Transverse $J$-holomorphic curves in nearly K\"ahler $\mathbb{CP}^3$

$J$-holomorphic curves in nearly K\"ahler $\mathbb{CP}^3$ are related to minimal surfaces in $S^4$ as well as associative submanifolds in $\Lambda^2_-(S^4)$. We introduce the class of transverse $J$-holomorphic curves and establish a Bonnet-type theorem for them. We classify flat tori in $S^4$ and construct moment-type maps from $\mathbb{CP}^3$ to relate them to the theory of $\mathrm{U}(1)$-invariant minimal surfaces on $S^4$.


Introduction
In symplectic geometry, the theory of J-holomorphic curves is a fundamental topic, which has for example led to the construction of Gromov-Witten invariants.From a Riemannian point of view, it is desirable to have examples of minimal surfaces in Einstein manifolds.One class of such examples comes from complex geometry by considering holomorphic curves in Einstein-Kähler manifolds.Nearly Kähler six-manifolds are Einstein with positive Einstein constant and J-holomorphic curves provide a class of examples of minimal surfaces inside them.However, they are neither symplectic nor complex and apart from local statements not much is known about J-holomorphic curves in general almost complex manifolds.One feature that is shared with the symplectic case is that the nearly Kähler structure equations guarantee that the volume remains constant on connected components of the moduli space of J-holomorphic curves [29].Nevertheless, a general, in-depth theory of J-holomorphic curves in nearly Kähler manifolds seems out of reach at the moment, which is why most work has been concerned with studying them in the homogeneous examples.Historically, M = S 6 has been the example studied the most, setoff by Bryant's construction of torsion-free J-holomorphic curves as integrals of a holomorphic differential system on Gr(2, R 7 ) [10].More recently, questions about certain components of the moduli space of J-holomorphic curves have been tackled [15].
J-holomorphic curves in the twistor spaces CP 3 and the flag manifold F are related to special submanifolds in two different geometric settings.Firstly, it is a general result that the cone of a Jholomorphic curve in a nearly Kähler manifold M gives an associative submanifold in the G 2 -cone C(M ).Remarkably, J-holomorphic curves in CP 3 and F also give rise to to complete associative submanifolds in the Bryant-Salamon spaces Λ 2 − (S 4 ) and Λ 2 − (CP 2 ) [21].Secondly, via the Eells-Salamon correspondence [14], J-holomorphic curves in the twistor spaces are in correspondence with minimal surfaces in S 4 and CP 2 .Somewhat similar to torsion-free curves in S 6 , the space CP 3 admits a particular class of J-holomorphic curves called superminimal curves.They can also be constructed via integrals of a holomorphic differential system and admit a Weierstraß parametrisation [9].In contrast, have been studied with methods of integrable systems [26] as minimal surfaces in S 4 .The aim of this article is to describe various results on non-superminimal surfaces in S 4 from the twistor perspective, building on F. Xu's work on J-holomorphic curves in nearly Kähler CP 3 [30].Adapting the twistor perspective means that dimension of the ambient space is increased while the second order PDE describing a minimal surface in S 4 is reduced to a first order PDE for J-holomorphic curves in CP 3 .Since the key geometric feature we exploit is the twistor fibration CP 3 → S 4 , it is to be expected that similar techniques can be applied to the flag manifold.

Summary of Results
By Cartan-Kähler theory, J-holomorphic curves in an almost complex manifold of dimension 6 are locally described by four functions of one variable.Two functions α − , α + of two variables that satisfy a Laplace-type equation are parametrised by four functions of one variable too.By using an appropriate adaption of frames, we distill such two functions α − and α + geometrically.The twistor fibration CP 3 → S 4 comes with a natural connection T CP 3 = H ⊕ V. We will focus on J-holomorphic curves which are going to be called transverse.In particular, they have the property that they are nowhere tangent to horizontal bundle H or vertical bundle V, see definition 4.1.
The material covered in section 2 provides background and context for the subsequent sections.Section 3 establishes relationships between geometric quantities associated to a minimal surface and those of its twistor lifts, featuring a twistor interpretation of Xu's correspondence of certain J-holomorphic curves in CP 3 .
In section 4, we define the two functions α − , α + : X → R for a transverse J-holomorphic curve which measure angles between different components of dΦ in T CP 3 .They always satisfy the 2D periodic Toda lattice equation for sp (2) which are equivalent to the system where γ 2 = (α − α + ) −1/2 is the conformal factor of the induced metric on X and ∆ 0 the Laplacian for the corresponding flat metric.Viewing the surface in the ambient space CP 3 means we can characterise additional data equipped to the curve, such as the first and second fundamental form in CP 3 .It turns out that both of them can be expressed through the functions α − and α + .As an application, we show in theorem 5.5 that any flat J-holomorphic torus is a Clifford torus.The second fundamental form II is a complex linear tensor and its differential ∂II can be computed neatly in terms of α − , α + and vanishes exactly in points where the curve is non-transverse.The normal bundle ν of a J-holomorphic curve carries a natural holomorphic structure.We show that the normal bundles of all transverse J-holomorphic tori are isomorphic to each other, in particular all of them admit a holomorphic section.
Section 5 features a proof of a Bonnet-type theorem for J-holomorphic curves stating that the first and second fundamental form on X are determined by α − and α + .The essence of theorem 5.2 is that if X is simply-connected then a C * -family of transverse J-holomorphic curves can be recovered from a solution to the 2D Toda lattice equation.Roughly speaking, this can be seen as a complex analogue to the statement that curves in R 3 are essentially classified by their curvature and torsion.
To get hold of examples of transverse J-holomorphic curves we impose an arbitrary U(1)symmetry on them in section 6, given by a certain element ξ ∈ sp (2).Such an action commutes with a T 2 -action of automorphisms on CP 3 .Given a T 3 -action on a torsion-free G 2 manifold M 7 the multi-moment maps give rise to a local homeomorphism M 7 /T 3 → R 4 [23,Theorem 4.5].The only known example of a nearly Kähler manifold admitting a T 3 -action is S 3 × S 3 whose geometry has been described with multi-moment maps by K. Dixon [13].Assuming a T 2 symmetry on a nearly Kähler manifold M 6 is less restrictive but the corresponding multimoment map ν : M 6 → R, introduced by G. Russo and A. Swann [27], is only real-valued.The question arises if there is a geometric construction of a map into R 4 which descends to a local homeomorphism to the T 2 quotient of M , at least away from a singular set.We construct a map p : T 2 \ (CP 3 \S) → R 4 for a certain singular set S and in theorem 6.10 it is shown that p descends to a branched double cover from T 2 \ CP 3 \S onto its image D ⊂ R 4 .p maps U(1)-invariant Jholomorphic curves in CP 3 to solutions of the 1D Toda lattice equation for sp(2) in D. Derived from the Lax representation of this equation one derives two preserved quantities, giving rise to a map u : D → R 2 .
If one equips CP 3 with its Kähler structure then a T 2 -action gives rise to a symplectic moment map whose image is a quadrilateral.Composing u with p gives rise to a T 2 invariant map In each case, the identity component of the group of nearly Kähler automorphisms is equal to G and only for S 3 × S 3 there is a finite subgroup Γ of G acting freely on M , giving rise to locally homogeneous examples [12].In addition, there are two known examples of compact, simplyconnected nearly Kähler manifolds which are not homogeneous but admit a cohomogeneity-one action.They have been constructed by L. Foscolo and M. Haskins via cohomogeneity one actions on S 3 × S 3 and S 6 [17].

