Energy of sections of the Deligne-Hitchin twistor space

We study a natural functional on the space of holomorphic sections of the Deligne-Hitchin moduli space of a compact Riemann surface, generalizing the energy of equivariant harmonic maps corresponding to twistor lines. We give a link to a natural meromorphic connection on the hyperholomorphic line bundle recently constructed by Hitchin. Moreover, we prove that for a certain class of real holomorphic sections of the Deligne-Hitchin moduli space, the functional is basically given by the Willmore energy of corresponding (equivariant) conformal map to the 3-sphere. As an application we use the functional to distinguish new components of real holomorphic sections of the Deligne-Hitchin moduli space from the space of twistor lines.


Introduction
The Deligne-Hitchin moduli space M DH (Σ, G C ) [27] for a compact Lie group G with complexification G C , is a complex analytic reincarnation of the twistor space of the hyper-Kähler moduli space M SD (Σ, G) of solutions of Hitchin's self-duality equations on a principal G-bundle over a compact Riemann surface Σ [17]. It is defined by gluing the Hodge moduli spaces of λ-connections on Σ and Σ via the monodromy representation, and it admits a holomorphic fibration over the projective line. The fibers are the moduli spaces of G C -Higgs bundles, flat G C -connections or G C -Higgs bundles over Σ. Holomorphic sections of the Deligne-Hitchin moduli space are interesting for various reasons: M DH (Σ, G C ) carries an anti-holomorphic involution τ covering the antipodal map λ → −λ −1 . By the twistor construction for hyper-Kähler manifolds [20], the hyper-Kähler moduli space M SD (Σ, G) can be identified with a certain component of the space of τ -real holomorphic sections of the fibration M DH (Σ, G C ) → CP 1 . These sections are called twistor lines. On the other hand, a solution of the self-duality equations corresponds to an equivariant harmonic map from the universal coverΣ into the symmetric space G C /G, which can be reconstructed from the associated twistor line by loop group factorization methods [26,3]. Apart from the twistor lines, holomorphic sections satisfying other types of reality conditions arise from equivariant harmonic maps ofΣ into different (pseudo-)Riemannian symmetric spaces related to the the group G C and its real forms. This paper is motivated by the work of some of the authors about the question of Simpson, whether all τ -real holomorphic sections are in fact twistor lines [27]. The answer turns out to be no [13,3], and leads to the problem of how to differentiate between the components of the space of τ -real holomorphic sections.
The most fundamental quantity associated to a harmonic map is its energy and the starting point of this paper is the simple observation that the energy of a harmonic map (defined on a compact Riemann surface) can be computed via its associated holomorphic section of the Deligne-Hitchin moduli space (see Theorem 2.4). This computation leads us to a well-defined energy functional on the space of holomorphic sections (see Proposition 2.1). The detailed investigation of this functional is the main objective of our work. It should be mentioned that the functional is defined in terms of the complex analytic structure of the Deligne-Hitchin moduli space and its definition does not involve the hyper-Kähler metric on M SD (Σ, G), i.e. the twistor lines.
We will mostly be concerned with the case G = SU (2), so that G C = SL (2, C). Twistor lines then correspond to equivariant harmonic maps ofΣ into the hyperbolic space H 3 = SL(2, C)/SU (2). This is the space of positive definite hermitian matrices, hence these harmonic maps are called harmonic metrics. One can also study equivariant harmonic maps fromΣ into the 3-sphere S 3 = SU(2) (the compact dual of SL(2, C)/SU(2)), into the anti de Sitter space AdS 3 = SL(2, R) or into the de Sitter space dS 3 = SL(2, C)/SL(2, R) via holomorphic sections of M DH (Σ, SL(2, C)) → CP 1 . To a conformal equivariant harmonic map one may, under certain circumstances, associate another harmonic map with in general different target space. This process, which we call twisting, played a central role in the construction of counterexamples to Simpson's question in [3] and we will give a more systematic treatment in this article. We study how the energy functional interacts with the two real structures on M DH (Σ, G C ). We will see that it takes real values on real holomorphic sections and is normalised in such a way that it takes non-positive values on twistor lines, while it is non-negative on holomorphic sections corresponding to equivariant harmonic maps into G. In the rank 2 case we then examine its behavior under twisting (see Proposition 2.7). The explicit relation between the energy of a section and its twist allows us to give an alternative proof that the τ -real sections constructed in [3] are not twistor lines by checking that these have positive energy.
Although the definition of the functional is motivated by the theory of harmonic maps and does not involve the hyper-Kähler structure of M SD (Σ, G), it can be given a natural interpretation in terms of the hyper-Kähler geometry of the moduli space M SD (Σ, G). The natural isometric circle action on M SD (Σ, G) plays a central role. It preserves one of the complex structures of and rotates the other two complex structures. We show that an analogous functional exists on the space of holomorphic sections of the twistor space of any hyper-Kähler manifold with an isometric circle action of this type. Building on work by Haydys [11], Hitchin [19] has shown that on the twistor space of such a hyper-Kähler manifold, one has a natural holomorphic line bundle with meromorphic connection. The pull-back of the meromorphic connection along a holomorphic section of M DH (Σ, G C ) → CP 1 has simple poles at λ = 0, ∞ only, and it turns out that the residue at λ = 0 coincides with the energy (Corollary 3.11). As a byproduct, we show that the residue evaluation along sections is always a complexification of the moment map of the S 1 -action (Theorem 3.9). Moreover, it automatically serves as a Kähler potential on all hyper-Kähler components of real holomorphic sections of the twistor space. Recently it has been shown [16] that there indeed exist such hyper-Kähler components of the space real holomorphic sections of M DH (Σ, SL(2, C)). The energy functional thus gives a Kähler potential on these new components and we hope to extract from it more information about the geometry of these components in the future.
The third main objective of the paper is the geometric interpretation of the energy for a class of τ -real holomorphic sections which are not twistor lines [13]. Recall that in the case of G = SU(2), twistor lines correspond to equivariant harmonic maps to hyperbolic 3-space H 3 . A holomorphic section of the type constructed in [13] is instead obtained from a Möbius equivariant Willmore surfaceΣ → S 3 . By decomposing the 3-sphere S 3 = H 3 ∪ S 2 ∪ H 3 into two hyperbolic balls separated by the boundary 2-sphere at infinity, one can show that such a holomorphic section defines a solution of the self-duality equations on an open dense subset of the Riemann surface Σ. The solution blows up in a well-behaved way near certain curves on the surface. The corresponding equivariant harmonic map into hyperbolic 3-space intersects S 2 , the boundary at infinity, along these curves and continues as a harmonic map on the other side. We prove that the energy of such a section is directly related to the Willmore energy of the surface, a conformally invariant measure of the roundness of an immersed surface. This relation allows us to prove our last main result: the sections constructed in [13] have positive energy. This gives a complex analytic way to distinguish the component of twistor lines from this newly discovered component of real holomorphic sections.
