key ingredient is that a harmonic map gives rise to a number of holomorphic differentials (the first being the (2, 0)-part of the complexified metric) which, of course, vanish since the domain is a Riemann sphere. In case M has higher genus, it may also happen that for a particular harmonic map, these differentials vanish and then the harmonic map may again be constructed algebraically from holomorphic maps. Such harmonic maps are variously called superminimal [4], pseudo-holomorphic [7] or isotropic [11] and represent the first class of harmonic maps of higher genus surfaces to be systematically treated.

The third author was supported by Alexander von Humboldt Foundation and Emory University Research Council The fourth author was supported by DFG project Pi 158/2-1

If M is a 2-torus then a harmonic map into S^{2} is either holomorphic (which we view as the superminimal case) or nowhere conformal. The nowhere conformal harmonic maps are the Gauss maps of constant mean curvature tori in ℝ^{3} and amount to doubly periodic solutions of the sinh– Gordon equation. These were studied by Pinkall–Sterling [28] who showed that all such could be obtained from solutions of a family of finite-dimensional completely integrable systems which linearise on the Jacobian of an algebraic curve, the spectral curve. These results enabled Bobenko [2], using methods from soliton theory, to provide an explicit description of these harmonic maps in terms of theta functions.

That such a picture persists for other symmetric targets was indicated by Hitchin’s study [18] of harmonic 2-tori in S^{3}—here also such maps are obtained from linear flows on the Jacobian of a spectral curve. Finally, in the work of Ferus–Pedit–Pinkall–Sterling [13] on minimal nonsuperminimal 2-tori in S^{4}, the link between harmonic maps and Hamiltonian systems on loop algebras was made explicit. It is this theme that we take up in the present paper: indeed, our main result is that the “generic” harmonic 2-torus in a compact Riemannian symmetric space may be obtained from such a Hamiltonian system.

When M has genus greater than one, harmonic maps into S^{2} (or any other symmetric space) are not understood in any systematic way although there are several classes of examples: the harmonic maps of rather symmetrical higher genus surfaces into S^{2} due to Lemaire [26], the minimal surfaces in S^{3} of Lawson [25] and Karcher–Pinkall–Sterling [21] and the Gauss maps of the constant mean curvature surfaces that Kapouleas [20] has obtained by analytic means.

Overview

Our main objects of study are pluriharmonic maps ϕ : ℂ^{n}/Γ → G/K of a complex torus ℂ^{n}/Γ into a symmetric space G/K, where G is a compact, usually semisimple, Lie group. In particular, when n = 1, these are precisely the harmonic 2-tori in G/K. We introduce a large class of such maps, the pluriharmonic maps of finite type, the description of which reduces to the integration of a family of commuting Hamiltonian flows on finite-dimensional subspaces of the algebra Ω𝔤 of based loops in the Lie algebra 𝔤 of G.

To obtain this description we reduce our problem in two ways: firstly, we lift our maps to the universal cover ℂ^{n} of ℂ^{n}/Γ. Secondly, we avail ourselves of the Cartan embedding of our symmetric space G/K into its group of isometries G. This is a totally geodesic immersion G/K ↪ G and therefore preserves pluriharmonicity. A large part of our study will therefore focus on pluriharmonic maps ϕ : ℂ^{n} → G.

Our analysis is based on the zero-curvature reformulation of the harmonic map equations due to Uhlenbeck [34] and extended to pluriharmonic maps by Ohnita–Valli [27]. To describe this, let θ be the Maurer–Cartan form of G and ϕ : ℂ^{n} → G a smooth map. Denote by α′ the (1, 0)-part of ϕ^{∗}θ and by α′′ the complex conjugate of α′. Now introduce a loop of gauge potentials (or, equivalently, an Ω𝔤-valued 1-form) by setting

A_{λ} = \frac{1 − λ}{2}α′ + \frac{1 − λ^{−1}}{2} α′′,  (1)

for λ ∈ S^{1}. The point of this construction is that ϕ is pluriharmonic if and only if d + A_{λ} is a flat connection for all λ ∈ S^{1}. Conversely, if d + A_{λ} is a loop of flat connections of the form (1) then, ℂ^{n} being simply connected, we may integrate the Maurer–Cartan equations to get an essentially unique pluriharmonic map ϕ : ℂ^{n} → G with ϕ^{∗}θ = α′ + α′′.

With this in hand, we are in a position to describe the class of pluriharmonic maps that will principally concern us. First, for d ∈ ℕ, let Ω_{d} be the finite dimensional subspace of Ω𝔤 consisting of loops of the form

ξ = ∑_{0<|n|≤d}(1 − λ^{n})ξ_{n},

with each ξ_{n} ∈ 𝔤^{ℂ}.

We say that a pluriharmonic map ϕ : ℂ^{n} → G is of finite type if, for some d ∈ ℕ, there is a map ξ : ℂ^{n} → Ω_{d} such that

dξ = [ξ, A_{λ}]  (2)
α′ = 4iV_{k}(ξ_{d})dz^{k},  (3)

where the V_{k} : 𝔤^{ℂ} → 𝔤^{ℂ} are Ad G^{ℂ} -equivariant polynomial maps. These two conditions, taken together, imply two things: firstly, ξ is a solution to a Lax equation and, secondly, α′ (and hence ϕ) can be recovered from ξ.

The Lax equations for ξ have a Hamiltonian formulation. From (3) we see that A_{λ} is constructed from suitable projections of gradients of adjoint invariants on Ω𝔤. Lax equations of this kind arise in the well-known Kostant–Adler–Symes theory of integrable systems (c.f., [1]) and the generalisation of that theory represented by the r-matrix formalism of Reyman–Semenov-Tian-Shansky [32]. In fact, the Lax equations (2) are obtained in a canonical way by applying the r-matrix formalism, which shows that ξ integrates a system of 2n commuting Hamiltonian flows on Ω𝔤.

We have therefore isolated a class of pluriharmonic maps into G that may be constructed by solving a finite dimensional system of commuting ordinary differential equations.

To find pluriharmonic maps of finite type into the symmetric space G/K, we must obtain conditions that ensure that pluriharmonic maps into G factor through the Cartan embedding and so give rise to pluriharmonic maps into G/K. This question has a pleasantly simple answer: it is necessary and sufficient that ξ be compatible with the involution defining the symmetric space.

Of course, the central issue is to discover sufficient conditions for a pluriharmonic map to be of finite type. Our main result in this direction is that a harmonic map of a (real) 2-torus into a symmetric space whose derivative along some holomorphic direction is regular semisimple at one (and hence every) point is of finite type. As a consequence of this, we prove that every non-conformal harmonic 2-torus in a rank one symmetric space is of finite type. In particular, taken together with [13], this accounts for all non-superminimal harmonic 2-tori in S^{4}.

For pluriharmonic maps of a complex n-torus, n > 1, our main result is also valid subject to a mild technical assumption—the surjection property—on the symmetric space which is satisfied by all group manifolds and all classical symmetric spaces.

To make contact with Algebraic Geometry and, in particular, recover Hitchin’s results [18], it remains to show that our Hamiltonian flows are algebraically completely integrable i.e., that our flows are equivalent to linear flows on the Jacobian of some spectral curve. Part of this programme has already been carried out: there is a standard construction of a spectral curve and a dynamical system on its Jacobian from a Lax equation on a matrix loop algebra and it is shown in [5] that, in our case, this dynamical system is linear. To complete the programme, one must invert this construction and we shall return to this elsewhere.

Outline

In preparing this paper, our exposition was influenced by several considerations. Firstly, we have, to a certain extent, treated the special case of harmonic maps ℂ /Γ → G/K separately since, in some ways, this is the case of primary interest. Moreover, in this setting, the arguments are simpler, the results more complete and require less technicalities then in the general case.

Secondly, our ideas have been strongly influenced by the previous works [13,28] and we have attempted to present the sometimes ad hoc methods of those papers in their essential form.

As a consequence, there is a risk that the Reader may find our exposition insufficiently concise and we therefore present here a detailed outline of the paper.

As we have already seen, pluriharmonic maps ℂ^{n} → G are essentially the same as loops of flat connections d+A_{λ} with A_{λ} of the form (1). The paper begins by describing a method of producing such connections from commuting flows on loop algebras. Choose Ad G^{ℂ} -equivariant polynomial maps V_{1}, …, V_{n} : 𝔤^{ℂ} → 𝔤^{ℂ}, d ∈ ℕ and define vectors fields X_{1}^{k}, X_{2}^{k}, 1 ≤ k ≤ n, on Ω_{d}, the linear space of loops in 𝔤 of the form ∑_{0<|n|≤d}(1 − λ^{n})ξ_{n}, by

\frac{1}{2}(X_{1}^{k} − iX_{2}^{k})(ξ) = [ξ, 2i(1 − λ)V_{k}(ξ_{d})].

It is then an immediate consequence of the r-matrix theory enunciated in theorem 1.3 that the X_{j}^{k} are a commuting set of complete vector fields on Ω_{d}. If we then fix a base-point ξ_{0} ∈ Ω_{d}, we may simultaneously integrate along these vector fields to obtain a map ξ : ℂ^{n} → Ω_{d} satisfying

\frac{∂ξ}{∂z^{k}} = [ξ, 2i(1 − λ)V_{k}(ξ_{d})]

ξ(0) = ξ_{0}.

Moreover, as a further consequence of (1.3), we learn that such a ξ defines a loop of flat connections d + A_{λ} of the form (1) by

A_{λ} = 2i(1 − λ)V_{k}(ξ_{d})dz^{k} − 2i(1 − λ^{−1})\overline{V_{k}(ξ_{d})}d\overline{z}^{k}

and thus a pluriharmonic map ϕ : ℂ^{n} → G with

α′ = 4iV_{k}(ξ_{d})dz^{k}.

The pluriharmonic maps so obtained are the pluriharmonic maps of finite type.

Matters simplify somewhat when n = 1 since then we can take the polynomial map V_{1} to be the identity id_{𝔤^{ℂ}} : 𝔤^{ℂ} → 𝔤^{ℂ}.

To treat pluriharmonic maps into a symmetric space G/K, we take the view that such a map is a pluriharmonic map into G which factors through the totally geodesic Cartan embedding G/K ↪ G. This embedding is given by

gK ↦ gτ(g)^{−1},

where τ is the involution of G fixing K. In section 1.4 we consider this situation and determine the image of the Cartan embedding. As a consequence, we give in theorem 1.12 conditions which ensure that pluriharmonic maps of finite type factor through the embedding. These conditions amount to demanding that

(i) the polynomial maps V_{k} defining the Lax equation are τ-compatible in the sense that V_{k} ∘ (−τ) = −τ ∘ V_{k}, for 1 ≤ k ≤ n;

(ii) the initial condition ξ_{0} ∈ Ω_{d} satisfies τξ_{0}(λ) = ξ_{0}(−λ), for λ ∈ S^{1}.

Again matters simplify when n = 1 since the first condition is always satisfied by V_{1} = id_{𝔤^{ℂ}}.

In the second and third chapters, which contain the heart of the paper, we discuss which pluriharmonic maps are of finite type. Clearly ϕ : ℂ^{n} → G is of finite type precisely when, for some d ∈ ℕ, there is a map ξ : ℂ^{n} → Ω_{d} satisfying

dξ = [ξ, A_{λ}]  (2)
α′ = 4iV_{k}(ξ_{d})dz^{k}  (3)

for some Ad G^{ℂ} -equivariant polynomial maps V_{k} : 𝔤^{ℂ} → 𝔤^{ℂ}. We call a map ξ : ℂ^{n} → Ω_{d} a polynomial Killing field for ϕ if ξ satisfies (2). Moreover, a polynomial Killing field is said to be adapted if (3) is replaced by the weaker condition that there exists a nowhere vanishing holomorphic vector field Z such that

α′(Z) = ξ_{d}.

It is clear that when n = 1 the existence of an adapted polynomial Killing field is sufficient for ϕ : ℂ^{n} → G to be of finite type. When n > 1, further conditions must be imposed in order that all of α′ can be reconstructed from knowledge of α′ along some holomorphic direction and thus from ξ_{d}.

