Simple factor dressing and the Lopez-Ros deformation of minimal surfaces in Euclidean 3-space

The aim of this paper is to give a new link between integrable systems and minimal surface theory. The dressing operation uses the associated family of flat connections of a harmonic map to construct new harmonic maps. Since a minimal surface in 3-space is a Willmore surface, its conformal Gauss map is harmonic and a dressing on the conformal Gauss map can be defined. We study the induced transformation on minimal surfaces in the simplest case, the simple factor dressing, and show that the well-known Lopez-Ros deformation of minimal surfaces is a special case of this transformation. We express the simple factor dressing and the Lopez-Ros deformation explicitly in terms of the minimal surface and its conjugate surface. In particular, we can control periods and end behaviour of the simple factor dressing. This allows to construct new examples of doubly-periodic minimal surfaces arising as simple factor dressings of Scherk's first surface.


Introduction
Minimal surfaces, that is, surfaces with vanishing mean curvature, first implicitly appeared as solutions to the Euler-Lagrange equation of the area functional in [Lag60] by Lagrange. The classical theory flourished through contributions of leading mathematicians including, amongst others, Catalan, Bonnet, Serre, Riemann, Weierstrass, Enneper, Schwarz and Plateau. By now, the class of minimal surfaces belongs to the best investigated and understood classes in surface theory. One of the reasons for the success of its theory is the link to Complex Analysis: since a minimal conformal immersion f : M → R 3 from a Riemann surface M into 3-space is a harmonic map, minimal surfaces are exactly the real parts holomorphic curves Φ : M → C 3 into complex 3-space. Due to the conformality of f , the holomorphic map Φ is a null curve with respect to the standard symmetric bilinear form on C 3 . A particularly important aspect of this approach is that the Enneper-Weierstrass representation formula, [Enn64], [Wei66], allows to construct all holomorphic null curves, and thus all minimal surfaces, from the Weierstrass data (g, ω) where g is a meromorphic function and ω a holomorphic 1-form. For details on the use of the holomorphic null curve and the associated Enneper-Weierstrass representation as well as historical background we refer the reader to standard works on minimal surfaces, such as [Nit89], [HK97], [LM99], [PR02], [DHS10], [MP12].
For the purposes of this paper, it is however useful to point out two obvious ways to construct new minimal surfaces from a given minimal surface f : M → R 3 and its holomorphic null curve Φ: firstly, multiplying Φ by e −iθ , θ ∈ R, one obtains the associated family of minimal surfaces f cos θ,sin θ = Re (e −iθ Φ) as the real parts of the holomorphic null curves e −iθ Φ. The associated family of minimal surfaces was introduced by Bonnet, [Bon53], in the study of surfaces parametrised by a curvilinear coordinate. An interesting feature of the associated family is that it is an isometric deformation of minimal surfaces which preserves the Gauss map. The converse was shown by  1 Schwarz, [Sch90]: if two simply-connected minimal surfaces are isometric, then, by a suitable rigid motion, they belong to the same associated family.
The second transformation, the so-called Goursat transformation [Gou87], is given by any orthogonal matrix A ∈ O(3, C): since A preserves the standard symmetric bilinear form on C 3 , the holomorphic map AΦ is a null curve, and Re (AΦ) is a minimal surface in R 3 . As pointed out by Pérez and Ros, [PR02], an interesting special case is known as the López-Ros deformation.
To show that any complete, embedded genus zero minimal surface with finite total curvature is a catenoid or a plane, López and Ros [LR91] used a deformation of the Weierstrass data which preserves completeness and finite total curvature. This López-Ros deformation has been later used in various aspects of minimal surface theory, e.g., in the study of properness of complete embedded minimal surfaces, [MPR04], the discussion of symmetries of embedded genus k-helicoids, [BB11], and in an approach to the Calabi-Yau problem, [FMUY14].
On the other hand, by the Ruh-Vilms theorem the Gauss map of a minimal surface is a harmonic map N : M → S 2 from a Riemann surface M into the 2-sphere, [RV70]. Harmonic maps from Riemann surfaces into compact Lie groups and symmetric spaces, or more generally, between Riemannian manifolds, have been extensively studied in the past. Harmonic maps are critical points of the energy functional and include a wide range of examples such as geodesics, minimal surfaces, Gauss maps of surfaces with constant mean curvature and classical solutions to nonlinear sigma models in the physics of elementary particles. Surveys on the remarkable progress in this topic may be found in [EL78], [EL88], [Gue97], [HW08], [Ohn10].
One of the big breakthroughs in the theory of harmonic maps was the observation from theoretical physicists that a harmonic map equation is an integrable system, [Poh76], [MZ78], [SZ79]: The harmonicity condition of a map from a Riemann surface into a suitable space can be expressed as a Maurer-Cartan equation. This equation allows to introduce the spectral parameter to obtain the associated family of connections. The condition for the map to be harmonic is then expressed by the condition that every connection in the family is a flat connection. This way, the harmonic map equation can be formulated as a Lax equation with parameter. Starting with the work of Uhlenbeck [Uhl89] integrable systems methods have been highly successful in the geometric study of harmonic maps from Riemann surfaces into suitable spaces, e.g., [Hit90], [Uhl92], [BFPP93], [BP95], [DPW98], [TU00]. In particular, the theory can be used to describe the moduli spaces of surface classes which are given in terms of a harmonicity condition, such as constant mean curvature surfaces, e.g., [PS89], [Bob91], isothermic surfaces, e.g., [CGS95], [BHJPP97], [Bur06], Hamiltonian Stationary Lagrangians, e.g., [HR02], [LR10], and Willmore surfaces, e.g., [Hél98], [Sch02], [Boh10], [BQ14].
We recall the methods of integrable systems which are relevant for our paper: given a C * -family of flat connections d λ of the appropriate form, one can construct a harmonic map from it. In particular, the associated family d λ of flat connections of a harmonic map gives an element of the associated family of harmonic maps by, up to a gauge by a d µ -parallel endomorphism, using the family d µλ for some fixed µ ∈ C * . The dressing operation was introduced by Uhlenbeck and Terng, [Uhl89], [TU00]: as pointed out to us by Burstall, in the case of a harmonic map N : M → S 2 the dressing is given by a gauged λ = r λ ·d λ of d λ by a λ-dependent dressing matrix r λ , [BDLQ13]. The dressing of N is then the harmonic mapN that hasd λ as its associated family of flat connections. In general, it is hard to find explicit dressing matrices and compute the resulting harmonic map. However, if r λ has a simple pole µ ∈ C * and is given by a d µ -parallel bundle, then the so-called simple factor dressing can be computed explicitly, e.g., [TU00], [DK05], [BDLQ13].
Parallel bundles of the associated family of flat connections also play an important role in Hitchin's classification of harmonic tori in terms of spectral data, [Hit90], and in applications of his methods to constant mean curvature and Willmore tori, e.g., [PS89], [Sch02]. The holonomy representation of the family d λ with respect to a chosen base point on the torus is abelian and hence has simultaneous eigenlines. From the corresponding eigenvalues one can define the spectral curve Σ, a hyperelleptic curve over CP 1 (which is independent of the chosen base point), together with a holomorphic line bundle over Σ, given by the eigenlines of the holonomy (these depend on the base point, and sweep out a subtorus of the Jacobian of the spectral curve). Conversely, the spectral data can be used to construct the harmonic tori in terms of theta-functions on the spectral curve Σ. This idea can be extended to a more general notion [Tai98] of a spectral curve for conformal tori f : T 2 → S 4 . Geometrically, this multiplier spectral curve arises as a desingularisation of the set of all Darboux transforms of f where one uses a generalisation of the notion of Darboux transforms for isothermic surfaces to conformal surfaces [BLPP12]. In the case when the conformal immersion is a constant mean curvature or Willmore surface, one obtains as special cases the so-called µ-Darboux transforms which are given by parallel sections of the associated family of flat connections. In particular, the (normalisations of the) eigenline spectral curve for the harmonic Gauss map of a constant mean curvature torus f is, [CLP13], the multiplier spectral curve of f . A similar result holds for (constrained) Willmore surfaces, [Boh10].