The structure of the paper is as follows. In Section 1 we set up some notation and recall basic notions associated with holomorphic sections of the Deligne-Hitchin moduli space over a compact Riemann surface. In Section 2 we define the energy functional and prove its basic properties. Section 3 then contains the natural interpretation of the energy functional in terms of the residue of the meromorphic connection on the hyperholomorphic line bundle over M DH (Σ, G C ). In Section 4, we relate the energy of the real holomorphic sections constructed in [13] to the Willmore energy of the related Möbius equivariant Willmore surfaces. In the final Section 5, we show that the energy functional can be used to distinguish different components of the space of real holomorphic sections. In particular, we prove that the new sections of [13] have positive energy.
1. The Deligne-Hitchin moduli space 1.1. λ-connections and the Deligne-Hitchin moduli space. Let (M 4k ; g, I 1 , I 2 , I 3 ) be a hyper-Kähler manifold. Recall that this means that g is a Riemannian metric and I 1 , I 2 , I 3 are orthogonal complex structures satisfying the quaternionic relations I 1 I 2 = I 3 = −I 2 I 1 such that the two-forms ω j = g(I j −, −), j = 1, 2, 3 are closed. It can be shown that ω C = ω 2 + iω 3 is a holomorphic symplectic form with respect to the complex structure I 1 .
Associated with a hyper-Kähler manifold we have the twistor space Z = Z(M ) which is a complex manifold of complex dimension 2k + 1 on which the hyper-Kähler structure is encoded in the following complex-geometric data [20]: • a holomorphic projection π Z : Z → CP 1 , • a holomorphic section ω ∈ H 0 (Λ 2 T * F (2)) inducing a holomorphic symplectic form on each fibre π −1 (λ) (here T F = ker dπ Z is the tangent bundle along the fibers), • an anti-holomorphic involution τ Z : Z → Z covering the antipodal map CP 1 → CP 1 and such that τ * Z ω = ω, • a family (parametrized by M ) of τ Z -real holomorphic sections with normal bundle isomorphic to O(1) 2k , the twistor lines.
We now briefly recall the construction of the Deligne-Hitchin moduli space, which may be interpreted as the twistor space of the hyper-Kähler moduli space of solutions to the self-duality equations on a Riemann surface Σ. For details we refer to [27], see also [3] for a more differential geometric account. The discussion of this subsection works for complex reductive Lie groups G as structure groups. Since we fully work out our concepts, e.g. twisting (Section 1.3), for SL(2, C), we choose G = SL(n, C) in this subsection for concreteness.
Let Σ be a compact Riemann surface and denote by E → Σ the trivial smooth rank n vector bundle. We endow E with an SL(n, C)-structure, i.e. a trivialisation det E ∼ = O Σ , which in the case of rank 2 is a complex symplectic form. We denote by sl(E) the subbundle of End(E) given by the endomorphisms of trace zero.
Denote by C(E) the space of holomorphic structures∂ on E that induce the trivial holomorphic structure on det E ∼ = O. It is an affine space for Ω 0,1 (Σ, sl(E)). To formulate the self-duality equations, we must reduce the structure group to the maximal compact subgroup SU(n), i.e. we choose a hermitian metric h on E. Then the self-duality equations are given by for a holomorphic structure∂ ∈ C(E) and Φ ∈ Ω 1,0 (Σ, sl(E)). As usual, ∇ h is the Chern connection with respect to∂ and h. Moreover, * h is the adjoint with respect to h, which we will sometimes just denote by * if confusion is unlikely. We denote by the space of solutions to (1.1). Then the moduli space of such solutions is given by with the special unitary gauge group G = Γ(SU(E)) = {g ∈ Γ(End(E) : u * u = id, det u = 1} acting by (∂, Φ).g = (g −1 •∂ • g, g −1 Φg). The smooth locus of M SD (Σ, SU(n)) is given by M irr SD (Σ, SU (n)) = H irr /G, where H irr denotes the set of irreducible solutions, i.e. those for which (∂, Φ).g = (∂, Φ) implies that g ∈ G is a constant multiple of id E .
• The differential operator D is holomorphic in the sense that The group of complex gauge transformations G C = Γ(SL(E)) = {g ∈ Γ(End(E)) : det g ≡ 1} acts on the space of λ-connections by A holomorphic λ-connection (∂, D) on E is called stable (resp. semi-stable) if any D-invariant holomorphic subbundle F ⊂ (E, ∂) satisfies deg F < 0 (resp. deg F ≤ 0). We call a holomorphic λ-connection polystable if it is isomorphic to a direct sum of stable λ-connections whose associated holomorphic bundles have degree zero.
Remark 1.2. The concept of holomorphic λ-connections gives a way of interpolating between flat SL(n, C)-connections and Higgs bundles.
(i) If λ = 0, then D is C ∞ -linear and holomorphic, hence defines a holomorphic section Φ ∈ H 0 (sl(E) ⊗ K). Hence a 0-connection is the same as an SL(n, C)-Higgs bundle. The Higgs bundle is stable (resp. semi-stable, resp. polystable) in the sense of [17] if and only if the 0-connection is stable (resp. semi-stable, resp. polystable) in the sense of the above definition. (ii) If λ = 0 and (∂, D) is a holomorphic λ-connection, then the condition (1.5) implies that we obtain a flat SL(n, C)-connection ∇ via Stability in this case means that there exist no ∇-invariant subbundles. A polystable λconnection corresponds to a completely reducible flat connection, i.e. a direct sum of irreducible flat connections. (iii) The action of the group of gauge transformations specialises to the usual action on the space of Higgs bundles and flat SL(n, C)-connections, respectively.
g ∈ G C is a constant multiple of the identity endomorphism id E , i.e. g is a constant map to the center of SL(n, C)). Since we work with vector bundles, irreducible λ-connections are equivalent to stable λ-connections. In particular, irreducible 0-connections are stable Higgs bundles.
There is a natural isometric circle action on M SD (Σ, SU(n)) induced by the circle action on C(E) × Ω 1,0 (sl(E)). It preserves the complex structure I 1 and rotates I 2 , I 3 . The action complexifies to a natural C * -action on M Hod (Σ, SL(n, C)) covering the standard C * -action on C. A given t ∈ C * acts on an element (∂, D, λ) by Remark 1.6. The projections from the respective Hodge moduli spaces to C glue to give a holomorphic projection π : M DH (Σ, SL(n, C)) → CP 1 . The Deligne-Hitchin moduli space is a complex space. Its smooth locus M s Hod (Σ, SL(n, C)) coincides with the twistor space of M irr SD (Σ, SU(n)) ([27, §4]).
The anti-holomorphic involution τ can be seen via the Riemann-Hilbert correspondence as follows. On M B (Σ, SL(n, C)) we have the natural anti-holomorphic involution which associates to a representation R : π 1 (Σ) → SL(n, C) its complex conjugate dual representation γ → R(γ) −1 t , i.e. R is composed with the Cartan involution corresponding to the compact real form SU(n). Under the Riemann-Hilbert correspondence M B (Σ, SL(n, C)) ∼ = M dR (Σ, SL(n, C)) this induces an antiholomorphic involution on the space of flat connections and we denote by ∇ * the flat connection associated to ∇ in this way. It can be interpreted as the connection on E * induced by ∇, hence the notation. We arrive at the following description of the anti-holomorphic involution on the Deligne-Hitchin moduli space (see also the discussion in (ii) We have an anti-holomorphic involution τ , covering the antipodal involution λ → −λ −1 .
