Isothermic constrained Willmore tori in 3-space

We show that the homogeneous and the 2-lobe Delaunay tori in the 3-sphere provide the only isothermic constrained Willmore tori in 3-space with Willmore energy below $8\pi$. In particular, every constrained Willmore torus with Willmore energy below $8\pi$ and non-rectangular conformal class is non-degenerated.


Introduction
The Willmore functional of an immersions f : M → S 3 from a oriented surface M into the 3-sphere is given by where H is the mean curvature and dA is the induced area form of f . Geometrically speaking W measures the roundness of a surface, physically the degree of bending, and in biology W appears as a special instance of the Helfrich energy for cell membranes. The Willmore functional is invariant under Moebius transformations (conformal transformations of the 3-sphere with its standard conformal structure). Critical points of the Willmore functional are Willmore surfaces. Examples are given by minimal surfaces in the Riemannian subgeometries of constant curvature of the conformal 3-sphere.
If M is equipped with a Riemann surface structure, it is natural to consider only conformal immersions f : M → S 3 , i.e., the complex structure is given by rotating tangent vectors by π 2 in the 3-space. Critical points of the Willmore functional restricted to a given conformal class are called constrained Willmore surfaces. The conformal constraint augments the Euler-Lagrange equation by a holomorphic quadratic differential ω ∈ H 0 (K 2 M ) paired with the trace-free second fundamental formÅ of the immersion H + 2H(H 2 + 1 − K) = < ω,Å >, see [7,27]. The first examples of these constrained Willmore tori are given by those of constant mean curvature (CMC) in a 3-dimensional space form.
It is well-known (and obvious by the holomorphicity of the Hopf differential) that CMC (constant mean curvature) surfaces admit conformal curvature line parametrizations away from their umbilical points. Surfaces with this property are called isothermic. Isothermic surfaces play an important role in conformal surface geometry, see [10,11], since the notion is independent of the specific metric in the conformal class of the ambient manifold.  where g round is the round metric on S 3 , and [.] denotes the conformal class in the Teichmüller space. The map π is a submersion except at isothermic immersions, see [7]. Hence, the Lagrange multiplier for isothermic constrained Willmore surfaces -the holomorphic quadratic differential -is no longer uniquely determined by the immersion.
In this paper, we restrict to compact Riemann surfaces of genus 1. We classify isothermic constrained Willmore tori with Willmore energy below 8π. Our main theorem is the following one. Theorem 1. Isothermic constrained Willmore tori in the conformal 3-sphere with Willmore energy below 8π are CMC surfaces in the round 3-sphere.
Strategy of proof. Richter [26] shows that isothermic constrained Willmore tori in the conformal 3-sphere are locally of constant mean curvature in a 3-dimensional space form. The solution of the Lawson and Pinkall-Sterling conjectures by Brendle [8] and Andrews-Li [2] further gives that embedded CMC tori in the 3-sphere are rotationally symmetric and thus consist only of the families of k-lobed Delaunay tori [19]. Moreover, the Willmore energy along every embedded family is monotonically increasing in the conformal class b. Thus since for k ≥ 3 the k-lobes bifurcates from the homogenous tori with Willmore energy above 8π, the 2-lobed family is the CMC-family with minimal Willmore energy in their respective conformal classes. The aim is to exclude the existence of constrained Willmore surfaces of constant mean curvature in R 3 or hyperbolic 3-space H 3 that can be compactified to a torus in S 3 with Willmore energy below 8π. By Li and Yau [21] these surfaces must be embedded.
The Alexandrov maximum principle [1] shows that there are no closed CMC tori with Willmore energy below 8π in R 3 or H 3 . The only non-closed CMC surfaces in R 3 that can be compactified to conformal embeddings in S 3 are minimal surfaces with planar ends (H = 0 is excluded by local analysis [20]), which have quantized energy 4πk, with k ≥ 2 being the number of ends. Thus, those surfaces have Willmore energy ≥ 8π. Similar arguments work for constant mean curvature surfaces in H 3 with mean curvature H = 1 giving quantized Willmore energy W = 4πk, where k ∈ N denotes the number of ends see [6] and one-punctured CMC 1 torus in H 3 does not exist by [23].