We therefore ask when a pluriharmonic map admits an adapted polynomial Killing field. Certainly there are necessary conditions: a polynomial Killing field solves the Lax equation (2) from which it follows that ξ_{d} has image in a single Ad G^{ℂ} -orbit in 𝔤^{ℂ}. Thus ϕ can only admit an adapted polynomial Killing field if α′(Z) has image in a single adjoint orbit.

The main point now is that for lifts of pluriharmonic maps of a complex torus ℂ^{n}/Γ such a condition is almost sufficient: say that ϕ is semisimple adapted if α′(Z) has image in a single Ad G^{ℂ} -orbit of semisimple elements of 𝔤^{ℂ}. We prove that a semisimple adapted pluriharmonic map ℂ^{n}/Γ → G admits an adapted polynomial Killing field. In particular, a semisimple adapted harmonic map ℂ /Γ → G is of finite type.

The proof of this result requires the introduction of an auxiliary object: an adapted formal Killing field for ϕ : ℂ^{n}/Γ → G is a formal Laurent series Y = ∑_{k≥i} λ^{−k}Y_{k}, with each Y_{k} : ℂ^{n}/Γ → 𝔤^{ℂ}, satisfying

dY = [Y, A_{λ}]
α′(Z) = Y_{i},

for some nowhere vanishing holomorphic vector field Z. If α′(Z) has image in a single adjoint orbit of semisimple elements then an adapted formal Killing field for ϕ can be constructed by a recursion argument (theorem 2.4). Moreover, each coefficient Y_{k} is a Jacobi field for ϕ (lemma 2.2) and so satisfies a linear elliptic partial differential equation. Since ℂ^{n}/Γ is compact we deduce that the Y_{k} span a finite-dimensional space and this enables us to construct an adapted polynomial Killing field for ϕ from some of the Y_{k}.

This analysis provides a sufficient condition that a harmonic map ϕ : ℂ /Γ → G/K be of finite type, namely: ϕ must be semisimple adapted. In general, when a pluriharmonic map ϕ : ℂ^{n}/Γ → G/K is semisimple adapted, we have now produced an adapted polynomial Killing field ξ for ϕ but to deduce from this that ϕ is of finite type requires additional assumptions. Firstly, we must further restrict the orbit type of α′(Z) = ξ_{d} by demanding that it have image in an orbit of regular (with respect to G/K) semisimple elements. We call such a map regular semisimple adapted. This condition is to ensure that α′ may be recovered as an Ad G^{ℂ} -equivariant polynomial in α′(Z). Secondly, we require that the symmetric space G/K have the surjection property which amounts to demanding that Ad G^{ℂ} -equivariant polynomials on 𝔤^{ℂ} generate the G-equivariant polynomial maps T (G/K) → T (G/K). When these extra conditions are imposed, one finds (theorem 2.8) Ad G^{ℂ} -equivariant polynomial maps V_{k} : 𝔤^{ℂ} → 𝔤^{ℂ} such that

α′ = 4iV_{k}(ξ_{d})dz^{k}

so that ϕ is of finite type.

Finally, in chapter three, we give the “regular semisimple adapted” condition a more geometrical flavour. The results of Kostant and Kostant–Rallis [23,24] identify the orbits of regular semisimple elements with certain level sets of G-invariant polynomial maps P : T (G/K)^{ℂ} → ℂ^{r}. Such maps, being parallel since invariant, pull back by a pluriharmonic ϕ to give holomorphic differentials ϕ^{∗}P^{(k,0)} on ℂ^{n}/Γ which are constant by Liouville’s theorem. This means that the condition that ϕ be regular semisimple adapted can be verified by the examination of polynomials in ∂ϕ at a single point of ℂ^{n}/Γ.

In particular, if G/K has rank one, the invariant polynomials are generated by the Killing form (the metric of G/K) and, in this case, ϕ : ℂ /Γ → G/K is regular semisimple adapted and so of finite type if and only if it is non-conformal.

Acknowledgements

During the course of this research, we have benefited from conversations with Allan Fordy and it is a pleasure to thank him here. Moreover, much of this work was done during visits of various of us to Warwick, Leeds and Berlin for which financial support was provided by the GADGET initiative of the EC. Also, the third author would like to thank the members of Fachbereich MA 8-3 and MA 8-4 of the Technische Universität Berlin for their hospitality.

Finally we would like to thank the referee for helpful comments that have enabled us to improve the exposition.

1 Harmonic maps and commuting Hamiltonians

1.1 The zero-curvature equations

We will study harmonic maps of a 2-torus into a compact Lie group G and, later on, pluriharmonic maps of higher dimensional tori into G. On two-dimensional domains, the harmonic map equations are conformally invariant and so it suffices to consider harmonic maps of ℝ^{2} into G which are doubly periodic with respect to some lattice.

The starting point of our analysis is the well-known observation that the harmonic map equations for maps of a (simply-connected) surface into a Lie group are equivalent to the flatness of a certain loop of connections. We begin by recalling this development.

Let 𝔤 be the Lie algebra of G and θ the (left) Maurer–Cartan form of G. Let ϕ : M → G be a map of a Riemannian manifold M into G and set α = ϕ^{∗}θ. Then α is a 𝔤-valued 1-form on M which satisfies the Maurer–Cartan equation:

dα + \frac{1}{2}[α ∧ α] = 0.  (4)

Otherwise said, the connection d + α on the trivial principal bundle M × G is flat.

Conversely, when M is simply-connected, any 𝔤-valued 1-form α on M satisfying (4) can be integrated to give a map ϕ : M → G, unique up to left translation by a constant element of G, for which α = ϕ^{∗}θ. Otherwise said, ϕ gauges the flat connection d + α to the trivial connection.

Fix, once and for all, a bi-invariant metric on G. Then ϕ : M → G is harmonic if and only if ϕ^{∗}θ is co-closed [29], i.e.

d^{∗}α = 0.  (5)

Thus the study of harmonic maps of (simply-connected) M into G reduces to the study of 𝔤-valued 1-forms on M satisfying (4) and (5).

Now we specialise to the case M = ℝ^{2}. Introduce a complex co-ordinate z = x + iy on ℝ^{2} and then we have a type decomposition:

α = α′ + α′′

where α′ is a 𝔤^{ℂ} -valued (1, 0)-form, α′′ = \overline{α′} where the conjugation in 𝔤^{ℂ} is with respect to the real form 𝔤. The exterior derivative also decomposes:

d = ∂ + \overline{∂}

and then (4) and (5) are equivalent to

\overline{∂}α′ + \frac{1}{2}[α′ ∧ α′′] = ∂α′′ + \frac{1}{2}[α′ ∧ α′′] = 0  (6)

The fundamental observation of Uhlenbeck [34] (c.f. also [37,38]) is that by introducing an auxiliary parameter λ (a spectral parameter), the equations (6) are equivalent to a single family of Maurer– Cartan equations.

Indeed, for λ ∈ ℂ^{∗}, define a 𝔤^{ℂ} -valued 1-form A_{λ} by

A_{λ} = \frac{1 − λ}{2}α′ + \frac{1 − λ^{−1}}{2} α′′.

Note that, for λ ∈ S^{1}, A_{λ} is in fact 𝔤-valued. Now compare coefficients to see that

dA_{λ} + \frac{1}{2}[A_{λ} ∧ A_{λ}] = 0

for all λ if and only if α = A_{−1} satisfies equations (6) or, equivalently, if α = ϕ^{∗}θ for a harmonic map ϕ : ℝ^{2} → G.

Warning Our λ is the reciprocal of that in [34].

We provide a setting for this result by introducing the Lie algebra Ω𝔤 of based loops in 𝔤. Thus

Ω𝔤 = {ξ : S^{1} → 𝔤: ξ(1) = 0}.

Ω𝔤 is a Lie algebra under pointwise bracket. Use the invariant inner product on 𝔤 to define the H^{1} inner product on Ω𝔤,

(ξ_{1}, ξ_{2})_{H^{1}} = ∫_{S^{1}}(ξ′_{1}, ξ′_{2}),

and then we have a Fourier decomposition on Ω𝔤. Indeed, any ξ ∈ Ω𝔤 may be written

ξ(λ) = ∑_{n≠0}(1 − λ^{n})ξ_{n}

with ξ_{n} ∈ 𝔤^{ℂ} and ξ_{−n} = \overline{ξ_{n}}. With this in mind, we introduce a filtration of Ω𝔤 by finite dimensional subspaces

Ω_{1} ⊂ Ω_{2} ⊂ ⋯ ⊂ Ω𝔤

setting

Ω_{d} = {ξ ∈ Ω𝔤: ξ_{n} = 0 for |n| > d}

for d ∈ ℕ.

We also define

Ω^{+} = {ξ ∈ Ω𝔤^{ℂ} : ξ_{n} = 0 for n < 0} Ω^{−} = {ξ ∈ Ω𝔤^{ℂ} : ξ_{n} = 0 for n > 0}.

Note that Ω^{±} are mutually conjugate subalgebras (with respect to the real form Ω𝔤) and

Ω𝔤^{ℂ} = Ω^{+} ⊕ Ω^{−}.

We may now summarize the above discussion in

Proposition 1.1 Let A = A_{λ} be an Ω𝔤-valued 1-form on ℝ^{2} such that (i) A^{(1,0)} has values in Ω^{+} ∩ Ω_{1}^{ℂ};

(ii) the connection d + A is flat.

Then there is a harmonic map ϕ : ℝ^{2} → G, unique up to left translation by a constant, such that

ϕ^{∗}θ = A_{−1}

and all harmonic maps of ℝ^{2} into G arise this way.

Our first result is that such A may be constructed by integrating a pair of ordinary differential equations.

1.2 Commuting flows

Fix d ∈ ℕ and define vector fields X_{1}, X_{2} on Ω_{d} by

\frac{1}{2}(X_{1} − iX_{2})(ξ) = [ξ, 2i(1 − λ)ξ_{d}].

Observe that the X_{i} are indeed tangent to Ω_{d} since the coefficient of 1 − λ^{d+1} on the right hand side is −2i[ξ_{d}, ξ_{d}] = 0. Concerning these vector fields we have

Theorem 1.2 The X_{i} are complete and commute. Thus, if X_{i}^{t} denotes the flow on Ω_{d} generated by X_{i}, then

(t^{1}, t^{2}) · ξ = X_{1}^{t^{1}} ∘ X_{2}^{t^{2}}(ξ)

defines an action of ℝ^{2} on Ω_{d}.

Fix an initial condition ξ_{0} ∈ Ω_{d} and let ξ : ℝ^{2} → Ω_{d} be given by

ξ(t^{1}, t^{2}) = (t^{1}, t^{2}) · ξ_{0}.

Then A_{λ} = 2i(1 − λ)ξ_{d}dz − 2i(1 − λ^{−1})ξ_{−d}d\overline{z} satisfies the Maurer–Cartan equation and so gives rise to a harmonic map ϕ : ℝ^{2} → G.

Otherwise said: if ξ : ℝ^{2} → Ω_{d} solves

\frac{∂ξ}{∂z} = [ξ, 2i(1 − λ)ξ_{d}]  (7)

then 4iξ_{d}dz = ϕ^{∗}θ^{(1,0)} for a harmonic map ϕ : ℝ^{2} → G and solutions of (7) exist on all ℝ^{2} for any choice of initial condition.

Proof We begin with an observation which is in any case fundamental to much of what follows; that is, the ordinary differential equations defining the flows of the X_{i} are of Lax type:

ξ̇ = [ξ, V_{i}(ξ)]

with V_{i} : Ω_{d} → Ω_{d}.

The L^{2} inner product on Ω𝔤 induces an invariant inner product on Ω_{d} so that

\frac{d}{dt^{i}} (ξ, ξ)_{L^{2}} = 2(ξ̇, ξ)_{L^{2}}
= 2([ξ, V_{i}(ξ)], ξ)_{L^{2}} = −2([ξ, ξ], V_{i}(ξ))_{L^{2}} = 0.

