Soliton Spheres

Soliton spheres are immersed 2-spheres in the conformal 4-sphere S^4=HP^1 that allow rational, conformal parametrizations f:CP^1->HP^1 obtained via twistor projection and dualization from rational curves in CP^{2n+1}. Soliton spheres can be characterized as the case of equality in the quaternionic Pluecker estimate. A special class of soliton spheres introduced by Taimanov are immersions into R^3 with rotationally symmetric Weierstrass potentials that are related to solitons of the mKdV-equation via the ZS-AKNS linear problem. We show that Willmore spheres and Bryant spheres with smooth ends are further examples of soliton spheres. The possible values of the Willmore energy for soliton spheres in the 3-sphere are proven to be W=4pi*d with d a positive integer but not 2,3,5, or 7. The same quantization was previously known individually for each of the three special classes of soliton spheres mentioned above.


Introduction
The study of explicitly parametrized surfaces is one of the oldest subjects in differential geometry. For more than two centuries the focus was essentially on local parametrizations. Towards the end of the 20th century, with the rise of the global theory of minimal surfaces [27,11,22,24] and the developments [1,30,21,2] initiated by Wente's solution [36] to the Hopf problem about constant mean curvature surfaces in R 3 , the focus shifted to rather global considerations. Nevertheless, our knowledge about explicit parametrizations of compact surfaces is still surprisingly rudimentary compared to, for example, the highly developed theory of complex algebraic curves.
An important source of explicitly parametrized spheres and tori is integrable systems theory in combination with complex algebraic geometry. The use of integrable systems methods in the study of general conformal immersions without special curvature properties has been pioneered by Konopelchenko [23] and Taimanov [32,33,34]. Their approach is based on the Weierstrass representation for conformal immersions into R 3 which provides an intimate connection between conformal immersions and Dirac operators, see the survey [35]. These ideas were of great influence in the Date: May 14, 2009May 14, . 2000 Mathematics Subject Classification. Primary: 53C42 Secondary: 53A30, 37K25. Both authors supported by DFG SPP 1154 "Global Differential Geometry". development of quaternionic holomorphic geometry [28,13] and the starting point of our investigations.
The present paper is devoted to the study of soliton spheres, a class of immersed 2-spheres in the conformal 4-sphere S 4 that admit explicit rational, conformal parametrizations f : CP 1 → HP 1 obtained by some twistorial construction from rational curves in CP 2n+1 and are characterized by the fact that equality holds in the quaternionic Plücker estimate [13] W ≥ 4π (n + 1)(n(1 − g) − d) + | ord H| , a fundamental estimate for the Willmore energy W of quaternionic holomorphic line bundles.
The term soliton spheres was introduced by Iskander Taimanov [34] for conformal immersions f : CP 1 → R 3 = Im H related to multi-solitons of the modified Korteweg-de Vries (mKdV) equation. With the more general notion of soliton spheres discussed in the present paper, all Willmore spheres and Bryant spheres with smooth ends are examples of soliton spheres. Section 2 gives an overview about the basic concepts of quaternionic holomorphic geometry. In particular, we explain the Möbius invariant representation of conformally immersed surfaces f : M → S 4 = HP 1 as quotients of quaternionic holomorphic sections. In Section 3 we discuss the relation between equality in the Plücker estimate and twistor holomorphic curves in HP n . In Section 4 we show how Taimanov's soliton spheres [34] can be treated using the quaternionic language. For this we review the quaternionic version [28] of the Weierstrass representation for conformal immersions into Euclidean 4-space R 4 = H.
In Section 5 we define soliton spheres using the Möbius invariant representation of conformal immersions. We derive an alternative characterization of soliton spheres in terms of the Weierstrass representation which shows that Taimanov's soliton spheres are soliton spheres in our sense.
In Sections 6 and 7 we use Darboux and Bäcklund transformations in order to show that Bryant spheres with smooth ends and Willmore spheres are examples of soliton spheres. In Section 8 we prove that all soliton spheres in R 3 with Willmore W ≤ 32π are Willmore spheres or Bryant spheres with smooth ends and show that the possible Willmore energies of immersed soliton spheres in R 3 are W = 4πd for d ∈ (N\{0, 2, 3, 5, 7}). This generalizes previously known results about the quantization of the Willmore energy for Willmore spheres in R 3 [9], Taimanov's soliton spheres [34], and Bryant spheres with smooth ends [5].

Quaternionic holomorphic geometry
The principal idea of quaternionic holomorphic geometry [28,13,10] is to approach conformal surface theory using the concept of quaternionic holomorphic line bundles. From this perspective the theory of conformal immersions appears as a "deformation" of the theory of holomorphic curves in CP n . This section gives a quick overview about the basic notions of quaternionic holomorphic geometry. Some special topics of surface theory in the conformal 4-sphere S 4 = HP 1 are discussed in the appendices.
2.1. Quaternionic vector spaces. The quaternions are the 4-dimensional real associative algebra H = R ⊕ Ri ⊕ Rj ⊕ Rk with multiplication rules i 2 = j 2 = k 2 = ijk = −1. A quaternion λ = a + bi + cj + dk is the sum of its real part Re(λ) = a and its imaginary part Im(λ) = bi + cj + dk. We writē λ = a − bi − cj − dk for the quaternionic conjugate. We identify R 4 with H and R 3 with the space of imaginary quaternions Im H = Ri ⊕ Rj ⊕ Rk.
All quaternionic vector spaces in this paper are right vector spaces. The dual V * of a quaternionic right vector space V , naturally a left vector space, is made into a right vector space by defining αλ := (x →λα(x)) for λ ∈ H and α ∈ V * .

2.2.
Quaternionic holomorphic line bundles. Let L be a quaternionic line bundle over a Riemann surface M , i.e., a vector bundle whose fibers are modeled on H. A complex structure on L, that is, a field J ∈ Γ(End L) of quaternionic linear endomorphisms with J 2 = − Id, makes L into a so called complex quaternionic line bundle. Denote by * the action on 1-forms of the complex structure of T M and by KL andKL the complex quaternionic line bundles whose sections are L-valued 1-forms ω that satisfy * ω = Jω or * ω = −Jω, respectively. A quaternionic holomorphic line bundle over a Riemann surface M is a complex quaternionic line bundle L together with a quaternionic holomorphic structure D, a differential operator D : Γ(L) → Γ(KL) satisfying the Leibniz rule D(ψλ) = (Dψ)λ + (ψdλ) for all ψ ∈ Γ(L) and λ : M → H, where ω := 1 2 (ω+J * ω). For the kernel of D one writes H 0 (L) and its elements are called holomorphic sections of L. The Leibniz rule implies that a nowhere vanishing holomorphic section uniquely determines a quaternionic holomorphic structure on a complex quaternionic line bundle.
The complex structure of a complex quaternionic line bundle L induces a decomposition A smooth map M → HP n is the same as a smooth line subbundle L of the trivial bundle H n+1 over M . In the following we will neither distinguish between L and the corresponding map M → HP n nor between the trivial vector bundle and the corresponding vector space. For example, a smooth map into HP n is usually denoted by L ⊂ H n+1 . The derivative of the map L is then given by δ = π∇ |L , where π : H n+1 → H n+1 /L is the canonical projection and ∇ denotes the trivial connection on H n+1 . A line subbundle L ⊂ H n+1 of the trivial H n+1bundle over a Riemann surface M is called a holomorphic curve if L admits a complex structure J ∈ Γ(End(L)), J 2 = − Id such that * δ = δJ.
A holomorphic curve HP n is immersed if δ is nowhere vanishing. Otherwise, a non-constant holomorphic curve is called branched, cf. Appendix A.1. 2.5. Holomorphic curves and linear systems. The so called Kodaira correspondence is a fundamental relation between holomorphic line bundles and holomorphic curves: let L ⊂ H n+1 be a holomorphic curve in HP n .
To avoid technicalities suppose that L is full, i.e., not contained in a linear subspace. The complex structure of L induces a complex structure on the dual bundle L −1 ∼ = (H n+1 ) * /L ⊥ which we again denote by J. Let π : (H n+1 ) * → (H n+1 ) * /L ⊥ be the canonical projection. Then L −1 carries a unique holomorphic structure D such that The isomorphism type of the holomorphic line bundle L −1 is a projective invariant of the holomorphic curve L ⊂ H n+1 in HP n , which we call the canonical holomorphic line bundle of the curve L. For an immersed holomorphic curve L ⊂ H 2 in HP 1 we therefore refer to L −1 as one of the Möbius invariant holomorphic line bundles of L, see also Section 5.1. Like in the complex case, the degree of a holomorphic curve L ⊂ H n+1 in HP n is defined as the degree of the corresponding quaternionic holomorphic line bundle L −1 . In other words, the degree of L ⊂ H n+1 seen as a holomorphic curve is minus the degree of L seen as a complex quaternionic line bundle.
The holomorphic structure on L −1 is the unique holomorphic structure with the property that all projections of constant sections of (H n+1 ) * are holomorphic. The linear system of holomorphic sections of L −1 obtained by projection from constant sections of (H n+1 ) * is called the canonical linear system of the curve L and denote it by (H n+1 ) * ⊂ H 0 (L −1 ). The canonical linear system of a holomorphic curve is always base point free, i.e., there are no simultaneous zeros of all holomorphic sections in (H n+1 ) * .
Let conversely H ⊂ H 0 (L) be a base point free linear system of a quaternionic holomorphic line bundleL. For p ∈ M denote L p ⊂ H * the 1dimensional subspace perpendicular to the hyperplane in H of sections vanishing at p. Then L is a holomorphic curve, the quaternionic holomorphic line bundleL is isomorphic to L −1 , and H corresponds to the canonical linear system of L, cf. [13,Section 2.6].
In case of holomorphic curves in HP 1 , Kodaira correspondence can be seen as a representation of conformal immersions as quotients of holomorphic sections: let L ⊂ H 2 be the holomorphic curve given by L = f 1 H, see Section 2.4, then the holomorphic sections e * 1 and e * 2 of L −1 obtained by projecting the standard basis of (H 2 ) * are related by Changing the basis of the canonical linear system amounts to a quaternionic fractional linear transformation of f , i.e., an orientation preserving Möbius transformation.
2.6. Weierstrass gaps and flag, dual curve. Let H ⊂ H 0 (L) be an (n + 1)-dimensional linear system of holomorphic sections of a holomorphic line bundle L. The Weierstrass gap sequence of H at a point p ∈ M is the sequence 0 ≤ n 0 (p) < n 1 (p) < ... < n n (p) of possible vanishing orders ord p (ψ) at p of sections ψ ∈ H. The Weierstrass points, i.e., the points at which the Weierstrass sequence differs from 0, 1, . . . , n, are isolated, see [13,Section 4.1]. The integer is the Weierstrass order of H at p and ord(H) is the Weierstrass divisor of H. If M is compact the Weierstrass degree | ord H| = p∈M ord p (H) is finite.
The members where tr R denotes real trace. The 2-form tr R (Q∧ * Q) is positive and W (L) = 0 is equivalent to Q ≡ 0. In other words, the Willmore energy W (L) of a quaternionic holomorphic line bundle L measures the deviation from the (double of the) underlying complex holomorphic line bundleL (Section 2.2). The Willmore energy of the Möbius invariant holomorphic line bundle L −1 of a holomorphic curve L ⊂ H 2 satisfies [10, Section 6.2] where H is the mean curvature, G the Gaussian curvature, and K ⊥ the curvature of the normal bundle of L with respect to any compatible metric on the conformal 4-sphere S 4 ∼ = HP 1 .