To prove Theorem 1 it is thus sufficient to show that isothermic constrained Willmore tori in S 3 , whose intersection with H 3 ⊂ S 3 is of constant mean curvature, cannot have Willmore energy below 8π. Those surfaces intersect the infinity boundary of H 3 ⊂ S 3 -a round 2-sphere -with an angle α satisfying cos(α) = H. In particular, the constant mean curvature must satisfy |H| < 1 or the surface is entirely contained in H 3 , and therefore cannot be embedded by maximum principle. It hence remains to show that CMC surfaces in H 3 with mean curvature |H| < 1 and Willmore energy below 8π cannot be embedded, see Theorem 3.
We will call isothermic constrained Willmore tori into S 3 which are CMC in H 3 with |H| < 1 on the intersection with the two hyperbolic balls Babich-Bobenko tori in the following. The first examples have been constructed by Babich and Bobenko [3] in the case of H = 0. The main idea of the proof is now to use the quaternionic Plücker estimate [13], which links lower bounds of the Willmore energy to the dimension of holomorphic sections of a certain quaternionic holomorphic vector bundle. This dimension is then related to the (necessarily odd) genus g of the spectral curve for Babich-Bobenko tori.
The paper is organized as follows: In Section 2 we study the spectral curve of Babich-Bobenko tori in detail. In Section 3, we use the special structure of the spectral curve to apply the Plücker estimate which yields a proof of Theorem 3.
Acknowledgements. The first author is supported by the DFG within the SPP Geometry at Infinity, and the second author is supported by RTG 1670 Mathematics inspired by string theory and quantum field theory funded by the DFG. The second author would also like to thank the International Centre for Theoretical Sciences (ICTS), Bangalore, for hospitality during the ICTS program on Analytic and Algebraic Geometry, where parts of the computations have been performed.

The constrained Willmore spectral curves of Babich-Bobenko tori
We consider two different approaches to the spectral curve theory of Babich-Bobenko tori. The aim of this section is to show that these two approaches towards the spectral curve are in fact equivalent. The lightcone model one is used to show that the spectral curve of a Babich-Bebenko torus -the Riemann surface parametrizing the eigenlines of d λ q -is hyperelliptic, while the Plücker estimate uses the multiplier spectral curve, which by [4] corresponds to the spectral curve of ∇ µ from the quaternionic approach. Subtleties arise from the non-uniqueness of the Lagrange multipliers.
2.1. Quaternionic geometry. The spectral curve theory for conformal immersions f from a 2-torus T 2 into the conformal 4-sphere has been developed in [9], where S 4 is considered as the quaternionic projective space HP 1 . To every conformal immersion f the quaternionic line bundle L = f * T ⊂ T 2 × H 2 given by the pull-back of the tautological bundle T of HP 1 is associated. Another quaternionic line bundle associated to f is V /L, where V = T 2 × H 2 . On V /L there exists a natural quaternionic holomorphic structure D (see [9,4] for a detailed definition and discussion) by demanding the projections of the constant sections (0, 1) and (1, 0) of V to be holomorphic. The immersion f is then recovered (up to conformal transformations) by The (multiplier) spectral curve Σ of a conformally immersed torus f is the normalization of the Riemann surface parametrizing all holomorphic sections of V /L with (complex) monodromy, i.e, every point of the spectral curve corresponds to a holomorphic section with monodromy [5]. Therefore, we can define maps from Σ to C -so-called monodromy maps ν i -by assigning to every point in Σ the monodromy of the underlying holomorphic section along generators γ i of the fundamental group π 1 (T 2 ).