Definition 1.8. We call a holomorphic section s : CP 1 → M DH (Σ, SL(n, C)) (i.e. a holomorphic map such that π • s = id CP 1 ) irreducible if the image of s is contained in M s DH (Σ, SL(n, C)). Remark 1.9. Note that we could also call such sections stable by Remark 1.2.
For every k ∈ N ∪ {∞}, λ-connections of class C k , instead of C ∞ , are defined in an obvious way. Also, the notion of holomorphic λ-connections of class C k is defined correspondingly. The next lemma shows their relation to (local) irreducible sections.  Remark 1.11. The proof in [3] is formulated for SL(2, C) and global irreducible sections but generalizes to the setup of Lemma 1.10. Note that if B is sufficiently small, any irreducible section s on B admits a lift to the space of holomorphic λ-connections of class C ∞ . We lose regularity when such local lifts are glued together over larger B though, see [3] for details.
We further observe that if s : B → M DH (Σ, SL(n, C)) is a local section around 0 ∈ CP 1 such that s(0) is a stable Higgs bundle, then there is an open neighborhood B ′ ⊂ B of 0 such that s |B ′ maps to M s DH (Σ, SL(n, C)). In particular, the germs of such sections always admit lifts to the space of holomorphic λ-connections.
Finally, the above lemma applies for sections locally defined around ∞ ∈ CP 1 in the obvious way.
Given an irreducible holomorphic section s with lifts s, − s over C and CP 1 \ {0} respectively, we will often work with the associated C * -family of flat connections and − ∇ defined similarly over CP 1 \ {0, ∞}. Here we write ∇ =∂ + ∂ in the notation of equation (1.8). One can show [3], that there exists a holomorphic C * -family g(λ) of GL(n, C)-valued gauge transformations, unique up to multiplication by a holomorphic scalar function, such that Irreducible holomorphic sections corresponding to solutions of the self-duality equations have the special property that we have lifts such that + ∇ = − ∇ on C * , i.e. we can arrange g ≡ id E in the above discussion. This is axiomatised as follows. In the rank 2 case the family of gauge transformations g(λ) such that + ∇.g = − ∇ can be used to define the following invariant of an irreducible section s. Definition 1.13. Let s be an irreducible section of M DH (Σ, SL(2, C)) with associated families + ∇ and − ∇ over C and CP 1 \ {0} respectively. Consider a holomorphic C * -family g(λ) of GL(2, C)valued gauge transformations such that + ∇ λ .g(λ) = − ∇ λ . The parity of s is the parity of the degree of the holomorphic function det g : C * → C * . Remark 1.14. (i) We remark that the parity of an irreducible section s is zero, if and only if we can arrange the family g(λ) gauging + ∇ to − ∇ to be SL(2, C)-valued. In particular, any admissible section has parity zero. In higher rank n > 2 one can construct a similar invariant given by deg(det g) mod n ∈ Z/nZ. For groups other than SL(n, C) it is not obvious what an appropriate generalisation of this invariant might be.
If we have a lift ∇ λ on C ⊂ CP 1 of a σ-real holomorphic section s, then for every λ ∈ C * there is a gauge transformation g(λ) such that the following equation holds If s is σ-real and irreducible of parity 0, we can choose the family of gauge transformations g(λ) in (1.9) to depend holomorphically on λ and may assume that it takes values in SL(2, C). By irreducibility, the holomorphic family g(λ) is then uniquely determined up to a sign. By [3, Lemma 2.15] the following definition makes sense. Definition 1.16. Let σ ∈ {τ, ρ} and consider an irreducible σ-real holomorphic section s : CP 1 → M DH (Σ, SL(2, C)) of parity 0. Let ∇ λ be a lift of s over C and let g(λ), λ ∈ C * , be a holomorphic family of SL(2, C)-valued gauge transformations such that (1.9) holds. Then Remark 1.17. (i) The signs in Definition 1.16 are chosen to be consistent with [3], where the fact that an SL(2, C) bundle is isomorphic to its dual is incorporated into the definition, see is a solution to the SU(2)-self-duality equations, the associated twistor line is given by the C * -family of flat SL(2, C)-connections It is shown in [3, Theorem 3.6] that the irreducible solutions of the self-duality equations correspond precisely to the admissible, irreducible τ -negative sections CP 1 → M DH (Σ, SL(2, C)).
By the non-abelian Hodge correspondence, these correspond to equivariant harmonic maps f :Σ → H 3 = SL(2, C)/SU(2). (iii) On the other hand, ρ-negative sections CP 1 → M DH (Σ, SL(2, C)) are automatically admissible and correspond to equivariant harmonic maps f :Σ → S 3 = SU(2). These are obtained from solutions to the harmonic map equations The associated sections are of the form Twisting. We briefly review the twisting or Gauß map procedure that played a central role in the construction of τ -positive holomorphic sections of M DH (Σ, SL(2, C)) in [3]. Starting from an irreducible solution (∇,Φ) to the SU(2)-harmonic map equations (1.10) with nilpotent Higgs field, one considers the associated family of flat connections Denote by L the kernel bundle ofΦ, so that we get a smooth splitting E = L ⊕ L ⊥ . To this familỹ ∇ λ of flat connections, one associates a new family ∇ λ of flat connections by twisting, which is given by is written with respect to the splitting E = L ⊕ L ⊥ . In [3] it was shown that the so defined C *family of flat connections extends to define an irreducible, admissible holomorphic section over all of CP 1 . In this section we study this procedure more systematically.
Let s : CP 1 → M DH (Σ, SL(n, C)) be a holomorphic section. Then we may use the C * -action to define a new holomorphic sections over C * : Since the C * -action on M DH (Σ, SL(n, C)) covers the obvious one on CP 1 = C ∪ {∞} it is clear thats is a holomorphic section over C * . It is a natural question to ask under what conditions on s the twisted sections extends to define a holomorphic sections : CP 1 → M DH (Σ, SL(n, C)). Definition 1.18. We call a holomorphic section s : CP 1 → M DH (Σ, SL(n, C)) twistable ifs : C * → M DH (Σ, SL(n, C)), λ → λ −1 .s(λ 2 ), extends to a holomorphic sections : CP 1 → M DH (Σ, SL(n, C)), which we call the twist of s. Remark 1.19. In terms of λ-connections, we can view the construction of the twist as follows. Write The construction of [3] suggests that there exists a transformation from the space of ρ-real twistable sections to the space of τ -real sections. The precise result is as follows.
section. Then the twists is N -invariant. ii) Suppose that s : CP 1 → M DH (Σ, SL(n, C)) is a ρ-real twistable holomorphic section. Then the twists is again ρ-real and moreover N -invariant, hence τ -real. iii) Suppose that s : CP 1 → M DH (Σ, SL(n, C)) is a τ -real and N -invariant twistable holomorphic section. Then the twists is again τ -real and moreover N -invariant.