As mentioned above, by the Ruh-Vilms theorem the Gauss map of a minimal immersion f : M → R 3 is harmonic, and thus, the various operations discussed above can be applied to its Gauss map. However, as opposed to the case of an immersion with constant non-vanishing mean curvature, the Gauss map does not uniquely determine the minimal surface. Thus, although the associated family and the dressing operation for the harmonic Gauss map of a minimal surface can be defined, [DPT07], the investigation of minimal surfaces with these dressed harmonic Gauss maps complicates. On the other hand, Meeks, Pérez and Ros [MPR14], [MP09], use algebrogeometric solutions to the KdV equation to show that the only properly embedded minimal planar domains with infinite topology are the Riemann minimal examples. The same Lamé potentials appear in the study of the spectral curve of an Euclidean minimal torus with two planar ends and translational periods [BT14]. This indicates that applying integrable system methods may lead to a further development of minimal surface theory. Conversely, getting a better understanding of the special case of minimal surfaces may also give insights into the more general methods from integrable systems.
The aim of our paper is to provide further evidence that concepts on minimal surfaces may in fact be special cases of the harmonic map theory: the López-Ros deformation is a special case of a simple factor dressing of a minimal surface.
To avoid the issue that a minimal surface is not uniquely determined by its Gauss map, we will work with the conformal Gauss map which determines a minimal surface in 3-space uniquely. Since minimal surfaces are Willmore the conformal Gauss map is harmonic, too. We will briefly recall the construction of the associated families d λ and d S λ of flat connections for both the harmonic Gauss map N and the conformal Gauss map S of a minimal surface in our setup. Both are closely related: parallel sections of d S λ can be expressed in terms of parallel sections of d λ and generalisations f p,q , p, q ∈ S 3 , of the associated family of minimal surfaces f cos θ,sin θ . It turns out that this new family f p,q , the right-associated family, is in fact a family of minimal surfaces in 4-space which contains the classical associated family. In view of this natural appearance of minimal surfaces in 4-space, we will develop our theory more generally for minimal surfaces in 4-space and restrict to the case of minimal surfaces in 3-space when appropriate. As in the case of a harmonic map N : M → S 2 one can define the associated family of harmonic maps of the harmonic conformal Gauss map of a Willmore surface, [BPP02]. In the case of a minimal surface, we show that the harmonic maps in the associated family of the conformal Gauss map are indeed the conformal Gauss maps of the associated family of minimal surfaces.
Moreover, due to the harmonicity of the conformal Gauss map of a Willmore surface, a dressing operation on Willmore surfaces can be defined [BQ14]. In particular, for the most simple dressing operation given by a dressing matrix with a simple pole, the so-called simple factor dressing, the new harmonic map can be computed explicitly and is the conformal Gauss map of a new Willmore surface in the 4-sphere, [Les11], [BQ14].
In the case of a minimal surface f : M → R 4 ⊂ S 4 we only consider simple factor dressings which preserve the Euclidean structure and show that in this case, the simple factor dressing of the conformal Gauss map of f is indeed the conformal Gauss map of a minimal surface in 4-space. In fact, the simple factor dressing can be given explicitly in terms of the minimal surface f , its conjugate and the parameters (µ, m, n) where µ ∈ C \ {0} is the pole of the simple factor dressing, and m, n ∈ S 3 determine the d S µ -stable bundle which is needed in the definition of the dressing matrix. Even for surfaces in 3-space, the simple factor dressing will in general give surfaces in 4-space. However, for n = m, the simple factor dressing of a minimal surface f : M → R 3 will be in 3-space and the Gauss map of a simple factor dressing is the simple factor dressing of the Gauss map of f . In the simplest case when n = m = 1 and µ ∈ R the simple factor dressing is the minimal surface (1) where s = − ln |µ| and f l , f * l are the coordinate functions of f and a conjugate f * of f . In this case, we see immediately that f µ is a Goursat transformation of f with holomorphic null curve L µ Φ where Φ = f + i f * is the holomorphic null curve of f and Indeed, we prove more generally that every simple factor dressing of a minimal surface f : M → R 4 with parameters (µ, m, n) is a Goursat transformation. In particular, we show that this implies that the simple factor dressing preserves completeness. If the Goursat transform is single-valued on M then finite total curvature is preserved, too.
In the case when m = n ∈ S 3 , the orthogonal matrix of the Goursat transformation is given as R m,m L µ R −1 m,m ∈ O(3, C) where the rotation matrix R m,m in 3-space is given by m ∈ S 3 ⊂ R 4 : decomposing m = (cos θ, q sin θ) with q ∈ R 3 , ||q|| = 1, the matrix R m,m is the rotation along the axis given by q about the angle 2θ. In other words, the simple factor dressing in R 3 with parameters (µ, m, m) is obtained from the simple factor dressing (1) with parameter µ applied to the (inverse of the) rotation given by m.
The López-Ros deformation of a minimal surface f : M → R 3 is usually given in terms of the Weierstrass data. We recall an explicit form of the López-Ros deformation in terms of the minimal surface and its conjugate surface: the López-Ros deformation f σ of f with parameter σ ∈ R, σ > 0, is indeed given by where s = ln σ. In other words, since with µ = − 1 σ and m = 1 2 (1, −1, −1, −1) ∈ S 3 , the Lopez-Ros deformation with parameter σ is the simple factor dressing of f with parameters (µ, m, m).
We investigate the periods of the simple factor dressings in terms of the periods of the holomorphic null curve, and give conditions on the parameters (µ, m, n) for a simple factor dressing to be singlevalued. We discuss the end behaviour of the simple factor dressing on minimal surfaces in 3-space with finite total curvature ends: the simple factor dressing preserves planar ends for all parameters and, due to the special form of the Goursat transformation, catenoidal ends if the parameters of the simple factor dressings are chosen so that the simple factor dressing is single-valued.
Previous results seemed to indicate that the µ-Darboux transformation, which is used in the geometric understanding of the spectral curve of conformal tori, preserves a surface class. For example, µ-Darboux transforms of constant mean curvature surfaces f : M → R 3 , i.e., Darboux transforms which are given by parallel sections of the flat connection d µ in the associated family of the Gauss map N of f , have constant mean curvature [CLP13] and similarly, µ-Darboux transforms of Hamiltonian Stationary Lagrangians are Hamiltonian Stationary Lagrangians, [LR10]. However, for minimal surfaces we show that µ-Darboux transforms are in general not minimal but are still given by complex holomorphic data. More precisely, a minimal surface has an associated Willmore surface which is the twistor projection of a holomorphic curve in complex projective 3-space, and a µ-Darboux transform of f is the associated Willmore surfaces of an element of the right-associated family f p,q of f . In case of a minimal surface f : M → R 3 the associated Willmore surface f is the conformal immersion in 4-space which is given by where N is the Gauss map of f and f * is a conjugate of f .