Bohle [4] gives an alternative approach to the spectral curve for constrained Willmore tori. For constrained Willmore surfaces f : M −→ S 3 ⊂ S 4 Bohle defined the following C * -family of flat SL(4, C)-connections Here d is the trivial connection on the trivial H 2 -bundle considered as a C 4 -bundle, where A is the Hopf field of the conformal immersion and q is the Lagrange-multiplier of the constrained Willmore Euler-Lagrange equation (which is not unique for isothermic surfaces). He showed that the flatness of an associated C * -family ∇ µ of SL(4, C)-connections defined on the trivial bundle V , considered as a C 4 -bundle, is equivalent to f being constrained Willmore. The (holonomy) spectral curve is then given by the Riemann surface parametrizing the eigenlines of the holonomy of ∇ µ . Bohle [4] showed that the (holonomy) spectral curve is always of finite genus and that both approaches to the spectral curve coincide. To be more precise, Bohle showed that ∇ µ -parallel sections with monodromy are the unique prolongations of the holomorphic sections with monodromy of the quaternionic holomorphic line bundle (V /L, D) to V. The genus g of the associated spectral curve is called the spectral genus of the immersion f .

Remark 1.
In the case of f mapping into the 3-sphere S 3 ⊂ S 4 , the spectral curve Σ admits an additional involution σ, see [16, Lemma 1]. Another involution ρ on σ arises from the quaternionic construction, i.e., by an appropriate multiplication by j. If the quotient Σ/σ is biholomorphic to CP 1 , there are two cases to distinguish depending on whether the real involution ρ • σ has fix points or not. In the first case the surface is of constant mean curvature in R 3 , S 3 or H 3 (with mean curvature |H| > 1). If ρ • σ has no fixed points then the corresponding immersion is of Babich-Bobenko type. We want to show the converse, i.e., that Σ/σ ∼ = CP 1 for Babich-Bobenko tori.
2.2. The light cone model. CMC surfaces in 3-dimensional space forms can also be described by associated families of flat SL(2, C)-connections ∇ λ , λ ∈ C * , on a rank 2 bundlẽ V → M [18,3]. In the case of tori, these families of flat connections can be described by (algebraic-geometric) spectral data consisting of a (compact) hyper-elliptic curveΣ (the spectral curve), two meromorphic differentials, and a holomorphic line bundle. In the case of Babich-Bobenko tori [3]Σ is the spectral curve of a finite gap solution of the Cosh-Gordon equation and admits a real involution covering λ → −λ −1 . ThereforeΣ hyperelliptic and of odd genus. In this alternate approach the light cone model as developed in [10,11] is used. Its relation to quaternionic holomorphic geometry can be found in [12, §5], details of the computations is also included in the thesis of Quintino [24] and in [25]. We only recall the main constructions here. The Plücker estimate cannot be applied to this approach directly, since∇ λ have singularities on M , corresponding to the intersection of the surface with the infinity boundary of H 3 , see [17].
As in [12] we start with C 4 equipped with a quaternionic structure, i.e., a complex antilinear map j : and det(e 1 , e 2 , je 1 , je 2 ) = 1 for {e 1 , e 2 , e 3 := je 1 , e 4 := je 2 } being the standard basis of C 4 . The quaternionic structure induces a real structure on Λ 2 C 4 (also denoted by j by abuse of notation) via v ∧ w → jv ∧ jw, and the determinant induces an inner product ., . on Λ 2 C 4 by
For a general (n + 2)-dimensional real vector space V with inner product of signature (n + 1, 1), the n-sphere can be naturally identified with projectivation PL of the light cone Moreover, PL is equipped with a natural conformal structure: For a lift l of π : L → PL the Riemannian metric g l is defined as The space of orientation preserving conformal transformations -the Moebius group -can be identified with For V being the real subspace of Λ 2 C 4 , a real non-zero lightlike vector of V is given by a complex 2-plane in C 4 (nullity) invariant under j (reality), i.e., it gives rise to a quaternionic line in (C 4 , j). This identifies the 4-sphere with the quaternionic projective line HP 1 , and relates the quaternionic holomorphic geometry to the lightcone model, see [12, §4].
Constant curvature subgeometries of the Moebius geometry (PL, SO(5, 1) + ) are specified by a choice v ∞ ∈ V \ {0}. Such a choice provides a natural lift l of PL onto the subset The corresponding group of orientation preserving isometries of the subgeometry is then given by To define the associated family of connections, we need the mean curvature sphere congruence S for the immersion f : M → S 4 . This is a map S from M into the space of oriented 2-spheres in S 4 , such that at every p ∈ M the corresponding 2-sphere S(p), touches the immersion at f (p) and has the same oriented tangent plane, and the same mean curvature.