Remark 1.21. (i) In part (iii) of Proposition 1.20 the assumption that the τ -real section s is moreover N -invariant is needed due to equation (1.7) with σ = τ . In general we get τ (s(λ)) = λ.s(−λ −2 ) and the N -invariance then ensures the τ -reality ofs.
(ii) Theorem 3.4 in [3] can be interpreted as the statement that, in the SL(2, C)-case, an irreducible admissible ρ-negative section s with nilpotent Higgs field is twistable and that the twists is τ -positive.
The following proposition describes a class of twistable sections in the SL(2, C)-case.
Proof. We need to prove that λ →s(λ) extends to λ = 0, ∞. Both cases work analogously, so we only deal with λ = 0. Let us consider a lift ∇ λ of s over {λ = ∞} ⊂ CP 1 given by Here Φ + ∈ Ω 1,0 , and Ψ k ∈ Ω 1 for k ≥ 1. Then by Remark 1.19 we get a lift ofs over C * bỹ The sections extends to λ = 0 if we can find a C * -family h(λ) of complex gauge transformations such that∇ λ kΨ k and the Higgs pairs(0) = (∂∇,Φ) is stable. If Φ + = 0 there is nothing to prove, so let us assume Φ + = 0. By assumption Φ + is nilpotent, so let us denote by L its kernel bundle, which must satisfy deg L < 0, since (∂ ∇ , Φ + ) is a stable Higgs pair by irreducibility of the section s (see Definition 1.8 and Remark 1.2). Take a complementary bundle L ⊥ . Then, with respect to the splitting E = L ⊕ L ⊥ , we can take with β ∈ Ω 1,0 (Hom(L, L ⊥ )). Then, writing With this the lift∇ λ transforms tõ It now follows just like in the proof of [3,Theorem 3.4.] that the sections extends to λ = 0 and that s(0) is a stable Higgs pair. Moreover, for any λ = 0 the connection∇ λ = ∇ λ 2 is irreducible, which implies that also∇ λ .h(λ) is irreducible. Altogether this shows thats is an irreducible section.
We expect that a similar construction works for n > 2 as well. The main difficulty is to verify the stability ofs(0) ands(∞) which is more involved for general n > 2.
to the space of λ-connections with associated family of flat connections Here Φ ∈ Ω 1,0 (sl(E)), Ψ ∈ Ω 0,1 (sl(E)) and ∇ =∂ + ∂ is an SL(n, C)-connection. We have seen in Lemma 1.10 that we can even find a global lift (i.e. B = C) if the section s is irreducible. Consider The function E : S → C is holomorphic in the following sense: if T is a complex manifold and s : T → S, t → s t is a holomorphic family of sections, then the function E • s : Proof. Write s = (∂ + λΨ + . . . , Φ + λ∂ + . . . , λ) as above. Let s.g be another lift of s, where g is a λ-dependent family of gauge transformations defined in a neighbourhoof of 0 ∈ C. We split g(λ) into the product of a constant gauge transformation and a gauge transformation which equals the identity for λ = 0: ). It is clear that E( s.g 0 ) = E( s), since Φ and Ψ are just conjugated by g 0 . Thus, we may assume that g 0 = 1. Then . . , λ) It then follows from Stokes' theorem and∂Φ = 0 that The holomorphicity of E as stated in the Proposition follows directly from the definition of E.  The name energy functional is motivated by the following observation.
Likewise, for a ρ-negative holomorphic section s of M DH (Σ, SL(2, C)) → CP 1 corresponding to an equivariant harmonic map to SU(2) (equipped with its constant sectional curvature 1 metric) we have Proof. Take the associated family of flat connections, which provides us with a natural lift of such a (τ -or ρ-)real holomorphic section (see Remark 1.17). The theorem then follows by interpreting the Higgs field as the (1, 0)-part of the differential of the map f , see [8,17].
In particular, if s is σ-real, then its energy E(s) is real.
The last statement is generalized to arbitrary holomorphic τ -real sections in Section 3 (cf. Lemma 3.6) and Corollary 3.11).
Proof. Since s is admissible, we may find a global lift s with associated C * -family ∇ λ of flat connections of the form It follows that

2.2.
The effect of twisting on the energy. In this paragraph we investigate how the energy functional behaves under the twisting construction introduced in Section 1.3.
Proof. By Proposition 1.22 we know that s is twistable. With the same notation as in the proof of Proposition 1.22 we have a lift of the form Let us for convenience relabel ψ = ψ 21 . The energy of s is given by The energy of the twist is Since∇ λ is flat for all λ ∈ C * , we see It follows that where 0 = Φ ∈ Ω 1,0 (sl(E)) is nilpotent and (∇, Φ) is an irreducible solution of the harmonic map equations (1.10). Then the energy of the twisted sections satisfies where L denotes the kernel bundle of Φ. In particular, the τ -real sections cannot be a twistor line.
Proof. By Theorem 2.4 we know that E(s) > 0, since Φ = 0. The energy of the twisted sections is then (note that deg L < 0 by irreducibility) We know from Proposition 1.20 thats is τ -real since s is ρ-real. But for a twistor line the energy is negative, while we have just seen that E(s) > 0. Hences cannot be a twistor line. More precisely, let (M 4k ; g, I 1 , I 2 , I 3 ) be a hyper-Kähler manifold with corresponding Kähler forms ω j , j = 1, 2, 3. Suppose that M comes equipped with an isometric circle action which preserves ω 1 and rotates ω 2 , ω 3 , i.e.
where X is the vector field induced by the circle action. Remark 3.1. We emphasize that for the existence of L M , and hence L Z , it is sufficient to work with i X ω 1 instead of the moment map (also see [23]).
where P Z is the principal C * -bundle corresponding to L Z and T P Z /C * is the vector bundle on Z whose sections correspond to the C * -invariant vector fields on P Z . For later reference we denote the extension class of (3.2) by η Z ∈ H 1 (Z, T * Z ). Since T P Z /C * is a Lie algebroid, η Z actually lies in (the image of) H 1 (Z, T * Z,cl ) for the closed 1-forms T * Z,cl . That is, η Z can be represented by aČech cocyle with values in the sheaf of closed one-forms.
Hitchin observed that η Z is of a special form if additionally H 1 (M, C) = 0. Namely, let Y ∈ H 0 (Z, T Z ) be the holomorphic vector field induced by the circle action lifted to the twistor space Z. After applying Möbius transformations, we may assume that the circle action satisfies If D := D 0 + D ∞ denotes the divisor determined by the fibers of π Z over 0 and ∞, then s yields the short exact sequence In his twistorial approach to L Z , Hitchin constructed a section ϕ ∈ H 0 (D, T * Z (2) |D ), assuming H 1 (M, C) = 0, which satisfies is the connecting homomorphism in the long exact sequence associated to (3.4), T F = ker dπ Z is the tangent bundle along the fibers of π Z and We next show that ϕ satisfying (3.5) is essentially unique. and non-singular otherwise.
The existence of a meromorphic connection ∇ ϕ with (3.7) already appeared in [19] but we include its proof for completeness.
As a preparation we give the proof of the following well-known lemma.