Equality in the Plücker estimate
In this section we describe the relation between equality in the Plücker estimate and twistor holomorphic curves.
gives a lower bound for the Willmore energy W (L) of a quaternionic holomorphic line bundle L of degree d over a compact Riemann surface of genus g with (n + 1)-dimensional linear system H ⊂ H 0 (L).

3.2.
Twistor holomorphic curves. The twistor projection is the map where vC denotes the complex line spanned by v ∈ H n+1 \{0} in the complex vector space (H n+1 , i) ∼ = C 2n+2 obtained by restricting the scalar field of the quaternionic right vector space H n+1 to C = R ⊕ Ri.
Definition. The twistor lift of a holomorphic curve L ⊂ H n+1 in HP n is the complex line subbundleL = { ψ ∈ L | Jψ = ψi } ⊂ (H n+1 , i). A holomorphic curve in HP n is called twistor holomorphic if it is the twistor projection of a complex holomorphic curve in CP 2n+1 . Proof. Let L ⊂ H n+1 be a holomorphic curve and E ⊂ (H n+1 , i) a complex holomorphic line subbundle that twistor projects to L. Holomorphicity of E means that every smooth section ψ ∈ Γ(E) satisfies * ∇ψ ≡ ∇ψi mod E. But then * δ = δJ, cf. Section 2.3, yields δψi = * δψ = δJψ such that Jψ = ψi for all ψ ∈ Γ(E). This implies that E is the twistor liftL of L.
3.4. Equality in the Plücker estimate. The link between equality in the Plücker estimate, Section 3.1, and twistor holomorphic curves is established by the quaternionic Plücker formula [13,Theorem 4.7]: let L be a quaternionic holomorphic line bundle of degree d over a compact Riemann surface of genus g and H ⊂ H 0 (L) an (n + 1)-dimensional linear system, then where L d ⊂ H denotes the dual curve of H, see Section 2.6. In particular, although the dual curve L d is only defined away form the Weierstrass points of H, the Willmore energy W ((L d ) −1 ) is finite. Equality in the Plücker estimate is thus equivalent to W ((L d ) −1 ) = 0 which, by Lemma 3.8 below, is equivalent to holomorphicity of the twistor lift of L d . As we will see in Lemma 3.10, the twistor lift of L d then extends continuously through the Weierstrass points. This yields the following theorem (cf. [13,Section 4.4]).
Theorem 3.5. A linear system of a quaternionic holomorphic line bundle over a compact Riemann surface has equality in the Plücker estimate if and only if its dual curve is twistor holomorphic. The twistor lift then extends holomorphically through the Weierstrass points of the linear system.
3.6. Example: degree formula. Applying the quaternionic Plücker formula to a 1-dimensional linear system yields the so called degree formula. The degree of a complex holomorphic line bundle over a compact Riemann surface equals the degree of the vanishing divisor of an arbitrary holomorphic section. In the quaternionic case the degree formula additionally involves the Willmore energy: let L be a quaternionic holomorphic line bundle over a compact Riemann surface and ϕ a holomorphic section of L. Denote by H the 1-dimensional linear system spanned by ϕ. The quaternionic holomorphic line bundle (L d ) −1 of the dual curve L d of H (Section 2.6) is isomorphic to L −1 restricted to M \ {zeros of ϕ} equipped with the holomorphic structure ∇ = 1 2 (∇ + J * ∇), where ∇ denotes the connection on L defined by ∇ϕ = 0. The Plücker formula applied to H thus becomes where | ord ϕ| is the zero divisor of ϕ.

3.7.
The canonical complex structure. The canonical complex structure [13] of an (n + 1)-dimensional linear system H of holomorphic sections of a quaternionic holomorphic line bundle L without Weierstrass points is the unique complex structure S ∈ Γ(End(H)), S 2 = −1 that respects the Weierstrass flag (Section 2.6), induces the given complex structure on L ∼ = H/H n−1 and satisfies H n−1 ⊂ ker(Q) and im(A) ⊂ H 0 = L d , where Q = 1 4 (S∇S − * ∇S) and A = 1 4 (S∇S + * ∇S) are the so called Hopf fields of S. It follows that S restricted to L d is the complex structure of the dual curve. Moreover, the restriction of Q to L ∼ = H/H n−1 coincides with the Hopf field of the holomorphic structure of L, cf. Section 2.5.
The last equivalence holds because −A * is the Hopf field of the canonical complex structure of the linear system (H n+1 ) * ⊂ H 0 (L −1 ) and hence induces the Hopf field of the quaternionic holomorphic line bundle L −1 .
3.9. Twistor lifts extend continuously through Weierstrass points. In order to complete the proof of Theorem 3.5 it remains to check that the twistor lift of the dual curve of a linear system extends continuously through the Weierstrass points. The proof of this fact given in [13] rests on the false claim that the canonical complex structure (Section 3.7) of a holomorphic curve in HP n extends continuously through the Weierstrass points. A counterexample is the holomorphic curve L = ψH in HP 1 defined by ψ = 1+jz z 2 whose canonical complex structure does not extend continuously into z = 0, because ψ(0) and ψ (0) are linearly dependent over H, cf. Lemma C.2.
We show now how to modify the arguments given in [13] in order to prove that the twistor lift of the dual curve extends continuously into the Weierstrass points. Proof. Let S be the canonical complex structure of H. We need to show that the twistor lift L d = { ψ ∈ L d | Sψ = ψi } of L d extends continuously through the Weierstrass points of H.
Let p be a Weierstrass point of H ⊂ H 0 (L). By Lemma 4.9 in [13] there exists a basis ψ k , k = 0, . . . , n of H that realizes the Weierstrass gap sequence n k (p) of H at p and has the following properties: Hence Sϕ = ϕ(i + O(1)) and ϕ(p)C ⊂ (H, i) extends the twistor lift L d of the dual curve continuously into the Weierstrass point p.
3.11. Two-dimensional linear systems with equality in the Plücker estimate. We have seen that the canonical complex structure of a linear system does not in general extend continuously through the Weierstrass points. However, in the special case of a base point free linear system with equality in the Plücker estimate it does extend smoothly into the Weierstrass points. We prove this only for 2-dimensional linear systems (which is what we need in Section 8.4). The extension to higher dimensional systems is straightforward, although slightly more involved.
Conversely, the extended mean curvature sphere congruence induces a complex structure on (L d ) ⊥ ⊂ (H 2 ) * which guarantees that (L d ) ⊥ is a holomorphic curve. Hence, L = H 2 /L d is, as explained in Section 2.5, a quaternionic holomorphic line bundle and the canonical linear system H 2 has equality in the Plücker estimate, by Theorem 3.5, because its dual curve L d is twistor holomorphic.