We conclude the paper by demonstrating our results for various well-known minimal surfaces, including the catenoid, surfaces with one planar end and Scherk's first surface.
In particular, the simple factor dressings of the catenoid which are again periodic are reparametrisations of the catenoid if they are surfaces in 3-space. This immediately follows from our result that the simple factor dressing of a catenoidal end is catenoidal, provided the simple factor dressing is single-valued. Since planar ends are preserved for any parameters, all simple factor dressings of surfaces with one planar end have one planar end, too.
Using our closing conditions, we show that the López-Ros deformation of Scherk's first surface gives doubly-periodic minimal surfaces. Moreover, for any rational number q > 0 we show that the simple factor dressing (1) with parameter µ = − 1 √ q is doubly-periodic, thus we obtain a family of new examples of doubly-periodic (non-embedded) minimal surfaces.
The authors would like to thank Wayne Rossman and Nick Schmitt for directing their attention towards the López-Ros deformation and the Goursat transformation. Parts of this research were conducted while the first author was visiting the Department of Mathematics at the University of Tsukuba and the OCAMI at Osaka University. The first author would like to thank the members of both institutions for their hospitality during her stay, and the University of Leicester for granting her study leave.

Minimal surfaces
We first recall some basic facts on minimal surfaces in Euclidean space which will be needed in the following whilst setting up our notation. Although we are mostly interested in minimal surfaces in R 3 , some of our transforms will be surfaces in R 4 . Therefore, we will study more generally minimal immersions in R 4 and specialise to the case of minimal surfaces in 3-space when appropriate.
2.1. Minimal surfaces in R 4 . Let f : M → R 4 be a conformal (branched) immersion from a Riemann surface M into 4-space. If f is minimal, then f is harmonic, i.e., where we put * ω(X) = ω(J T M X) for a 1-form ω ∈ Ω 1 (T M), X ∈ T M. Here, J T M is the complex structure of the Riemann surface M, thus, * is the negative Hodge star operator. In particular, * d f is closed if f is harmonic and there exists a conjugate surface f * on the universal coverM of M, given up to translation by Note that f * is minimal, and so is the associated family, e.g. [Eis12], We model Since H is normal we have NH = HR. We put H = −RH and denote by the (1, 0) and (0, 1)-part of dR with respect to the complex structure R. Then the equation of the mean curvature vector becomes (3) Similarly, there is also an equation for the mean curvature vector in terms of the left normal: . In particular, both the left and right normal N and R of a minimal surface are conformal and harmonic.
Next, we observe that a conjugate surface f * of a minimal surface f has the same left and right normal as f since is a holomorphic curve in C 4 , that is, * dΦ = i dΦ. Here we use "i " to denote the complex structure of the complexification C 4 = R 4 + i R 4 to avoid confusion with the imaginary quaternion i. If f : M → R 4 is a minimal conformal immersion, then the holomorphic curve is a null curve in C 4 , i.e., dΦ 0 ⊗ dΦ 0 + dΦ 1 ⊗ dΦ 1 + dΦ 2 ⊗ dΦ 2 + dΦ 3 ⊗ dΦ 3 = 0. In fact, every holomorphic null curve Φ :M → C 3 gives rise to a conformal minimal immersion by setting f = Re (Φ) :M → R 3 . Note that the holomorphic null curve of the associated family f cos θ,sin θ of a minimal surface f is given by Φ cos θ,sin θ = e −i θ Φ where Φ is the holomorphic null curve of f .
Our choice of Enneper-Weierstrass representation is so that it specialises to the standard Enneper-Weierstrass representation in R 3 whenever f : M → H is a minimal surface with Re ( f ) = 0. We allow f to be branched which happens whenever the order of ω at p is bigger than the maximum order of poles of g 1 , g 2 , and g 2 1 + g 2 2 at p ∈ M. Note that f is in general only defined on the universal coverM of M. We say that a minimal surface f :M → R 4 is single-valued on M if f • π −1 : M → R 4 is well-defined where π :M → M is the canonical projection of the universal coverM to M. In this case, we will identify f and f • π −1 and write, in abuse of notation, from now on f : M → R 4 .
Finally we recall the following result due to Chern and Osserman: [Mor98]). Let f : M → R 4 be a complete (branched) minimal immersion with holomorphic null curve Φ : M → C 4 . Then f has finite total curvature if and only if M is conformally equivalent to a compact Riemann surfaceM punctured at finitely many points p 1 , . . . , p r such that dΦ extends meromorphically into punctures p i .

2.2.
Minimal surfaces in R 3 . We identify the Euclidean 3-space with the imaginary quaternions R 3 = Im H. If f : M → R 3 is conformal then the left and right normal coincide and are given by the Gauss map N : M → S 2 of f . In this case, the function H given in (4) is real-valued, and indeed, H is the mean curvature function of f . For a minimal immersion in R 3 , the Gauss map is thus both harmonic and conformal, that is, * dN = −NdN , and d(dN) = 0 , where as before (dN) = 1 2 (dN − N * dN) is the (1, 0)-part of dN with respect to the complex structure N.
Conversely, a meromorphic g and a holomorphic 1-form ω, such that if g has a pole of order m at p then ω has a zero of order at least 2m, give a minimal surface f :M → R 3 as f = Re (Φ) where Φ is given by the Enneper-Weierstrass representation, [Enn64], [Wei66], For convenience, we also will occasionally use in the case of a surface in R 3 the Weierstrass data (g, dh) given in terms of the height differential dh = gω. Written in terms of (g, dh) the Enneper-Weierstrass representation becomes The Gaussian curvature of f is given in terms of the Weierstrass data [Kar89] as and the Gauss map is given by g via stereographic projection (7) N = 1 |g| 2 + 1 (2 Re g, 2 Im g, |g| 2 − 1) .
Note that if we consider a minimal surface f : M → R 3 as a surface in 4-space, that is f : M → H with Re ( f ) = 0, then its Weierstrass data in R 4 is under our choices given by (g 1 = g, g 2 = 0, ω) and its holomorphic null curve 0, 1 2 (1 − g 2 )ω, i 2 (1 + g 2 )ω, gω in C 4 is given by the embedding of the curve Φ into C 4 . Moreover, we see that in this case the Gauss map G = (G 1 , G 2 ) : M → S 2 × S 2 takes values in the diagonal of S 2 × S 2 and is given by G 1 = G 2 = g which is by [HO80], [Tai06] the condition for the Enneper-Weierstrass representation to take values in 3-space.
As before, complete minimal immersions of finite total curvature can be characterised by the holomorphic null curve Φ: Oss64]). Let f : M → R 3 be a complete (branched) minimal immersion with holomorphic null curve Φ : M → C 3 .
Then f has finite total curvature if and only if M is conformally equivalent to a compact Riemann surfaceM punctured at finitely many points p 1 , . . . , p r such that dΦ extends meromorphically into punctures p i .
We will now give a description of embedded finite total curvature ends in terms of the holomorphic null curve. Although the result seems to be known for vertical ends, we include the argument for completeness.
Theorem 2.3. Let f : M → R 3 be a minimal surface with complete end at p. Let z be a conformal coordinate of M at the end p which is defined on a punctured disc D * = D \ {0} and is centered at p.
Then the following statements are equivalent: (i) f has an embedded finite total curvature end at p. (ii) dΦ has order −2 at z = 0 and res z=0 dΦ is real.
If res z=0 dΦ = 0 then the end is planar, otherwise, it is catenoidal.
is the holomorphic null curve of f and ord z=0 dΦ is the minimum of ord z=0 dΦ i for i = 1, 2, 3.