Twistor spaces
We briefly review a few fundamental results on four-dimensional twistor theory, see [14].To each even-dimensional Riemannian manifold N one can associate a twistor space Z ± (N ), which is a fibre bundle over N .The fibre of this bundle over x is given by Unless specified otherwise, by twistor space Z(N ) we refer to the convention of orientation reversing endomorphisms, i.e.Z − (N ).From now on, let N be four-dimensional such that the twistor space Z(N ) can be identified with elements in Λ 2 − (N ) of unit length.It is a sphere bundle over N which inherits the Levi-Civita connection from Λ 2 (N ).Hence T Z(N ) splits into a vertical and horizontal subbundle T Z(N ) = H ⊕ V.It turns out that Z(N ) possesses a natural almost complex structure J 1 known as the Atiyah-Hitchin-Singer almost complex structure.It splits into a sum of almost complex structures on H and V. Reversing J 1 on V defines the Eells-Salamon almost complex structure J 2 .The two almost complex structures J 1 and J 2 are fundamentally different.It turns out that the twistor space of N = S 4 with the round metric is CP 3 .The fibration CP 3 → S 4 is constructed by composing the map with a diffeomorphism HP 1 ∼ = S 4 .The twistor space of N = CP 2 is the flag manifold SU(3)/T 2 .In both cases, J 1 gives rise to the well-known Kähler structures on the spaces while J 2 belongs to nearly Kähler structures.The splitting T Z(N ) = H ⊕ V is parallel with respect to ∇ when N = S 4 or CP 2 .For any immersion f : X → N 4 the differential df defines a lift, called the Gauss lift, φ from X into the oriented Grassmannian bundle Gr 2 (T N ).This bundle in turn projects to Z(N ) such that by composition with φ one obtains a map ϕ : X → Z(N ) which is called the twistor lift of f .
There is a general relation between the Riemannian geometry of N and (almost) complex geometry of Z(N ) which is exemplified by the following result.
Proposition 2.1 (Eells-Salamon).[14, Corollary 5.4] Let X be a Riemann surface and N be a four-dimensional Riemannian manifold.Then f : X → N is a minimal branched immersion if and only if ϕ : X → Z ± (N ) is a J 2 -holomorphic non-vertical curve.
Note that if f : X → N is a branched minimal immersion, i.e. minimal immersion off a discrete set of points, then there is a rank two subbundle E ⊂ f * (T N ) which contains df (T X) so the Gauß lift is still well defined in this case.Furthermore, it should be mentioned that, since the domain is two-dimensional, a branched minimal immersion is the same thing as a conformal harmonic map.Via this correspondence, branched minimal surfaces in S 4 are identified with J 2 -holomorphic curves in the nearly Kähler twistor spaces CP 3 .

The nearly Kähler structure on CP 3
As indicated in the previous section, the nearly Kähler structure on CP 3 can be defined via twistor structure.For explicit computations it is convenient to define the nearly Kähler from the homogeneous space structure This identification gives an action of Sp(2) on C 4 which descends to CP 3 and acts transitively on that space.The stabiliser of the element (1, 0, 0, 0) is More specifically, consider the Maurer-Cartan form on Sp(2) which can be written in components as Here, ω 1 , ω 2 , ω 3 and τ are complex-valued and ρ 1 as well as ρ 2 are real-valued one-forms on Sp(2).The equation Let finally The nearly Kähler structure on CP 3 is defined by declaring the forms s * (ω 1 ), s * (ω 2 ) and s * (ω 3 ) to be unitary (1, 0) forms for any local section s of the bundle Sp(2) → CP 3 .The resulting almost complex structure and metric do not depend on the choice of s.The matrix A ω is in fact the connection matrix of the nearly Kähler connection ∇ in the frame dual to the local co-frame (ω 1 , ω 2 , ω 3 ).Note that the forms (ω 1 , ω 2 , ω 3 , τ ) are complex-valued invariant forms on Sp( 2) and as such they can be seen as elements in sp(2) ∨ ⊗ C.They span different root spaces with respect to the maximal torus S 1 × S 1 , see fig. 1.

Properties of twistor lifts
As seen in eq.(2.1), there is a notion of a negative and positive twistor space for a Riemannian four-manifold N 4 .In general, they can be different fibre bundles over N 4 and proposition 3.1 expresses how the second fundamental form of a surface in N 4 is related to the lift into both twistor spaces.However, for N = S 4 both twistor space are isomorphic as fibre bundles which we exploit to give a twistor interpretation of Xu's correspondence.

Encoding the second fundamental form
Let N 4 be a Riemannian manifold and f : X 2 → N 4 be an isometric immersion.The second fundamental form II encodes local geometric information about f .The aim of this section is to relate the norm of II to quantities defined for the twistor lift of f .Let ν be the normal bundle of T X in f * (T N ).Locally, fix an oriented orthonormal frame {e 1 , e 2 , e 3 , e 4 } such that {e 1 , e 2 } is an oriented basis of f * (T X).Denote by ω ij (v) = g(∇ v e i , e j ) the locally defined connection one-forms of the Levi-Civita connection ∇ on N .Furthermore, let ω ijk = ω ij (e k ).
Then ω ijk = −ω jik and if i, k ∈ {1, 2} and j ∈ {3, 4} then ω ijk = ω kji holds by torsion-freeness.The vertical component of the twistor lift can be computed in the following way [18] (dϕ − ) V = ω 13 + ω 24  Proof.Let ν be the normal bundle of X in Y .Since X, Y are compact, Y is the image of exp : ν → Z. exp is G-invariant since G acts by isometries, which implies the statement.

Twistor interpretation of Xu's correspondence
We give a brief overview of Xu's work on J-holomorphic curves in CP 3 and provide a twistor interpretation of his correspondence of superminimal curves and curves with vanishing torsion [30].A superminimal curve is a J-holomorphic curve ϕ : X → CP 3 which is tangent to H, i.e. dϕ : maps T X to ϕ * H.These curves are special since are holomorphic for J 1 too and R. Bryant found a Weierstraß parametrisation for them, i.e. each of them is a projective line or parametrised by for f, g meromorphic functions on X with g being non-constant [9, Theorem F].For a general J-holomorphic curve ϕ there is a unique holomorphic line bundle L ⊂ ϕ * H which contains the projection of dϕ(T X) to ϕ * H. Let N be the quotient bundle ϕ * H/L, which naturally carries a holomorphic structure.Define which is an S 1 × S 1 sub-bundle of ϕ * Sp(2).The bundle P can be equipped with a connection such that the forms τ and ω 2 restrict to basic forms on P .By multiplying τ by an appropriate section in L ∨ ⊗ N one obtains the form Π ∈ Ω 1,0 (X, L ∨ ⊗ N ) which is shown to be holomorphic [31,Lemma 3.2.3].
Proposition 3.5.In CP 3 , there is a one-to-one correspondence between horizontal holomorphic curves and J 2 -holomorphic curves on which Π vanishes.
The correspondence of proposition 3.5 can be entirely described in terms of twistor lifts.All statements from section 2.2 apply to the negative as well as to the positive twistor space.The twistor theory of S 4 has the particularity that both the negative and the positive twistor space can be identified with CP 3 .To distinguish the two spaces as bundles over S 4 we denote them by CP 3  ± .Both spaces are quotients of Sp(2) by different but conjugate subgroups. .
For a J 2 -holomorphic curve ϕ − : X → CP 3 − Xu constructs a lift φ− : X → Sp(2)/S 1 × S 1 and then considers the projection onto CP 3  + which yields a map X → CP 3 + .He constructs a similar lift φ+ when starting with a curve in CP 3  + and shows that both constructions are inverse to each other.Observe that Sp(2)/S 1 × S 1 is nothing but Gr 2 (S 4 ).Let π ± be the projection of CP 3 ± onto S 4 .It turns out that the lifts ϕ ± equal the Gauß lift of π ± • ϕ into Sp(2)/S 1 × S 1 .This immediately shows that the two constructions are inverse to each other since the transformation leaves the underlying map into S 4 unchanged.In fact, this procedure gives a way to pass between J 2 -holomorphic curves in CP 3  + and CP 3 − due to proposition 2.1.However, when starting with a horizontal J 2 -holomorphic curve in CP 3  − the resulting curve in CP 3 + need not be horizontal.In fact the tangent bundle of F admits a natural splitting which is a connection of the fibration Sp(2)/S 1 × S 1 → S 4 such that If {f 1 , f 2 , f 3 , f 4 } denotes a frame dual to {ω 1 , ω 2 , ω 3 , τ } then H is locally spanned by f 1 and f 2 , V − by f 3 and V + by f 4 .Consider a J 2 -holomorphic curve ϕ − : X → CP 3 − and the Gauß lift φ− : X → Sp(2)/S 1 × S 1 .The curve is horizontal if the lift does not have a component in V − .eq. (3.4) describes the splitting of sp(2)/t 2 into root spaces.In fact, eq.(2.2) reveals that the component in V + vanishes if and only if ϕ * τ = 0 ⇔ Π = 0 on X which proves proposition 3.5.Finally, observe that eq.(2.3) implies that Π is equal to the second fundamental form of L in ϕ * H which is an element in Ω 1,0 (X, Hom(L, N )) because L is a holomorphic sub bundle of ϕ * H.The holomorphicity of Π can then be related to properties of the curvature tensor of ϕ * H.