Example: Taimanov soliton spheres
The term soliton sphere was introduced by Iskander Taimanov [34] for immersions into R 3 with rotationally symmetric Weierstrass potential corresponding to mKdV-solitons. In this section we treat Taimanov's soliton spheres in the framework of quaternionic holomorphic geometry using Pedit and Pinkall's Weierstrass representation [28] for conformal immersions into R 4 = H.

Weierstrass representation for conformal immersions into R 4 .
Taimanov's approach [34] to soliton spheres is based on the generalization of the Weierstrass representation for minimal surfaces to arbitrary conformal immersion into R 3 . This representation describes the differential of the immersion as the "square" of a Dirac spinor. In contrast to the Kodaira correspondence (Section 2.5), the Weierstrass representation of conformal immersions is not Möbius invariant, but only invariant under similarity transformations.
The Weierstrass representation of a conformal immersion f : M → R 4 = H into Euclidean 4-space then reads as follows: where H denotes the mean curvature vector and dσ the area element of f .

Weierstrass representation for conformal immersions into R 3 .
The image of the differential df of a conformal immersion lies in Im where * denotes the complex structure on T * M and N : M → S 2 ⊂ Im H is the Gauss map of f . This implies that df = (α, ψ) is Im(H)-valued if and only if α → ψ induces a holomorphic bundle isomorphism KL −1 ∼ = L. A quaternionic holomorphic line bundle L that is isomorphic to KL −1 is called a quaternionic spin bundle [28].
The differential of the minimal immersion f is given by where ψ i ⊗ ψ j stands for the complex valued (1,0)-form (ψ i j, ψ j ). This is the well known spinor Weierstrass representation of minimal surfaces.

4.7.
Taimanov soliton spheres. Let L be a quaternionic spin bundle over M = CP 1 and z : CP 1 \ {∞} → C an affine chart. Then dz is a meromorphic section of the canonical bundle K with a second order pole at ∞ and no zeroes. The section ϕ above is thus a meromorphic section ofL without zeroes and with a first order pole at ∞. A function q : C → R is then a coordinate representation of a smooth Hopf field Q of L if and only if |w| −2 q(w −1 ) is smooth at w = 0. Similarly, µ 1 , µ 2 : C → C are the coordinate representation of a globally smooth section of L if and only if |µ Suppose that q is rotationally symmetric, i.e., q(z) = q(|z|). Taimanov then proves [34,Lemma 4] that there are N + 1 integers 0 ≤ n 0 < . . . < n N and a basis {ψ j } 0≤j≤N of H 0 (L) which is "rotationally symmetric" in the sense that where ν j : R → C 2 is a rapidly decaying solution of the ZS-AKNS linear problem The eigenvalues of L are preserved under the mKdV hierarchy and the trace formula for L implies Equality in this estimate holds if and only if U (x) = q(e x )e x is a reflectionless potential of L, i.e., a multi-soliton of the mKdV equation, and i 2 (2n j + 1) are all points in the spectrum of L with positive imaginary part, cf. [34]. On the other hand, equality in this estimate, is equivalent to equality in the Plücker estimate (Section 3.1) for the full linear system H 0 (L) of holomorphic sections of the spin bundle L, because (2n j + 1), since g = 0, deg(L) = g − 1 = −1, ord 0 (ψ j ) = ord ∞ (ψ) = n j , and 0 and ∞ are the only possible Weierstrass points.
Theorem 4.8 (Taimanov [34]). Let L be a holomorphic spin bundle L over CP 1 whose potential q is rotationally symmetric with respect to some affine coordinate z : CP 1 \ {∞} → C. Then equality in the Plücker estimate holds for H 0 (L) if and only if the Willmore energy satisfies where 0 ≤ n 0 < . . . < n N are the integers such that there exists a basis of H 0 (L) consisting of rotationally symmetric holomorphic sections {ψ j } 0≤j≤N with ord 0 (ψ j ) = ord ∞ (ψ j ) = n j .
We call a conformal immersion f : CP 1 → R 3 a Taimanov soliton sphere if its Euclidean holomorphic line bundle has a rotationally symmetric potential with equality in the Plücker estimate for the full linear system. A special example of Taimanov soliton spheres are Dirac spheres [31] which have the most symmetric reflectionless potentials q(z) = N +1 1+|z| 2 . They are soliton spheres with n j = j, j = 0,...,N such that dim H 0 (L) = N + 1 and W (L) = 4π(N + 1) 2 . Taimanov [34] gives explicit rational formulae for all q and ψ j that may arise in Theorem 4.8. For every (N + 1)-tuple 0 ≤ n 0 < . . . < n N of integers there is an R N +1 -parameter family of q's and corresponding ψ j 's. Since we are only interested in immersed spheres, we need to start with a base point free linear system such that we have to assume that n 0 = 0. Because the integers that may be written as the sum of 1 with other pairwise distinct odd integers are N \ {0, 2, 3, 5, 7} we obtain the following corollary.   [34] for the meaning of λ j ) are the catenoid cousins that have smooth ends, see Example 6.11 for images.     Figure 5 shows a deformation of the first into the third surface in Figure 1 through a family of surfaces (ψ, ψ) where ψ is a linear combination of ψ 0 and ψ 2 . The last surface in the figure is branched, all others are immersed.

Soliton spheres
We define soliton spheres in terms of the Möbius invariant holomorphic line bundles of a conformal immersion f : CP 1 → HP 1 . We also give a Euclidean characterization of soliton spheres based on the Weierstrass representation. This in particular implies that Taimanov's soliton spheres are examples of soliton spheres in our sense.

Definition.
A conformal immersion f : CP 1 → HP 1 into the conformal 4-sphere is called a soliton sphere if one of the two Möbius invariant holomorphic line bundles admits a linear system with equality in the Plücker estimate that contains the canonical linear system. Two fundamental properties of soliton spheres are immediate consequences of the definition: firstly, the notion of soliton spheres is Möbius invariant. Secondly, given a soliton sphere L ⊂ HP 1 , Theorem 3.5 implies that either L or L ⊥ is the projection to HP 1 of a holomorphic curve in HP n that is the dual curve (Section 2.6) of a twistor holomorphic curve in HP n (Section 3.2). In other words, every soliton sphere is obtained from a rational curve in CP 2n+1 via twistor projection CP 2n+1 → HP n , dualization and projection to HP 1 (possibly followed by a dualization in HP 1 ), i.e., by or the same sequence followed by a dualization in HP 1 , depending on whether H 0 (L −1 ) or H 0 (H 2 /L) has a linear system with equality. The dual curve of a holomorphic curve L in HP n is the solution of a system of quaternionic linear equations whose coefficients are the (n − 1) th derivatives of a generic section of L. Hence, every soliton sphere admits a rational, conformal parametrization.
Definition. The soliton number of a soliton sphere that is not the round sphere is defined as the minimal number n for which one of the canonical linear systems is contained in an (n + 1)-dimensional linear system with equality in the Plücker estimate. The soliton number of the round sphere is defined to be 0.
The soliton number n is the smallest number for which a soliton sphere can be obtained via the above construction from a rational curve in CP 2n+1 . The only 0-soliton sphere is the round sphere. A 1-soliton sphere is either twistor holomorphic or the dual of a twistor holomorphic curve, that is, all 1-soliton spheres are superconformal Willmore spheres (cf. Section 8.2 of [10]) and vice versa. Examples of 2-soliton spheres are Bryant spheres with smooth ends (Section 6) and non-superconformal Willmore spheres (Section 7).