Proof. We first assume that the end is vertical. By [HK97] an embedded complete finite total curvature vertical end has logarithmic growth α ∈ R satisfying res z=0 dΦ = −(0, 0, 2πα). The end is planar if α = 0 and catenoidal otherwise.
Moreover, if f has a catenoidal end then the Gauss map g of f has a simple pole or zero, and the height differential dh has a simple pole. If f has a planar end then g has a pole or zero of order m > 1 and dh has a zero of order m − 2. In both cases, (6) shows that ord z=0 dΦ = −2.
If the end is not vertical, we can apply a rotation R ∈ SO(3, R) on f to obtain a minimal surfacẽ f = R f with a vertical end. Then the holomorphic null curve off isΦ = RΦ and thus, ord z=0 dΦ = ord z=0 dΦ = −2. Moreover, the residues at z = 0 vanish at a planar end, whereas res z=0 dΦ = R −1 res z=0 dΦ is real for a catenoidal end.
Conversely, we will show that if ord z=0 dΦ = −2 and res z=0 dΦ is real, then we can assume f has an embedded vertical end at p and its Gauss map g has a pole or zero of order m ≥ 1 and ord z=0 dh = m − 2 holds for the height differential. From this we conclude that the end has finite total curvature.
If res z=0 dΦ = (0, 0, 0) then we can rotate the end so that we have a vertical end without changing the order and the residues of dΦ at the end. In particular, we can assume that the Gauss map g has a pole or zero at p of order m ≥ 1. Since the residue of dΦ at z = 0 vanishes, we see that dh cannot have a simple pole. But then ord z=0 dΦ = −2 implies that dh is holomorphic with ord z=0 dh = m − 2, m ≥ 2, so that with the Enneper-Weierstrass representation (6) After possible translation, we thus have Since the end is at z = 0, we have that z ∈ C r tends to 0 for r → ∞. In particular, the asymptotic behaviour of ψ r = f r : C r → S 2 is governed by From [JM83] we know that ψ r converges to a horizontal circle with multiplicity when r → ∞ and that the end is embedded if the multiplicity is one. From the above asymptotic behaviour we see that the multiplicity of lim r→∞ ψ r is indeed one. Hence, the end is embedded but then (8) shows that p is a planar end.
We first show that the end is vertical by contradiction: if g has neither a zero nor a pole at the end, then ord z=0 gdh = ord z=0 dh g = ord z=0 dh. Using ord z=0 dΦ = −2 and the Enneper-Weierstrass representation (6), the order of dh at the end is −2. We write and res z=0 dΦ 1 = res z=0 dΦ 2 = 0, we have Hence αb 0 = 0 which contradicts α 0 and b 0 0. Therefore, g has a zero or a pole at the end, but then the end has vertical normal, the zero or pole of g is simple and dh has a simple pole at z = 0, that is, the holomorphic null curve Φ can be written as Since res z=0 dΦ = (0, 0, −2πα) we have a −1 = −2πα ∈ R and, after possibly translation, Let as before C r = {z ∈ D * | || f (z)|| = r} then the asymptotic behaviour of ψ r = f r : C r → S 2 is determined by Since the multiplicity does not depend on |z| we see that in the limit r → ∞ the multiplicity is again 1, and the end is embedded by [JM83]. Following the arguments in the proof of [HK97, Prop. 2.1], f is a graph over (the exterior of a bounded domain of) the ( f 1 , f 2 ) plane with asymptotic behaviour for R = f 2 1 + f 2 2 , and the end has finite total curvature, [Sch83] .
The López-Ros deformation of a minimal surface f : M → R 3 with Weierstrass data (g, ω) is [LR91] the minimal surface f r :M → R 3 given by the new Weierstrass data (rg, ω r ) with r ∈ R * . Obviously this can be extended to a deformation f σ with complex parameter σ ∈ C * = C \ {0} by using the Weierstrass data (σg, ω σ ). Since any matrix in O(3, C) = {A ∈ GL(3, C) | A t = A −1 } preserves the standard symmetric bilinear form on C 3 , we obtain new minimal surfaces via the Goursat transformation, [Gou87]: if f : M → R 3 is minimal with holomorphic null curve Φ = f + i f * then AΦ is again a holomorphic null curve and Re (AΦ) is a minimal surface in R 3 .
As pointed out by Pérez and Ros [PR02], the López-Ros deformation is a special case of the Goursat transformation: Theorem 2.4. The López-Ros deformation f σ with complex parameter σ = e s+i t ∈ C * is given by a straightforward computation shows thatf = Re (L σ Φ) where the holomorphic map L σ Φ, is a null curve in C 3 . Thus,f is a Goursat transformation of f . Indeed, if (g, ω) denotes the Weierstrass data of f then the Weierstrass data off computes tõ This shows thatf = f σ is the López-Ros deformation of f with parameter σ ∈ C * .
2.3. Willmore surfaces. Using the one-point compactification of R 4 we consider a conformal immersion f : M → R 4 as a conformal immersion into the 4-sphere. We identify the 4-sphere S 4 = HP 1 with the quaternionic projective line where the oriented Möbius transformations are given by GL(2, H). In particular, a map f : M → HP 1 can be identified with a line subbundle L ⊂ H 2 = M × H 2 of the trivial H 2 bundle over M whose fibers at p ∈ M are given by .
For an immersion f : M → R 4 the line bundle L is given by when choosing the point at infinity as ∞ = 1 0 H ∈ HP 1 . Oriented Möbius transformations on In particular, the group of oriented Möbius transformations acts on the line bundle L given by an immersion f : M → R 4 via L → BL, B ∈ GL(2, H). Every pair of unit quaternions m, n ∈ S 3 gives an element R m,n ∈ SO(4, R) by v ∈ H → R m,n v = mvn −1 ∈ H , and conversely, every element of the special orthogonal group arises this way, [Cay89]. The corresponding action on the line bundle L of an immersion f : M → R 4 is given by Here, the Hopf field A of S is the 1-form given by where (dS ) = 1 2 (dS − S * dS ) is the (1, 0)-part of the derivative of S with respect to the complex structure S .
In affine coordinates, the conformal Gauss map of a conformal immersion f : M → R 4 is given by the complex structure, see [BFL + 02, p. 42], Thus, A is indeed a 1-form with values in L. Note that the Hopf field A is holomorphic [BFL + 02, p. 68] with im A ⊂ L. In particular, if A 0, the Hopf field A gives the line bundle L by holomorphically extending im A into the isolated zeros of A. Therefore, when fixing the point at ∞, the immersion f : M → R 4 is uniquely determined by A.
In particular, if f : M → R 4 is minimal then * dR = −RdR and thus d f ∧dR = 0 by type arguments. Therefore, the Hopf field is harmonic, that is, d * A = 0. From this characterisation of Willmore surfaces and the fact that the conformal Gauss map of a minimal surface is harmonic we see: Consider the right normal R of a minimal surface f : M → R 4 which satisfies * dR = −RdR = dRR. Thus, if the right normal is not constant, we can consider R : M → S 2 as a (branched) conformal immersion whose right normal is −R. Since * d(−R) = (−R)d(−R), we see from (10) that its Hopf field vanishes on the line bundle of R, that is, R is the twistor projection of a holomorphic curve in CP 3 : In this case, f is Willmore and the twistor lift of F is the holomorphic curve F : M → CP 3 which is given by the line subbundle E ⊂ L by where E is the +i eigenspace of the conformal Gauss map S | L restricted to L.