Thus the flows evolve on spheres in Ω_{d} and so are complete.

Set Z = \frac{1}{2}(X_{1} − iX_{2}). The X_{i} commute if and only if [Z, \overline{Z}] vanishes. For this we compute:

d_{Z} \overline{Z} = d_{Z}[ξ, −2i(1 − λ^{−1})ξ_{−d}] = [Z, −2i(1 − λ^{−1})ξ_{−d}] + [ξ, −2i(1 − λ^{−1})Z_{−d}].

Using (1 − λ^{n})(1 − λ^{m}) = (1 − λ^{n}) + (1 − λ^{m}) − (1 − λ^{n+m}), we see that

Z_{−d}(ξ) = 2i[ξ_{−d}, ξ_{d}]

so that

d_{Z} \overline{Z} = 4\{(1 − λ)(1 − λ^{−1})[[ξ, ξ_{d}], ξ_{−d}] + (1 − λ^{−1})[ξ, [ξ_{−d}, ξ_{d}]]\}.

Conjugating gives

d\overline{Z}Z = 4\{(1 − λ)(1 − λ^{−1})[[ξ, ξ_{−d}], ξ_{d}] + (1 − λ)[ξ, [ξ_{d}, ξ_{−d}]]\}

and then an application of the Jacobi identity shows that [Z, \overline{Z}] = d_{Z} \overline{Z} − d\overline{Z}Z vanishes.

Finally let ξ : ℝ^{2} → Ω_{d} be the orbit map through an initial condition ξ_{0}. Thus

\frac{∂ξ}{∂z} = [ξ, 2i(1 − λ)ξ_{d}]  (8′)

and, since ξ is real, we have the conjugate equation

\frac{∂ξ}{∂\overline{z}} = −[ξ, 2i(1 − λ^{−1})ξ_{−d}].  (8′′)

Otherwise put,

dξ = [ξ, A].

Now take the (1 − λ^{−d}) coefficient in (8′) and the (1 − λ^{d}) coefficient in (8′′) to get

\frac{∂ξ_{−d}}{∂z} = 2i[ξ_{−d}, ξ_{d}] \frac{∂ξ_{d}}{∂\overline{z}} = −2i[ξ_{d}, ξ_{−d}]

which are easily checked to be equivalent to (6) for α = 4iξ_{d}dz − 4iξ_{−d}d\overline{z} and thus the Maurer– Cartan equation for A. □

1.3 r-matrices, higher flows and pluriharmonic maps

Theorem (1.2), although easily proved, is somewhat lacking in motivation. To remedy this, and to extend our theory to higher dimensional domains, let us briefly recall a development which is well known in the integrable systems literature (c.f. [32,33]).

Let G be a (possibly infinite-dimensional) Lie algebra. A linear map R : G → G is called an r-matrix if the bracket defined by

[ξ, η]_{R} =^{def} [Rξ, η] + [ξ, Rη]

satisfies the Jacobi identity. This is certainly the case if R satisfies the (modified) classical Yang– Baxter equation:

R[ξ, η]_{R} − [Rξ, Rη] = α[ξ, η]  (9)

for some fixed α ∈ ℂ.

Let V : G → G be a vector field on G. We say that V is ad-equivariant if

dV_{ξ}([ξ, η]) = [V(ξ), η]

for all ξ, η ∈ G. We note that this forces V(ξ) to lie in the centre of the centraliser of ξ and so, in particular, any two such vector fields commute pointwise.

We remark that if G is the Lie algebra of a Lie group, then ad-equivariance is just the infinitesimal version of the condition that V should commute with the adjoint action of the group i.e. Ad gV(ξ) = V(Ad gξ). If, in addition, G is equipped with an invariant inner product, then the gradients of functions on G invariant under the adjoint action furnish examples of ad-equivariant vector fields.

An abstract setting for the results of the previous section is now provided by

Theorem 1.3 [13] Let R be a solution to the modified classical Yang–Baxter equation (9) and V_{1}, …, V_{n} be ad-equivariant vector fields on G. Let μ_{ij} be real constants, 1 ≤ i, j ≤ n and define vector fields X_{i} on G by

X_{i}(ξ) = [ξ, RV_{i}(ξ)] = [ξ, RV_{i}(ξ) + μ_{ij}V_{j}(ξ)].

Then the vector fields X_{i} mutually commute.

If, moreover, the X_{i} are complete, let X_{i}^{t} denote the flow on G generated by X_{i}. Then

(t^{1}, …, t^{n}) · ξ = X_{1}^{t^{1}} ∘ ⋯ ∘ X_{n}^{t^{n}}(ξ)

defines an action of ℝ^{n} on G. Fix an initial condition ξ_{0} ∈ G and let ξ : ℝ^{n} → G be the orbit map through ξ_{0}. Then the G-valued 1-form

A = (RV_{i} + μ_{ij}V_{j})dt^{i}

satisfies the Maurer–Cartan equation.

To apply this to the case at hand, take G = Ω𝔤 and define an r-matrix on Ω𝔤 by

R = \begin{cases}i & on Ω^{+}, \\ −i & on Ω^{−}. \end{cases}

That R satisfies the modified classical Yang–Baxter equation (9) is a simple consequence of the fact that Ω^{±} are subalgebras.

Define ad-equivariant vector fields V_{1}, V_{2} by

\frac{1}{2}(V_{1} − iV_{2})(ξ) = λ^{1−d}ξ

and let the μ_{ij} be given by

(μ_{ij}) = (\begin{matrix}0 & 1 \\ −1 & 0 \end{matrix})

to get vector fields X_{1}, X_{2} given by

\frac{1}{2}(X_{1} − iX_{2})(ξ) = [ξ, (R + i)λ^{1−d}ξ]

which on Ω_{d} become

\frac{1}{2}(X_{1} − iX_{2})(ξ) = [ξ, 2i(1 − λ)ξ_{d}]

since R + i annihilates Ω^{−}.

Thus we may view (1.2) as a special case of (1.3) but in fact we can do more. Let V : 𝔤^{ℂ} → 𝔤^{ℂ} be an Ad G^{ℂ} -equivariant polynomial map and let

V = V^{(1)} + ⋯ + V^{(k)}

be its decomposition into homogeneous maps with V^{(i)} homogeneous of degree d_{i}. Each V^{(i)} is also Ad G^{ℂ} -equivariant. We define ad-equivariant vector fields V_{1}, V_{2} on Ω𝔤 by

\frac{1}{2}(V_{1} − iV_{2})(ξ) = ∑_{i=1}^{k}(−1)^{d_{i}+1}λ^{1−dd_{i}}V^{(i)}(ξ)

and then, with the same choice of μ_{ij} as before, we obtain vector fields X_{1}, X_{2} which on Ω_{d} are given by

\frac{1}{2}(X_{1} − iX_{2})(ξ) = [ξ, 2i(1 − λ)V (ξ_{d})].

(We note that the X_{i} are tangent to Ω_{d} since V (ξ_{d}) lies in the centre of the centraliser of ξ_{d}.)

Just as in (1.2), the X_{i} are complete on Ω_{d} since their flows preserve the L^{2} norm and so (1.3) gives

Theorem 1.4 Let V_{1}, …, V_{n} : 𝔤^{ℂ} → 𝔤^{ℂ} be Ad G^{ℂ} -equivariant polynomial maps and d ∈ ℕ. Then for any ξ_{0} ∈ Ω_{d} there is a map ξ : ℂ^{n} → Ω_{d} solving ξ(0) = ξ_{0} and

\frac{∂ξ}{∂z^{k}} = [ξ, 2i(1 − λ)V_{k}(ξ_{d})]  (10)

and then the Ω𝔤-valued 1-form on ℂ^{n} defined by

A_{λ} = 2i(1 − λ)V_{k}(ξ_{d})dz^{k} − 2i(1 − λ^{−1})\overline{V_{k}(ξ_{d})}d\overline{z}^{k}

satisfies the Maurer–Cartan equation.

We have seen that for n = 1, such A give rise to harmonic maps ϕ : ℝ^{2} → G. For higher dimensions, there is a similar interpretation due to Ohnita–Valli [27].

Let M be a complex manifold and N a Riemannian manifold. A map ϕ : M → N is pluriharmonic if

∇ϕ ; \overline{X}∂ϕ(Y) − ∂ϕ(\overline{∂}\overline{X}Y) = 0  (11)

for all X, Y ∈ T^{1,0}M. Here ∇^{ϕ} is the pull-back of the Levi-Civita connection on N, \overline{∂} is the Cauchy–Riemann operator on T^{1,0}M and ∂ϕ : T^{1,0}M → T N^{ℂ} is the (1, 0)-part of the derivative of ϕ. If M is Kähler, this reduces to the condition

∇dϕ^{(1,1)} = 0.

It is well known that ϕ is pluriharmonic if and only if the restriction of ϕ to any holomorphic curve in M is harmonic.

Now take N to be a compact Lie group G, set α = ϕ^{∗}θ with type decomposition α = α′ + α′′. Then Ohnita–Valli prove:

Theorem 1.5 [27] ϕ : M → G is pluriharmonic if and only if

dα′ + \frac{1}{2}[α′ ∧ α′′] = dα′′ + \frac{1}{2}[α′ ∧ α′′] = 0
[α′ ∧ α′] = [α′′ ∧ α′′] = 0.

Further, in this case, the Ω𝔤-valued 1-form A on M given by

A_{λ} = \frac{1 − λ}{2}α′ + \frac{1 − λ^{−1}}{2} α′′

satisfies the Maurer–Cartan equation.

Conversely, if M is simply connected and A is an Ω𝔤-valued 1-form on M such that

(i) A^{(1,0)} has values in Ω^{+} ∩ Ω_{1}^{ℂ};

(ii) the connection d + A is flat;

then there is a pluriharmonic map ϕ : M → G, unique up to left translation by a constant, such that ϕ^{∗}θ = A_{−1}.

Corollary 1.6 In the situation of (1.4), we have

4iV_{k}(ξ_{d})dz^{k} − 4i\overline{V_{k}(ξ_{d})}d\overline{z}^{k} = ϕ^{∗}θ

for some pluriharmonic map ϕ : ℂ^{n} → G unique up to left translation by a constant.

Thus the flows on each Ω_{d} that produce harmonic maps of ℝ^{2} are part of a larger commuting family of flows that produce pluriharmonic maps of ℂ^{n}.

We now give a Hamiltonian interpretation of the ordinary differential equations that we have been discussing. For this, we return to the abstract setting of a Lie algebra G and an r-matrix R : G → G.

An r-matrix equips G with a second Lie algebra structure and thus G^{∗} acquires a second Poisson structure. For f, g ∈ C^{∞}(G^{∗}), set

{f, g}_{R}(x) = ⟨x, [df_{x}, dg_{x}]_{R}⟩

to get a Poisson bracket and thus a Lie algebra homomorphism f ↦ X_{f}, C^{∞}(G^{∗}) → C^{∞}(T G^{∗}) given by X_{f}g = {f, g}_{R}.

Say that f is invariant if it is invariant under the co-adjoint action on G^{∗}. We have [32]

(i) if f, g are invariant they Poisson commute so that [X_{f}, X_{g}] vanishes;

(ii) if f is invariant then, for x ∈ G^{∗},

X_{f}(x) = − ad^{∗}(Rdf_{x})(x).

Suppose now that G has an invariant inner product and use it to identify G^{∗} with G. Then f is invariant if and only if its gradient is ad-equivariant and then the corresponding Hamiltonian vector field is given by

X_{f}(ξ) = [ξ, R∇f(ξ)].

Thus, when our ad-equivariant vector fields are gradients, the vector fields X_{i} discussed above are Hamiltonian.

In the case at hand, take

(f_{1} − if_{2})(ξ) = ∫_{S^{1}} λ^{1−d}(ξ, ξ)

to get Hamiltonians defining the flows of theorem (1.2). Similarly, if P : 𝔤^{ℂ} → ℂ is an Ad(G^{ℂ})- invariant homogeneous polynomial of degree r with gradient V, we take

\frac{1}{2}(f_{1} − if_{2})(ξ) = ∫_{S^{1}}(−1)^{r+1}λ^{1−dr}P (ξ)

to get Hamiltonian flows with vector fields

\frac{1}{2}(X_{1} − iX_{2})(ξ) = [ξ, 2i(1 − λ)V (ξ_{d})].