Characterization in terms of the Weierstrass representation.
We give now a Euclidean characterization of soliton spheres in terms of the Weierstrass representation. As an application we show that Taimanov's soliton spheres are also soliton spheres as in Section 5.1.
Let L ⊂ H 2 be an immersed holomorphic curve and fix a point ∞ ∈ HP 1 that does not lie in the image of L. Without loss of generality we may assume that ∞ = [e 1 ], where e 1 , e 2 ∈ H 2 denotes the standard basis. This basis of H 2 defines the affine chart (Section 2.4) The affine chart is in fact defined uniquely up to similarity transformation by the choice of ∞ = [e 1 ] ∈ HP 1 . The immersed holomorphic curve L can then be written as Let e * 1 , e * 2 ∈ Γ(L −1 ) be the projections to the first and second coordinate of H 2 seen as sections of L −1 = (H 2 ) * /L ⊥ . The canonical linear system (Section 2.5) of L is then spanned by the holomorphic sections e * 1 and e * 2 whose quotient isf , i.e., e * 1 = e * 2f . Hence f : CP 1 → H is a soliton sphere if and only if f orf is the quotient of two quaternionic holomorphic sections that are contained in a linear system with equality in the Plücker estimate, because replacing f byf is equivalent to replacing L by L ⊥ and interchanging the Möbius invariant and the Euclidean holomorphic line bundles.
The choice of ∞ ∈ HP 1 defines a unique flat connection ∇ on L −1 which satisfies ∇e * 2 = 0 (where e * 2 is perpendicular to ∞). The induced connection ∇ on L then satisfies ∇ψ = 0. Moreover d ∇ induces a quaternionic holomorphic structure on KL −1 (in the same way as d defines the complex holomorphic structure on complex valued (1, 0)-forms) and ∇ = 1 2 (∇ + J * ∇) induces a quaternionic holomorphic structures on L with respect to which α = ∇e * 1 = e * 2 df and ψ are holomorphic sections of KL −1 and L, respectively. The holomorphic structures thus defined make KL −1 and L into paired holomorphic lines bundles and is the Weierstrass representation of f . Theorem 4.2 implies that the line bundles L and KL −1 with holomorphic structures d ∇ and ∇ are the Euclidean holomorphic line bundles of f .  Proof of Theorem 5.3. The holomorphic structure of the Möbius invariant holomorphic line bundle L −1 = (H 2 ) * /L ⊥ is given by D = ∇ , where as above ∇ is the connection defined by ∇e * 2 = 0. It is sufficient to show that L −1 admits a linear system that contains e * 1 and e * 2 and has equality in the Plücker estimate if and only if KL −1 admits a linear system with equality that contains α. This follows from Lemma 5.5 below, because ∇e * 2 = 0, ∇e * 1 = α, and CP 1 is simply connected. Lemma 5.5. Let L be a quaternionic holomorphic line bundle over a compact Riemann surface with a nowhere vanishing holomorphic section ϕ 0 . Let ∇ be the flat connection on L defined by ∇ϕ 0 = 0 and d ∇ the induced quaternionic holomorphic structure on KL.
Then ∇ induces a linear map from H 0 (L) to H 0 (KL) which maps every (n + 1)-dimensional linear system H ⊂ H 0 (L) containing ϕ 0 to an n-dimensional linear system ∇H ⊂ H 0 (KL). The linear system H has equality in the Plücker estimate if and only if ∇H has equality.
Plugging all these identities into the Plücker estimate (Section 3.1) shows that equality for H is equivalent to equality for ∇H.

Example: Bryant spheres with smooth ends
We characterize 2-soliton spheres using the Darboux transformation [4] for conformal immersions into S 4 . Because the hyperbolic Gauss map of a Bryant surface is a totally umbilic Darboux transform [19] we conclude that Bryant spheres with smooth ends [5] are soliton spheres.
6.1. 2-Soliton spheres and Darboux transformations. Let L ⊂ H 2 be an immersed holomorphic curve. The canonical projection π : The formula in Section 2.5 for the holomorphic structure of H 2 /L shows that Proof. The theorem is a direct consequence of the following proposition. The prolongation ψ of ϕh is then given by The Darboux transform L ⊂ H 2 corresponding to ϕh is defined away from the zeros of ψ as the line subbundle spanned by ψ . Its affine part f is defined away from the zeros of g and satisfies This proves the following lemma. Unless ϕh is contained in the canonical linear system, away from the isolated zeros of g, h, and dg, the affine part f of L is a conformal immersion. Proof. From df = h d(g −1 ) we obtain that the right normal vectors (cf. Appendix A. 3) of f and g −1 coincide On the other hand d( The Proof of Proposition 6.3 continued. We have to show that L = f 1 H is twistor holomorphic if and only if the linear system H spanned by ϕ, ϕf , and ϕh has equality in the Plücker estimate. Applying Lemma 5.5 to the nowhere vanishing holomorphic section ϕ shows that equality in the Plücker estimate for H is equivalent to equality in the Plücker estimate for the 2dimensional linear system ∇H of K(H 2 /L) spanned by ϕdf , ϕdh = −ϕdf g. Theorem 3.5 now implies that equality for ∇H is equivalent to g being twistor holomorphic, since ( g 1 )H ⊂ H 2 is the dual curve of ∇H. This proves the claim, because g is twistor holomorphic if and only if f is twistor holomorphic (Lemma 6.5).
Remark 6.6. Proposition 6.3 holds verbatim for compact Riemann surfaces of higher genus if one allows for linear systems with monodromy.
Remark 6.7. In general, a Darboux transform of a conformal immersion L ⊂ H 2 may not extend smoothly through the isolated zeros of the corresponding holomorphic section of H 2 /L. We show now that, in the situation of Proposition 6.3, the Darboux transform L extends smoothly through the zeros of the defining holomorphic section of H 2 /L and has a globally smooth twistor lift (which is hence a rational curve in CP 3 ).
Let L ⊂ H 2 be a conformal immersion and ϕh a holomorphic section of H 2 /L that, together with the canonical linear system, spans a 3-dimensional linear system H ⊂ H 0 (H 2 /L) with equality in the Plücker estimate. Then L = ψ H, ψ = f 1 g + ( 1 0 )h is defined and smooth away from the common zeros of g and h. Moreover, L is a holomorphic curve with complex structure J ψ = −ψ R g for R g the right normal of g, because ∇ψ = ψ dg.
By Theorem 3.5, the curve ( g 1 )H has a globally defined holomorphic twistor lift that locally is of the form g 1 +jg 2 g 3 +jg 4 C ⊂ (H 2 , i) with complex holomorphic functions g 1 , . . . , g 4 . Let p be a common zero of g and h.
Without loss of generality we may assume that g 3 + jg 4 does not vanish at p, because g has no "poles". Now R g = −(g 3 + jg 4 )i(g 3 + jg 4 ) −1 implies that the twistor lift of L is locally given as the complex line spanned by If n is the vanishing order of g at p, then h vanishes to order n + 1 at p, because df g = −dh. Since g 3 + jg 4 does not vanish, n is the vanishing order of g 1 + jg 2 . The twistor lift of L can be extended continuously through p, because the limit of ψ (g 3 + jg 4 )z −n at p exists and is not zero, where z is a local holomorphic chart centered at p. The claim now follows from Riemann's removable singularity theorem.
6.8. Bryant spheres with smooth ends are 2-soliton spheres. Bryant spheres with smooth ends [5] are surfaces of mean curvature one in hyperbolic space that compactify to immersed spheres by adding points on the ideal boundary of hyperbolic space. In [19] it is shown that Bryant surfaces are characterized by the existence of a totally umbilic Darboux transform which is then the hyperbolic Gauss map, see also [5,Theorem 9]. In order to apply Theorem 6.2 it remains to check that the holomorphic section defining this Darboux transform extends smoothly through smooth Bryant ends. Remark 6.9. It seems worthwhile to note that the characterization of Bryant surfaces [19,5] by the existence of a totally umbilic Darboux transform requires that both the surface and its Darboux transform take values in the same round 3-sphere in S 4 = HP 1 . This holds automatically for the "classical" Darboux transform in the isothermic surface sense as used in [19,5], but not for the Darboux transform of [4] as used in Section 6.1.
An immersion L ⊂ H 2 of CP 1 into HP 1 is a Bryant sphere with smooth ends if and only if, up to Möbius transformation, L = ψH, ψ = F k 1 , for a rational null immersion F into SL(2, C) such that all poles of dF F −1 have order 2, cf. [5]. Here null immersion means that det(dF ) = 0 and dF has no zeros. The kernels and images of dF F −1 then coincide and extend holomorphically through the poles of F . The holomorphic map      Figure 6 shows f :        7.1. Mean curvature sphere congruence. In case of immersed holomorphic curves in HP 1 , the canonical complex structure defined in Section 3.7 can be interpreted as the mean curvature sphere congruence.
The oriented totally umbilic 2-spheres in the conformal 4-sphere HP 1 are in one-to-one correspondence with the quaternionic linear complex structures on H 2 : let S ∈ End(H 2 ) such that S 2 = −1, then is a totally umbilic 2-sphere in HP 1 . The complex structures S and −S define the same 2-sphere, but different orientations on the tangent spaces T p S = H 2 /p, p ∈ S. One therefore calls a map S : M → End(H 2 ) with S 2 = −1 a sphere congruence. Its derivative may be decomposed which is called the mean curvature sphere congruence of L and coincides with the canonical complex structure defined in Section 3.7. The name mean curvature sphere congruence reflects the fact that the sphere S p at a point p ∈ M is the unique sphere that touches the curve at L p and has the same mean curvature vectors with respect to any compatible metric of the conformal 4-sphere S 4 , cf. [10, Section 5.2]. The Hopf fields A and Q measure the change of S along the curve. The integrals 2 A ∧ * A and 2 Q ∧ * Q measure the global change of S and coincide with the Willmore energies of the quaternionic holomorphic line bundles L −1 and (L ⊥ ) −1 = H 2 /L which Kodaira correspond, as in Section 2.5, to L and L ⊥ , respectively.
It can be shown, e.g. [10,Section 6], that an immersed holomorphic curve in HP 1 is Willmore, i.e., a critical point of the Willmore energy, if and only if its mean curvature sphere is harmonic. This is equivalent to Special examples of Willmore surfaces are twistor holomorphic curves which are characterized by A ≡ 0, see Lemma 3.8, and curves with Q ≡ 0 for which the dual curve L ⊥ is twistor holomorphic.