. Hence if we restrict α (as a section) to the image s m (CP 1 ) ⊂ Z, we obtain α |sm(CP 1 ) = 0. By varying m ∈ M , we conclude α = 0. The same argument shows H 0 (Z, Λ k T * Proof of Proposition 3.2. First of all, we consider for each F ∈ T * Z , T * F , π * Z T * CP 1 = π * Z O(−2) the short exact sequence These fit into the diagram 0 with exact rows and columns. Next we consider (parts of) the corresponding long exact sequences.
For the last statement, assume ϕ ∈ H 0 (D, T * Z (2)| D ) with (3.5) exists. This is the case, for example, if H 1 (M, C) = 0. Then choose an appropriate open covering U of Z such that ϕ |U ∩D lifts to ϕ U ∈ H 0 (U, T * U (2)) for every U ∈ U . By (3.5), the cocycle so that ϕ U s are connection 1-forms of a meromorphic connection ∇ ϕ on L Z with the claimed properties.
For the uniqueness of ∇ ϕ , let ∇ 1 , ∇ 2 be two meromorphic connections on L Z with res D (∇ 1 ) = res D (∇ 2 ) and holomorphic otherwise. Then ∇ := ∇ 1 ⊗∇ * 2 is a holomorphic connection on L Z ⊗L * Z = O Z . Hence ∇ is of the form d + α for a global holomorphic 1-form α on Z which must vanish. Consequently, ∇ 1 and ∇ 2 are equal.
As a next step, we examine how such ϕ interact with the real structure τ Z . First observe that for every ϕ satisfying (3.5), the section ϕ r := 1 2 (ϕ + τ * Z ϕ) again satisfies (3.5) and is further real, i.e.
Since δ Z commutes with τ Z * , it follows that τ * Z (η Z ) = η Z and consequently τ * Z L Z ∼ = L Z . For later reference, we record the following observation.

3.2.
Residues. For the next proposition, we assume H 1 (M, C) = 0 so that real sections ϕ ∈ H 0 (D, T * Z (2)) as in Corollary 3.5 and correspondingly meromorphic connections ∇ ϕ on L Z exist as in Proposition 3.2. Let S be the complex-analytic space of holomorphic sections of π Z : Z → CP 1 and define the function res ϕ : S → C, res ϕ (s) := res 0 (s * ∇ ϕ ). This is well-defined because s * ϕ 0 ∈ H 0 ({0}, O) = C. It is immediate that res ϕ is holomorphic in the following sense: if T is a complex manifold and (s t : CP 1 → Z) t∈T a holomorphic family of sections of π Z , then T → C, t → res ϕ (s t ) is holomorphic. We further observe that res ϕ (s) is defined for any local holomorphic section around 0 ∈ CP 1 .
In case s is a real section defined on all of CP 1 , then we obtain the following relation: Lemma 3.6. If s ∈ S R is a real holomorphic section of π Z , then res ϕ (s) = deg(s * L Z ) − res ϕ (s). (3.13) In particular, if s is a real holomorphic section with deg(s * L Z ) = 0, then res ϕ (s) ∈ iR.
The previous lemma reflects the fact that res ϕ yields a moment map on all connected components of S R , see Section 3.4. To show this and the relation of res ϕ to the previously defined energy functional, we need an explicit formula for res ϕ . We begin with the following lemma (see [19, Lemma 8]).
Then ψ |D is a holomorphic section of T Z (2) |D and satisfies (3.5).
Proof. Let X be the C ∞ -vector field on M induced by the circle action which we identify with a C ∞ -vector field on Z = M × CP 1 . We denote by X 1,0 λ the (1, 0)-part of X with respect to the complex structure I λ on M . The holomorphic structure∂ Z on T * Z (2) = T * F (2) ⊕ π * Z O CP 1 (with respect to the natural C ∞ -splitting) is given bȳ cf. equation (8) after Lemma 7 in [19]. Here we abuse notation and write∂ λ also for the induced complex structure on O(2)-valued one-forms etc.
It remains to prove that ψ satisfies ψ |T F = 1 2i i Y ω along D (again for ω as in (3.6)). Since ψ is real, it suffices to check this equality at λ = 0: Here we have used that ω 2 + i ω 3 is of type (2, 0) along D 0 and X 1,0 0 = Y .  Proof. We first prove the statement about the additive constant. Let ϕ, ϕ ′ ∈ H 0 (D, T * Z (2)) satisfy (3.5). As we have seen in Corollary 3.5, we have for any holomorphic section s of π Z .
Hence it is sufficient to prove Now ψ is given in the dual splitting θ * sm so that Lemma 3.7 and (3.22).
Hence the residue res ϕ is natural in several ways. Not only is it essentially independent of ϕ or the base point (i.e. 0 or ∞) but it is also the analytic continuation of the moment map µ : M → iR to the space of all holomorphic sections, where we identify M with space of the twistor lines.
The moment map (with respect to ω 1 ) is given by (3.24) as follows easily from the explicit form of the metric g((γ, β), (γ, β)) = 2i Σ tr(γ * ∧ γ + β ∧ β * ). (3.25) Recall moreover that the holomorphic symplectic 2-form ω C = ω 2 + iω 3 (with respect to I 1 ) is is an arbitrary tangent vector then (3.23) and (3.26) combine to give Hence Theorem 3.9 implies (3.31) Remark 3.12. (i) We have formulated Corollary 3.11 for the SL(n, C)-case. However, Corollary 3.11 makes sense for any complex reductive group G C and the previous proof stills works once we replace tr by an appropriate non-degenerate invariant form on g C . In the semi-simple case we take (an appropriate multiple of) the Killing form.
(ii) To our best knowledge, meromorphic connections ∇ ϕ in terms of determinant line bundles have only been given for M = M irr SD (Σ, C * ) in [21,Theorem 5.13] via their theory of intersection connections on Deligne pairings. Our results could be useful to extend [21,Theorem 5.13] to higher rank.
3.4. The energy as a moment map. Let Z = Z(M ) be the twistor space of a connected hyper-Kähler manifold M with circle action as before with [ω 1 /2π] ∈ H 2 (M, Z) and H 1 (M, C) = 0. We assume that there exists a component N of real holomorphic sections of Z → CP 1 which is different from the component M of twistor lines. We further assume that the normal bundle for any section s ∈ N is the direct sum of O(1) → CP 1 and that the twistor construction [20] yields a positive definite Riemannian metric g N induced by ω. This implies that the evaluation map for any λ ∈ CP 1 of real normal sections is a local diffeomorphism. Hence, by [20], (N, g N ) extends to a hyper-Kähler manifold (N ; g N , I N 1 , I N 2 , I N 3 ). The circle action on the twistor space induces a circle action on N . Indeed, for c ∈ S 1 ⊂ C, and the corresponding biholomorphic map Φ c : Z → Z, we define for a given section s the new section . Clearly, s c is real holomorphic if s is real holomorphic, and because S 1 is connected s and s c are in the same component of real holomorphic sections. This circle action is again rotating. . Then N has a rotating circle action, and the residue res ψ : S → C of the natural meromorphic connection ∇ ψ on L Z → Z restricted to N yields a moment map for the circle action with respect to ω N 1 . In particular, res ϕ is a Kähler potential for (N, g, I N 2 ).