This space is uniquely determined by its orthogonal complement, V N := V ⊥ S , which is a oriented real 2-plane with positive definite inner product, and therefore admits a unique compatible complex structure A conformal immersion f : M → PL is naturally equipped with the real rank 4 subbundle of the trivial rank 6 bundle V , with complexification locally given by for some local liftf : U ⊂ M → L (and where f z = ∂f ∂z etc.), see [10,11,12]. The bundle V S has induced signature (3, 1) and a natural orientation. Therefore V S gives rise to a sphere congruence, i.e., to a smooth map into the space of oriented 2-spheres in S 4 . It can be computed (see [10,11]) that the sphere congruence V S is the mean curvature sphere congruence, i.e., (V S ) p is the unique oriented 2-sphere in S 4 which touches (with orientation) the surface at f (p) to second order. Analogous to the classical case of surface geometry in euclidean 3-space R 3 , we consider the induced splitting of the trivial connection d with respect to Another related vector bundle Z is the bundle of skew-symmetric maps of (V, ., . ) which map Rf to Span{f ,f z ,fz} and vice versa vanishing on other components.
With these notations we list a few further important properties of the mean curvature sphere congruence: • (see [10]) f is isothermic if and only if there exists η ∈ Ω 1 (M, Z) with • (see [11]) the Willmore energy of f is given by [7,10,11] • (see [10] or [12, §3.3]) a surface f has parallel mean curvature vector H in the constant sectional curvature subgeometry In particular, the Lagrange multiplier q of a constrained Willmore surface f is unique if and only if f is non-isothermic, as for two Lagrange multipliers q 1 , q 2 the 1-form The following theorem reduces the constrained Willmore property of a given immersion f to the flatness of an associated family of flat connections in the language of the light cone model. 0) and (0, 1) are the complex linear and complex anti-linear parts of a 1-form.

2.3.
Compatibility of the quaternionic and the lightcone theory. The two approaches, the quaternionic and the lightcone one, towards the associated family of flat connections are in fact equivalent, as both associated families are gauge equivalent, when choosing suitable parameters. In order to provide a link between these families we need to relate the two different ways to obtain the mean curvature sphere congruence S. On the other hand, an oriented 2-sphere S ⊂ PL is determined by an oriented 4-dimensional vector space V S ⊂ V of signature (3, 1) via S = PV S ∩ PL. Moreover, V N := V ⊥ S is a oriented real 2-plane with positive definite inner product, and therefore admits a compatible complex structure We therefore obtain a decomposition ±iv} are complex null lines that are complex conjugated to each other, i.e., In particular, V ± N gives rise to complex planes W ± in C 4 satisfying Hence, there exists a uniqueS ∈ SL(4, C) with which is a 2-sphere in HP 1 in the quaternionic sense.
Conversely, every quaternionic 2-sphereS determines its ±i eigenspaces W ± S which are interchanged via j. They define complex null-lines V ± N satisfying jV ± N = V ∓ N , and therefore define a real oriented 2-plane of signature (2, 0). Its orthogonal complement in V is a real oriented vector space V S of signature (3, 1), hence a 2-sphere in PL. for η corresponding to q under the above identifications.
The following Proposition is proven in [24, § 9], and is used below to determine the structure of the spectral curves of the Babich-Bobenko tori: Proposition 2. The family of SL(4, C)-connections ∇ µ as in (2.1) is gauge equivalent to d λ η,S for µ = λ 2 .

2.4.