Adapting frames on J-holomorphic curves
Recall from section 2.3 that there are three distinguished classes of J-holomorphic curves in CP 3 .The first are curves which are always tangent to the vertical bundle V.They are twistor lines and are parametrised by elements in S 4 .Since the horizontal bundle H is of complex rank two it is less restrictive to require a curve being tangent to H.These curves are called superminimal and classified in eq.(3.3).There is a third class of curves, namely those on which Π vanishes identically.Such curves are in one to one correspondence with superminimal curves by proposition 3.5.Since all of these classes are relatively well understood we are interested in studying J-holomorphic curves which do not belong to these three classes.A general Jholomorphic curve will have isolated points in which it is either horizontal, vertical or satisfies Π = 0.These points will appear as singularities in our treatment and the statements we will show are for J-holomorphic curves without such singularities which we will call transverse.Definition 4.1.A J-holomorphic curve ϕ : X → CP 3 is called transverse if one of the equivalent conditions is satisfied • Π = 0 everywhere and ϕ is nowhere tangent to the horizontal bundle H or the vertical bundle V • Both twistor lifts of π − • ϕ : X → S 4 are nowhere horizontal or vertical If one is willing to compromise on completeness one can consider the open set away from the singularities.If X is compact with genus g then the curve is automatically superminimal or satisfies Π = 0 if g = 0, has non-transverse points if g ≥ 2 and is either superminimal, satisfies Π = 0 or transverse if g = 1 [31,Remark 4.11.].For this reason, we are mainly interested in the case when X has genus one.
In the case of the nearly Kähler S 3 × S 3 , the almost product structure gives a natural choice of an SU(3) frame along any J-holomorphic curve.This allows to study the structure equations in terms of a reduced set of functions, classify and identify sub-classes of J-holomorphic curves in S 3 × S 3 [8].The aim of this section is to provide a similar analysis for J-holomorphic curves in CP 3 .Instead of an almost product structure, the parallel splitting T CP 3 = H ⊕ V plays a key role in this case.By an appropriate frame adaption we will describe transverse J-holomorphic curves by two functions α − , α + : X → R which carry local geometric information such as the curvature and second fundamental form of the curve.

The action of H on sp(2)
The choice of such an A is unique up to multiplication by an element in the subgroup K = {diag(e iϑ , e −2iϑ , e iϑ )} ⊂ SU(3).Define the double cover u : H = U(1) × Sp(1) → U(2) where u(λ, q) acts on The action of H on h is the adjoint action while H acts on V 1 by ρ.
We have proven the following.
Lemma 4.2.For any The adjoint action of K on h splits into one-dimensional subspaces.Let The action of K on V 3 is trivial while it acts on V 2 by multiplication of e −6iθ .On the other hand, K acts on (z 1 , 0, z 3 ) ∈ W by multiplication of e 2iθ in each component.Define We conclude

Transverse J-holomorphic curves in CP 3
Consider a general J-holomorphic curve ϕ : X → CP 3 .Denote by g the nearly Kähler metric on CP 3 which splits as g H + g V since the splitting T CP 3 = H ⊕ V is orthogonal.We will consider the pull-back metrics of g, g H , g V to X via ϕ and denote them with the same symbols.Note that Sp(2) pulls back to an S 1 × S 3 bundle over X.The structure equations are formulated with respect to differential forms on Sp(2).To simplify matters, one reduces the S 1 × S 3 bundle ϕ * (Sp(2)) which ensures that additional relations of the differential forms are satisfied.To begin with, one can reduce the bundle ϕ * (Sp(2)) to an S 1 × S 1 bundle P by imposing the equation ω 2 = 0, see [30].This gives a lift of ϕ into Sp(2)/S 1 × S 1 .On this reduction, τ becomes a basic form of type (1, 0).Since ϕ is J-holomorphic dϕ(T 1,0 ) takes values in the subbundle corresponding to the root spaces {(2, 0), (0, −2), (−1, 1)} which are associated to (ω 3 , τ, ω 1 ) under the isomorphism Ω Given a semi-simple Lie group with maximal torus T the flag manifold G/T carries a natural m-symmetric structure τ where m is the height of the Lie algebra of G. τ gives rise to a splitting T C (G/T ) = ⊕[M k ] and a map ψ : X → G/T is called τ -primitve if it satisfies a compatibility condition between the complex structure on X and τ , see [7].In the case of which is τ -primitive in the sense of [7].Conversely, composing any τ -primitive map Note that under this identification, the formula [31,Remark 4.11.]becomes an application of the more general Pluecker formula for τ -primitive maps [6].
We will now explain how ϕ * (Sp(2)) can in fact be reduced to a discrete bundle for a transverse J-holomorphic curve ϕ : X → CP 3 .We can define the function α where ξ is any non-zero vector in T x X.Here | • | H and | • | V denote the norms with respect to the metric g H and g V on X.The value of α − does not depend on this choice because ϕ is J-holomorphic.The function α − is a measure for the angle in which T X lies between H X and V X .In accordance with lemma 4.2 we now adapt frames in the following way.Over X, define the principal bundle Q by the relations The bundle Q has structure group K ∼ = S 1 .From now on, we will consider the restrictions of all differential forms to Q without changing the notation.Note that, since the structure group of Q is K, the forms τ, 3ρ 1 − ρ 2 and ω 1 ∧ ω 1 become basic, i.e. they descend to forms on CP 3 .The structure equations then yield Combining all of the equations gives that τ and −dlog(α − ) + i(−3ρ 1 + ρ 2 ) are (1, 0)-forms since their wedge product with ω 1 vanishes.Since ρ 1 and ρ 2 and −dlog(α − ) are real-valued we get that dlog(α for any tangent vector ξ.In other words, Note that a-priori this is an equation on Q but it also holds on X since −3ρ 1 + ρ 2 is a basic form.eq. ( 4.4) implies that Here, ∆ denotes the positive definite Laplace operator Ω 0 (X) → Ω 2 (X).