Thus pluriharmonic maps can be constructed from Hamiltonian flows given by invariant functions on an infinite-dimensional Lie algebra: such flows are a ubiquitous feature of completely integrable systems theory [1,30,31].

Remark This method of producing Poisson commuting functions on the dual of a Lie algebra is very similar to that of Kostant–Adler–Symes (see, in particular, [1, theorem 4.2]) but there are differences. In particular, in our case, the r-matrix is not skew with respect to an invariant inner product. As a consequence, the finite-dimensional subspaces Ω_{d} are not Poisson submanifolds for the R-Poisson structure although they are invariant under the Hamiltonian flows of invariant Hamiltonians.

In conclusion then, we have seen how pluriharmonic maps of ℂ^{n} into compact Lie groups may be obtained from commuting Hamiltonian flows. Following [28] we christen such maps in the following definition.

Definition A pluriharmonic map ϕ : ℂ^{n} → G is of finite type if there are Ad G^{ℂ} -equivariant polynomial maps V_{1}, …, V_{n} : 𝔤^{ℂ} → 𝔤^{ℂ}, d ∈ ℕ and ξ : ℂ^{n} → Ω_{d} such that ξ satisfies (10) and

ϕ^{∗}θ = 4iV_{k}(ξ_{d})dz^{k} − 4i\overline{V_{k}(ξ_{d})}d\overline{z}^{k}.

We shall address below the question of which pluriharmonic maps are of finite type but make some preliminary observations here.

First observe that the ordinary differential equations defining an orbit map ξ : ℂ^{n} → Ω_{d} are of Lax type and so have many conserved quantities. Indeed, if P : 𝔤^{ℂ} → ℂ is an Ad G^{ℂ} -invariant polynomial, then P (ξ) is a constant element of ℂ [λ, λ^{−1}]. Comparing coefficients then gives that P (ξ_{d}) is constant. In fact, a slightly stronger result is true.

Proposition 1.7 If ξ : ℂ^{n} → Ω_{d} satisfies

\frac{∂ξ}{∂z^{k}} = [ξ, 2i(1 − λ)V_{k}(ξ_{d})]

for Ad G^{ℂ} -equivariant polynomial maps V_{1}, …, V_{n} then ξ_{d} takes values in a single Ad G^{ℂ} orbit in 𝔤^{ℂ}.

Proof Comparing coefficients, we see that

dξ_{d} = −2i[ξ_{d−1}, V_{k}(ξ_{d})]dz^{k} + 2i[ξ_{d}, \overline{V_{k}(ξ_{d})}]d\overline{z}^{k}.

Since V_{k}(ξ_{d}) lies in the centre of the centraliser of ξ_{d}, it is easy to see that [V_{k}(ξ_{d}), 𝔤^{ℂ}] ⊂ [ξ_{d}, 𝔤^{ℂ}] so that dξ_{d} takes values in [ξ_{d}, 𝔤^{ℂ}]. Otherwise said, dξ_{d} takes values in the tangent space to the Ad G^{ℂ} orbit through ξ_{d} so that the assertion now follows from the uniqueness of solutions of ordinary differential equations. □

Corollary 1.8 If ϕ : ℂ^{n} → G is a pluriharmonic map of finite type then each ϕ^{∗}θ(\frac{∂}{∂z^{k}}) takes values in a single Ad G^{ℂ} orbit in 𝔤^{ℂ}.

Proof ϕ^{∗}θ(\frac{∂}{∂z^{k}}) = 4iV_{k}(ξ_{d}) and V_{k} sends orbits to orbits. □

In particular, a non-constant harmonic map of ℝ^{2} into G has nowhere vanishing derivative if it is of finite type.

1.4 Maps into symmetric spaces

In Differential Geometry, interest is more often focussed on harmonic maps into symmetric spaces rather than just Lie groups. However, the (pluri)harmonic map equations are preserved by postcomposition by a totally geodesic map and, as is well known [9], any symmetric space may be totally geodesically immersed in its group of isometries. Thus the symmetric space problem reduces to the Lie group problem. On the other hand, the above development produces pluriharmonic maps of ℂ^{n} into a Lie group and it is natural to ask when these maps factor through a symmetric space.

In this section, we shall provide a simple necessary and sufficient condition for this to be the case. We begin by recalling the details of the Cartan embedding. Let N be a compact Riemannian symmetric space and set G = I_{0}(N) with Lie algebra 𝔤. Fix x ∈ N and let σ_{x} ∈ I(N) be the involutive isometry with isolated fixed point x. Then g ↦ σ_{x}gσ_{x} is an involution of G which we call the involution at x and denote by τ_{x}. We now have a symmetric decomposition

𝔤 = 𝔨_{x} ⊕ 𝔭_{x}

of 𝔤 into ±1-eigenspaces of (the derivative of) τ_{x} satisfying the familiar relations

[𝔨_{x}, 𝔨_{x}] ⊂ 𝔨_{x}  [𝔨_{x}, 𝔭_{x}] ⊂ 𝔭_{x}  [𝔭_{x}, 𝔭_{x}] ⊂ 𝔨_{x}.

Moreover, 𝔨_{x} is the Lie algebra of the isotropy group K_{x} of x and (G^{τ_{x}})_{0} ⊂ K_{x} ⊂ G^{τ_{x}}.

The map 𝔤 → T_{x}N given by η ↦ \frac{d}{dt} |_{t=0} exp tη · x is an isomorphism on 𝔭_{x}. Inverting this at each point gives a 𝔤-valued 1-form β on N which provides an isomorphism of T N with a subbundle [𝔭] of the trivial bundle N × 𝔤 (c.f. [6]).

Now fix a basepoint o ∈ N, set τ = τ_{o}, K = K_{o} etc. and define a map i_{o} : N → G by

i_{o}(g · o) = g^{τ}g^{−1}

which is certainly well defined. Concerning i_{o} we have

Proposition 1.9 i_{o} : N → G is a totally geodesic immersion with

i_{o}^{∗}θ = −2β.

Let N̂ be the image of i_{o}. Then N̂ is a globally Riemannian symmetric space totally geodesically embedded in G. Moreover N̂ = exp 𝔭 and further, N̂ is the component of {g ∈ G: g^{τ}g = e} containing the identity e of G.

Proof That i_{o} is a totally geodesic immersion is well known [9] and the formula for i_{o}^{∗}θ is proved in [6, Lemma 8.1]. As for the remaining assertions, temporarily denote the component of {g ∈ G: g^{τ}g = e} containing e by F. Then F is the component containing e of the fixed set of the isometry g ↦ g^{−τ} and so is a totally geodesic submanifold of G. Moreover, the derivative at e of this isometry is η ↦ −τη with fixed set 𝔭 so that F = exp 𝔭. Thus F is a global Riemannian symmetric space by [16]. Finally N̂ is a compact subset of F and both F and N̂ have the same dimension dim 𝔭 so that N̂ and F coincide. □

Using this, we obtain a simple criterion for a map ϕ : M → G to factor through N̂:

Lemma 1.10 Let ϕ : M → G be a map of a connected manifold M with ϕ(m) = e for some m ∈ M. Then ϕ has image in N̂ if and only if

τ Ad ϕ ϕ^{∗}θ + ϕ^{∗}θ = 0.

Proof By (1.9), it suffices to show that ψ = ϕ^{τ}ϕ is constant. However, a simple calculation gives

ψ^{∗}θ = Ad ϕ^{−1}τϕ^{∗}θ + ϕ^{∗}θ

whence the result follows. □

Let us now return to our orbit maps ξ : ℂ^{n} → Ω_{d}, that is to solutions of

dξ = [ξ, 2i(1 − λ)V_{k}(ξ_{d})dz^{k} − 2i(1 − λ^{−1})\overline{V_{k}(ξ_{d})}d\overline{z}^{k}].  (12)

To adapt this to the symmetric space setting, we make a

Definition Let V : 𝔤^{ℂ} → 𝔤^{ℂ} be an Ad G^{ℂ} -equivariant map and τ an involution of 𝔤^{ℂ}. Then V is τ-compatible if for all η ∈ 𝔤^{ℂ}

V (−τη) = −τV (η).

We note that if 𝔤 = 𝔨 ⊕ 𝔭 is the eigenspace decomposition of τ then, for τ-compatible V, we have V (𝔭) ⊂ 𝔭. In general, any Ad G^{ℂ} -equivariant V has a decomposition into Ad G^{ℂ} -equivariant maps

V = V_{+} ⊕ V_{−}

with V_{+} τ-compatible and V_{−} anti-commuting with −τ. Thus V_{+} |_{𝔭} is the 𝔭-part of V |_{𝔭} . We also note that if τ is an inner automorphism then τ-compatibility reduces to the demand that the homogeneous components of V have odd degrees.

With this in mind, we have

Proposition 1.11 Let d ∈ ℕ be odd, τ an involution of 𝔤, V_{1}, …, V_{n} : 𝔤^{ℂ} → 𝔤^{ℂ} τ-compatible Ad G^{ℂ} -equivariant maps and ξ : ℂ^{n} → Ω_{d} a solution of (12). Let ϕ : ℂ^{n} → G satisfy ϕ^{∗}θ = 4iV_{k}(ξ_{d})dz^{k} − 4i\overline{V_{k}(ξ_{d})}d\overline{z}^{k}. Define ξ̃ by

ξ̃(λ) = τ Ad ϕξ(−λ) , λ ∈ S^{1}.

Then ξ̃ also satisfies (12) i.e.

dξ̃ = [ξ̃, 2i(1 − λ)V_{k}(ξ̃_{d})dz^{k} − 2i(1 − λ^{−1})\overline{V_{k}(ξ̃_{d})}d\overline{z}^{k}].

Remark ξ̃ need not take values in Ω_{d} since ξ̃(1) need not vanish but it does take values in Λ_{d} = \{∑_{|n|≤d} ξ_{n}λ^{n}: ξ_{n} ∈ 𝔤^{ℂ} , \overline{ξ_{n}} = ξ_{−n}\} on which our vector fields are also defined.

Proof We compute:

dξ̃(λ) = τd(Ad ϕ)ξ(−λ) + τ Ad ϕ dξ(−λ)
= τ Ad ϕ [ϕ^{∗}θ, ξ(−λ)] + τ Ad ϕ dξ(−λ)

using

d(Ad ϕ) = Ad ϕ ∘ ad ϕ^{∗}θ.

Now substitute from (12) to get

dξ̃(λ) = τ Ad ϕ[4iV_{k}(ξ_{d})dz^{k} − 4i\overline{V_{k}(ξ_{d})}d\overline{z}^{k}, ξ(−λ)]

+ τ Ad ϕ[ξ(−λ), 2i(1 + λ)V_{k}(ξ_{d})dz^{k} − 2i(1 + λ^{−1})\overline{V_{k}(ξ_{d})}d\overline{z}^{k}]

= τ Ad ϕ[ξ(−λ), 2i(λ − 1)V_{k}(ξ_{d})dz^{k} − 2i(λ^{−1} − 1)\overline{V_{k}(ξ_{d})}d\overline{z}^{k}].

However, since d is odd we have

ξ̃_{d} = −τ Ad ϕξ_{d}

while −τ Ad ϕ commutes with both V_{k} and \overline{V_{k}} (since both τ and Ad ϕ are real). Thus

dξ̃ = [ξ̃, 2i(1 − λ)V_{k}(ξ̃_{d})dz^{k} − 2i(1 − λ^{−1})\overline{V_{k}(ξ̃_{d})}d\overline{z}^{k}].

as required. □

Before applying this, we briefly pause to introduce some notation. Let τ : 𝔤 → 𝔤 be an involution and set

Λ(𝔤, τ) = {ξ : S^{1} → 𝔤: ξ(−λ) = τξ(λ) for all λ ∈ S^{1}}.