7.2.
Willmore spheres in the 4-sphere. Bryant's classification [7] of Willmore spheres in the conformal 3-sphere has the following extension to the conformal 4-sphere [12,26,25,10]: an immersed Willmore sphere L ⊂ H 2 in the conformal 4-sphere S 4 = HP 1 is either • twistor holomorphic, which is equivalent to A ≡ 0, • its dual L ⊥ is twistor holomorphic, which is equivalent to Q ≡ 0, • or it is Euclidean minimal, where we call a holomorphic curve L in HP 1 Euclidean minimal if it is minimal in the Euclidean space H = HP 1 \ {∞} for some point ∞ ∈ HP 1 . This is equivalent to the Möbius invariant condition that all mean curvature spheres of L intersect in one point. If a Euclidean minimal curve in HP 1 is immersed, the corresponding minimal immersion into H = HP 1 \{∞} has planar ends [7] at the points where the curve goes through ∞. Theorem 3.5 immediately implies that the first two cases are soliton spheres with equality in the Plücker estimate for the canonical linear system: if A ≡ 0, then L itself is twistor holomorphic and equality in the Plücker estimate holds for the canonical linear system of (L ⊥ ) −1 = H 2 /L. If Q ≡ 0, then L ⊥ is twistor holomorphic and equality holds for the canonical linear system of L −1 = H 2 /L ⊥ . It therefore remains to show that Euclidean minimal spheres are soliton spheres. Proof. As seen in Section 7.2 it suffices to show that every immersed Euclidean minimal sphere L ⊂ H 2 whose Hopf field A does not vanish identically is a soliton sphere. We fix a point ∞ = [e 1 ] ∈ HP 1 such that L does not go through ∞ and, using the notation of Section 5.2, write L = ψH with ψ = f 1 for f : Because CP 1 is simply connected, there is a globally defined 1-step forward Bäcklund transform g : CP 1 → H of f that satisfies dg = e * 2 (2 * Ae 1 ) (see Appendix B.2). It is non-constant, because f is not Euclidean minimal. By assumption there is a ∈ H such that (f − a) −1 is Euclidean minimal. Theorem B.4 iii) thus implies that g is twistor holomorphic. By Theorem 3.5 this yields that the linear system H = Span{ψ, ψg} ⊂ H 0 (L) of the Euclidean holomorphic structure on L defined by ∞ has equality in the Plücker estimate. Hence L is a soliton sphere by Theorem 5.3.
Using Proposition 3.12 and Corollaries B.5 and B.6, the proof of Theorem 7.4 gives rise to the following representation of Willmore spheres in the conformal 3-and 4-sphere in terms of twistor holomorphic curves. Appendix B.7 explains how this representation is related to the Weierstrass representation of minimal surfaces. Corollary 7.5. Let f : CP 1 → H be a conformally immersed sphere. Suppose that neither f norf is twistor holomorphic. Then f is Willmore if and only if there is a twistor holomorphic curve L : CP 1 → HP 1 with smoothly immersed mean curvature sphere congruence S such that f = e * 2 (Se 1 ) + c, for some c ∈ H, e 1 ∈ H 2 \{0}, and e * 2 ∈ (H 2 ) * \{0} such that e * 2 (e 1 ) = 0. The Willmore sphere f takes values in R 3 = Im H if and only if the twistor holomorphic curve L is hyperbolic superminimal with respect to the hyperbolic geometry defined by the Hermitian form ( x 1 x 2 ), ( y 1 y 2 ) =x 2 y 1 +x 1 y 2 (see Appendix C.9) and f = a, Sa + c for some c ∈ Im H and a ∈ H 2 \{0} with a, a = 0.
Remark 7.6. The Willmore energy of a Willmore sphere f obtained as in Corollary 7.5 from a twistor holomorphic curve L is Example 7.7 (Willmore spheres in S 3 with Willmore energy 16π). As an application of Corollary 7.5 we derive a formula for Willmore spheres in the conformal 3-sphere with Willmore energy 16π, the lowest critical value of the Willmore energy for spheres in S 3 above the minimum 4π. Remark 7.6 implies that d = 3 and b = 0 for Willmore spheres in S 3 with Willmore energy 16π. By Proposition C.11, the twistor projection L of the holomorphic curveL = [ϕ] : CP 1 → CP 3 given by ϕ := e 1 z + e 1 j 1 6 z 3 − e 2 + e 2 j 1 2 z 2 is hyperbolic minimal with respect to the Hermitian form in Corollary 7.5, because in the basisê 1 , . . . ,ê 6 of Appendix B.7 the curveL has the tangent line congruenceL 1 = [Ŝ] : which is polar to the space like vectorê 1 . Figure 20 shows the Willmore spheres f = a, Sa in R 3 obtained for a = e 2 + e 2 j (left) and a = −e 1 + e 2 (right). The left image in Figure 21 shows f = a, Sa with ϕ replaced by ϕ + e 2 j3z and a = e 2 + e 2 j; the right images is obtained for a = −e 1 + e 2 when ϕ is replaced by ϕ + e 2 j7z.