Further reduction of the structure group
Since ϕ is assumed to be transverse, τ is nowhere vanishing.On Q, both τ and ω 1 reduce to forms on X with values in the same line bundle.So we can define for any ξ ∈ T x X which does not depend on ξ ∈ T x X since τ is a (1, 0)-form.In fact, by proposition 4.4 transverse J-holomorphic curves correspond to τ -primitive maps φ : X → Sp(2)/S 1 × S 1 .As seen in eq.(3.4) Note that α − is a measure of the angle of T X between φ * (H) and φ * (V − ) while α + is a measure of the angle of T X between φ * (H) and φ * (V − ).This is why we will refer to α − and α + as angle functions.By lemma 4.2, we can adapt frames further.The bundle Q restricts to a K F bundle R which is characterised by the equation τ = α + ω 1 .From the structure equations we now get Combining eq.(4.7) and eq.(4.6) one infers that −dlog(α This implies Let us summarise the results so far Lemma 4.5.Let ϕ : X → CP 3 be a transverse J-holomorphic curve.Then the bundle ϕ * (Sp(2)) restricts to a K F bundle R on which the following equations hold Reducing the bundle Sp(2) over a transverse J-holomorphic curve can be summarised in the following table.

Bundle
Structure Group Reduction characterised by Restriction implies Table 1: Stepwise reductions of the bundle ϕ * Sp(2).Note that P is defined for any J-holomorphic curve while Q and R need the assumption of transversality.Q is not strictly a reduction of P but this is only due to the convention ω 1 = 0 in [30] instead of ω 2 = 0.This does not change any of the results.
By the uniformisation theorem, any metric on a Riemann surface is conformally equivalent to a constant curvature metric which is being made explicit by the following proposition.
Proposition 4.6.The metrics g H and g are conformally flat for any transverse J-holomorphic curve ϕ : X → CP 3 .The conformal factor is given by γ Proof.Since the metrics g H and g only differ by the conformal factor (1 + α 2 − ) it suffices to prove this statement for g H . First assume that X is simply-connected, i.e.X ∼ = D or C. In this case, the bundle R admits a global section s.Then s * ω 1 is a unitary (1, 0)-form on X, satisfying the equation d(s ) is a (1, 0)-form.This means that the metric γ −2 g H has a closed, unitary (1, 0)-form γ −1 s * ω 1 , hence it is flat.If X is not simply-connected we can show the statement by passing to the universal cover.
Putting eq.(4.5), eq.(4.9), eq.(4.7) and eq.(4.8) together gives If we equip X with the metric g H then − 1 2i ω 1 ∧ ω 1 becomes the volume form vol H on X and we may rewrite the equations as The curvature form on X is then given by Let γ = (α − α + ) −1/4 as in proposition 4.6, i.e. γ −2 g H is flat.Denote by ∆ 0 the Laplace operator on functions with respect to γ −2 g H .
Theorem 4.7.Let ϕ : X → CP 3 be a transverse J-holomorphic curve.Then the functions α − , α + satisfy (4.13) These equations are equivalent to the affine 2D Toda lattice equations for sp (2).The induced Gauß curvature on X is Proof.Eq 4.13 is a direct consequence of the discussion preceding the theorem.Note that the curvature two-form of the metric g H is equal to − 1 2i ∆(logγ 2 ).Combining this with eq.(4.12) gives ∆log(γ )γ 2 which also follows from eq. (4.13).Hence, the induced Gauß curvature on X is If we define α± = γα ± such that γ = (α − α+ ) 1/2 and eq.(4.13) becomes ) these equations are equivalent to the 2D affine Toda equations for sp(2).Remark 4.8.Note that the result is consistent with proposition 4.4 since a τ -primitive map into G/T is described by the Toda lattice equations for g [7].Furthermore, the relationship between Toda lattice equations and minimal surfaces in S 4 has already been observed in [26].

The second fundamental form of transverse curves
As a nearly Kähler manifold, CP 3 comes equipped with two natural connections.The Levi-Civita connection ∇ and the nearly Kähler connection ∇.It turns out that the second fundamental form of a J-holomorphic curve is the same for ∇ and ∇.Despite ∇ having torsion it is the connection that is easier to work with since it preserves the almost complex structure J.For a fixed J-holomorphic curve ϕ : X → CP 3 consider the map Θ ϕ : T X → ν which is defined as Θ = −α 2 − Id H + Id V .Observe that Θ is injective and let N 1 = Θ(T X).Denote by N 2 the orthogonal complement of N 1 in ν.It turns out that N 2 is in fact equal to the kernel of the orthogonal projection ν → V X .In other words, ϕ * (T CP 3 ) splits into an orthogonal sum of complex line bundles where N 1 ∼ = T X and N 2 ∼ = (T X) −2 .This splitting is related to the reduction of ϕ * (Sp(2)) to and s is a (local) section s : X → Q, s * (u 1 , u 2 , u 3 ) is a unitary frame as well.Let (f 1 , f 2 , f 3 ) be the dual frame of s * (u 1 , u 2 , u 3 ).Then f 1 always takes values in T X, f 2 in N 1 and f 3 in N 2 .A frame with this property will be called a Q-adapted frame from now on.The connection matrix A u for the frame {f 1 , f 2 , f 3 } is then computed via applying the base change eq.(4.16) to the connection matrix A ω in eq. ( 2.3) and using eq.(4.4) . (4.17) Lemma 4.9.For a Q-adapted frame f 1 , f 2 , f 3 and co-frame u 1 , u 2 , u 3 , the second fundamental form of X is equal to for . We see that transverse points of a Jholomorphic curve are never totally geodesic.Conversely, the frame {f 1 , f 2 , f 3 } and the first and second fundamental form of a transverse curve determine α − and α + .
Putting all together gives that which is a constant by proposition 4.6.However, this is only possible if C = 0 or if α − is constant.But solutions with α − constant force α 2 − = α 2 + = 1/2 and hence C = 0. Different Q-adapted frames are related by an action of K ∼ = S 1 .To work out tensors which are invariant under this action, let λ = (e iθ , e 3iθ ) be an element in K and denote −2θ = ϑ.The action of λ introduces a gauge transformation leading to a transformed set of tensors (ω 1 , ω 2 , ω 3 , τ ) on Q.Note that (ω 1 , ω 2 , ω 3 ) = (e iϑ ω 1 , e −2iϑ ω 2 , e iϑ ω 3 ) Consequently, the tensors τ ⊗ u i ⊗ f 3 , u i ⊗ f j , u 3 ⊗ f 3 for i, j = 1, 2 are all invariant under K and hence correspond to tensors on X. Regard the second fundamental form II as a section in Ω 1,0 (X, Hom(T X, ν)).The connection ∇ induces connections ∇ T and ∇ ⊥ on T X and ν and thus there is a well-defined object ∂∇ II = d ∇ II ∈ Ω 1,1 (X, Hom(T X, ν)).We have seen that a transverse J-holomorphic curve has no points which are totally geodesic.Having holomorphic second fundamental form is a natural generalisation of being totally geodesic.
Theorem 4.10.The differential of the second fundamental form is equal to In particular, there is no transverse J-holomorphic curve with holomorphic second fundamental form.
Proof.Using eq.(4.18), the differential of II is computed by This expression takes values in Ω 1,1 (X, T X ∨ ⊗ ν) and we compute the components in u 1 ⊗ f 2 and u 1 ⊗ f 3 separately.Note that Furthermore, the Finally, the Observe that various terms cancel and that the The u 1 ⊗ f 3 component is computed in an analogous way and equal to since both II 2 and dα − − id C α − are (1, 0) forms.This proves the formula For a transverse J-holomorphic curve, α − is always positive.So the second fundamental form is holomorphic if and only if α − is constant to 2/3.However, no such solution exists for eq.(4.13).