Thus, if ξ = ∑ λ^{n}ξ_{n}, then ξ ∈ Λ(𝔤, τ) if and only if ξ_{n} ∈ 𝔭^{ℂ} for n odd and ξ_{n} ∈ 𝔨^{ℂ} for n even.

We now come to the main result of this section:

Theorem 1.12 Let N be a compact Riemannian symmetric space with I_{0}(N) = G. Fix a basepoint o ∈ N, set τ = τ_{o} and recall the immersion i_{o} : N → N̂ ⊂ G.

Let d ∈ ℕ be odd, V_{1}, …, V_{n} : 𝔤^{ℂ} → 𝔤^{ℂ} τ-compatible Ad G^{ℂ} -equivariant maps and ξ_{0} ∈ Ω_{d}∩Λ(𝔤, τ). Let ξ : ℂ^{n} → Ω_{d} be a solution of (12) with ξ(0) = ξ_{0} and let ϕ : ℂ^{n} → G be the pluriharmonic map satisfying ϕ(0) = e and ϕ^{∗}θ = 4iV_{k}(ξ_{d})dz^{k} − 4i\overline{V_{k}(ξ_{d})}d\overline{z}^{k}.

Then ϕ has image in N̂ and moreover, ξ(x) ∈ Λ(𝔤, τ_{ϕ(x)}) for each x ∈ ℂ^{n} where τ_{ϕ(x)} is the involution at ϕ(x) ∈ N̂.

Proof Set ξ̃(λ) = τ Ad ϕ ξ(−λ) then by (1.11), ξ̃ also solves (12). But since ξ_{0} ∈ Λ(𝔤, τ)

ξ̃(λ)(0) = τξ(−λ)(0) = ξ(λ)(0)

so that, by uniqueness of solutions of ordinary differential equations, we get ξ = ξ̃. Comparing coefficients at λ^{d} then gives

−τ Ad ϕ ξ_{d} = ξ_{d}

whence, for each k

V_{k}(ξ_{d}) = −τ Ad ϕ V_{k}(ξ_{d})

and hence,

τ Ad ϕ ϕ^{∗}θ + ϕ^{∗}θ = 0.

Thus ϕ has image in N̂ by (1.10).

Finally, if ϕ(x) = i_{o}(g · o) then the involution at ϕ(x) is given by

τ_{ϕ(x)} = Ad g τ Ad g^{−1} = τ Ad(g^{τ}g^{−1}) = τ Ad ϕ(x)

so that ξ = ξ̃ means that ξ ∈ Λ(𝔤, τ_{ϕ}). □

Thus we conclude that the question of whether our pluriharmonic maps factor through a symmetric space is simply a question of choosing τ-compatible flows and the right initial condition.

For the case where the domain is ℝ^{2}, the situation is even simpler, there we take V = id_{𝔤^{ℂ}} as our Ad G^{ℂ} -equivariant map which is always τ-compatible and then we produce harmonic maps into symmetric spaces whenever we have chosen an appropriate initial condition.

2 Polynomial Killing fields and harmonic maps of finite type

2.1 Polynomial Killing fields

Let ϕ : ℂ^{n} → G be a pluriharmonic map and recall the Ω𝔤-valued 1-form A_{λ} given by

A_{λ} = \frac{1 − λ}{2}α′ + \frac{1 − λ^{−1}}{2} α′′

where ϕ^{∗}θ = α′ + α′′. We have seen that ϕ is of finite type precisely when, for some d ∈ ℕ, there is a map ξ : ℂ^{n} → Ω_{d} such that

dξ = [ξ, A_{λ}]  (13)

and

α′ = 4iV_{k}(ξ_{d})dz^{k}  (14)

for some Ad G^{ℂ} -equivariant polynomial maps V_{k} : 𝔤^{ℂ} → 𝔤^{ℂ}.

This motivates the following definition in which our terminology is taken from [13].

Definition Let ϕ : M → G be a pluriharmonic map of a complex manifold into G with A_{λ} as above. A polynomial Killing field for ϕ is a map ξ : M → Ω_{d}, some d ∈ ℕ, satisfying (13).

If, in addition, there is a nowhere vanishing holomorphic vector field Z on M such that

ξ_{d} = α′(Z),

we say that ξ is an adapted polynomial Killing field.

Of course, the situation we have in mind is when Z is a holomorphic co-ordinate vector field on a complex torus or Euclidean space. In particular, if M is ℝ^{2} or a 2-torus, we may assume that Z = \frac{∂}{∂z} and then an adapted Killing field is an orbit map with V = id_{𝔤^{ℂ}} and we conclude:

Lemma 2.1 If ϕ : ℝ^{2} → G admits an adapted polynomial Killing field then ϕ is of finite type.

In fact, it will be shown in section (2.4) that a similar result is true for a large class of pluriharmonic maps. The question therefore arises as to which pluriharmonic maps admit such a polynomial Killing field. For maps that factor through a complex torus, it suffices to consider a more general kind of field which we now describe.

2.2 Formal Killing fields

The idea is to consider an analogue of polynomial Killing fields taking values in a space of 𝔤^{ℂ} -valued formal Laurent series rather than in some Ω_{d}.

Definition Let ϕ : M → G be a pluriharmonic map of a complex manifold into G with A_{λ} as above. A formal Killing field for ϕ is a formal Laurent series Y = ∑_{k≥i} λ^{−k}Y_{k} with each Y_{k} : M → 𝔤^{ℂ} satisfying

dY = [Y, A_{λ}].  (15)

If, in addition, there is a nowhere vanishing holomorphic vector field Z on M for which the top term of Y satisfies

Y_{i} = α′(Z)

we say that Y is adapted.

It will be convenient to write our equations in a slightly different way: if we identify ϕ^{−1}T G with M × 𝔤 via ϕ^{−1}θ, then the pull-back of the Levi-Civita connection on G is given by (c.f. [6])

∇ = d + \frac{1}{2} ad α.

In terms of ∇, the formal Killing field equations (15) read

∇Y = \frac{1}{2}[λα′ + λ^{−1}α′′, Y]  (16)

while the pluriharmonic map equations of theorem (1.5) read

d_{∇}α′ = d_{∇}α′′ = 0  (17)
[α′ ∧ α′] = [α′′ ∧ α′′] = 0  (18)

where d_{∇} is the induced exterior derivative on 𝔤-valued forms.

The technical fact that will allow us to pass from formal Killing fields to polynomial Killing fields is that the coefficients of a formal Killing field are solutions of a linear elliptic partial differential equation.

Definition Let ϕ : M → G be a pluriharmonic map of a Hermitian manifold. A map v : M → 𝔤^{ℂ} is a Jacobi field for ϕ if it solves the linear elliptic equation

J_{ϕ}v = (∂_{∇}\overline{∂}_{∇} − \overline{∂}_{∇}∂_{∇})v(Z_{k}, \overline{Z}_{k}) − \frac{1}{4}[α(\overline{Z}_{k}), [α(Z_{k}), v]] − \frac{1}{4}[α(Z_{k}), [α(\overline{Z}_{k}), v]] = 0

where {Z_{k}} is a local unitary frame for T^{1,0}M.

Remark If M is Kähler, then J_{ϕ} is the Jacobi operator of ϕ, i.e. the linearization of the harmonic map equation, whence our terminology.

Lemma 2.2 Let ϕ : M → G be a pluriharmonic map of a Hermitian manifold and Y a formal Killing field for ϕ. Then each coefficient Y_{k} is a Jacobi field.

Proof We prove a stronger result: we have from (16)

∂_{∇}\overline{∂}_{∇}Y = \frac{1}{2}∂_{∇}[λ^{−1}α′′, Y] = \frac{1}{2}λ^{−1}[∂_{∇}α′′, Y] − \frac{1}{2}[λ^{−1}α′′ ∧ ∂_{∇}Y]
= \frac{1}{2}λ^{−1}[∂_{∇}α′′, Y] − \frac{1}{4}[α′′ ∧ [α′, Y]]
= −\frac{1}{4}[α′′ ∧ [α′, Y]],

since ∂_{∇}α′′ vanishes by the pluriharmonic map equation (17). Similarly

\overline{∂}_{∇}∂_{∇}Y = −\frac{1}{4}[α′ ∧ [α′′, Y]]

so that

(∂_{∇}\overline{∂}_{∇} − \overline{∂}_{∇}∂_{∇})Y + \frac{1}{4}[α′′ ∧ [α′, Y]] − \frac{1}{4}[α′ ∧ [α′′, Y]] = 0.

Taking the appropriate trace in this and comparing coefficients now establishes the lemma. □

With this understood, we now have

Theorem 2.3 Let ϕ : M → G be a pluriharmonic map of a compact complex manifold. If ϕ admits a non-zero formal Killing field Y then ϕ admits a polynomial Killing field ξ. Moreover, if Y is adapted so is ξ.

Finally, if ϕ factors through a symmetric space N̂ ⊂ G as in section (1.4) then

ξ(x) ∈ Λ(𝔤, τ_{ϕ(x)})

for all x ∈ M where τ_{ϕ(x)} is the involution at ϕ(x) ∈ N̂.

Proof We begin by introducing some notation: if Y = ∑_{k≥i} λ^{−k}Y_{k} is a formal Laurent series, let

Y^{+} = ∑_{0≥k≥i} λ^{−k}Y_{k}

denote its polynomial part. We define a vector space K^{+} by

K^{+} = {Y^{+} : Y a formal Killing field for ϕ}.

Further, we define linear maps on K^{+} by

DY^{+} = ∇Y^{+} − \frac{1}{2}[λα′ + λ^{−1}α′′, Y^{+}]
EY^{+} = ev_{1} ∘Y^{+}

where ev_{1} is evaluation at λ = 1.

We claim that both D and E have finite rank. For this, introduce a Hermitian metric on M so that, by lemma (2.2), the coefficients of any formal Killing field are Jacobi fields and thus lie in a finite-dimensional space by the ellipticity of J_{ϕ} and the compactness of M. Now

EY^{+} = ∑_{0≥k≥i} Y_{k}

is a Jacobi field so that E has finite rank. Moreover,

∇Y^{+} = (∇Y)^{+} = \frac{1}{2}[λα′ + λ^{−1}α′′, Y]^{+}
= \frac{1}{2}[λα′ + λ^{−1}α′′, Y^{+}] + \frac{1}{2}[α′, Y_{−1}] − \frac{1}{2}[λ^{−1}α′′, Y_{0}].

Thus DY^{+} lies in the sum of the images of the Jacobi fields under the linear maps \frac{1}{2} ad α′ and \frac{1}{2} ad λ^{−1}α′′ and so D also has finite rank and the claim is proved.

Consider now a non-zero formal Killing field Y. For each N ∈ ℕ, λ^{N}Y is also a formal Killing field and, moreover, for N large enough, the truncations (λ^{N}Y)^{+} are linearly independent. Thus some linear combination of these, ξ′ say, lies simultaneously in the kernels of D and E. Since these equations are both real, the Laurent polynomial ξ = ξ′ + \overline{ξ′} satisfies

ξ(1) = 0, ∇ξ = \frac{1}{2}[λα′ + λ^{−1}α′′, ξ].

The first equation means that ξ lies in some Ω_{d} while the second says that ξ is a polynomial Killing field.

Moreover, if Y is adapted, so is each λ^{N}Y and, since the top term of ξ is a non-zero multiple of that of Y, after multiplication by a suitable constant, we conclude that ξ is also adapted.

Finally, suppose ϕ factors through a symmetric space N̂ ⊂ G and let τ_{ϕ} be the field of involutions where τ_{ϕ(x)} is the involution at ϕ(x) ∈ N̂. Then τ_{ϕ} is ∇-parallel and τ_{ϕ}α = −α. From this it is easy to check that if Y = ∑_{k≥−1} λ^{−k}Y_{k} is a formal Killing field, so is

Ŷ = ∑_{k≥−1} λ^{−k}(−1)^{k}τ_{ϕ}Y_{k}

and thus also Ỹ = \frac{1}{2}(Y +Ŷ). Then each (λ^{2N} Ỹ)^{+} lies in Λ(𝔤^{ℂ} , τ_{ϕ}) and we apply the above argument to these to get a polynomial Killing field ξ with each ξ(x) ∈ Λ(𝔤, τ_{ϕ(x)}) as required. □

2.3 Existence of formal Killing fields

Now we come to the heart of the matter and introduce a hypothesis for pluriharmonic maps that ensures the existence of a formal Killing field.