Note that H 1 (N, R) might not be zero so that general arguments do not even guarantee the existence of a moment map on N .
Proof. For every s ∈ N , there exist open neighborhoods U ⊂ N , V ⊂ M = Z 0 of s and s(0) respectively such that there is a biholomorphism of the twistor spaces of U and V . It is compatible with the fibrations to CP 1 , the real structures and the twisted relative symplectic forms. Even though U might not be S 1 -invariant, there is a holomorphic line bundle L Z(U ) -induced by a hyperholomorphic line bundle L U over U -with a meromorphic connection ∇ ϕ U as before, cf. Remark 3.1 and 3.8. Theorem 3.9 implies that res ϕ U :

The energy and the Willmore functional
We have seen in Theorem 2.4 that for twistor lines the energy E is directly related to the harmonic map energy of the corresponding equivariant harmonic map. In [13], non-admissible τ -negative real holomorphic sections of the rank 2 Deligne-Hitchin moduli spaces have been constructed. These sections correspond to equivariant Willmore surfaces, for definitions see Section 4.2 below. We will exhibit an explicit formula relating the Willmore energy of the surface with the energy of the corresponding section of M DH → CP 1 . Before we can state the main results, we need an auxiliary tool: the dual surface construction. In the following sections we restrict to rank 2 Deligne-Hitchin moduli spaces.
4.1. The dual surface construction. Consider a holomorphic section s of the Deligne-Hitchin moduli space. We assume that s(0) is a stable Higgs pair with nilpotent Higgs field. The section s admits a (local) lift ∇ λ = λ −1 Φ + ∇ + λΨ + . . . such that Φ is nilpotent.
By assumption, the kernel bundle L of Φ has negative degree. Choose a complementary subbundle L and apply the gauge transformation h(λ) = diag(λ −1 , 1), written with respect to L⊕L * to ∇ λ , cf. the proof of Proposition 1.22. In this way, we obtain a new C * -family of flat SL(2, C)-connectionŝ (4.1) With respect to L ⊕ L * we may write . By a computation analogous to the one in the proof of Proposition 1.22 we see that Note that although the corresponding family of λ-connections has a limit as λ → 0, this family is not the lift of any holomorphic sectionŝ : C → M DH , as the Higgs pair (∂∇,Φ) at λ = 0 is unstable: Indeed, we have Thus, the holomorphic subbundle L * is the kernel bundle ofΦ and has positive degree. Still, we can interpret λ →∇ λ as a mapŝ into the space of holomorphic λ-connections, and consider its energy E(ŝ) as defined in (2.1). This is well-defined, and invariant under holomorphic families of gauge transformations λ → g(λ) which extend holomorphically to λ = 0 (see the proof of Proposition 2.1). WithΨ = ψ 11 α * ψ 22 a computation analogous to the proof of Proposition 2.7 yields the following formula relating the energy ofŝ to that of s. Note that on the right hand side of the formula E(s) appears with a factor 1 as opposed to the formula in Proposition 2.7.
where (∇, Φ) is a solution of (1.10). The Higgs field Φ is nilpotent as the surface is given by a conformal harmonic map, and we can apply the construction (4.1). Denote the kernel bundle of Φ by L and write with respect to E = L ⊕ L ⊥ The dual surface construction then yields the familŷ which satisfies the same reality condition as ∇ λ , i.e. both are unitary for λ ∈ S 1 . Moreover∇ λ has nilpotent Higgs field as well. It therefore gives another conformal harmonic map into S 3 = SU (2) which is branched at the zeros of the Hopf differential of the surface f . This construction is wellknown in classical surface theory, and is sometimes called the parallel or dual surface of the initial minimal surface f , see [22] and the references therein.
The dual surface construction yields the familŷ We observe that Thus,∇ λ satisfies a different reality condition than ∇ λ . In fact, it follows that the family∇ λ does not give an equivariant harmonic map to H 3 but an equivariant harmonic mapΣ → dS 3 = SL(2, C)/SU(1, 1) into the de Sitter space, see [3,Section 3]. Because the Higgs field 0 0 α * 0 of the family∇ λ is also nilpotent the corresponding equivariant harmonic map into de Sitter space is conformal as well.
The de Sitter space dS 3 has the identification as the space of oriented circles on a fixed 2sphere. We consider the 2-sphere as the equatorial 2-sphere S ∞ in the 3-sphere which separates two hyperbolic 3-balls. The space of oriented circles C in the 2-sphere can be identified with the space of oriented 2-spheres S in the 3-sphere which intersect S ∞ perpendicularly, i.e. C = S ∩ S ∞ as oriented submanifolds of S 3 . In this interpretation, the equivariant conformal harmonic map into de Sitter 3-space yields a map into the space of oriented 2-spheres in the 3-sphere. We will see in Section 4.3 below, that the latter map is the mean curvature sphere of the minimal surface f in H 3 ⊂ S 3 , i.e., the map which associates to a point p of the surface the best touching 2-sphere of f at p.
Let s be a twistor line given by a nilpotent Higgs pair s(0), and apply the dual surface construction. From (4.2) we can directly compute that E(ŝ) ≥ 0 with equality if and only if ∇ is reducible, i.e. α = 0. As an application of Proposition 4.1 we reobtain the well-known energy estimate where g is the genus of the surface Σ.

4.2.
The Willmore functional and the energy of higher sections. A solution (∇, Φ) of the self-duality equations with nilpotent Higgs field Φ = 0 gives rise to a branched conformal harmonic map, i.e., an equivariant minimal surface f :Σ → H 3 with branch points. The basic invariant of the equivariant minimal surface is the area of a fundamental piece, which is determined by the energy of the harmonic map, i.e., where s is the τ -real holomorphic section of the Deligne-Hitchin moduli space corresponding to the solution (∇, Φ) of the self-duality equations.
The Willmore energy of a conformal immersion f :Σ → M into a Riemannian 3-manifold M is given by where dA is the induced area form, K is the curvature of the induced metric, H = 1 2 tr(II) is the mean curvature, i.e., the half-trace of the second fundamental form II, and for p ∈Σ the quantityK p is the sectional curvature of the tangent plane T f (p) f (Σ) ⊂ T f (p) M . It was known already to Blaschke that the Willmore functional for surfaces in R 3 or S 3 is invariant under Möbius transformations of the target space. It was first shown in [7] that the Willmore integrand is actually invariant under conformal changes of the metric on M .