Holomorphic structure of the normal bundle
The nearly Kähler connection ∇ preserves J and ν is a complex subbundle of T CP 3 .Hence ∇ ⊥ gives rise to a holomorphic structure on ν.Let ϕ : X → CP 3 be a transverse torus.The degree of ν is zero since the first Chern class of any nearly Kähler manifold vanishes, i.e.
Let Bun(r, d) be the space of indecomposable holomorphic bundles of rank r and degree d over X.By Atiyah's classification of holomorphic vector bundles over elliptic curves [2], Bun(2, 0) is isomorphic to a two-torus.For any element E ∈ Bun(2, 0) the line bundle Λ 2 (E) is trivial.This is consistent with the fact that Λ 2 (ν) = T X.The space Bun(2, 0) has a distinguished element E 0 , the unique non-trivial extension of the sequence Based on this, there are a-priori three possibilities for ν.Either ν is decomposable, isomorphic to E 0 or another element in Bun(2, 0).In the following we will see that for a transverse Jholomorphic torus in CP 3 , ν is always isomorphic to E 0 .Let σ ij be the components of the connection matrix A u .Assume that the frame {f 1 , f 2 , f 3 } is R-adapted, such that eq.(4.10) hold.Then s = s 2 f 2 + s 3 f 3 describes a general section in the normal bundle for s 2 , s 3 ∈ Ω(X, C).
By the Leibniz rule for ∂, holomorphic sections are solution of the equation Since σ 32 is of type (1, 0) it annihilates ∂ ∂ z .Note that on X, u 1 = 1 + α 2 − ω 1 and hence by proposition 4.6 we can find a local coordinate z on X such that dz = (α Then eq.(4.21) reduces to Hence, all solutions of eq.(4.21) are given by for a constant c ∈ C. Furthermore, we have and this in turn results in + is non-vanishing which forces c = 0 and s 2 to be a constant.This computation implies the following two lemmas.Lemma 4.11.Let X be a transverse torus then h 0 (X, ν) = 1 Lemma 4.12.The holomorphic section s 2 f 2 induces an exact sequence of holomorphic bundles Proof.Since s 2 is non-vanishing the section gives an injection C → ν.The quotient bundle admits a non-vanishing holomorphic section because σ 0,1 33 is ∂-exact.Hence the quotient is trivial.
Summarising the two statements above gives Proposition 4.13.As a holomorphic bundle, the normal bundle ν of a transverse torus is the unique non-trivial extension of the sequence 0 → C → ν → C → 0.

A Bonnet-type theorem
The uniqueness part of the classical Bonnet theorem says that two surfaces Σ, Σ ⊂ R 3 with the same first and second fundamental form I and II necessarily differ by an isometry of R 3 .The existence part states that given a simply-connected surface Σ with the tensors I, II defined on Σ satisfying the Gauss and Codazzi equation there is an immersion Σ → R 3 with induced first and second fundamental form equal to I and II.One way how to prove this statement is via the theorem of Maurer-Cartan: Let G be a Lie Group with Maurer-Cartan form Ω M C , let N be connected and simply-connected, equipped with η ∈ Ω 1 (N, g) satisfying dη + 1 2 [η, η] = 0. Then there exists a smooth map f : N → G, unique up to left translation in G, such that f * Ω M C = η.In this section we will show, also using the theorem of Maurer-Cartan, an analogue of Bonnet's theorem for J-holomorphic curves in CP 3 .
We have seen that given a transverse J-holomorphic curve ϕ : X → CP 3 there are functions α − , α + , γ that satisfy eq.(4.13),where γ is determined by α − and α + .Up to a constant factor, the functions (α − , α + , γ) determine the first and second fundamental form of X. Composing ϕ with an element in an automorphism in Sp(2) leaves the quantities (α − , α + ) invariant since they are defined via components of Ω M C .Besides, the data (α − , α + ) determines the first and second fundamental form up to a constant.
If a Bonnet-theorem holds for J-holomorphic curves in CP 3 then the first and second fundamental form determine the curve up to isometries.Furthermore it would say that such a curve exists if the Gauss and Codazzi equations are satisfied.In our setting, the system eq.(4.11) plays the role of these equations.This raises the following question.Are the solutions of eq.(4.11) in one to one correspondence to transverse J-holomorphic curves up to isometries?Later, we will see that the answer to this question is no because there are periodic solutions of eq.(4.11) which do not descend to a two-torus.However, there is a positive result for when X is simply-connected.Lemma 5.1.Let X be a simply-connected Riemann surface equipped with a metric k.Let furthermore α − , α + : X → R >0 such that eq.(4.13) are satisfied for γ = (α − α + ) −1/4 and that γ −2 k is flat.Then there is a J-holomorphic immersion ϕ : X → CP 3 such that the induced metric satisfies g H = k.The angle functions of ϕ are exactly α − and α + .The immersion is unique up to isometries of CP 3 and an element in S 1 /Z 4 ∼ = S 1 which parametrises a choice of a unitary (1, 0)-form ω 0 on X such that d(γ −1 ω 0 ) = 0.
Proof.Since X is simply-connected and conformally flat, it is isomorphic to C as a complex manifold by the uniformisation theorem.In particular, Ω 1,0 (X) is trivial as a bundle, let ω 0 be a unitary (1, 0)-form on X. Arguing similarly as in the proof of proposition 4.6 we see that, since γ −2 k is flat, dω 0 = id C log(γ) ∧ ω 0 . (5.1) Now, define a sp(2)-valued one-form on X by .
Theorem 5.2.Let X be a simply-connected Riemann surface and let (α − , α + ) be solutions of eq.(4.13) for some flat metric on X.Then there is a J-holomorphic immersion ϕ : X → CP 3 such that the angle functions ϕ are exactly α − and α + .The immersion is unique up to isometries of CP 3 , a choice λ > 0 and an element in S 1 /Z 4 ∼ = S 1 which parametrises a choice of a unitary (1, 0)-form ω 0 on X such that d(γ −1 ω 0 ) = 0.
In other words, a solution (α − , α + ) of eq.(4.13) specifies the immersion up to isometries and a choice of a constant (λ, ω 0 ) ∈ C * .In the case when the solutions (α − , α + ) and hence g have symmetries, the different embeddings ϕ λ,γ might come from reparametrisations of X.Consider for example, X = C and let L λ : C → C be the multiplication by λ.Define furthermore 4.11) for a metric g on C then (α − λ , α + λ ) is a solution of eq.(4.11) for the metric λ 2 g λ .This solution comes from the J-holomorphic curve ϕ • L λ .In particular, if g λ = g then the different immersions ϕ λ come from reparametrisations of X by L λ .Similarly, if the induced g has radial symmetry, different choices of ω 0 correspond to reparametrisations by rotations.

Special solutions of the Toda equations
Note that eq.(4.13) are symmetric in α − and α + , meaning a distinguished set of solutions is of the form α − = α + .This reduces eq.(4.13) to which becomes the Sinh-Gordon equation after rescaling the metric with a constant factor.This is somewhat similar to the situation in S 3 × S 3 where J-holomorphic curves with Λ = 0 are locally described by the same equation.In particular, Proposition 5.3.Transverse J-holomorphic curves in CP 3 with α − = α + are locally described by the same equation as constant mean curvature tori in R 3 .
Geometrically, the condition α − = α + is satisfied for a J-holomorphic curve if the corresponding minimal surface X → S 4 lies in a totally geodesic S 3 , see corollary 3.3.Hence the twistor lifts of the surfaces T k,m ⊂ S 3 ⊂ S 4 from section 6.1 give rise to examples of J-holomorphic tori with α − = α + .The Clifford torus T 1,1 plays a special because it is a T 2 group orbit.We refer to Clifford tori to all J-holomorphic curves isometric to the twistor lift of T 1,1 .Since isometries leave α − and α + invariant this means that α − and α + are constant on Clifford tori.Observe, that there is in fact only one solution for α − and α + constant.
Lemma 5.4.If either α − or α + is constant then both must be constant and equal to 1/ √ 2.
Theorem 5.5.Let ϕ : X → CP 3 be a transverse J-holomorphic curve such that the induced metric g H is flat and X is equal to T 2 or C as a complex manifold.Then ϕ parametrises a Clifford torus for X = T 2 and its universal cover if X = C.
Proof.By passing to the universal cover it suffices to assume X = C. Furthermore, by eq. ( 4.12), flatness of g H is equivalent to 1 = α 2 − + α 2 + and so eq.( 4.13) implies that log(α 2 − , α 2 + > 0 and hence constant.Since α − and α + must be constant on Clifford tori It follows from lemma 5.4 that α 2 − = α 2 + = 1/2.Now, theorem 5.2 can be applied to prove uniqueness.Note that since C carries the flat metric, a different choice of (ω 0 , λ) amounts to applying an isometry of X.