Willmore numbers of soliton spheres in 3-space
In [9] Bryant shows that the possible Willmore energies W = |H| 2 of Willmore spheres in R 3 are W = 4πd with d ∈ (N \ {0, 2, 3, 5, 7}). The same quantization holds for Bryant spheres with smooth ends [5] and for Taimanov soliton spheres, Corollary 4.9. In the present section we show that this quantization more generally holds for all immersed soliton spheres in 3-space. The main ingredient in the proof of this is Theorem 8.11 which says that all soliton spheres in 3-space with Willmore energy W ≤ 32π are Willmore spheres or Bryant spheres with smooth ends. 8.1. Equality in the Plücker estimate for spin bundles over CP 1 . In order to investigate soliton spheres in the conformal 3-sphere we apply the characterization in terms of Euclidean holomorphic line bundles given in Section 5.2. The advantage of the Euclidean point of view is that, if a conformal immersion L ⊂ H 2 into HP 1 takes values in a totally umbilic 3-sphere S 3 ⊂ HP 1 , the Euclidean holomorphic structure on L defined by a point ∞ ∈ S 3 not on L makes L into a quaternionic spin bundle, see  [5], and Willmore spheres in the conformal 3-sphere, see [9]. It is therefore sufficient to show that W ∈ 4π{2, 3, 5, 7} does not occur as Willmore energy of immersed soliton spheres in 3-space. Proof. If W (L) < 16π, then n = 0 and the linear system H ⊂ H 0 (L) above is 1-dimensional. Because it is base point free, it has no Weierstrass points. Thus W (L) = 4π which implies that the immersion is the round sphere [37].
This shows that the Willmore energies 8π and 12π do not occur so that it remains to check that 20π and 28π are impossible. 8.4. Soliton spheres in 3-space related to superminimal curves. If L is the spin bundle of a soliton sphere in 3-space with 16π ≤ W (L) ≤ 32π, the base point free linear system H with equality in the quaternionic Plücker estimate in Section 8.1 is 2-dimensional and "full" in the sense that H = H 0 (L). In particular we are in the situation described by Proposition 3.12, i.e., the dual curve L d of H ∼ = H 2 is twistor holomorphic, extends through the Weierstrass points of H, and has a mean curvature sphere congruence which is everywhere defined and smooth. In order to derive a condition on the dual curve L d which guaranties that L is spin, i.e., KL −1 ∼ = L, we make use of a bundle isomorphism induced by L d : let L d ⊂ H ∼ = H 2 be a twistor holomorphic curve with everywhere defined and smooth mean curvature sphere congruence S, cf. Proof. In order to check that the above holomorphic structure on H * /L is well defined we have to distinguish two cases: ifL is constant, the holomorphic structure is given by D = ∇ = 1 2 (∇ + * J∇) where J denotes the complex structure on H * /L induced by −S * and ∇ the trivial connection of H * /L. IfL is non-constant, then H * /L is the canonical holomorphic line bundle of the holomorphic curveL ⊥ as defined in Section 2.5, because 0 = d * Q * |L = − * Q * ∧δ impliesδ = −S * δ . Holomorphicity of the bundle homomorphism induced by * Q * also follows from d * Q = 0: by definition of the holomorphic structure on KL −1 , a section α ∈ Γ(KL −1 ) is holomorphic if and only if α(a) is closed for all a ∈ H. But * Q * y ∈ Γ(KL −1 ) is closed for every y ∈ H * and hence * Q * maps the projection to H * /L of every constant section of H * to a holomorphic section of KL −1 , cf. Section 4.1.
Assume now that L is a spin bundle, i.e., KL −1 ∼ = L. The composition of the bundle homomorphism 2 * Q * with the isomorphism KL −1 ∼ = L is then a holomorphic bundle homomorphism  To prove the quantization of the Willmore energy for soliton spheres in 3-space it therefore remains to exclude type b) curves of degree 4 and 6. A direct proof of this turns out to be very technical. Below such curves are excluded by a Möbius geometric argument, see the proof of Theorem 8.11.
Proof. From Proposition 3.12 we know that a base point free, 2-dimensional linear system H ⊂ H 0 (L) with equality in the Plücker estimate has a globally defined, twistor holomorphic dual curve L d ⊂ H whose mean curvature sphere congruence S extends smoothly through its branch points.
Assume that L is spin and B(H * ) ⊂ H. We distinguish the cases of non-constant and constant Bäcklund transformL ⊂ ker(Q * ). a) IfL is non-constant, then B maps the space H * of holomorphic sections of H * /L onto H ⊂ H 0 (L). Because by assumption H is base point free, the bundle homomorphism B is an isomorphism. Since ∇S = 2 * Q, Lemma 3.8, this implies that the mean curvature sphere congruence S of L d is immersed.
The bundle isomorphism B induces an isomorphismB : H * → H of vector spaces between the 2-dimensional linear systems H * and H. It mapsL ⊂ H * onto L d and thus maps the mean curvature sphere congruence −S * of L onto the mean curvature sphere congruence S of L, i.e., SB = −BS * . Differentiating the last equation one obtains thatB maps (L d ) ⊥ = im(Q * ) ontoL ⊥ = im(Q). This implies that the adjointB * ofB also induces a bundle isomorphism H * /L → L ∼ = H/L d . Because every automorphism of a quaternionic holomorphic line bundle with non-trivial Hopf field acts by multiplication with a real constant, we obtain thatB * = ±B. NowB * = −B is impossible: for every x ∈ H * \{0} the non-trivial 1-form * Q * x,Bx = * Q * x(Bx) had to be real-valued, because SB = −BS * implies QB = −BQ * and hence * Q * x,Bx = x,B * * Q * x = − x,B * Q * x = x, * QBx = * Q * x,Bx .
But this contradicts * * Q * x,Bx = − * Q * S * x,Bx = N * Q * x,Bx with N ∈ Im H defined by S * x = xN modL. HenceB * =B andB defines a non-degenerate Hermitian form on H with respect to which S is skew. Thus L is spherical or hyperbolic superminimal, by Lemma C.10. b) IfL is constant, then ∞ =L ⊥ ⊂ H lies on all mean curvature spheres of L d . Hence L d is Euclidean superminimal. The statement about the intersection divisor may be seen as follows: the vanishing divisor of a nontrivial holomorphic section x ∈ H ⊂ H 0 (L) with x ∈ ∞ is the intersection divisor of L d with ∞. Let y ∈ H * with y(x) = 1. Because y ∈L = ∞ ⊥ it induces a nowhere vanishing holomorphic section of H * /L. The vanishing divisor of By thus equals the branching divisor of S, since ∇S = 2 * Q.
We prove now that the holomorphic sections By and x differ by a real constant only and hence have the same vanishing divisor: because S is smooth and S∞ = ∞, there is a map N : CP 1 → S 2 with Sx = xN . On the other hand, since y(x) = 1 we have −S * y ≡ yN modL and S(By) = (By)N . Thus, away from the isolated zeros of By and x there exist real valued functions λ 1 , λ 2 with By = x(λ 1 + λ 2 N ). Since Q is non-trivial, the Leibniz rule in Section 2.2 implies that λ 1 is constant and λ 2 ≡ 0. Hence By = xλ 1 with λ 1 ∈ R such that By and x have the same vanishing divisor.
We show now that, conversely, if L d ⊂ H is a superminimal curve as in a) or b), then L = H/L d is a quaternionic spin bundle, i.e., KL −1 ∼ = L, which by construction satisfies B(H * ) ⊂ H. a) Let L d be spherical or hyperbolic superminimal. Then there exists a non-degenerate Hermitian form ·, · with respect to which S = −S * , see Lemma C.10, and ∇S = 2 * Q because L d is twistor holomorphic. Identifying H and H * via ·, · , this implies Q = −Q * and henceL = ker(Q * ) = ker(Q) = L d . Since im(Q) =L ⊥ = (L d ) ⊥ ∼ = (H/L d ) −1 = L −1 , the Hopf field * Q induces a quaternionic holomorphic bundle homomorphism L → KL −1 which is an isomorphism because by assumption S is an immersion. b) If L d is Euclidean superminimal, then ∞ =L ⊥ = im(Q) is a constant point. As above, let x ∈ ∞ and y ∈ (H 2 ) * such that y(x) = 1. Then * Q * y is a holomorphic section of KL −1 whose vanishing divisor by assumption coincides with the vanishing divisor of x seen as a holomorphic section of L. As before, there exists a quaternion valued map N defined away from the zeros of x and * Q * y, such that Jx = xN and J * Q * y = * Q * yN , where J denotes the complex structures of the respective quaternionic holomorphic line bundles L and KL −1 . This proves L ∼ = KL −1 , because the identification of two holomorphic sections with the same vanishing divisor and the same "normal vector" N gives rise to a holomorphic line bundle isomorphism. This follows from Riemann's removable singularity theorem, because holomorphic homomorphisms between two quaternionic holomorphic line bundles are in particular complex holomorphic sections of the line bundle of homomorphisms between the underlying complex holomorphic line bundles. 8.10. Classification of soliton spheres in 3-space with Willmore energy 16π ≤ W ≤ 32π. Let L ⊂ H 2 be a soliton sphere with 16π ≤ W ≤ 32π in the conformal 3-sphere S 3 ⊂ HP 1 . For every ∞ ∈ S 3 not on L, the induced quaternionic spin structure on L then admits a 2-dimensional linear system related to a superminimal curve as described in Proposition 8.8. Investigation of the corresponding 3-dimensional linear system of the Möbius invariant holomorphic line bundle H 2 /L shows that L is either a Willmore sphere or a Bryant sphere with smooth ends. Theorem 8.11. Let L ⊂ H 2 be a soliton sphere in the conformal 3-sphere with Willmore energy 16π ≤ W ≤ 32π. Then the full space of holomorphic sections H 0 (H 2 /L) of the Möbius invariant holomorphic line bundle H 2 /L is a 3-dimensional linear system with equality in the Plücker estimate that also contains a (unique) 1-dimensional linear system H ⊂ H 0 (H 2 /L) with equality in the Plücker estimate. Moreover, either • L is a Willmore sphere or • L is a Bryant sphere with smooth ends, depending on whether H is contained in the canonical linear system or not. In the Bryant case, the Darboux transform corresponding to H is the hyperbolic Gauss map of L. In the Willmore case, the Darboux transform is constant and coincides with the point ∞ for which L is Euclidean minimal in HP 1 \ {∞}.