In the case of an equivariant, immersed minimal surface f :Σ → H 3 into hyperbolic 3-space, H = 0 and the Willmore functional therefore equals to We apply the dual surface construction (4.1) to the corresponding holomorphic section s with nilpotent Higgs field Φ. We obtain a new family of λ-connectionsŝ. Let L be the kernel bundle of Φ. Because Φ can be interpreted as the (1, 0)-part of the differential of the minimal surface, the zeroes of the Higgs field are branch points of f . Since we assume that f is not branched, we must have deg(L) = 1 − g. Hence, Proposition 4.1 implies The conformally invariant Willmore integrand H 2 − K +K dA can be generalized to a class of branched conformal maps into the conformal 3-sphere. The extra assumption is that the mean curvature sphere (the exact definition is given in Section 4.3 below) extends through the branch points of the conformal map, see [5] and related literature. We will see in Section 4.3 that this assumption holds for branched minimal surfaces in hyperbolic 3-space, yielding (4.3) in this more general situation. In fact, we obtain an equality for the Willmore integrand: where∇ λ := ∇ λ .h(λ) =∇ + λ −1Φ + λΨ is given by the dual surface construction. In particular whereŝ is the family of λ-connections determined by∇ λ .
A proof is given in Section 4.3 below using notions from conformal surface geometry.
In [13] it was shown that there exist compact Riemann surfaces Σ whose associated Deligne-Hitchin moduli spaces admit τ -negative holomorphic sections s with the following properties: (1) the Higgs field Φ is nilpotent, where s(0) = [∂, Φ]; (2) the section s is not admissible: for a lift ∇ λ with ∇λ −1 = ∇ λ .g(λ) the Birkhoff factorization g = g + g − fails along a real analytic (not necessarily connected) curve γ ⊂ Σ (see Remark 1.14); (3) On Σ \ γ the section s gives rise to an (equivariant) conformal harmonic map which extends through the boundary 2-sphere at infinity of the hyperbolic 3-space, yielding a Möbius equivariant Willmore surface f : It is a natural guess that the energy E(s) is related to the Willmore energy of a fundamental piece of f . We remark that [2,Theorem 9] can be interpreted as this relation in the case of Σ being of genus 1. Our main result here is the following theorem, whose proof we postpone to section 4.4.  [24,1] for more details): Consider a minimal surface in a totally geodesic H 3 ⊂ AdS 4 which intersects the boundary at infinity. If the surface extends to a Willmore surface in S 3 , giving rise to a τ -negative holomorphic section of the Deligne-Hitchin moduli space of a compact Riemann surface, the finite part of the area functional is given by the Willmore energy of the surface. If we additionally have a symmetry between the two pieces of the minimal surface in the two components of H 3 in S 3 = H 3 ∪ S 2 ∪ H 3 , the Willmore energy is given in terms of the energy of the section. A similar relation holds for space-like minimal surfaces in AdS 3 .

4.3.
The lightcone approach to conformal surface geometry. Our proofs of Proposition 4.4 and Theorem 4.5 will use some concepts of conformal surface geometry in the lightcone model, which we recall here. We refer to [6,4,25] for details.
Consider R 4,1 with the standard Minkowski inner product restricts to a natural diffeomorphism between the 3-sphere S 3 and the projectivization P L ⊂ P R 4,1 . There exists a natural conformal structure on P L induced by ., . , which contains the round metric on S 3 . If σ is a (local) section of π : L → P L then the conformal structure is represented by the Riemannian metric g σ defined as g σ (X, Y ) := dσ(X), dσ(Y ) . The round metric is obtained from the lift σ([x]) = x x 0 , [x] ∈ P L. The space of orientation preserving conformal diffeomorphisms of S 3 ∼ = P L is then given by P SO(4, 1) (via its natural action on P R 4,1 ). Those transformations are also called Möbius transformations. We will often consider the conformal 3-sphere as the union S 3 = H 3 ∪ S 2 ∪ H 3 , i.e. as the union of two hyperbolic balls separated by an equatorial S 2 . In the lightcone model a 2-sphere S 2 can be written as P (v ⊥ ∩ L) for a space-like vector v ∈ R 4,1 . It is known that the complement {[x] ∈ P L : x, v = 0} is conformal to H 3 ∪ H 3 . In particular, we note that a 2-sphere S 2 ⊂ S 3 corresponds to a subspace of R 4,1 of signature (3,1).
Consider a conformal immersion f : Σ → P L from a Riemann surface. There exists a real rank 4 vector bundle S ⊂ R 4,1 locally defined with respect to a holomorphic coordinate z and a local liftf of f to R 4,1 as where for a function g we denote g z := ∂g ∂z and so on. The real rank 4 bundle is well-defined, and ., . restricts to an inner product of type (3,1). Under the correspondence between 2-spheres in S 3 and subspaces of signature (3,1) in R 4,1 the bundle S can be interpreted as a family of 2-spheres. It is called the mean curvature sphere congruence associated with f . Its orthogonal complement is denoted by N and we obtain an induced decomposition of the trivial connection on R 4,1 = S ⊕ N into diagonal and off-diagonal parts d = D S + β, where β is tensorial and D S is a connection. The Willmore energy of the surface is then given by which is equivalent to the flatness of the family of SO(4, 1) C = SO(5, C) connections λ ∈ C * → D λ = D S + λ −1 β (1,0) + λβ (0,1) . The equivariant Willmore surfaces constructed in [13] have the additional property that they are minimal in two hyperbolic balls separated by the boundary at infinity S 2 ⊂ S 3 = P L. This condition is equivalent to the fact that there exists a space-like vector v of length 1 which is contained in S p for all p ∈ Σ, see [6,4,25] for a proof. Therefore v is also parallel with respect to D λ for all λ ∈ C * . After applying a Möbius transformation we can assume that v = e 4 .
In order to compare the SL(2, C)-family ∇ λ with the SO(5, C)-family D λ of flat connections coming from the Willmore surface, we make use of the following model of An isometry Ψ : R 4,1 → V is given by Let Σ be a Riemann surface. Consider a C * -family ∇ λ of flat SL(2, C)-connections of the selfduality form (on the trivial C 2 bundle over Σ with standard hermitian metric) corresponding to an equivariant minimal surface f :Σ → H 3 on the universal covering, i.e.
Note that (4.8) yields an induced frame of the flat rank 5 bundle V by extending the mean curvature sphere bundle by a constant length 1 section of its orthogonal complement. We want to describe the connection D λ with respect to this frame.
Locally, on open sets where F is well-defined and where we have a holomorphic coordinate z, we can find an SU(2)-frame such that The locally defined function u is determined by the induced metric g (from the hyperbolic minimal surface) by g = e 2u dz ⊗ dz, and u z , uz are determined by u z dz + uzdz = du and q is a holomorphic function (representing the Hopf differential q(dz) 2 of the surface).
Proof. ObviouslyD λẽ 4 = 0 and for i = 1, 2, 3, 5 we haveẽ i = (E i , 0) with a constant matrix E i ∈ gl(2, C). Thus,D A direct calculation then yields the connection matrix. Note that in the notation of Remark 4.8 we have thatẽ i is obtained from e i by multiplying the matrix part by diag(1, −1) and leaving the scalar part unchanged. Denote this map e i →ẽ i by S. Then S −1 = S and we have with the notation of Lemma 4.7 Clearly, the gauge transformationĜ =F • S is independent of λ and satisfiesĜ −1 • D λ •Ĝ =D λ as can be checked in the frame {ẽ i }.

4.4.