U(1)-invariant J-holomorphic curves
U(1)-actions on the nearly Kähler CP 3 have been studied on the resulting G 2 -cone by Atiyah and Witten in the context of dimensional reductions of M-theory [3].Closed expressions for the induced metric, curvature and the symplectic structure on R 6 have recently been found by Acharya, Bryant and Salamon [1] for one particular U(1)-action.
While superminimal curves can be parametrised very explicitly our description of transverse curves has been, with the exception of Clifford tori, rather indirect so far.Imposing U(1)symmetry on the curves reduces a system of PDE's to a system of ODE's.In terms of α − and α + , the 2D Toda lattice equation will reduce to the 1D Toda lattice equation.Killing vector fields on CP 3 are in one to correspondence with Killing vector fields on S 4 .We will provide a twistor perspective on the work of U(1)-invariant minimal surfaces [16].The geodesic equation on S 3 for the Hsiang-Lawson metric is replaced by the computationally less involved 1D Toda equation.The twistor perspective establishes relationship between U(1)-invariant curves and the toric nearly Kähler geometry of CP 3 .

U(1)-invariant minimal surfaces in S 3
In [22], B. Lawson developed a rich theory of minimal surfaces in S 3 .For coprime positive integers m ≥ k he finds the minimal surface which represent Klein bottles for 2|(mk) and tori otherwise.The surface T 1,1 is the Clifford torus T .They are geodesically ruled and are invariant under the U(1)-action ρ given by e iϑ (z, w) = (e kiϑ z, e miϑ z).
It is furthermore shown that closed minimal surfaces invariant under this action are in one to one correspondence with closed geodesics in the orbit space S 3 /U(1) equipped with an ovaloid metric.By studying this metric explicitly, a rationality condition on the initial values for the geodesic to be closed is obtained.This gives rise to a countable family of tori (T k,m,a ) a∈A k,m where A k,m is a certain countable dense subset of (0, π/2).This family is bounded by T k,m corresponding to the boundary case a = 0 and the Clifford torus T for a = π/2.Similarly, for either m = 0, k = 0 or m = ±k there is a countable family C a of minimal tori bounded by the Clifford torus and a totally geodesic two-sphere [20,Theorem 8].
Any cohomogeneity-one minimal surface in S 3 belongs to the families C a or T k,m,a [20,Theorem 9].Furthermore, Lawson constructs for any minimal surface in S 3 an associated minimal surface in S 5 which he calls bipolar surface.For the family T k,m the bipolar surfaces lie in S 4 but not in a totally geodesic S 3 .Extend ρ to an action on S 4 ⊂ C 2 ⊕ R by letting U(1) act trivially on the R component.The Hsiang-Lawson family T k,m,a is contained in the totally geodesic three sphere x = 0. Furthermore, the corresponding surfaces of this family are also U(1)-invariant but under the modified action ( k, m) = (m − k, m + k).Denote by T k,m,a the family of bipolar surfaces constructed from T k, m,a .The two families T k,m,a and T k,m,a will play a special role in the following discussion.

Killing vector fields
Since the nearly Kähler CP 3 has isometry group Sp(2) any element ξ ∈ sp(2) gives rise to the Killing vector field Assume that there is t > 0 such that exp(tξ) equals the identity e ∈ Sp(2), i.e. ξ corresponds to an action ρ of U(1) on CP 3 .Acting via ρ on integral curves of JK ξ gives rise to U(1)-invariant J-holomorphic curves in CP 3 .Another way of stating this is that [K ξ , JK ξ ] = 0 and that the span of K ξ and JK ξ defines an integrable distribution V ξ on M = CP 3 \ (K ξ ) −1 (0).The integral submanifold are exactly the ρ invariant J-holomorphic curves in CP 3 which foliate M .Due to the isomorphism between sp(2) ∼ = so(5), the element ξ also gives rise to a Killing vector field K ξ S 4 .This results in a uniqueness statement for minimal surfaces in S 4 .Lemma 6.1.Let x ∈ S 4 such that V ⊂ T x S 4 be a two-dimensional subspace containing K ξ S 4 (x) and let p ∈ π −1 (x) ⊂ CP 3 be the twistor lift of V .If K ξ p is non-vertical there is a locally unique minimal surface Σ with x ∈ Σ and V = T x Σ.
Let X be an integral submanifold of the distribution V ξ .Since it is a J-holomorphic curve the bundle Sp(2)| X reduces to the Z 8 bundle R X , see lemma 4.5.There is an Z 8 bundle R over M which restricts to R X on each integral submanifold.Note that the construction of R is linked to the subspace r ⊂ sp(2) which becomes apparent in the following lemma.Lemma 6.2.If a section s : In other words, s satisfies eq.(4.10) and hence s also has values in R. Since the sections s and s are joined by a continuous path and R has discrete structure group it follows that s = s, i.e.

s(gx) = gs(x).
This implies The point of the previous lemma is that it gives a more explicit description of the bundle R.Over U , define the K F bundle R = {g ∈ Sp(2) | g −1 ξg ∈ r}.By lemma 6.2, R = R .The point of the previous lemma is that it gives a more explicit description of the bundle R. Corollary 6.3.A section s : U → Sp(2) takes values in R if and only if s −1 ξs ∈ r.

The presence of the vector field K ξ means we can define the functions
Let X be a transverse integral submanifold.Since K ξ and JK ξ commute, these vector fields give rise to coordinates (u, t) such that ∂ ∂u = K ξ and ∂ ∂t = JK ξ .The induced metric g H on X is equal to h(du 2 + dt 2 ).In particular, due to proposition 4.6 and since h is the conformal factor of the metric, the quantity eq. (4.13) reduces to It is clear that the equations of lemma 4.5 hold when the forms are restricted to R. However, the fact that K ξ is a Killing vector field guarantees that the following additional equations are satisfied ) for a constant λ ∈ S 1 .Note that eq. ( 6.3) are satisfied because isometries preserve the quantities α − and α + .eq. ( 6.4) holds because (u, t) are isothermal coordinates for g H on integral submanifold.Lastly, eq. ( 6.5) and eq.(6.6) follow from eq. ( 6.4) and the definition of v − , v + , i.e. eq. ( 6.1) as well as the equations defining R, i.e. eq.(4.10).Note that we have abused notation slightly here.By how K F acts on r the forms ω 1 , . . ., ω 3 are basic on R but take values in a possibly non-trivial line bundle on X whose structure group reduces to K F .In other words, λ is only well-defined up to multiplication of a fourth root of unity.However, this poses no problem since the results which follow will only depend on µ = λ 4 .