Definition Let ϕ : M → G be a pluriharmonic map of a complex manifold with ϕ^{∗}θ = α. We say that ϕ is semisimple adapted if there is a non-zero holomorphic vector field Z on M such that α(Z) takes values in a single Ad G^{ℂ} -orbit of semisimple elements of 𝔤^{ℂ}.

Remark In general, the notion of semisimplicity of an element of a Lie algebra depends on a representation of that algebra. However, in what follows, we shall only consider semisimple 𝔤^{ℂ} where the notion is unambiguous or take 𝔤^{ℂ} = gl(n, ℂ) with its fundamental representation. For semisimple subalgebras of gl(n, ℂ), the two notions coincide (c.f. [19]).

We shall see below that a large class of pluriharmonic maps of complex n-tori are semisimple adapted.

The theorem we shall prove is

Theorem 2.4 Let ϕ : M → G be a semisimple adapted pluriharmonic map of a complex manifold where G is semisimple or a unitary group. Then ϕ admits an adapted formal Killing field.

The proof of this will occupy us for the rest of this section. Before embarking on this, let us pause to extract an important corollary: from (2.1), (2.3) and (2.4) we get Corollary 2.5 Let ϕ : T^{2} → G be a semisimple adapted harmonic map of a 2-torus into a compact semisimple Lie group. Then ϕ is of finite type.

Now let us turn to the proof of (2.4): it will transpire that it is enough to prove the result for the case that G is a unitary group so that is where we shall begin.

Let ϕ : M → U(n) be a pluriharmonic map with Z a holomorphic vector field on M such that α(Z) takes values in a single Ad GL(n, ℂ)-orbit of semisimple elements of gl(n, ℂ). We seek a formal Laurent series Y = ∑_{k≥0} λ^{−k}Y_{k} with Y_{k} : M → gl(n, ℂ); Y_{0} = α(Z) and

dY + [A, Y] = 0,

where as usual A = \frac{1−λ}{2}α′ + \frac{1−λ^{−1}}{2} α′′.

Set B = α(Z) : M → gl(n, ℂ). We define a bundle decomposition

M × gl(n, ℂ) = V ⊕ V^{⊥}

by setting V = ker ad B, V^{⊥} = im ad B and note the relations

V V ⊂ V,  V^{⊥}V ⊂ V^{⊥},  V V^{⊥} ⊂ V^{⊥}.  (19)

Since B has image in a single conjugacy class, dB takes values in V^{⊥} whence ∇B = dB + \frac{1}{2}[α, B] also takes values in V^{⊥}. Now ad B is invertible on V^{⊥} so we may define a V^{⊥}-valued 1-form Q by

∇B = \frac{1}{2}[B, Q]

and thus a new connection D = ∇ + \frac{1}{2} ad Q for which B and hence V and V^{⊥} are parallel. In fact, it follows from the pluriharmonic map equations (17) that Q is a (1, 0)-form. We use D to write the formal Killing field equations (16) as

DY = \frac{1}{2}[Q + λα′ + λ^{−1}α′′, Y].  (20)

Remark The (1, 0)-part of this equation is essentially that considered in integrable systems theory for constructing hierarchies of nonlinear evolution equations (c.f. [14]).

We will construct Y via the following ansatz (c.f. [12,13]): we seek Y of the form

Y = (1 + W)^{−1}B(1 + W)

where the formal gauge transformation W = ∑_{k≥1} λ^{−k}W_{k} has coefficients which are sections of V^{⊥}. Now

(1 + W)D((1 + W)^{−1}σ(1 + W))(1 + W)^{−1} = Dσ + [σ, DW (1 + W)^{−1}]

so that such a Y will satisfy (20) if and only if

[DW (1 + W)^{−1} + \frac{1}{2}(1 + W)(Q + λα′ + λ^{−1}α′′)(1 + W)^{−1}, B] = 0

since DB vanishes. Thus we seek W such that

DW (1 + W)^{−1} + \frac{1}{2}(1 + W)(Q + λα′ + λ^{−1}α′′)(1 + W)^{−1} = ω

for a 1-form ω taking values in (formal Laurent series in) V. Otherwise put, we want

DW + \frac{1}{2}(1 + W)(Q + λα′ + λ^{−1}α′′) = ω(1 + W).

We can find such a W for which this holds at least in the Z direction: evaluate the above on Z to get

D_{Z}W + \frac{1}{2}(1 + W)(Q(Z) + λB) = ω(Z)(1 + W)

and project onto V and V^{⊥} to get

(W Q(Z))^{V} + λB = 2ω(Z)  (21)
2D_{Z}W + Q(Z) + (W Q(Z))^{⊥} + λW B = 2ω(Z)W.  (22)

Here ^{V}, ^{⊥} denote the projections onto V and V^{⊥} respectively and we have used (19) and the fact that V and V^{⊥} are D-parallel. Now substitute (21) into (22) to get an equation for W:

λ[B, W] = 2D_{Z}W + Q(Z) + (W Q(Z))^{⊥} − (W Q(Z))^{V} W

which can be recursively solved. Indeed, in terms of coefficients, it reads

[B, W_{1}] = Q(Z)
[B, W_{k+1}] = 2D_{Z}W_{k} + (W_{k}Q(Z))^{⊥} − ∑_{i+j=k}(W_{i}Q(Z))^{V} W_{j},

for k ≥ 1, which uniquely define the W_{k} since ad B is invertible on V^{⊥}. With W so defined, we take

Y = (1 + W)^{−1}B(1 + W)

and conclude that

D_{Z}Y = \frac{1}{2}[Q(Z) + λα(Z), Y]

so that Y satisfies the formal Killing field equations (20) in the Z direction.

It remains to show that Y satisfies (20) in all other directions. This can be accomplished by a direct but extremely lengthy computation, however, we shall present a shorter indirect argument.

Let Z̃ be some local vector field on M commuting with Z. In terms of the flat Ω𝔤-connection, D = d + ad A, the Killing field equations we wish to establish read

DY = 0.

We have D_{Z}Y = 0 so that, by the flatness of D, we get

D_{Z}D_{Z}Y = D_{Z}D_{Z}(Y) = 0.

Setting D_{Z}Y = (1 + W)^{−1}σ(1 + W) this means

D_{Z}σ = [ω(Z), σ]

with ω(Z) given by (21). Using (1 + W)Y (1 + W)^{−1} = B, we see that

σ = D_{Z}B − [D_{Z}(1 + W)^{−1} + \frac{1}{2}(1 + W)(Q(Z̃) + λα′(Z̃) + λ^{−1}α′′(Z̃))(1 + W)^{−1}, B]

so that, since B is D-parallel, we conclude that the coefficients of σ are sections of V^{⊥}. That D_{Z}Y vanishes is now a consequence of the following lemma:

Lemma 2.6 Let σ = ∑_{k≥i} λ^{−k}σ_{k} be a solution of

D_{Z}σ = [ω(Z), σ]  (23)

with each coefficient σ_{k} a section of V^{⊥}. Then σ vanishes identically.

Proof Recall that ω(Z) = \frac{1}{2}(λB + (W Q(Z))^{V}) and observe that W Q(Z) involves only negative powers of λ. If σ does not vanish then the top term σ_{i} is not identically zero but comparing coefficients of λ^{−i+1} in (23) gives

0 = \frac{1}{2}[B, σ_{i}]

so that σ_{i} is simultaneously a section of V and V^{⊥} and so vanishes identically. □

Thus we conclude that D_{Z}Y = 0 whence Y is a formal Killing field. Moreover, Y_{0} = B = α(Z) so that Y is adapted.

This establishes theorem (2.4) in the case that G = U(n). If G is semisimple, we choose a faithful unitary representation and thus an inclusion G ⊂ U(n). If we view ϕ : M → G as a map into U(n) then ϕ remains pluriharmonic and semisimple adapted. Thus we have a gl(n, ℂ)-formal Killing field Ŷ adapted to ϕ. However, 𝔤 being semisimple, we can find an ad_{gl(n,ℂ)} 𝔤^{ℂ} -invariant complement to 𝔤^{ℂ} ⊂ gl(n, ℂ) projection of Ŷ along which produces a 𝔤^{ℂ} -formal Killing field adapted to ϕ. This completes the proof of (2.4).

2.4 Reconstruction and factorisation

In lemma (2.1), we saw that for a harmonic map ϕ : M → G to be of finite type, it is necessary and sufficient that ϕ admit an adapted polynomial Killing field. In this section, we address the corresponding question for pluriharmonic maps into Lie groups and symmetric spaces where the issue is more subtle. Throughout this section, we take G to be semisimple.

Recall that a pluriharmonic map ϕ : ℂ^{n} → G is of finite type if we can find a polynomial Killing field ξ : ℂ^{n} → Ω_{d} from which we can reconstruct the derivative ϕ^{∗}θ = α′ + α′′ by

α′ = 4iV_{k}(ξ_{d})dz^{k}

for some Ad G^{ℂ} -equivariant polynomial maps V_{1}, …, V_{n} : 𝔤^{ℂ} → 𝔤^{ℂ}. Moreover, when ϕ factors through a symmetric space, we further require that ξ take values in Λ(𝔤, τ_{ϕ}), d be odd and that the V_{k} be τ-compatible (c.f. section (1.4)). To achieve such a reconstruction, we will have to place conditions on the orbit in which ξ_{d} : ℂ^{n} → 𝔤^{ℂ} has its image (c.f. (1.7)).

In what follows, most of the technical difficulties arise in treating the symmetric space case. To avoid dealing with this case separately, we will unify the exposition by changing our viewpoint somewhat. That is, we shall regard Lie groups G as type II symmetric spaces G × G/ΔG (c.f. [16]) and restrict attention to pluriharmonic maps into symmetric spaces.

So let G be a compact semisimple Lie group and N = G/K be a compact symmetric space with involution τ and symmetric decomposition 𝔤 = 𝔨 ⊕ 𝔭 at the basepoint o = eK ∈ N. We begin by recalling some generalities about the orbit structure of the Ad K^{ℂ} -action on 𝔭^{ℂ}. Let ℂ [𝔭] denote the polynomial ring over 𝔭^{ℂ} and let ℂ [𝔭]^{K} denote the ring of Ad K^{ℂ} -invariant polynomials. Then ℂ [𝔭]^{K} is a polynomial ring on r = rank N homogeneous generators, see [17]. In the case that N is a group manifold, this reduces to the more familiar assertion that the ring ℂ [𝔤]^{G} of Ad G^{ℂ} -invariant polynomials on 𝔤^{ℂ} is a polynomial ring on rank G generators.

Following Kostant–Rallis [24], we call an element ξ ∈ 𝔭^{ℂ} regular if its Ad K^{ℂ} -orbit has maximal possible dimension. Such orbits turn out to be easy to characterise, at least when ξ is semisimple (in the sense that ad ξ is semisimple on 𝔤^{ℂ}). Indeed, let O = Ad K^{ℂ} · ξ be the orbit of a regular semisimple element ξ ∈ 𝔭^{ℂ} and let ℂ [𝔭]^{K} = ℂ [p_{1}, …, p_{r}]. From [24] we have

(i) the centraliser C_{𝔭} (ξ) of ξ in 𝔭^{ℂ} is a maximal abelian subspace of 𝔭^{ℂ};

(ii) O is the level set of a regular value of the map (p_{1}, …, p_{r}) : 𝔭^{ℂ} → ℂ^{r}.

From this, we see immediately that C_{𝔭} (ξ) (which is the Killing orthogonal complement to T_{ξ}O = [ξ, 𝔨^{ℂ}]) has basis given by the gradients of the p_{j} at ξ.