The structure of the spectral curve of a Babich-Bobenko torus. The aim of this section is to show the holonomy spectral curve of a Babich-Bobenko torus defined by the rank 4 family ∇ µ has the same properties as the Cosh-Gordon spectral curve by taking the Lagrange multiplier η corresponding to q ∞ as defined in [24, page 130]. Note that we consider the immersions maps into S 3 ⊂ S 4 ∼ = HP 1 (to make the relation to quaternionic holomorphic surface geometry transparent). We first study the structure of the spectral curve for the case of H = 0, which is equivalent to the vanishing of the Langrange multiplier η = q = 0. The case 0 < |H| < 1 is morally the same, though the details are slightly different, see Section 2.5. The application of the Plücker estimate in Section 3 below works totally analogous in both cases.
with µ = λ 2 through a λ-dependent family of complex gauge transformations, where Moreover, d + ω(−λ) and d + ω(λ −1 ) are gauge equivalent for all λ ∈ C * , and the monodromies of d + ω(−λ) along non-trivial elements of the first fundamental group have neither unimodular nor real eigenvalues for generic λ ∈ S 1 .
Proof. Because the Babich-Bobenko surface is minimal in the intersection with the hyperbolic space S ∞ , the parallel vector v ∞ is space-like and is contained in the mean curvature sphere bundle V for all p ∈ M. In particular, we have N (v ∞ ) = 0. Hence, by Proposition 1 and (2.7) together with q = 0 v ∞ is parallel with respect to Λ 2∇λ for all λ ∈ C * . Recall that the 3-sphere S 3 ⊂ S 4 is determined by a space-like vector v via Thus, v is contained in V N for all p ∈ M and hence v is also parallel with respect to Λ 2∇λ for all λ ∈ C * . Note also that v and v ∞ are perpendicular. There exists a conformal transformation of S 4 given by a real element of SO(Λ 2 C 4 , det) which transforms the (real and space-like) 2-vectors v and v ∞ (which are perpendicular to each other as they define perpendicular 3-spheres in the 4-sphere) as follows.
Hence, we can assume without loss of generality that v =ṽ and v ∞ =ṽ ∞ are parallel for all λ ∈ C * . For a connection d + A with A ∈ Ω 1 (M, sl(4, C)) the 2-vectors v, w ∈ Γ(M, Λ 2 C 4 ) are parallel if and only if it e 1 ∧ e 2 and e 3 ∧ e 4 are parallel. This is equivalent to A being of the form Hence, with q =η = 0 we see that∇ λ has the form where ω(λ) is as stated in (2.8). By [24, Lemma 9.14] and [4, Equation (2.11)],∇ λ is gauge equivalent to ∇ µ as defined in (2.1) (with µ = λ 2 ).
If the monodromies of d + ω(−λ) along non-trivial elements γ i of π 1 (M ) would have either unimodular or real eigenvalues for generic λ ∈ S 1 then [4, Proposition 3.2] shows that the eigenvalues of ∇ µ must be all equal to 1 for all µ ∈ C * and therefore this case can be excluded by [4, Theorem 5.1].
It remains to prove that d + ω(−λ) and d + ω(λ −1 ) are gauge equivalent for all λ ∈ C * . We make use of the fact that∇ λ and∇ −λ are gauge equivalent (as both are gauge equivalent to ∇ µ=λ 2 ), and want to determine the gauge as explicit as possible. i.e., S + p = span(e 1 − e 4 , e 3 − e 2 ) and S − p = span(e 1 + e 4 , e 2 + e 3 ). By [12, §3.2 ] or [24,Lemma 9.14] where H : Using the standard basis of C 4 , H p is given by Now, let (twice) the normal of f at the point q ∈ M be arbitrary, i.e. N q is in the real part of Λ 2 C 4 perpendicular toṽ andṽ ∞ and of length 2. There is a conformal transformation Ψ q of S 4 which fixes the 3-sphere and the sphere at infinity, and maps N p to w. It must be (considered as a SL(4, C)-matrix commuting with j) of the form where P q is a 2 by 2 matrix of unimodular determinant. Denote Then, Because the space of SL(4, C) matrices commuting with j and fixing v,ṽ ∞ and e 1 ∧ e 3 − e 2 ∧ e 4 is given by  where where a, b ∈ C, α ∈ S 1 with aā − bb = 1, P q is unique up to Note thatᾱ As P can be locally chosen to be smooth on M we find a well-defined global gauge transformation g : M → GL(2, C) with unimodular determinant which is locally given by g q =P −1 q KP q and satisfies (d + ω(λ)).g = d + ω(λ −1 ). Moreover, due to the quadratic factor α 2 , one can deduce that g can actually be chosen to be SL(2, C)-valued.