Lax Representation and Toda Lattices
Since a general transverse curve satisfies the 2D periodic Toda lattice equation, U(1)-invariant curves will satisfy the 1D periodic Toda lattice equations for the same Lie algebra, i.e. eq.(6.2).These equations admit a Lax representation [5].In the following, we will work out this Lax representation directly from the formalism of adapted frames.Consider the restriction of Ω M C to the reduced bundle R, which will still be denoted by Ω = Ω M C .We have Note that eq. ( 6.3)-eq.(6.6) imply Consequently, K ξ (Ω(JK ξ )) = 0 because h, v − , v + are constant along K ξ .With this in mind, let us evaluate both two-forms at Evaluating the right hand side of eq.(6.9) gives Hence, eq.(6.9) is equivalent to eq. ( 6.2).In other words, we have found a Lax representation of the ODE system.This proves the following lemma.
Introduce the variables where the dot denotes the derivative with respect to d dt , i.e. along JK ξ .Then eq. ( 6.2) is equivalent to This system is Hamiltonian with The Hamiltonian is in the form of [5,Theorem 1].In other words, eq. ( 6.10) are the equations for a generalised, periodic Toda lattice for the Lie algebra sp(2).Bogoyavlensky's Lax representation for such a system coincides with eq.(6.9).Thus we have proven Proposition 6.5.Simply connected, embedded, transverse J-holomorphic curves with a onedimensional symmetry give rise to solutions to the 1D periodic Toda lattice equations for the Lie algebra sp(2) with Lax representation Define Then one can check explicitly that the eigenvalues of Ω(K ξ ) are given by We have seen that a choice of a Killing vector field on CP 3 gives rise to the above ODE system.The converse statement is also true.Proposition 6.6.Let X ∼ = C be a Riemann surface equipped with coordinates (u, t) a metric k = Cγ 2 (du 2 + dt 2 ) and α − , α + : X → R >0 satisfy the ODE system eq.( 6.2) for some C > 0.
Then there is an element ξ ∈ sp(2) such that X is an integral manifold of the distribution V ξ and such that α − , α + are the angle functions and the tautological embedding X → CP 3 is J-holomorphic and isometric.

The T 2 -action
In this subsection, we will investigate describe the geometry of a general U(1)-action on CP 3 with respect to the vector field JK ξ and make use of the fact that such an action commutes with a subgroup of Sp (2), which is generically a two-torus.To that end, we fix which integrates to the U( 1 We will refer to the first two cases as degenerate and to the last case as non-degenerate.Due to the presence of a larger symmetry group, the degenerate cases are simpler and have been treated by the author elsewhere.This is why, in this article, we restrict ourselves to the degenerate case in which Stab(ξ) is equal to the standard two-torus in Sp(2) i.e. {diag(e iθ , e iφ )} to which we will simply refer as T 2 unless stated otherwise.Let ξ 1 , ξ 2 be the elements in sp(2) corresponding to the action of e iθ and e iφ respectively.This torus action gives rise to the multi-moment map ν = ω(K ξ1 , K ξ2 ), introduced in [27].
Proposition 6.7.The nearly Kähler multi-moment map on CP 3 is given in homogeneous coordinates by There are two Clifford tori which arise as orbits of T 2 and are equal to the preimages of the extremal values of ν.Hence they are T 2 orbits of the points [1, 1, 1, i] and [1, 1, 1, −i].When investigating ρ-invariant J-holomorphic curves, it proves worthwhile to study lower-dimensional subsets Y of CP 3 which are both invariant under ρ and the flow of JK ξ .In other words, the distribution V ξ is then tangent to Y and the problem of finding ρ-invariant J-holomorphic curves can be done separately on each such Y .Since dν = ψ + (K ξ1 , K ξ2 , •), V ξ is tangent to each preimage ν −1 (c).The value 0 as well as extrema of ν have a distinguished geometrical importance, there are the following sets which arise in a natural geometric way and to which V ξ is tangent • Q, the quadric associated to ξ under the identification sp(2) with the real part of S 2 (C 4 ) • S, the set where T 2 does not act freely • T , a distinguished S 1 bundle over S 3 0 ⊂ R 4 ⊕ {0} (for (k, m) non-degenerate) • T , a four-dimensional submanifold constructed from T via Lawson's bipolar construction.
In the following we will define and outline the properties of each of the subsets.
Note that L 1 and L 6 are twistor lines, i.e. h = 0, while L 2 , . . ., L 5 are superminimal, i.e. v − = 0. Since L 2 , . . ., L 5 project to totally geodesic two-spheres they furthermore satisfy v + = 0, see proposition 3.1.Via the isomorphism sp(2) ⊗ C ∼ = S 2 (C 4 ), ξ is identified with the polynomial i(kZ 0 Z 1 + mZ 2 Z 3 ).(6.13) Furthermore, since sp(2) ∼ = so(5), ξ defines a vector field K ξ S 4 on S 4 .Since it is a Killing vector field, ∇(K ξ S 4 ) can be identified with a two form on S 4 .Its anti-self-dual part satisfies the twistor equation and gives thus rise to a quadric on CP 3 = Z − (S 4 ), holomorphic with respect to J 1 , see [4, ch. 13].This quadric is given by and coincides with the vanishing set of the quadratic expression eq.(6.13).For x ∈ CP 3 , let X x be the unique ρ-invariant embedded J-holomorphic curve containing x and Σ x be the corresponding minimal surface in S 4 .Define T = {x ∈ CP 3 | Σ x is contained in a totally geodesic S 3 }.Proof.For k = m = 1 the action on S 4 only rotates the first two components of R 2 ⊕ R 3 .This action commutes with SO(2) × SO(3).Let Σ be a minimal surface containing the orbit O x for some x.Let ν be the normal bundle of O x in Σ and v ∈ ν x .By an action of SO(2) × SO(3) we can assume that x and v lie in R 4 ⊕ {0}.In particular, x is contained in S 3 0 ⊂ R 4 ⊕ {0} and v is tangent to it.Because ν and S 3 are ρ-invariant we have that O x ⊂ S 3 and ν x ⊂ T S 3 .By [28], there is an embedded, ρ-invariant minimal surface Σ such that O x ⊂ Σ ⊂ S 3 .Now, lemma 6.1 implies that Σ = Σ , i.e.Σ ⊂ S 3 0 .Lemma 6.9.If k = m and Σ is a ρ-invariant minimal surface which is contained in a totally geodesic N ∼ = S 3 then N = S 3 0 = R 4 ⊕ {0} ∩ S 4 .Proof.One can check that R 4 ⊕ {0} ∩ S 4 is the only totally geodesic S 3 on which ρ acts.The statement follows from lemma 3.4.
From now on, assume k = m.Over the totally geodesic S 3 0 we introduce the reduced Grassmannian bundle 0 ) is in fact an S 1 -bundle over S 3 0 which is mapped diffeomorphically to T ⊂ CP 3 = Z − (S 4 ) by the projection Gr 2 (S 4 ) → Z − (S 4 ).Note that T is also invariant under ρ and JK ξ is tangent to T .For a minimal surface Σ denote by Σ the bipolar surface from section 6.1.Let T = {x ∈ CP 3 | Σ x = S for S a minimal surface lying in a totally geodesic S 3 }.
For both T and T we obtain explicit expressions in section 6.7.