Remark When N is a group manifold, these results describe the Ad G^{ℂ} -orbits of regular semisimple elements of 𝔤^{ℂ} and are due to Kostant [23].

Our analysis of pluriharmonic maps of finite type involves τ-compatible gradients of Ad G^{ℂ} - invariant polynomials on 𝔤^{ℂ} rather than Ad K^{ℂ} -invariant polynomials on 𝔭^{ℂ} and this prompts the following definition.

Definition A symmetric space has the surjection property if the restriction map ℂ [𝔤]^{G} → ℂ [𝔭]^{K} is surjective.

We see from the following examples that almost all symmetric spaces have this property:

(i) all rank one symmetric spaces have the surjection property: indeed, ℂ [𝔭]^{K} is generated by the restriction of the Killing form;

(ii) all symmetric spaces of type II (i.e. group manifolds) have the surjection property: indeed, if p is a homogeneous Ad G^{ℂ} -invariant polynomial of degree d on 𝔤^{ℂ}, then p is the restriction of the Ad(G × G)^{ℂ} -invariant polynomial on (𝔤 × 𝔤)^{ℂ} given by

p̂((ξ_{1}, η_{1}), …, (ξ_{n}, η_{n})) = \frac{1}{2}(p(ξ_{1}, …, ξ_{n}) + (−1)^{d}p(η_{1}, …, η_{n})).

(iii) Finally, Helgason [15] shows by a case by case analysis that if G and K are classical then G/K has the surjection property. He also shows that the surjection property fails for certain exceptional symmetric spaces.

We now globalise this situation by translating it around our symmetric space N = G/K. For this, recall from section (1.4) that we may identify T N with a subbundle [𝔭] of the trivial bundle N × 𝔤 where

[𝔭]_{x} = 𝔭_{x}.

Under this identification, the G-action on T N becomes the restriction of the adjoint action of G on N × 𝔤. Let O be an Ad K^{ℂ} -orbit in 𝔭^{ℂ} = T_{o}N^{ℂ}. We define the G-translate of O, denoted O_{G} ⊂ T N^{ℂ}, by

(O_{G})_{g·o} = Ad g(O).

This is clearly well-defined since Ad K(O) = O and O_{G} is a sub-fibre-bundle of T N^{ℂ} invariant under the G-action. We note that, viewing O_{G} as a subset of N × 𝔤^{ℂ}, its image under projection onto 𝔤^{ℂ} lies in a single Ad G^{ℂ} -orbit.

Example If N is a Lie group G and O ⊂ 𝔤^{ℂ} is an Ad G^{ℂ} -orbit, then, identifying T N with G × 𝔤 via the left Maurer–Cartan form, O_{G×G} is just G × O.

Now let f ∈ ℂ [𝔭]^{K}. We define a symmetric form f̃: T N^{ℂ} → ℂ by

f̃_{g·o} = f ∘ Ad g^{−1}.

Again, the Ad K^{ℂ} -invariance of f means that f̃ is well-defined and G-invariant (and hence parallel for the Levi-Civita connection on N).

Example If N has rank one, then ℂ [𝔭]^{K} is generated by the Killing form whose extension to T N^{ℂ} is just the (complexified) metric of N.

With this in hand, we reformulate the above discussion of orbit structure in

Proposition 2.7 Let N = G/K be a symmetric space of rank r, ℂ [𝔭]^{K} = ℂ [p_{1}, …, p_{r}] and O ⊂ 𝔭^{ℂ} a regular semisimple Ad K^{ℂ} -orbit. Then

(i) O_{G} is a level set of the map (p̃_{1}, …, p̃_{r}) : T N^{ℂ} → ℂ^{r};

(ii) if, in addition, N has the surjection property, there exist Ad G^{ℂ} -equivariant, τcompatible polynomial maps J_{1}, …, J_{r} : 𝔤^{ℂ} → 𝔤^{ℂ} such that, for η ∈ (O_{G})_{x}, the centraliser of η in 𝔭_{x}^{ℂ} has basis J_{1}(η), …, J_{r}(η).

We have now assembled what we need to discuss whether a pluriharmonic map is of finite type. So let ϕ : M → N be a pluriharmonic map of a complex manifold. Suppose that ϕ admits a polynomial Killing field ξ : M → Ω_{d} with d odd and ξ(x) ∈ Λ(𝔤, τ_{ϕ(x)}) for all x ∈ N. (We recall from (2.3) and (2.4) that such a ξ exists if ϕ is semisimple adapted and M is compact). Since d is odd, ξ_{d} : M → 𝔤^{ℂ} is a section of [𝔭]^{ℂ} along ϕ and we say ξ is of regular semisimple orbit type if ξ_{d} has image in the G-translate of a single Ad K^{ℂ} -orbit of regular semisimple elements of 𝔭^{ℂ}.

Theorem 2.8 Let M = ℂ^{n}/Γ be a complex torus, N a compact symmetric space with the surjection property and ϕ : M → N a pluriharmonic map admitting a polynomial Killing field ξ : M → Ω_{d} of regular semisimple orbit type. Then ϕ is of finite type, that is, there exist Ad G^{ℂ} -equivariant, τ-compatible maps V_{1}, …, V_{n} : 𝔤^{ℂ} → 𝔤^{ℂ} such that

α′ = 4iV_{k}(ξ_{d})dz^{k}

where ϕ^{∗}θ = α′ + α′′.

Proof Comparing coefficients in the Killing field equations (13) gives

0 = [ξ_{d}, α′]  (24)

dξ_{d} = [ξ_{d}, \frac{1}{2}α′′] − [ξ_{d−1}, \frac{1}{2}α′].  (25)

Since ξ is of regular semisimple orbit type and N has the surjection property, we know from (2.7) that there are r = rank N Ad G^{ℂ} -equivariant, τ-compatible polynomial maps J_{1}, …, J_{r} : 𝔤^{ℂ} → 𝔤^{ℂ} such that for each x ∈ M, the {J_{k}(ξ_{d}(x))} span the centraliser of ξ_{d}(x) in 𝔭_{ϕ(x)}^{ℂ}. Thus, from (24) we see that

α′ = J_{j}(ξ_{d})c^{j}  (26)

for (1, 0) forms c^{1}, …, c^{r} on M. We claim that the c^{j} are closed: indeed

dα′ = d(J_{j}(ξ_{d})) ∧ c^{j} + J_{j}(ξ_{d})dc^{j}.

Using the Ad G^{ℂ} -equivariance of J_{j} we have

d(J_{j}(ξ_{d})) = (dJ_{j})_{ξ_{d}}(dξ_{d})
= (dJ_{j})_{ξ_{d}}([ξ_{d}, \frac{1}{2}α′′]) − (dJ_{j})_{ξ_{d}}([ξ_{d−1}, \frac{1}{2}α′])
= [J_{j}(ξ_{d}), \frac{1}{2}α′′] − (dJ_{j})_{ξ_{d}}([ξ_{d−1}, \frac{1}{2}α′]).

Now, d being odd, for x ∈ M we have ξ_{d−1}(x) ∈ 𝔨_{ϕ(x)}^{ℂ} while α′_{x} takes values in the (abelian) centraliser of ξ_{d}(x) so that

[ξ_{d−1}(x), α′_{x}] ∈ [𝔨_{ϕ(x)}^{ℂ}, C_{𝔭} (ξ_{d}(x))] = [ξ_{d}(x), 𝔨_{ϕ(x)}^{ℂ}]

whence

(dJ_{j})_{ξ_{d}(x)}([ξ_{d−1}(x), \frac{1}{2}α′_{x}]) ∈ [J_{j}(ξ_{d}(x)), 𝔨_{ϕ(x)}^{ℂ}] = [ξ_{d}(x), 𝔨_{ϕ(x)}^{ℂ}]

where we have used the fact that, since C_{𝔭} (ξ_{d}(x)) is abelian,

[𝔨_{ϕ(x)}^{ℂ}, C_{𝔭} (ξ_{d}(x))] = [𝔨_{ϕ(x)}^{ℂ}, ξ_{d}(x)].

Thus,

dα′ = [J_{j}(ξ_{d}), \frac{1}{2}α′′] ∧ c^{j} − [ξ_{d}, ω] + J_{j}(ξ_{d})dc^{j}

for some [𝔨]^{ℂ} -valued 1-form ω, or, using (26),

dα′ + \frac{1}{2}[α′ ∧ α′′] = −[ξ_{d}, ω] + J_{j}(ξ_{d})dc^{j}.

However, by theorem (1.5), the left hand side of this vanishes giving

[ξ_{d}, ω] = J_{j}(ξ_{d})dc^{j}

in which the right hand side takes values in C_{𝔭}(ξ_{d}) and the left in im ad ξ_{d}, which are complementary since ξ_{d} is semisimple. Thus J_{j}(ξ_{d})dc^{j} vanishes, whence, since J_{j}(ξ_{d}) are linearly independent, the c^{j} are closed.

We remark that so far we have used no facts concerning the nature of M. Now using the fact that M is a complex torus, we conclude that the c^{j} are constant:

c^{j} = a_{k}^{j}dz^{k},

for constants a_{k}^{j} ∈ ℂ, so that defining Ad G^{ℂ} -equivariant τ-compatible polynomial maps V_{1}, …, V_{n} : 𝔤^{ℂ} → 𝔤^{ℂ} by

V_{k} = \frac{1}{4i}J_{j}a_{k}^{j}

we get

α′ = 4iV_{k}(ξ_{d})dz^{k}

as required. □

We are now in a position to state the analogue of (2.5) for pluriharmonic maps. First a definition.

Definition Let ϕ : M → N be a pluriharmonic map of a complex manifold into a compact symmetric space. Say ϕ is regular semisimple adapted if there is a holomorphic vector field Z on M such that ϕ_{∗}Z takes values in the G-translate of a single regular semisimple Ad K^{ℂ} -orbit in 𝔭^{ℂ}.

Example In case N = G, this reduces to the demand that α(Z) take values in a single regular semisimple Ad G^{ℂ} -orbit in 𝔤^{ℂ}.

We remark that if ϕ : M → N is regular semisimple adapted then α(Z) takes values in a single semisimple orbit in 𝔤^{ℂ} i.e. the map M → N̂ ⊂ G is semisimple adapted so that (2.4) applies. Then an application of (2.3) provides a polynomial Killing field of regular semisimple orbit type and (2.8) gives:

Corollary 2.9 Let ϕ : ℂ^{n}/Γ → N be a regular semisimple adapted pluriharmonic map of a complex torus into a compact symmetric space with the surjection property. Then ϕ is of finite type.

In the next section we shall see that the “generic” pluriharmonic map of a torus is regular semisimple adapted. In the meantime we conclude this section with a reformulation of (2.8) which gives a factorisation theorem for pluriharmonic maps of simply connected domains with a polynomial Killing field of regular semisimple orbit type. It is easy to see that such a map has rank not exceeding 2 rank N but in fact a stronger result is true: such a map factors holomorphically through ℂ^{r}. This result is reminiscent of those obtained by Carlson and Toledo for pluriharmonic maps into (quotients of) negatively curved symmetric spaces [8].

Theorem 2.10 Let M be a simply connected complex manifold, N a compact symmetric space of rank r with the surjection property and ϕ : M → N a pluriharmonic map admitting a polynomial Killing field of regular semisimple orbit type.

Then there exists a holomorphic map h : M → ℂ^{r} and a pluriharmonic map of rank 2r ψ : ℂ^{r} → N of finite type such that ϕ = ψ ∘ h.

Proof The argument of (2.8) gives Ad G^{ℂ} -equivariant, τ compatible polynomial maps J_{1}, …, J_{r} and closed (1, 0) forms c^{1}, …, c^{r} such that

α′ = J_{j}(ξ_{d})c^{j}.

Fix a base point x_{0} in M with ϕ(x_{0}) = o ∈ N. Then we can integrate to get a holomorphic map h = (h^{1}, …, h^{r}) : M → ℂ^{r} with

dh^{j} = c^{j}, h(x_{0}) = 0.