As an immediate corollary the spectral curve Σ has the same properties as a Cosh-Gordon spectral curve.
• Σ has a anti-holomorphic involution ρ covering λ → −λ −1 with ρ * ν i =ν i , for i = 1, 2 • µ : Σ −→ CP 1 is a four-fold covering, i.e., for generic µ ∈ C * the connection ∇ µ has 4 distinct eigenvalues along the generators γ k k = 1, 2 of π 1 (T 2 ) given by the four elements of the set Proof. By the previous proposition, the spectral curve Σ is given by the holonomy spectral curve of the family of flat connections∇ λ = d + ω(λ). Since∇ λ is SL(2, C), the hyperelliptic involution σ maps an eigenvalue of the monodromy to its inverse. The other involution ρ is induced by the quaternionic multiplication j, which covers λ → −λ −1 and (complex) conjugates the eigenvalues of the monodromy. Moreover, the parameter covering µ = λ 2 is unbranched over C * . Thus the quotient Σ/σ is biholomorphic to CP 1 .

2.5.
Non-minimal Babich-Bobenko tori. We show a modified version of Corollary 1 for Babich-Bobenko surfaces f : M −→ S 3 with mean curvature H = 0 (and |H| < 1) in the hyperbolic space H 3 ⊂ S 3 . Again we use the notations as introduced in Section 2.3 (or [24] for more details) and consider the C * -associated family of flat connections d λ η,S on the trivial C 4 -bundle for the Lagrange multiplier η given by where η ∞ is defined in [24,Theorem 8.16]. For further references see [11,12,25]. The connections d λ η,S induce the family of flat connections 1) , on the Λ 2 C 4 with Lagrange multiplier η. By [24, Lemma 9.14] and [4, Equation (2.11)] the connections d λ η,S and the constrained Willmore associated family of flat connections ∇ µ defined in (2.1) are gauge equivalent for µ = λ 2 .
The surface f is an isothermic constrained Willmore torus by assumption and admits a conserved quantity [24,Proposition 8.20]. Since we are considering surfaces in S 3 ⊂ S 4 , there is for every λ ∈ C * a complex 2-dimensional subspace of Λ 2 C 4 on which d λ η acts trivially, see (2.7). Applying a suitable SL(4, C)-transformation (depending on λ and p ∈ M ) we can assume without loss of generality that the invariant subspace is spanned by v = e 1 ∧ e 2 , e 3 ∧ e 4 .
Proof. LetΣ be the constrained Willmore spectral curve given as the parametrization of the (generically 4 distinct) eigenvalues of ∇ µ . Since ∇ µ is gauge equivalent to (2.9), it is the direct sum of two flat SL(2, C)-connections d + A(λ) and d + B(λ) for λ 2 = µ. Let h be an eigenvalue of the monodromy of ∇ µ . We assume without loss of generality that it is a eigenvalue of d + A(λ). Since d + A(λ) is a SL(2, C)-connection, h −1 is also a eigenvalue of d + A(λ). Thus we can define an involution (note that σ holomorphically extends to µ = 0, ∞ and therefore is well-defined on Σ). Since the decomposition into blocks is valid for all µ ∈ C * , the quotient Σ/σ is CP 1 . The remaining properties can be easily proved using [4, Proposition 3.1] together with the reality conditionĀ = B.
Remark 3. Note that for H = 0, the connections d λ η,S and the connections given by (2.9) are gauge equivalent by a λ-dependend gauge transformation. Thus, the map λ : Σ −→ Σ/σ is not necessarily branched over 0 and ∞ as in the H = 0 case.