The surfaces which are both minimal and twistor projections are indeed given by holomorphic maps into C 2 : Then R is constant if and only if f is minimal and the twistor projection of a holomorphic curve in CP 3 .
In this case, we can identify R 4 = H with C 2 via the complex structure which is given by right multiplication by the constant −R. Then f : M → C 2 is holomorphic.
Proof. If dR = 0 then we see by (3) that H = 0 and f is minimal. But then (11) shows that the Hopf field A vanishes identically, that is, f is the twistor projection of a holomorphic curve.
Conversely, if f is minimal and A| L = 0 we see again by (11) that dR = 0.
If R is constant, then the right multiplication by −R gives a complex structure on R 4 = H. Then Note that if f is a minimal surface with constant right normal R then For general minimal surfaces f the map f R − f * is a Willmore surface in R 4 : [DT07]). If f : M → R 4 is a minimal surface with conjugate surface f * and (non-constant) right normal R then is a twistor projection of a holomorphic curve in CP 3 . We call f an associated Willmore surface of f .
Conversely, let f : M → R 4 be a twistor projection of a holomorphic curve in CP 3 on the simply connected M and let R be its non-constant right normal. Away from the isolated zeros of dR , the immersion f is, up to translation, given by either f = cR , c ∈ H, or by a minimal surface Proof. If f is minimal then its right normal R satisfies * dR = −RdR = dRR and and thus f is a twistor projection of a holomorphic curve in CP 3 by Theorem 2.9 and (10).
Conversely, let f : M → R 4 be the twistor projection of a holomorphic curve in CP 3 , and let R be its right normal. From Theorem 2.9 and (10) we see that * dR = R dR = −dR R .
Since R is conformal and not constant, the zeros of dR are isolated. Since

Harmonic maps and their associated families of flat connections
It is well-known [Uhl89] that a harmonic map gives rise to a family of flat connections. There are various transformations on harmonic maps whose new harmonic maps are build from parallel sections of the associated family of flat connections: e.g., the associated family, the simple factor dressing [TU00] and Darboux transforms [CLP13], [BDLQ13]. In this paper, we investigate the links between these transformations, when applied to the left and right normals and to the conformal Gauss map of a minimal surface f : M → R 4 .
3.1. The harmonic right normal and its associated family. We equip H with the complex structure I which is given by the right multiplication by the unit quaternion i. This way, we identify C 2 = (H, I). It is worthwhile to point out that this complex structure I differs from the complex structure i we used before. We will use the symbol C = span R {1, I} to indicate that we use the complex structure I whereas C = span R {1, i }.
With this at hand, the C * -family of flat connections of a harmonic map R : M → S 2 on the trivial C 2 bundle C 2 = H over M can be written as where d is the trivial connection on H, λ ∈ C * = C \ {0}, and Q = − 1 2 ( * dR) = 1 4 (RdR − * dR) is the Hopf field of R. Moreover, are the (1, 0) and (0, 1)-parts of Q with respect to the complex structure I. The flatness of d λ is obtained from the harmonicity (5) of R, that is, from d * Q = 1 2 d(dR) = 0. Note that our choice of associated family differs from the associated family we see that our family of flat connections d λ is the associated family in [CLP13,BDLQ13] of the harmonic map −R. Indeed, both families are gauge equivalent [CLP13]. We choose the d λ family since it is closely related to the associated family of flat connections of the conformal Gauss map.
Since R is conformal, i.e., * dR = −RdR, we have 2 * Q = dR , and, for µ ∈ C * and β ∈ Γ(H), We first note that for µ = 1 the connection d µ = d is the trivial connection on H, and all parallel sections are constants. Thus, we will from now on assume that µ 1. For a d µ -parallel section β ∈ Γ(H) put Then m is constant since a 2 + b 2 = 0 and thus In particular, if β is d µ -parallel then (15) shows where we used a 2 + b 2 = 1. Therefore −1 = 1+a 1−a which gives a contradiction. Conversely, every β given by the equation (17) is d µ -parallel. Since we can summarise: Lemma 3.1. Let f : M → R 4 be minimal and d λ the associated family of the right normal R of f . For µ ∈ C \ {0, 1} every (non-trivial) d µ -parallel section β ∈ Γ(H) is given by In a similar way, the associated family of the left normal of a minimal surface can be discussed.
3.2. The conformal Gauss map and its associated family. Again, we identify C 4 = (H 2 , I) where I is given by right multiplication by the unit quaternion i. If the conformal Gauss map of a conformal immersion f : M → S 4 is harmonic, that is d * A = 0, by the same arguments as in the case of harmonic maps into the 2-sphere, the C * -family of connections We consider the case when S is the conformal Gauss map of a minimal immersion in R 4 . We fix µ ∈ C * and compute all parallel sections of d S µ . If µ = 1 then d S µ = d is trivial, and every constant section is parallel. Assume from now on that µ 1, and let Here we used that * dR = −RdR = dRR by the minimality of f and thus from (11) we see In particular, β is a d µ -parallel section by (14), and by Lemma 3.1 it is given by If β = 0 is trivial, that is, m = 0, then (21) shows that α is constant. For m 0, we see that is the general solution of (21) where f * is a conjugate minimal surface of f , that is, We summarise:  In particular, the associated family f cos θ,sin θ = f cos θ + f * sin θ can easily be extended to a family with p, q ∈ H, (p, q) (0, 0). Since p + Rq is a quaternionic holomorphic section it has only isolated zeros, and shows that f p,q is a branched conformal immersion. The right normal R of f gives, away from the isolated zeros of p + Rq, via the right normal of f p,q whereas the left normal N of f is the left normal Thus, by (4) d f p,q H p,q = (dN p,q ) = dN = 0 and f p,q is a (branched) minimal immersion. Similarly, we have a family of (branched) minimal immersions f p,q = p f + q f * , (p, q) (0, 0), with right normal R p,q = R and left normal Note that for p, q ∈ R, (p, q) (0, 0), we obtain the usual associated family of a minimal surface up to scaling. Moreover, f pn,qn = f p,q n is given by a scaling of f p,q and an isometry on R 4 for n ∈ H * .
Theorem 4.2. The right (left) associated family is a S 3 -family of minimal surfaces in R 4 which contains the classical associated family f cos θ,sin θ , θ ∈ R, of minimal surfaces.
The right (left) associated family preserves the conformal class, and a surface f p,q (or f p,q ) is isometric to f if and only if it is an element of the classical associated family, up to an isometry of R 4 .
Proof. It only remains to show that f p,q is isometric to f if and only if f p,q = f n cos θ,n sin θ for some θ ∈ R and n ∈ S 3 . Assume first that f p,q is isometric to f . By multiplying with a unit quaternion from the right we may assume that p ∈ R. Then |d f p,q | = |d f | implies by (24) p 2 + |q| 2 + 2pRe (Rq) = |p + Rq| 2 = 1 and thus Re (Rq) is constant. Since the stereographic projection of R is a meromorphic function, the right normal R can only take values in a plane if R is constant. In other words, if R is not constant then Re (Rq) = Re (R Im q) = − < R, Im q > is constant only if Im q = 0. But then the above equation gives p 2 + q 2 = 1 with q ∈ R and (p, q) = (cos θ, sin θ) for some θ ∈ R. If R is constant then Conversely, f p,q = f n cos θ,n sin θ , n ∈ S 3 , then |p + Rq| = 1 and thus |d f p,q | = |d f | by (24).
A similar argument shows the statement for the left associated family.