Separating T 2 -orbits
As highlighted in the introduction, it is desirable to have a geometric construction of a map into R 4 which descends to a local homeomorphism to the T 2 quotient of CP 3 , at least away from a singular set.In general, one candidate for such a map is (ν, K ξ1 , K ξ2 , g(K ξ1 , K ξ2 )).For S 3 × S 3 such a map cannot have four-dimensional image however, due to the presence of a unit Killing vector field on S 3 × S 3 .This is a special case, because S 3 × S 3 is the only nearly Kähler manifold admitting a unit Killing vector field [24].Nevertheless, it seems difficult to compute the differentials of g(K ξi , K ξj ) in a general setting.
For the case M = CP 3 we use a variant p of this map, based on the splitting T CP 3 = H ⊕ V.It turns out that, up to a sign, the multi-moment map ν can be expressed as a function of ν, corollary 6.17.The quotient T 2 \ CP 3 \{L1∪•••∪L6} is smooth and will be denoted by M F .Note that the functions (v + , v − , r + , r − ) are all T 2 -invariant.Theorem 6.10.The functions map to a bounded set D ⊂ R 4 over which p is a branched double cover.The two different points in the fibres of p are complex conjugates of each other and the branch locus is equal to We will now define each of the maps individually and establish its properties.
F then the orbit of x under the action of S 1 × S 3 intersects r F in a K F orbit.In other words, there is an injective map Π : sp(2) F /S 1 × S 3 → r F /K F .On the other hand, the inclusion r µ), motivated by eq.(6.11), we define the functions Denote by D ⊂ R 6 the set defined by the equations The image of r F under ζ is D. This follows because eq.(6.15) describe how eigenvalues of elements in r are calculated, i.e. eq.(6.12).Furthermore, since sp( 2) is semi-simple, conjugacy classes are uniquely characterised by their eigenvalues.Note that ζ descends to a double cover r/K F , branched over R 3 + × R 2 × {−1, 1}.The two preimages are obtained by switching between λ and λ. pr: Since eq. ( 6.15) can be solved uniquely for h and µ the projection pr from R 3 + × R 2 × [0, 1] to the first four components maps D diffeomorphically onto its image We have shown that eq.(6.14) restricts to and that ζ is a branched double cover while each other map is a diffeomorphism.Hence, p is a branched double cover.By eq.(6.7) and lemma 6.2, p = (v − , v + , r − , r + ).Observing that p stays invariant under the map δ : The proof of theorem 6.10 gives an explicit description of the branch locus of ζ.The involution δ preserves the metric and reverses the almost complex structure on CP 3 .The fixed point set of δ is the standard RP 3 which is hence a special Lagrangian submanifold for the nearly Kähler structure [31].Lemma 6.11.The set RP 3 is special Lagrangian for the nearly Kähler structure on CP 3 .
Given a nearly Kähler manifold with a T 2 -action the possibly singular space T 2 \ ν −1 (0) arises naturally.In the case of CP 3 this space is an orbifold.Lemma 6.12.The set T 2 \ ν −1 (0) is homeomorphic to RP 3 /{±1} where the action of −1 is given by

A torus fibration of D
The image of p is equal to D and explicitly described by eq.(6.16).To understand the set more conceptually, we show that D itself admits a two-torus fibration over a rectangle R in R 2 .The torus fibres degenerate over the edges of the rectangle.Consider (H 2 , 64C 2 ) as a map from D to R 2 .Recall that (ik, im) are purely imaginary and equal to In fact, the image of u = (H 2 , 64C 2 ) is equal to the rectangle, with one corner point missing Note that R is bounded by the following line segments.While the previous proposition shows that fibres over R 0 are regular it is to be expected that the preimages of the boundary of R correspond to subsets with special geometric properties.In section 6.4, we have considered the distinguished subsets S, T , T , C and B. The following proposition is visualised in fig. 3 and summarises how the preimages of l 1 ∪ • • • ∪ l 4 are related to these subsets.In the case k = m the rectangle R degenerates to the line l 2 , which is a manifestation of lemma 6.8.If m = 0 then R degenerates to the line l 3 .
Note that the map u • p extends to the singular set, i.e. to the map P : T 2 \ CP 3 → R = R ∪ {(H 2 , 64C 2 ) = (16(k 2 − m 2 ) 2 , 0)} mentioned in the introduction and P maps T 2 \ S to the point (16(k 2 − m 2 ) 2 , 0).To prove theorem 6.14 we first establish a more explicit understanding of the map p.

Relation between p and moment maps
If one is just interested in a homeomorphism or cover T 2 \ CP 3 → R 4 the functions (v − , v + , r − , r + ) are an unnecessarily complicated choice from a topological point of view.The reason for using these functions is that JK ξ takes a simple form + (h − 2v + ) ∂ ∂r + .
To get an idea of what the functions (v − , v + , r − , r + , h) look like in homogeneous coordinates consider the following set of T 2 invariant functions (6.17) Clearly, 1 = f 1 + f 3 and the functions are also invariant under δ.Note that the Kähler structure on CP 3 admits a T 3 action and after choosing an appropriate basis for t 3 ∨ , (f 1 , f 2 , f 4 ) are multiples of the symplectic moment map on CP 3 .We can furthermore deduce the relation between the functions f 1 , . . ., f 5 and ν.Note that ν is not invariant under δ but satisfies ν • δ = −ν and can thus not be expressed in terms of f 1 , . . ., f 5 .However, observe that the square of ν can be expressed in terms of the f i via ). (6.18) Denote by D f the image of (f 1 , f 2 , f 4 , f 5 ) in R 4 .By writing down an explicit section one can prove that away from the branch locus B the T 2 × {±1} principal bundle (f 1 , f 2 , f 4 , f 5 ) : CP 3 → R 4 is trivial.Our aim is to express p = (v − , v + , r − , r + ) in terms of the functions f 1 , . . ., f 5 .For a quaternion q = z + jw ∈ C ⊕ jC let q C = z and q jC = w. where Proof.The functions (v − , r − , v + , r + ) can be computed by conjugating σ with an element in S 1 × S 3 to an element in r.Note that r F /K F = f 1 /K for f = q 1 −q 3 q 3 q 2 | q 3 ∈ C ⊂ sp(2), f 1 = f ∩ sp(2) F .
Furthermore, p is invariant under the group K so it suffices to find g ∈ Sp(2) such that g −1 σg ∈ f 1 .Hence, we can conjugate σ with the element diag(1, q2 |q2| ) and the result follows from the definition of ζ.
We now pick a section Consequently, Denote by π T 2 the quotient map CP 3 \ S → M F by the T 2 -action.The equations eq. ( 6.20) can in fact be solved for (f 1 , f 2 , f 4 , f 5 ).This implies together with the triviality of the bundle f : CP 3 \ S → D f the following proposition.
Proposition 6.16.There is a homeomorphism D f → D which is a diffeomorphism in the smooth points such that the following diagram commutes In particular p • π T 2 is a trivial T 2 × {±1} bundle when restricted to CP 3 \ (S ∪ B).
Note that eq.(6.18) yields the following corollary of proposition 6.16.
Corollary 6.17.The square of the multi-moment map ν is a function of (v − , v + , p − , p + ) The rest of the subsection consists of proofs of some of the statements of theorem 6.14.Note that the preimage of l 1 , l 2 in D are exactly all points with µ = 1 or µ = −1 respectively.Hence, the preimage of l 1 ∪ l 2 is equal to the branch locus of ζ.We can check explicitly Note that on u −1 (l 3 ) the equation 2h = E + E − is satisfied.Observe that X L is totally geodesic if and only if L = 0.This is equivalent to saying that v + vanishes on the curve as well.X 0 is part of the singular set of the torus action.
c F : Consider the map c : Sp(2) → sp(2), g → g −1 ξg.Denote the image of c by O ξ .Note that c descends to a diffeomorphism from T 2 \ Sp(2) onto its image.Quotienting both spaces by the right action of S 1 × S 3 gives a homeomorphism between T 2 \ CP 3 and O ξ /S 1 × S 3 .Denote the restriction of this map to M F by c F and observe that c F maps M F diffeomorphically onto

Figure 3 :
Figure 3: The image of u : D → R 2 .theorem 6.14 expresses how the preimage of the boundary points is related to special subsets of CP 3 , compare with [16, Fig. 1].