Now let u^{1}, …, u^{r} be the co-ordinates on ℂ^{r} and η : ℂ^{r} → Ω_{d} the solution to

dη = [η, \frac{1}{2}(1 − λ)J_{j}(η_{d})du^{j} + \frac{1}{2}(1 − λ^{−1})\overline{J_{j}(η_{d})}d\overline{u}^{j}]
η(0) = ξ(x_{0}).

Then, by (1.12), there is a pluriharmonic map ψ : ℂ^{r} → N̂ ∈ G satisfying

(ψ^{∗}θ)′ = J_{j}(η_{d})du^{j}, ψ(0) = e.

Note that ψ has rank 2r since the J_{j}(η_{d}) are linearly independent at 0 and hence everywhere by (1.7).

We claim that ϕ = ψ∘h. Certainly these maps agree at x_{0} so it suffices to show that ϕ^{∗}θ = (ψ∘h)^{∗}θ (we are of course viewing ϕ as a map to N̂ ⊂ G). Now

(ϕ^{∗}θ)′ = J_{j}(ξ_{d})c^{j}

while

((ψ ∘ h)^{∗}θ)′ = h^{∗}(J_{j}(η_{d})du^{j}) = J_{j}(η_{d} ∘ h)h^{∗}du^{j} = J_{j}(η_{d} ∘ h)c^{j}

so it suffices to prove that η ∘ h = ξ. However,

dξ = [ξ, \frac{1}{2}(1 − λ)J_{j}(ξ_{d})c^{j} + \frac{1}{2}(1 − λ^{−1})\overline{J_{j}(ξ_{d})}\overline{c}^{j}]

while

d(η ∘ h) = h^{∗}dη = h^{∗}[η, \frac{1}{2}(1 − λ)J_{j}(η_{d})du^{j} + \frac{1}{2}(1 − λ^{−1})\overline{J_{j}(η_{d})}d\overline{u}^{j}]
= [η ∘ h, \frac{1}{2}(1 − λ)J_{j}(η_{d} ∘ h)c^{j} + \frac{1}{2}(1 − λ^{−1})\overline{J_{j}(η_{d} ∘ h)}\overline{c}^{j}].

Moreover η ∘ h(x_{0}) = η(0) = ξ(x_{0}) so that ξ = η ∘ h by the uniqueness of solutions to ordinary differential equations. □

3 Conclusions

In this section, we shall tie together the preceding results and, in particular, show how simple geometric conditions ensure that a (pluri)harmonic map from a complex torus is of finite type and so arises from commuting flows. Thus we obtain an explicit construction of a substantial class of pluriharmonic tori in Lie groups and symmetric spaces. As in section (2.4), we shall unify the exposition by viewing a Lie group G as a symmetric G × G-space.

Recall that a harmonic map of a 2-torus into a symmetric space is of finite type if it is semisimple adapted (2.5) while a pluriharmonic map of a higher dimensional complex torus into a symmetric space with the surjection property is of finite type if it is regular semisimple adapted (2.9). Thus we need a criterion for detecting when a map is (regular) semisimple adapted. For this we use (2.7) and a familiar holomorphic differentials argument going back to [10]: recall that an Ad K^{ℂ} - invariant polynomial p : 𝔭^{ℂ} → ℂ gives rise to a G-invariant (and hence covariant constant) map p̃: T N^{ℂ} → ℂ. Concerning such maps, we have

Lemma 3.1 Let ϕ : M → N be a pluriharmonic map of a complex manifold into a Riemannian manifold and let P be a covariant constant section of S^{d}(T^{∗}N^{ℂ}). Then (ϕ^{∗}P)^{(d,0)} is a holomorphic differential of degree d on M.

Proof Let Z_{1}, …, Z_{d} be (local) holomorphic vector fields on M. We must show that

\overline{∂}(ϕ^{∗}P (Z_{1}, …, Z_{d})) = 0.

But if ^{ϕ}∇ denotes the pull-back of the Levi-Civita connection on N then, since P is covariant constant, we have

\overline{∂}(ϕ^{∗}P (Z_{1}, …, Z_{d})) = ∑_{i=0}^{d} P (ϕ_{∗}Z_{1}, …,^{ϕ}∇^{(0,1)}ϕ_{∗}Z_{i}, …, ϕ_{∗}Z_{d})

which vanishes since each ^{ϕ}∇^{(0,1)}ϕ_{∗}Z_{i} vanishes by the pluriharmonic map equations (11). □

We apply this to the case at hand and obtain the main result of the paper:

Theorem 3.2 Let ϕ : ℂ^{n}/Γ → N = G/K be a pluriharmonic map of a complex torus into a compact symmetric space which has the surjection property if n > 1.

Let Z be a holomorphic vector field on ℂ^{n}/Γ and suppose that, for some x_{0} ∈ ℂ^{n}/Γ, ϕ_{∗}Z_{x_{0}} ∈ 𝔭_{ϕ(x_{0})}^{ℂ} is regular semisimple. Then ϕ is of finite type.

Proof It suffices to show that ϕ is regular semisimple adapted. Now ϕ_{∗}Z_{x_{0}} is an element of the G-translate O_{G} of some Ad K^{ℂ} -orbit of regular semisimple elements. From (2.7), we see that there are r = rank N covariant constant homogeneous maps (p̃_{1}, …, p̃_{r}) : T N^{ℂ} → ℂ^{r} for which O_{G} is a common level set. From (3.1), we conclude that each p̃_{j}(ϕ_{∗}Z) is holomorphic on ℂ^{n}/Γ and thus constant by Liouville’s Theorem. Thus ϕ_{∗}Z takes values in the level set of (p̃_{1}, …, p̃_{r}) containing ϕ_{∗}Z_{x_{0}} which is O_{G} whence ϕ is regular semisimple adapted as required. □

Thus, for pluriharmonic maps of complex tori, it suffices to verify the regular semisimple adapted condition at a single point. We remark that the regular semisimple elements form an open dense subset of 𝔭^{ℂ} so that in some sense the regular semisimple adapted pluriharmonic tori are “generic”.

Let us extract a corollary of (3.2) concerning harmonic 2-tori in rank one symmetric spaces: in this case, the Ad K^{ℂ} -invariant polynomials are generated by the Killing form whose extension to T N^{ℂ} is just the complexified metric. Moreover, ξ ∈ 𝔭^{ℂ} is regular semisimple if and only if the Killing form is non-zero on ξ. Thus a harmonic map ϕ : T^{2} → N is regular semisimple adapted if and only if it is non-conformal at a single point (and hence everywhere). This gives

Corollary 3.3 Let ϕ : T^{2} → N be a non-conformal harmonic map of a 2-torus into a compact rank one symmetric space. Then ϕ is of finite type.

Taken together with the results of Calabi [7] and Bryant [4] for superminimal 2-tori in S^{4} and those of Ferus–Pedit–Pinkall–Sterling [13] for non-superminimal minimal 2-tori in S^{4}, the above result accounts for all harmonic 2-tori in S^{4} and thus in S^{3} = SU(2). The Reader is invited to compare our results with those of Hitchin [18] concerning harmonic 2-tori in S^{3}.

For other rank one symmetric spaces, it remains to deal with the conformal harmonic (i.e. branched minimal) 2-tori. So far, a complete theory exists only for certain minimal tori in S^{n} and ℂ P^{n}. For these, the isotropic ones are dealt with by Calabi [7] in the S^{n} case and by Eells–Wood [11] in the ℂ P^{n} case. Another class, characterised by the vanishing of all but one of a series of holomorphic differentials, have been dealt with in [3] using methods similar to our own. These latter maps correspond to doubly periodic solutions of the Toda field equations for SO(n + 1) and SU(n + 1). As special cases of these results, all almost complex tori in S^{6} and all minimal non-isotropic 2-tori in ℂ P^{2} are shown to be of finite type.

When the rank of N is greater then one, it is not so easy to characterise the condition on the derivative that leads to ϕ being regular semisimple adapted. However, a result of Kostant implies that, for higher rank Lie groups, there are minimal tori that are regular semisimple adapted. In fact, let I_{1}, …, I_{r} be homogeneous algebraically independent generators of ℂ [𝔤]^{G} arranged in increasing order of degree. If ξ ∈ 𝔤^{ℂ} satisfies

I_{1}(ξ) = ⋯ = I_{r−1}(ξ) = 0
I_{r}(ξ) ≠ 0

then we have from [22, corollary 9.2] that ξ is regular semisimple. Following Kostant we call such ξ cyclic. If we now take I_{1} to be the Killing form, we get

Corollary 3.4 Let ϕ : T^{2} → G be a minimal torus in a Lie group of rank ≥ 2. If ϕ^{∗}θ(∂/∂z) is cyclic at a single point, it is of finite type.

Finally, using (2.10) rather than (2.9), we get a mild generalisation of (3.2):

Theorem 3.5 Let ϕ : M → N be a pluriharmonic map of a compact complex manifold into a compact symmetric space of rank r which has the surjection property if dim_{ℂ} M > 1.

Let Z be a holomorphic vector field on M and suppose that, for some x_{0} ∈ M, ϕ_{∗}Z_{x_{0}} ∈ 𝔭_{ϕ(x_{0})}^{ℂ} is regular semisimple. Then the lift ϕ̃ : M̃ → N to the universal cover of M factors as ϕ̃ = ψ ∘ h where h : M̃ → ℂ^{r} is holomorphic and ψ : ℂ^{r} → N is a rank 2r pluriharmonic map of finite type.

Proof As in (3.2), we see that ϕ is regular semisimple adapted and so admits a polynomial Killing field of regular semisimple orbit type. Thus ϕ̃ also admits such a polynomial Killing field so that we may apply (2.10). □

We remark that under the hypotheses of (3.5), the holomorphic vector field Z is nowhere vanishing which imposes severe restrictions on the nature of M.

We conclude with another, simpler, version of (3.5) for harmonic 2-tori which shows that any regular semisimple adapted harmonic 2-torus may be extended on the universal cover to a pluriharmonic map. One may view this map as a “coherent” collection of harmonic maps of ℝ^{2} commuting with the original one.

Theorem 3.6 Let ϕ : ℂ /Γ → N be a regular semisimple adapted harmonic map of a 2-torus into a compact symmetric space of rank r with the surjection property. Then there is a pluriharmonic map of finite type and rank 2r ϕ̃: ℂ^{r} = ℂ × ℂ^{r−1} → N such that

ϕ̃|_{ℂ} = ϕ.

Proof From (2.5), we know that there is a map ξ : ℂ → Ω_{d}, for some odd d, satisfying

\frac{∂ξ}{∂z} = [ξ, 2i(1 − λ)ξ_{d}]

4iξ_{d}dz = α′.

Moreover, since N has the surjection property, there are r Ad G^{ℂ} -equivariant, τ-compatible polynomial maps V_{1}, …, V_{r} : 𝔤^{ℂ} → 𝔤^{ℂ} with {V_{j}(ξ_{d})} linearly independent. Without loss of generality, we may take V_{1} = id_{𝔤^{ℂ}} . Now let ξ̃: ℂ^{r} → Ω_{d} solve

\frac{∂ξ̃}{∂z^{k}} = [ξ̃, 2i(1 − λ)V_{k}(ξ̃_{d})]

ξ̃(0) = ξ(0).

Then we conclude that ξ̃|_{ℂ} = ξ and take ϕ̃ to be the pluriharmonic map arising from ξ̃ in (1.6). □

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(F.E. Burstall) School of Mathematical Sciences, University of Bath, Bath, BA2 7AY, UK E-mail address: f.e.burstall@maths.bath.ac.uk

(D. Ferus) Fachbereich Mathematik, MA 8-3, Technische Universität Berlin, Strasse des 17. Juni 135, 1-Berlin 33, FRG E-mail address: ferus@willmore.math.tu-berlin.de

(F. Pedit) Department of Mathematics and Computer Science, Emory University, Atlanta, GA 30322, USA E-mail address: pedit@mathcs.emory.edu

(U. Pinkall) Fachbereich Mathematik, MA 8-4, Technische Universität Berlin, Strasse des 17. Juni 135, 1-Berlin 33, FRG E-mail address: pinkall@willmore.math.tu-berlin.de