Remark 4.3. For any immersion f : M → R 3 = Im H in Euclidean 3-space, the left and right normal coincide. A surface in the right associated family f p,q of a minimal surface f : M → R 3 has left normal N p,q = N and right normal R p,q = (p + Nq) −1 N(p + Nq). In particular, we have N p,q R p,q in general and thus, elements of the right associated families of a minimal surface f : M → R 3 are not necessarily minimal in 3-space but are minimal surfaces in R 4 .

4.2.
The associated family of the harmonic conformal Gauss map. We now give a derivation of the associated family of minimal surfaces in terms of the associated family of harmonic maps. Recall that in the case of a constant mean curvature surface f : M → R 3 the Gauss map N : M → S 2 of f is by the Ruh-Vilms theorem [RV70] harmonic and its associated familyd λ of flat connections (13) gives rise to the associated family of constant mean curvature surfaces: for µ ∈ C * the map N is harmonic with respect to the quaternionic connectiond µ for all µ ∈ S 1 , see e.g. [BDLQ13] for details. Thus, for anyd µ -parallel section ϕ ∈ Γ(H) the map N µ = ϕ −1 Nϕ is harmonic with respect to d. Note that ϕ is unique up to a right multiplication by a constant quaternion. This family N λ , λ ∈ S 1 , is called the associated family of N. The Sym-Bobenko formula [Bob91] relies on the link (4) between the differentials of f and N to obtain the constant mean curvature surface f by differentiating a family ϕ λ of parallel sections of d λ with respect to the parameter λ. This way, one can obtain from the associated family N λ of N a family of constant mean curvature surfaces, the associated family of f .
In the case of a minimal surface f : M → R 4 with right normal R, we have seen in Lemma 3.1 that every parallel section β ∈ Γ(H) of the associated family of flat connections of R is, using (18), given by where a = µ+µ −1 2 , b = i µ −1 −µ 2 , and µ ∈ C \ {0, 1}. If µ ∈ S 1 , µ 1, then a = Re µ, b = Im µ ∈ R, and β = (R + b a−1 )m with m ∈ H * . Thus, the associated family of harmonic maps is given by In particular, the harmonic map R µ does not depend on the parameter µ. In the case of a minimal surface, the equation (4) does not allow to reconstruct the minimal surface from the harmonic Gauss map. In particular, there is no Sym-Bobenko formula to obtain a minimal surface with right normal R µ via differentiation of parallel sections with respect to the parameter µ.
To obtain a non-trivial family of minimal surfaces with right normal R µ for µ ∈ C * , we consider instead the harmonic conformal Gauss map S of f :  ϕ 2 ) is an invertible endomorphism with d µ -parallel columns ϕ 1 , ϕ 2 , and µ = cos(2θ) − i sin(2θ) ∈ S 1 . Moreover, S φ = φ −1 S φ is the conformal Gauss map of f cos θ,sin θ .
Proof. For µ ∈ S 1 and invertible φ andφ with d µ -parallel columns, we haveφ = φB with B constant so that Lφ = B −1 L φ is given by a Möbius transformation. Thus, we can assume without loss of generality that for µ 1 φ = (e, ϕ) where ϕ = eα + ψβ is a d µ -parallel section with nowhere vanishing β. Then so that L φ is the line bundle of the (branched) conformal immersion f φ = −α : M → R 4 . By Proposition 3.2 and (18) we have We consider α up to Möbius transformations, and may thus assume without loss of generality that m = − sin ϑ 2 so that f φ = f cos θ,sin θ for θ = − ϑ 2 . For µ = 1 we can choose φ as the identity matrix, and obtain f φ = f 1,0 = f . By definition S φ leaves L φ invariant. The final statement follows by a similar argument as in [BDLQ13]. In fact, this is a special case of a corresponding statement for (constrained) Willmore surfaces f : M → R 4 , see [BPP02], [BQ14].
We also get an interpretation of the right associated family: If we consider φ = (e, ϕ) with d S µ ϕ = 0 for some µ ∈ C \ {0, 1}, then the line bundle L φ = φ −1 L gives by the same argument as above a member of the generalised associated family, up to Möbius transformation, Note however, that φ is not a d S µ -parallel endomorphism since d S µ is a complex but not a quaternionic connection.

Simple factor dressing
As we have seen, the harmonic left and right normals and the harmonic conformal Gauss map of a minimal surface give rise to families of flat connections. Conversely, if a family of flat connections is of an appropriate form, it can be used to construct a harmonic map. In particular, if d λ is the associated family of flat connections of a harmonic map, one can gauge d λ with a λ-dependent dressing matrix r λ to obtain a new family of flat connectionsd λ = r λ · d λ . If r λ satisfies appropriate reality and holomorphicity conditions, thend λ is the associated family of a new harmonic map, a so-called dressing of the original harmonic map, see [Uhl89], [TU00]. 5.1. Simple factor dressing of the left and right normals. We first consider the case when R : M → S 2 is the harmonic right normal of a minimal surface in R 4 and choose the simplest possible dressing matrix: If d λ is the associated family (12) of the harmonic map R then the simple factor dressing matrix r λ is obtained by choosing µ ∈ C \ {0, 1} and a parallel section β of the flat connection d µ of the associated family as In [BDLQ13] it is shown thatd λ = r λ · d λ is the associated family of a harmonic mapȒ : M → S 2 , the simple factor dressing of R, andȒ =T −1 RT whereT = 1 2 (−Rβ(a − 1)β −1 + βbβ −1 ) is given explicitly in terms of β and µ, where a = µ+µ −1 2 , b = i µ −1 −µ 2 . In our case, any d µ -parallel section β is given by Lemma 3.1 and (18) as Since β is nowhere vanishing, we see that and the simple factor dressing of R is In particular, a simple factor dressing is determined by the choice µ ∈ C \ {0, 1}, giving the pole of the simple factor, and m ∈ H * , giving the parallel bundle of d µ . A straight forward computation shows that dȒ = (1 + ρ 2 )(R + ρ) −1 dR(R + ρ) −1 . Since The right hand side of this equation commutes with R so that also Therefore, which shows thatȒ is conformal and harmonic. Indeed, the simple factor dressing of the right normal of a minimal surface in R 4 is the right normal of a minimal surface: Theorem 5.1. Let f : M → R 4 be a minimal surface with right normal R.
Then every simple factor dressing of R is the right normal R p,q of a minimal surface f p,q in the right associated family of f .
Thus, the right normal of the associated minimal surface h is (27) the dressing of R.
Remark 5.2. Note that the simple factor dressing of the harmonic right normal does not single out a canonical minimal surface with this right normal: the element fb ,â−1 is one example of such a minimal surface but so is p fb ,â−1 + q f * b,â−1 for any p, q ∈ H * .
An analogue theorem holds for the left normal N of a minimal surface f : M → R 4 : Theorem 5.3. The simple factor dressing of N is the left normal N p,q of an element f p,q in the left associated family of f .
As noted before, if f : M → R 3 is a minimal surface in R 3 then the left and right associated families give in general minimal surfaces in R 4 . In particular: Corollary 5.4. Let f : M → R 3 be a minimal surface with Gauss map N and assume that f is not a plane. Let fb ,â−1 be the minimal surface in the associated family of f whose right normal Rb ,â−1 is the simple factor dressing of N given by µ ∈ C \ {0, 1}, m ∈ H * , whereâ = m µ+µ −1 2 m −1 ,b = mi µ −1 −µ 2 m −1 .
Then fb ,â−1 is a minimal surface in R 3 if and only if µ ∈ S 1 , µ 1.