Plücker estimates
We show that all isothermic constrained Willmore tori of Babich-Bobenko type have Willmore energy above 8π. The following Plücker estimate is relating the dimension of the holomorphic sections of V /L (without monodromy) to the Willmore energy of the corresponding immersion.  Proof. The spectral curve Σ is the surface parametrizing the eigenlines of ∇ µ -the constrained Willmore associated family of flat connections. It admits two involutions: σ and ρ. While the involution ρ corresponds to the quaternionic multiplication by j and is fixpoint free, the involution σ maps a holomorphic section ψ with monodromy h to a holomorphic section with monodromy h −1 . Therefore the branch points of Σ correspond to those ∇ µ -parallel sections ψ of V with Z 2 -monodromy, i.e., prolongations of holomorphic sections of V /L with Z 2 -monodromy. It is thus crucial to show that these ψ are non-constant sections of V , which clearly holds whenever the monodromy of the section ψ is non-trivial. Thus we restrict to the case where ψ has trivial monodromy.
It remains to show that also for the case µ 0 = λ 2 0 = 1 there exists a non-constant sectioñ ψ, given by a prolongation of a holomorphic section of V /L, with trivial monodromy. Let ∇ µ be the CW associated family, and µ 0 = 1 be a branch point of the spectral curve Σ. Because of the ρ-symmetry (interchanging the two points over µ 0 = 1) Σ is not totally branched at µ 0 , i.e., we can use a local coordinate ξ on Σ with ξ 2 = µ − 1. Assume that the Willmore energy is below 8π. By [5,Theorem 4.3 (iii)] there is a smooth family of ∇ µ(ξ) -parallel sections ψ ξ parametrized on an open subset of Σ around ξ = 0 depending smoothly on ξ, i.e., we have ∇ µ(ξ) ψ ξ = 0.
Differentiating this equation with respect to ξ (denoted by () ) at ξ = 0 (and therefore µ = 1) gives Differentiating once more and evaluating at ξ = 0 thus gives , dψ is contained in L as well showing that ψ is the prolongation of a (locally) holomorphic section of V /L. Since the monodromy takes the value 1 with at least second order (because the monodromy is trivial at the branch point µ = 1), ψ has also trivial monodromy. If ψ would be constant then 0 = (−2A 1,0 • + 2A 0,1 • )ψ, which yields that KerA • is constant giving a contradiction. Lemma 2. Two holomorphic sections of V /L with non-trivial Z 2 -monodromy corresponding to different branch points of λ : Σ → Σ/σ = CP 1 not lying over 0 or ∞, which are not interchanged by the involution ρ, are quaternionic linear independent.
Proof. Letψ 1 andψ 2 be two holomorphic sections of V /L with Z 2 -monodromy. If these sections have different Z 2 -monodromies, then they are clearly quaternionic linear independent. Thus let the ψ i have the same non-trivial Z 2 -monodromy in the following.
If µ 1 = µ 2 we obtain thatψ 1 andψ 2 are complex linear independent. Assume that they are not independent as quaternionic sections, then we would have w.l.o.g. ψ 1 = aψ 2 + bψ 2 j for some a, b ∈ C. Moreover, from (3.1) By type decomposition (see [13, Section 2.1]) we obtain A • ψ 2 = 0. Since ψ 2 is ∇ µ 2 -parallel this implies ψ 2 being constant, which is a contradiction by [4,Theorem 5.3] Lemma 3. Let f : T 2 −→ S 3 be a constrained Willmore torus of Babich-Bobenko type with spectral genus g ≥ 3. Then, either one of the branch points of the spectral curve over the unit disc D ⊂ C corresponds to the trivial monodromy or at least two of the branch points of the spectral curve on the punctured unit disc D * correspond to the same (non-trivial) Z 2 -monodromy.
Proof. For g > 3 we have at least 5 branch points over the punctured unit disc The claim follows from the fact that there exist only 3 different non trivial spin structures of the torus.
It remains to show the Lemma in the case of g = 3, where we have 4 branch points over the unit disc (that are not interchanged by ρ). Assume that none of the 4 branch points on the unit disc corresponds to the trivial spin structure and moreover, for µ = 0 the other 3 branch points correspond to different spin structures. Then the spin structure at µ = 0 must coincide with the one at P k for a k ∈ {1, 2, 3}, since there exist only 3 different non-trivial spin structures of a torus. Without loss of generality we can assume k = 1. We want to show that the spin structures corresponding to P 2 and P 3 must then coincide.