Remark 8.12. The assumption 16π ≤ W ≤ 32π in Theorem 8.11 may be replaced by the weaker assumption B(H * ) ⊂ H of Proposition 8.8. In the proof of Theorem 8.11 we show that a soliton sphere L in R 3 = Im(H) whose spin bundle is of type a) in Proposition 8.8 is a Willmore sphere or a Bryant sphere with smooth ends (in the ball model of hyperbolic space). A soliton sphere in R 3 whose spin bundle is of type b) in Proposition 8.8 is a Bryant sphere with smooth ends (in the half space model of hyperbolic space).
For the proof of Theorem 8.11 we need to derive some properties of 1dimensional linear systems with equality in the Plücker estimate. If ϕ is an arbitrary quaternionic holomorphic section, then N with Jϕ = ϕN is continuous at the zeros of ϕ and smooth elsewhere, cf. the appendix to [3]. The following lemma together with Riemann's removable singularity theorem implies that N is everywhere smooth in case ϕ spans a 1-dimensional linear system with equality in the Plücker estimate. Proof. i) Equality in the Plücker estimate for the 1-dimensional linear system spanned by ϕ is equivalent to W (L −1 , ∇ ) = 1 2 dN ∧ * dN = 0, see Section 3.6. This is equivalent to dN ≡ 0, because dN ∧ * dN is a positive real valued 2-form. ii) follows from i), because * df = N df and dN = df H, see Appendix A.4. iii) Assume ϕ and ψ ∈ H 0 (L) span two different 1-dimensional linear systems with equality. Then f and f −1 defined by ψ = ϕf are both Euclidean minimal. This implies that f is planar and, consequently, the Willmore energy of L vanishes, cf. Section 7.3.
Proof of Theorem 8.11. Let L = ψH ⊂ H 2 , ψ = f 1 be a soliton sphere in the conformal 3-sphere S 3 = Im H ∪ {∞} ⊂ HP 1 with affine representation f : CP 1 → Im H. As explained in Section 8.1, the choice ∞ = ( 1 0 )H makes L into a quaternionic spin bundle for which ψ ∈ H 0 (L). Using the Plücker estimate and Theorem 5.3, the assumption 16π ≤ W (L) ≤ 32π implies that the full space of holomorphic sections H 0 (L) of L is 2-dimensional and has equality in the Plücker estimate. Since H = H 0 (L), we obtain in particular that the assumption B(H * ) ⊂ H of Proposition 8.8 is satisfied.
Denote by ϕ ∈ H 0 (H 2 /L) the projection of e 1 to the Möbius invariant holomorphic line bundle H 2 /L of L. The canonical linear system of L is then spanned by ϕ, ϕf , cf. Section 2.5. Let ∇ be the flat connection of H 2 /L such that ∇ϕ = 0. Then ϕdf is a holomorphic section of KH 2 /L when KH 2 /L is, as in Section 5.2, equipped with the holomorphic structure d ∇ . The derivative δ : L → KH 2 /L of L is then a complex quaternionic line bundle isomorphism, cf. Section 2.3, which is holomorphic because δ maps the holomorphic section ψ ∈ H 0 (L) to ϕdf ∈ H 0 (KH 2 /L). Using this isomorphism, Lemma 5.5 applied to ϕ implies that H 0 (H 2 /L) is 3-dimensional and has equality in the Plücker estimate, because CP 1 is simply connected. a) Suppose that the dual curve L d of H 0 (L) is of type a) in Proposition 8.8, i.e., the mean curvature sphere congruence S of L d is everywhere defined, immersed, and skew with respect to a non-degenerate Hermitian form ·, · . Since L d is twistor holomorphic one has ∇S = 2 * Q. In the proof of Proposition 8.8 it is shown that the isomorphism of L and KL −1 is provided by ∇S = 2 * Q, i.e., df = (ψ, ψ) = 2 ψ, * Qψ . Thus, without loss of generality f = ψ, Sψ . If ψ, ψ = 0, then f −1 is Euclidean minimal by Corollary B.6, and Lemma 8.13 implies that the 1-dimensional linear system spanned by ϕf has equality in the Plücker estimate.
If ψ, ψ = 0 we may assume that ψ, ψ = 1. Let e 1 , e 2 be a basis of H 0 (L) such that e 2 = ψ, e 1 , e 2 = 0, and L d = [ g 1 ] with nowhere vanishing g. From the formula for S in Appendix A. 4 we then obtain Since L d is twistor holomorphic, dR g = dR = H g dg and hence df = dH g g. By Lemma 6.4, this implies that f = f −H g g = −R g is a Darboux transform of f . This shows that f is a Bryant sphere with smooth ends: because f = −R g is totally umbilic and both f and f = −R g take values in Im(H), we obtain from [19,5] (see also Remark 6.9) that, away from the isolated points where f and f intersect, f is a Bryant surface. Since the immersion f is defined on all of CP 1 , the isolated intersection points are smooth Bryant ends.
Moreover, by Lemma 8.13 the 1-dimensional linear system of H 2 /L that is spanned by ϕH g has equality in the Plücker estimate, since H −1 g is Euclidean minimal by Corollary B.5.
b) Suppose now that the dual curve L d of H 0 (L) is of type b) in Proposition 8.8, i.e., there exists a line ∞ ⊂ H 0 (L) such that L d is Euclidean superminimal in P H 0 (L) \ {∞}. Let e 1 , e 2 ∈ H 0 (L) be a basis such that e 1 = ψ and e 2 ∈ ∞ and let L d = [ g 1 ] with respect to this basis. Then g −1 is Euclidean superminimal. The description of the spin pairing KL −1 ∼ = L in the proof of Proposition 8.8 implies (e 2 , e 2 ) = e * 2 (2 * Qe 2 ) = dH g g and df = (ψ, ψ) = (e 2 g −1 , e 2 g −1 ) =ḡ −1 dH g = −dH g g −1 , where the last equation holds because df is Im(H)-valued. Thus df g+dH g = 0 and Lemma 6.4 implies that f = f +H g g −1 is a Darboux transform of f . Moreover,H −1 g is Euclidean minimal by Corollary B.5 and twistor holomorphic, by iii) of Theorem B.4, becauseH g is a 1-step forward Bäcklund transform ofḡ −1 which is Euclidean minimal. Thus f is planar by Lemma 6.5, because both g −1 andH −1 g are Euclidean superminimal. By Lemma 8.13, the 1-dimensional linear system spanned by ϕH g has equality in the Plücker estimate.
To see that f is a Bryant sphere with smooth ends, by [19,5] it remains to proof that the planar Darboux transform f is Im(H)-valued (see Remark 6.9): by assumption e 2 H is a fixed line of S such that e * 2 (Se 2 ) is a quaternionic that squares to −1 and is hence imaginary. Because on the other hand e * 2 (Se 2 ) = H g g − R g we obtain that H g g takes values in Im(H). This implies f = f +H g g −1 is Im(H)-valued.
cf. [18]. We write b p (f ) = k for the branching order of f at p and b(f ) for the branching divisor of f . A map f is called a branched immersion if all points at which df fails to be injective are branch points. If M is a Riemann surface, a branched immersion is called conformal if it is conformal away from its branch points. The following proposition relates the branching divisor of a holomorphic curve in HP 1 to the Weierstrass divisor of its canonical linear system, cf. Sections 2.5 and 2.6. because L ⊥ = (e * 1 − e * 2f )H ⊂ (H 2 ) * . The maps N and R are called left normal and right normal of f , respectively. If f is a conformal immersion, both normal vectors exist and are smooth. One can prove (cf. appendix to [3]) that if f is a non-constant map for which one of the two normals exists and is smooth, the other normal vector can be globally defined as a continuous map which is smooth away from the branch points of f .
Recall from Section 2.5 that if L is a holomorphic curve then L −1 = (H 2 ) * /L ⊥ is a Möbius invariant holomorphic line bundle. The constant sections of (H 2 ) * project to the canonical linear system of holomorphic sections which is spanned by e * 1 = e * 2f and e * 2 . In particular, the complex structure J of L −1 satisfies Je * 2 = e * 2 R. Similarly, if L ⊥ is holomorphic, then H 2 /L = (L ⊥ ) −1 is a Möbius invariant holomorphic line bundle whose canonical linear system is spanned e 1 and e 2 = −e 1 f and whose complex structure J satisfies Je 1 = e 1 N .
A.4. Mean curvature sphere congruence. Assume now that L is immersed. Then both Möbius invariant quaternionic holomorphic line bundles are defined. The left and right normals N and R are both smooth and * df = N df = −df R. In [10,Section 7.1] it is shown that the mean curvature vector H of f is related to N and R via where H = −RH = −HN . In particular f is Euclidean minimal if and only if N and R are anti-holomorphic, i.e., dN = dR = 0. The mean curvature sphere congruence of L (Section 7.1) is given by One can check [10,Section 7] that w is closed if and only if A or, equivalently, Q is co-closed which in turn is equivalent to f being Willmore (see Section 7).
If f is a Willmore immersion, a smooth map g : M → H that satisfies dg = w = e * 2 (2 * Ae 1 ) is called a 1-step forward Bäcklund transform of f . Up to similarity transformation g is uniquely determined by the choice of ∞. If w does not vanish identically, then g is a branched conformal immersion. Similarly, a map h with dh = w − dH = e * 2 (2 * Qe 1 ) is called a 1-step backward Bäcklund transform. i) the 1-step forward Bäcklund transform g is a branched conformal Willmore immersion. Away from its branch points df = w g − dH g . In particular, f is a 1-step backward Bäcklund transform of g and f + H g is a 1-step forward Bäcklund transform of g. ii) the 2-step forward Bäcklund transform f of f coincides with the 1-step forward Bäcklund transform f = f + H g of g. ] is the 2-step forward Bäcklund transform of f which, by i) and ii), is equivalent to w g ≡ 0. Because w g ≡ 0, the assumption A g ≡ 0 would imply that g is Euclidean minimal, i.e., H g ≡ 0, which contradicts a = f = f + H g . Thus, for the 1-step Bäcklund transformation g we have that w g ≡ 0 is equivalent to A g ≡ 0.