Proofs. We will now use the theory of the previous section to give the proofs of the results formulated in section 4.2. Proof of Theorem 4.5: Let s be a section satisfying the assumptions in Theorem 4.5, with associated equivariant Willmore surface f :Σ → S 3 = P L. We start with a lift of s where η λ is a λ-family of sl(2, C)-valued 1-forms on Σ. There exists a curve γ ⊂ Σ such that on M = Σ \ γ we have a holomorphic λ-family of gauge transformations g + (λ) which extends to λ = 0 and which gauges ∇ λ into self-duality form. That is, on M we have ∇ λ .g + (λ) = λ −1 φ + ∇ 0 + λφ * , where (∇ 0 , φ) solves the self-duality equations and φ is still nilpotent. Denote by L the kernel bundle of the Higgs field φ with orthogonal complement L ⊥ .
Moreover, by Lemma 4.9,D λ .Ĝ(λ) and D λ are gauge equivalent by a λ-independent gauge transformation. The mean curvature sphere family extends smoothly through the singularity set of the equivariant minimal surface f (or likewise g + ). Therefore alsoĜ extends smoothly through this singularity set as a positive gauge transformation. The Theorem now follows from Remark 4.10 and Proposition 4.4.

Energy estimates
Corollary 2.8 gives us a possibility to distinguish the space of twistor lines, i.e., the space of τ -negative admissible holomorphic sections, from the space of τ -positive admissible sections, by looking at the value range of E. Note that this criterion is much easier to handle in practice than determining whether a τ -real section is τ -positive or τ -negative. We shall be able to use E also to distinguish the recently discovered new components of τ -negative sections [13] from the component of twistor lines. We emphasize that these τ -negative sections cannot be admissible. In view of Simpson's question [27], such a complex-analytic tool to distinguish those components seems desirable.
The first indication that the function E does help can be seen in the case of tori, i.e., for Σ of genus 1. In this case, the SL(2, C) Deligne-Hitchin moduli space has a 2-fold covering of the C * Deligne-Hitchin moduli space. Note that the E-function is still well-defined in this situation, even if we do not have any irreducible λ-connections at all: It is well-known that on a torus solutions of the self-duality equations are totally reducible. Applying Hitchin's spectral curve approach [18] to this situation, we easily see that twistor lines correspond to spectral data of spectral genus 0. Other components of the space of τ -negative holomorphic sections are given by spectral data for spectral curves of positive genus, compare with [2,13,17]. While the spectral genus distinguishes the different components of τ -negative sections, the following theorem indicates the use of the E-function in this context. Theorem 5.1. Let s be a holomorphic section of the (singular) Deligne-Hitchin moduli space over a Riemann surface of genus 1 which is τ -negative and has a nilpotent Higgs field. Assume that the spectral genus is bigger than 1. Then E(s) ≥ 1 4 .
Proof. Such sections give rise to Möbius equivariant Willmore surfaces f :Σ → S 3 into the conformal 3-sphere, see [2] or also [13]. Because the kernel bundle of the nilpotent Higgs field on a torus has degree 0, we obtain from Theorem 4.5 that where W(f ) is the Willmore energy of a fundamental piece of f . The theorem follows from an application of the quaternionic Plücker estimate, see [9,Equation (89)]: That the spectral genus is at least 2 (in fact it must be odd) implies that there are two quaternionic holomorphic linearly independent sections on an unbranched 4-fold covering of the torus of a quaternionic holomorphic line bundle. The Willmore energy of this quaternionic holomorphic line bundle coincides with the Willmore energy of f on a fundamental piece.
Note that holomorphic sections with nilpotent Higgs field on a torus cannot be totally reducible and therefore are not twistor lines. They therefore lie in a different component of the space of τ -negative sections than the twistor lines. The assumption on the spectral genus in Theorem 5.1 leaves open the case of spectral genus 1. In that case, as the solutions are equivariant, one can make the energy E(s) to be arbitrarily close to 0 by changing the conformal type of the torus Σ. On the other hand, it does not seem possible to fix the Riemann surface Σ of genus 1 and then find, for each ǫ > 0, a τ -negative holomorphic section s in the Deligne-Hitchin moduli space with nilpotent Higgs field such that E(s) < ǫ.
In general, one might try to use the energy to distinguish different components of τ -negative holomorphic sections of the Deligne-Hitchin moduli space. A first result is given in the following theorem, where we show that the energy is positive for the τ -negative holomorphic sections constructed in [13].
In particular, these sections cannot be twistor lines.
Proof. The non-admissible τ -real holomorphic sections have been constructed by a deformation of finite gap solutions of the cosh-Gordon equation of spectral genus 1 on a torus Σ. The initial section on the torus Σ yields an equivariant Willmore surface f . By Theorem 4.5, the Willmore energy of a fundamental piece is the energy of the section, since the degree of the kernel bundle L is necessarily 0. Because the Hopf differential q(dz) 2 does not vanish, the Willmore integrand is positive, which implies that the Willmore energy of f is positive.
The τ -negative holomorphic sections s on surfaces of high genus have been constructed as follows (see [13, Theorem 4.5] for details): There is a q-fold covering Riemann surfaceΣ → Σ of the initial torus, branched over the four half-lattice points with branch order q−1. On Σ, there is a holomorphic family of connections with regular singularities at the half-lattice points and local monodromies in the conjugacy class of e 2πi/q 0 0 e −2πi/q .
The pull-back of this family of flat connections toΣ can be desingularized, and yields a lift of a τ -negative holomorphic section s onΣ. This gives rise to a branched equivariant Willmore surfacê f which is minimal in H 3 away from its intersection with the boundary at infinity [13, Section 5].
The counting of branch orders in [14,Theorem 3.3] also holds in the case of (equivariant) minimal surfacesf constructed by the τ -negative holomorphic sections s, as it only depends on the local analysis near the singular points, and branch orders are given by the vanishing order of the Higgs field. In particular, (for odd q), this yields that (with the notations of [14, Theorem 3.3]) p q = 2/q + 1 4 = 2 + q 4q , wherep = 2 + q andq = 4q are coprime since q is odd. Then g(Σ) = 2q − 1.
Hence, as the differential of the surface is a holomorphic section of where L is the kernel bundle of the Higgs field of s onΣ, we compute deg(L) = 1 2 (2 − 2g(Σ) + 4(q − 3)) = −4.
By Theorem 4.5 it remains to show that the Willmore energy off is bigger than 16π. This can be seen as follows: The family of regular singular connections on the torus Σ yields a equivariant Willmore surface f on the 4-punctured torus by the reconstruction method in [13, Section 5]. Putting q many Möbius-congruent pieces of f together in the conformal 3-space yields the (equivariant) Willmore surfacef . By construction f is close to f away from two branch cuts between the singular points on the torus Σ. It follows from [13, Section 5] that for every ǫ > 0 there exists δ > 0 such that for all q with 1 q < δ we have |W (f ) − W (f )| < ǫ. Take ǫ small such that 1 2 W(f ) > ǫ. As the Willmore energy of f is positive (independent of q) we obtain for q large enough. Remark 5.3. Alternative proofs of the theorem can be given by making use of the special coordinates introduced in [12].