In this case, the closed non-trivial curve, the green curve in Figure 2, through the branch points P 2 and P 3 denoted by γ 23 , is homologous to the difference of the closed (red) curve γ S through the Sym-points S 1 and S 2 and the closed (blue) curve γ 01 connecting 0 and P 1 . Let θ i = d log ν i (i = 1, 2) be the logarithmic differentials of the monodromy maps ν i and consider integrals of θ i along these curves. Using the hyper-elliptic symmetry we want to show that 2 P 3 Since 0 and P 1 correspond to the same spin structure by assumption we first show that γ 01 θ k ∈ 4πiZ for k = 1, 2. As θ k has trivial residue, we can interpret the above integral as the integral of θ k along any curve homotopic to γ 01 which does not pass through 0 ∈ Σ.
In order to analyze the integral, we apply a renormalization: For k = 1, 2 there exist a closed 1-form η k on Σ \ {0} with support in a small neighborhood of 0 which satisfies σ * η k = −η k and such that θ k + η k extends smoothly through 0, compare with the limiting analysis of [18,Proposition 3.10]. Note that γ 01 η k = 0. Using an analogous computation as in [18,Proposition 3.10] again, we can associate to 0 ∈ Σ renormalized eigenvaluesν 1 andν 2 which take values in {±1} and encode the spin structure of the surface. The sign is encoded in the parity of the constant part of the expansion in [18,Proposition 3.10]. Then, we obtain where γ + 01 is given by a part of γ 01 which goes from 0 to P 1 , and the last equality follows from the fact that the values ν k (0) and ν k (P 1 ) coincide as 0 and P 1 correspond to the same spin structure.
Thus it remains to prove that the integral of θ i along the red curve γ S satisfies γ S θ k ∈ 4πiZ.
This follows from the ρ-symmetry of the spectral data: the integral of θ k along the red curve γ S is twice the integral along the curveγ which is defined to be the part of γ S from the point S 1 lying over the Sym point µ 1 to the point S 2 lying over −μ −1 1 , i.e., where the last equality uses the fact that the integral takes imaginary values. The welldefinedness of f then gives γ θ k ∈ 2πiZ proving the claim.
Theorem 3. The Willmore energy of a constrained Willmore torus f : T 2 −→ S 3 of Babich-Bobenko type is at least 8π.
Proof. Since the spectral curve Σ admits an involution ρ covering λ → −λ −1 , it must be of odd genus g, or λ is unbranched. For g ∈ {0, 1} the surface f is equivariant, see [15]. For g = 0 the surface must be homogenous. This case cannot appear, since it would be a surface entirely contained in hyperbolic 3-space. For g = 1 the surface is rotational symmetric and obtained by rotating a closed wavelike elastic curve in the hyperbolic plane H 2 around the infinity boundary of H 2 . The only periodic solution in this class is the family of elastic figure-8 curves in H 2 . These surfaces are non-embedded, see for example [14] or [28], and therefore they have Willmore energy above 8π by [21].
Let g ≥ 3. By Lemma 1 we can associate to every branch point of Σ a holomorphic section ψ with Z 2 -monodromy of V /L. There exist exactly 4 possible Z 2 -monodromies for ψ arising from the 4 different spin structures of T 2 . To be more concrete, the two monodromy maps ν i of ψ satisfies: (ν 1 (ψ), ν 2 (ψ)) ∈ {(1, 1), (1, −1), (−1, 1), (−1, −1)}. Every ψ with ±1 monodromy gives rise to a proper holomorphic section of V /L considered as a bundle over a suitable double coverT 2 of T 2 . Thus the theorem follows from the previous Lemma by applying the Plücker estimate (Theorem 2). If the trivial monodromy arises over λ = 0, the immersion f has trivial spin structure and by [22] the surface cannot be embedded and hence its Willmore energy is at least 8π.