Using the formula for S in Section A.4 we obtain: Corollary B.5. Let f : M → H be a nowhere vanishing conformal immersion of a simply connected Riemann surface. Its inversion f −1 is Euclidean minimal if and only if there is a branched twistor holomorphic immersion g with mean curvature sphere congruence S g such that f = −H g = e * 2 (S g e 1 ) away from the branch points of g.
In other words, f −1 is Euclidean minimal if and only iff = −N g H g = −H g R g , i.e.,f is the rotation of the mean curvature vector H g of a twistor holomorphic curve g by minus π/2 in the normal bundle of g. Note that locally every branched twistor holomorphic immersion g into H that is neither totally umbilic nor Euclidean minimal is a 1-step forward Bäcklund transformation of f as in the preceding corollary.
By Richter's theorem [10, Theorem 9], a Willmore surface f : M → H that is not Euclidean minimal takes values in R 3 = Im H if and only if it has a 1-step forward Bäcklund transform g that is minimal with respect to the hyperbolic geometry defined by the Hermitian form ( x 1 x 2 ), ( y 1 y 2 ) =x 2 y 1 +x 1 y 2 . (By Lemma C.10 the latter is equivalent to S g being skew with respect to ·, · . The hyperbolic minimal Bäcklund transforms are then characterized by the property that Re(g) = 1 2 H.) We obtain: Corollary B.6. Let f : M → R 3 = Im H be a nowhere vanishing conformal immersion of a simply connected Riemann surface. Its inversion f −1 is Euclidean minimal if and only if there is a branched twistor holomorphic immersion g with mean curvature sphere congruence S g that is hyperbolic minimal with respect to ·, · and a null vector e 1 ∈ H 2 such that f = −H g = e 1 , S g e 1 away from the branch points of g.
In other words, f −1 is Euclidean minimal if and only if f = N g H g = H g R g , i.e., f is the rotation of the mean curvature vector H g of a twistor holomorphic curve g by π/2 in the normal bundle of g. B.7. Weierstrass representation and 1-step Bäcklund transformation of twistor holomorphic curves. The formula f = −H g = e * 2 (Se 1 ) in Corollary B.5 can be seen as an integral free version of the Weierstrass representation of the minimal surface f −1 : we show that f −1 = (e * 2 (Se 1 )) −1 is the imaginary part of a holomorphic null curve in C 4 obtained from g.
Right multiplication by i makes H 2 into a complex 4-dimensional vector space C 4 ∼ = (H 2 , i). Let [ g 1 ] ⊂ H 2 be a twistor holomorphic curve as in Corollary B.5 (i.e., g is neither Euclidean minimal nor totally umbilic). Away from the branch points of [ g 1 ], the i-eigenspaces of its mean curvature sphere congruence S coincide with the tangent line congruence (or first osculating curve) of its twistor lift, cf. Appendix C. Because by assumption H g is an immersion, the vector e 1 is only at isolated points an eigenvector of S. The tangent line congruence of the twistor holomorphic curve [ g 1 ] is, away from these isolated points, the holomorphic null immersion where e 1 , e 2 is the standard basis of H 2 and where null means that the tangent lines of [Ŝ] are contained in Q 4 . Consider the real structure on Λ 2 (H 2 , i) defined by x ∧ y → xj ∧ yj. A real basis of Λ 2 (H 2 , i) is then given bŷ e 1 = e 1 ∧ e 2 j − e 1 j ∧ e 2 ,ê 2 = (e 1 ∧ e 2 j + e 1 j ∧ e 2 )i, e 3 = e 1 ∧ e 2 + e 1 j ∧ e 2 j,ê 4 = (e 1 ∧ e 2 − e 1 j ∧ e 2 j)i, e 5 = e 1 ∧ e 1 j,ê 6 = e 2 ∧ e 2 j.
Appendix C. Twistor holomorphic curves in HP 1 This appendix relates properties of the mean curvature sphere congruence S of a twistor holomorphic curve to properties of the osculating curves of its twistor lift.
A twistor holomorphic curve in HP 1 is a holomorphic curve whose Hopf field A vanishes identically (Lemma 3.8) and which hence is in particular Willmore (cf. Section 7.1). Equivalently, a holomorphic curve in HP 1 is twistor holomorphic if and only if the derivative ∇S of its mean curvature sphere congruence S satisfies ∇S = 2 * Q.
C.1. Mean curvature sphere congruence. The k-th osculating curvê L k of a holomorphic curveL ⊂ C n+1 is the analytic continuation of the rank k + 1 subbundle spanned by the k-th derivatives of sections ofL, cf. [17, p. 262]. The first osculating curve or tangent line congruenceL 1 of the twistor liftL = { ψ ∈ L | Jψ = ψi } ⊂ (H 2 , i) of a twistor holomorphic curve L in HP 1 is the i-eigenspace of the mean curvature sphere congruence S of L, because S∇ϕ = (∇S)ϕi+∇ϕi = ∇ϕi for all ϕ ∈ Γ(L). Although S is not defined at branch points of L, the tangent line congruenceL 1 is, because all osculating curves of complex holomorphic curves extend through their Weierstrass points. We obtain the following characterization of the branch points of L through which the mean curvature sphere congruence extends smoothly.
Lemma C.2. The mean curvature sphere congruence of a twistor holomorphic curve L extends smoothly through a branch point p of L if and only if at p the tangent line congruenceL 1 of its twistor lift is not tangent to the fiber of the twistor projection, i.e., if H 2 =L 1| p ⊕L 1 j |p .
C.3. 2-step Bäcklund transformation. The complex dual curve of the twistor lift of a twistor holomorphic curve L in HP 1 can again be projected to HP 1 . We prove now that the curve thus obtained is the dual curve of the 2-step backward Bäcklund transform L of L (Appendix B.1).
Lemma C.4. Let L ⊂ H 2 be a non totally umbilic twistor holomorphic curve in HP 1 with twistor liftL. Its 2-step backward Bäcklund transform L = im(Q) extends smoothly through zeros of Q and branch points of L (where the mean curvature sphere S of L and hence Q is not defined) and the dual L ⊥ ⊂ (H 2 ) * is a twistor holomorphic curve with mean curvature sphere congruence −S * whose twistor lift is the complex dual curveL d ofL.
As in [10,20] we use quaternionic Hermitian forms in order to describe the space form subgeometries of 4-dimensional Möbius geometry in the quaternionic projective framework: let ·, · be a non-trivial quaternionic Hermitian form on H 2 and denote I = { [v] ∈ HP 1 | v, v = 0 } its set of null lines. Depending on whether ·, · is definite, indefinite, or degenerate, the set I is empty, a round 3-sphere in HP 1 , or a point.
If ·, · is non-degenerate one can define the Riemannian metric on HP 1 \ I, where x ∈ H 2 and v, w ∈ T [x] HP 1 = Hom([x], H 2 /[x]). The Riemannian metric is compatible with the conformal structure on HP 1 and the Riemannian manifold (HP 1 /I, ±g) is isometric to either S 4 or two copies of hyperbolic 4-space depending on the signature of ·, · .
Lemma C.10. A holomorphic curve in HP 1 is minimal with respect to the space form geometry defined by a Hermitian form ·, · if and only if its mean curvature sphere congruence is skew Hermitian with respect to ·, · . Proof. A surface in a space form is minimal if and only if all its mean curvature spheres are totally geodesic. The 2-sphere described by the eigenlines of an endomorphism S of H 2 that squares to −1 is totally geodesic if and only if the corresponding Möbius transformation is an isometry of the space form defined by ·, · which is equivalent to S being skew Hermitian.
We call a twistor holomorphic curve in HP 1 all of whose mean curvature spheres are skew with respect to some Hermitian form on H 2 spherical, hyperbolic, or Euclidean superminimal depending on the type of the Hermitian form. Note that compact superminimal curves in HP 1 exist in all three cases, although in the hyperbolic and Euclidean case they go through infinity I.
Because Ω is a real linear form on Λ 2 (H 2 , i) it can be realized as the scalar product with a real element Ω ∈ Re(Λ 2 (H 2 , i)). One can check that Ω is time like, space like, or light like depending on whether the Hermitian form is definite, indefinite, or degenerate. Moreover, it can be proven that a twistor holomorphic curve L in HP 1 is superminimal with respect to the Hermitian form ·, · on H 2 if and only if the tangent line congruenceL 1 of its twistor liftL is polar to Ω . Proposition C.11. A holomorphic curve L ⊂ H 2 is spherical, hyperbolic, or Euclidean superminimal if and only if its twistor liftL ⊂ (H 2 , i) is holomorphic and its tangent line congruenceL 1 ⊂ Λ 2 (H 2 , i) is polar to a time like, space like, or light like vector in Re(Λ 2 (H 2 , i)).
Remark C.12. The 2-form Ω corresponding to Ω ∈ Re(Λ 2 (H 2 , i)) with Ω ∧ Ω = 0 induces an isomorphism between the complex vector space (H 2 , i) and its dual. The tangent line congruenceL 1 ofL is polar to Ω if and only if this isomorphism mapsL to its complex dualL d : take a local holomorphic section ψ ofL. ThenL 1 is polar to Ω if and only if Ω(ψ, ψ ) = 0. But this implies Ω(ψ, ψ ) = 0 and is hence equivalent to ker(Ω(ψ, .)) = span C {ψ, ψ , ψ }, i.e., toL being self-dual in the sense that L ∼ =L d with respect to the isomorphism induced by Ω. This shows that the twistor lift of a spherical or hyperbolic superminimal curve in HP 1 is self dual.