We will use Remark 5.2 to obtain a minimal surface in 3-space with a given simple factor dressing as its Gauss map. This operation turns out to be the surface obtained by applying a corresponding simple factor dressing on the conformal Gauss map.

5.2.
Simple factor dressing of a minimal surface. The conformal Gauss map of a Willmore surface is harmonic and one can define a dressing on it [Qui09], [BQ14]. Since the conformal Gauss map determines a conformal immersion (if the Hopf field is not zero), this induces a transformation, a dressing, on Willmore surfaces. (Actually, Burstall and Quintino define more generally a dressing on constrained Willmore surfaces).
We will again only discuss the special case of simple factor dressing by choosing the simplest possible dressing matrix. As before, the simple factor dressing of the conformal Gauss map S of a Willmore surface f : M → S 4 is given explicitly by parallel sections of a connection d S µ of the associated family of flat connections [Les11]: for µ ∈ C \ {0, 1} let W µ be a d S µ -stable, complex rank 2 bundle overM with W µ ⊕ W µ j = H 2 . For two d S µ -parallel sections ϕ 1 , ϕ 2 ∈ Γ(W µ ) with φ = (ϕ 1 , ϕ 2 ) regular, define a conformal immersionf :M → S 4 bŷ where b a−1 is the left multiplication by the quaternion b a−1 on H 2 and as usual a = µ+µ −1 2 , b = i µ −1 −µ 2 . Then the conformal Gauss mapŜ off is the simple factor dressing of S . In particular,Ŝ is harmonic, andf is a Willmore surface. It is known thatL, and thusf , is independent of the choice of basis for W µ , [Les11]. We call the Willmore surfacef a simple factor dressing of f . Note that b a−1 ∈ R for µ ∈ S 1 so thatf = f in this case. Note that this gives a conformal theory. However, in the case of a minimal surface f : M → R 4 we are only interested in the Euclidean theory, that is, simple factor dressings which are again surfaces in the same 4-space. Thus, we will restrict to simple factor dressings such that S + φ b a−1 φ −1 stabilises the point at infinity ∞ = eH. Because S e = eN by (9) where N is the left normal of f , we have to restrict to φ with φ b a−1 φ −1 ∞ = ∞. Since b a−1 ∈ R if and only if µ ∈ S 1 we can assume that b a−1 is not real as otherwisef = f . We recall that any d S µ -parallel section ϕ en, n ∈ H, is given by Proposition 3.2 and (18) as 1−µ m −1 and m ∈ H * . In particular, for regular φ = (ϕ 1 , ϕ 2 ) with two such d S µ -parallel sections ϕ 1 , ϕ 2 we write Since β 1 , β 2 are nowhere vanishing so is ζ 1 and the equation above shows that ζ 2 ζ −1 1 commutes with the complex number b a−1 R. Therefore, we have ζ 2 = qζ 1 with q : M → C * and, because φφ −1 = Id and ζ 1 is nowhere vanishing, we conclude β 1 + β 2 q = 0.
If R is constant, that is, if f is the twistor projection of a holomorphic curve in CP 3 , then f * = f R + c with some c ∈ H, so that with (31) and (32) we obtain ϕ 1 + ϕ 2 q = en with n = −c(m 2 q + m 1 ) and q ∈ C * . In other words, en is a d S µ -parallel section of W µ and we can replace φ in the definition ofŜ byφ = (en, ϕ 2 ).
If R is not constant, we use again the explicit forms (32) to obtain from β 1 + β 2 q = 0, q ∈ C * , that R(m 1 + m 2 q) is constant, which implies that m 1 = −m 2 q. But then (31) and (32) show that ϕ 1 = −ϕ 2 q which contradicts the assumption that φ is regular.
Thus, from now on we can restrict to regular endomorphism φ of the form Thenf is a minimal surface, andf is preserved when changing the parameters (µ, m, n) to (µ, mz, nw) or (μ −1 , m j, n j) for z, w ∈ C * . For µ ∈ S 1 the simple factor dressingf = f is trivial.
is an element of the left associated family of the minimal surface in the right associated family of f . In particular,f is minimal. From the explicit formula above we see thatf is preserved when changing (µ, m, n) to (µ, mz, nw) with z, w ∈ C * . Sinceμ =b and (m j)z(m j) −1 = mzm −1 for all z ∈ C, m ∈ H * , we obtain the same simple factor dressing for the parameters (µ, m, n) and (μ −1 , m j, n j). The final statement follows from the fact that b a−1 ∈ R for µ ∈ S 1 .
Remark 5.6. Note that the last statements in the above corollary are special cases of more general facts for simple factor dressings of Willmore surfaces: since the simple factor dressing is independent of the choice of basis of W µ and the family of flat connections satisfies a reality condition [BDLQ13], the surface is preserved under the given changes of parameter. The last statement holds for general simple factor dressings with µ ∈ S 1 .
In particular, we emphasise again that in contrast to the simple factor dressing of the right and left normal, the simple factor dressing of the conformal Gauss map associates a unique minimal surface: Definition 5.7. The simple factor dressing of a minimal surface f : M → R 4 with parameters (µ, m, n) is the minimal surfacef :M → R 4 given by where m, n ∈ S 3 , µ ∈ C \ {0, 1} and a = µ+µ −1 2 , b = i µ −1 −µ 2 . If m = n = 1 then we refer to f µ =f as the simple factor dressing of f with parameter µ.
The simple factor dressing with parameter µ of the rigid motionf = n −1 f m of f is given bỹ wheref is the simple factor dressing (33) of f with parameters (µ, m, n). Thus, all simple factor dressings are build from rigid motions of the simple factor dressings with parameter µ: Proposition 5.8. Letf be a simple factor dressing of a minimal surface f : M → R 4 with parameters (µ, m, n). Thenf = R n,m ((R −1 n,m ( f )) µ ) where (R −1 n,m ( f )) µ is the simple factor dressing of the rotated surface R −1 n,m ( f ) = n −1 f m with parameter µ.
Since the associated families of the left and right normals and the conformal Gauss maps are related, we also have a correspondence between the resulting simple factor dressings: Proof. The differential of the simple factor dressingf with parameters (µ, m, n) is given by In particular, the right normal Moreover, the choice of a different conjugate surface results in a translation of the simple factor dressing in 4-space.

Simple factor dressing and the López-Ros deformation
Given a minimal surface f : M → R 4 in 4-space with Weierstrass data (g 1 , g 2 , ω) denote, in analogy to the case of a minimal surface in R 3 , by f σ the López-Ros deformation of f with complex parameter σ ∈ C, that is, the minimal surface given by the Weierstrass data (σg 1 , σg 2 , ω σ ). Similarly, the Goursat transformation is defined by Re (A( f + i f * )) where A ∈ O(4, C) and f * is a conjugate surface of f . In this section, we will show that the López-Ros deformation is a special case of the simple factor dressing. Indeed, all simple factor dressings are (special) Goursat transformations.
6.1. The López-Ros deformation in R 4 . Since by Proposition 5.8 any simple factor dressing is given in terms of the simple factor dressing with parameter µ, we will first show that these simple factor dressings are Goursat transformations.
Theorem 6.1. Let f : M → R 4 be a minimal surface in R 4 . Then the simple factor dressing with parameter µ of f = f 0 + f 1 i + f 2 j + f 3 k is given by In particular, f µ is a Goursat transform of f whose holomorphic null curve is where Φ is the holomorphic null curve of f and Proof. Let µ ∈ C \ {0, 1} and put, as usual, a = µ+µ −1 2 , b = i µ −1 −µ 2 . The simple factor dressing of f with parameter µ is given by Next, we observe for v ∈ C = span R {1, i} that where we used that a 2 + b 2 = 1. To compute T 1 (v), T 2 (v) for v ∈ C j = span{ j, k} we recall that Therefore, since wv = vw for every w ∈ C and v ∈ C j, we see , the simple factor dressing of f with parameter µ is then given by sinh s which gives (35). The final statement follows by a straight forward computation of the holomorphic null curve.
By Proposition 5.8 we immediately see that the general simple factor dressing is a Goursat transformation, too.
Theorem 6.2. The simple factor dressing of a minimal surface f : M → R 4 is a Goursat transformation of f .
Proof. Let µ ∈ C \ {0, 1}, then by Proposition 5.8 the simple factor dressingf with parameters (µ, m, n) is given byf = R n,m ((R −1 n,m ( f )) µ ) where R n,m ∈ SO(4, R) is the map v → nvm −1 and (R −1 n,m ( f )) µ is the simple factor dressing of f = n −1 f m with parameter µ. If Φ denotes the holomorphic null curve of f then the null curve of R −1 n,m ( f ) is R −1 n,m Φ since R n,m is real. But then the holomorphic null curve of the simple factor dressing of R −1 n,m ( f ) with parameter µ is L µ R −1 n,m Φ by (36). Thus, the holomorphic null curve of the simple factor dressingf with parameters (µ, m, n) is given bŷ But R n,m L µ R −1 n,m ∈ O(4, C) so thatf is a Goursat transformation of f .
Note that the simple factor dressing is a special case of the Goursat transformation: its matrix A ∈ O(4, C) has det A = 1 and special behaviour of the eigenspaces.
As before the López-Ros deformation can be given in terms of the surface and its conjugate which immediately shows that it is a special case of the simple factor dressing: Theorem 6.3. Let f : M → R 4 be a minimal surface in R 4 with conjugate surface f * and let σ = e s+i t ∈ C * . Then the López-Ros deformation f σ of f is given by where f l and f * l are the coordinates of In particular, the López-Ros deformation f σ of f with parameter σ = e s+i t ∈ C * , |σ| 1, is the simple factor dressingf of f with parameters (µ, m, m) where µ = 1−e −(s+it)

From this we see again that the López-Ros deformation is a trivial rotation in the
Proof. Let f σ be the Lopez-Ros deformation of f with parameter σ = e s+i t ∈ C * , |σ| 1. The first equation (37) is an analogue computation as in the proof of Theorem 2.4.
, and the simple factor dressing off with parameter µ is given by Theorem 6.1 as Remark 6.4. In particular, with Proposition 5.8 we see that all simple factor dressings of a minimal surface are given, up to rotations, by the López-Ros deformation applied to a rigid motion of f .
In particular, f µ is closed along γ ∈ π 1 (M), that is, γ * f µ = f µ , if and only if From this, we can immediately compute the periods of all simple factor dressings by Proposition 5.8.
In particular, assume that f : M → R 4 is single-valued on M, and that there exist m, n ∈ H * such that all periods of the conjugate surface f * can be rotated simultaneously into the 1, i-plane, that is, Then all minimal surfaces in the complex 1-parameter family given by the simple factor dressings with parameters (µ, m, n), µ ∈ C \ {0, 1}, are singlevalued on M.
Finally, since a simple factor dressing of a finite total curvature minimal surface is given by a Goursat transformation, it has again finite total curvature: Theorem 6.6. If f : M → R 4 has finite total curvature and if the simple factor dressingf : M → R 4 of f with parameters (µ, m, n) is single-valued on M thenf has finite total curvature.
Proof. Since f has finite total curvature, we can assume by Theorem 2.1 that M =M \{p 1 , . . . , p r } whereM is a Riemann surface punctured at finitely many p i . Moreover, if Φ denotes the holomorphic null curve of f then we can assume that dΦ extends meromorphically into the p i . Since the simple factor dressing is a Goursat transformation, the holomorphic null curveΦ off is given byΦ = AΦ with A ∈ O(4, C). Thus, dΦ extends meromorphically into the punctures p i . 6.2. Simple factor dressing in R 3 . Given a minimal surface in R 3 we now discuss when the simple factor dressingf is a minimal surface in 3-space. Considering a surface in R 3 = Im H as a surface in H with vanishing real part, we immediately see with Theorem 6.1 that a simple factor dressing of f with parameter µ ∈ C \ {0, 1} gives a minimal surface (1−µ) . Moreover, f j and f * j are the coordinates of f = i f 1 + j f 2 +k f 3 and f * = i f * 1 + j f * 2 + k f * 3 respectively. Since any simple factor dressingf of f with parameters (µ, m, n) is given by a simple factor dressing with parameter µ and an operation of R n,m ∈ SO(4, R), we see from (35) thatf is in 3-space if R n,m stabilises C = span R {1, i}. In particular: Theorem 6.7. Let f : M → R 3 be minimal. The simple factor dressingf with parameters (µ, m, n) with m = nλ, λ ∈ C * , is a minimal surfacef : M → R 3 in 3-space.
In particular, f µ is closed along γ if and only if τ 1 = 0 and We can also investigate the behaviour of simple factor dressings in R 3 at ends: Proof. Let p be an complete, embedded, finite total curvature end of f . We can assume that the end of f at p is vertical: if the end is not vertical, let n ∈ H * such thatf = R −1 n,n f has vertical normal at p. Since R m,m = R n,n • R n −1 m,n −1 m andf = R m,m ((R −1 m,m ( f )) µ ) by Proposition 5.8, the simple factor dressing of f is up to rotation given by the simple factor dressing off with parameters (µ, n −1 m, n −1 m).
By [HK97] the periods of the conjugate surface f * around the end are given by res z=0 dΦ. Therefore, if f has a planar end then f * is single-valued on M, and if f has a catenoidal end then the periods of f * are given by −2παk. By Proposition 5.8 and Corollary 6.8, the simple factor dressing is single-valued for all parameters if p is a planar end. Otherwise, it is single-valued for parameters (µ, m, m) such that m −1 km = ±i, that is, m = (1 ∓ j)λ with λ ∈ C.
We know from Corollary 5.9 that the simple factor dressing preserves completeness, that is, the end off at p is complete. Since the simple factor dressingf is a Goursat transformation, the holomorphic null curve off is given byΦ = AΦ with A ∈ O(3, C) and thus, ord z=0 dΦ = −2. At a planar end we have res z=0 dΦ = res z=0 dΦ = (0, 0, 0). Therefore,f has a planar end by Theorem 2.3.

From Proposition 5.8 and (36) we know that
At a catenoidal end, a single-valued simple factor dressing has parameters (µ, m, m) with m = (1 ∓ j)λ, λ ∈ C. Thus, R −1 m,m res z=0 dΦ is an eigenvector with eigenvalue 1 of the matrix L µ . But then res z=0 dΦ = res z=0 dΦ is real, and the end of the simple factor dressing is catenoidal by Theorem 2.3.
Again, we obtain from Theorem 6.3 the link to the López-Ros deformation: Note that if f is a minimal surface with vertical catenoidal end at p, then the proof of Theorem 6.9 shows that the López-Ros deformation is single-valued since 2m In particular, the López-Ros deformation of f has a catenoidal end at p, too.

Darboux transforms of minimal surfaces
We now connect the simple factor dressing of a minimal surface with its Darboux transform. Previous results [CLP13], [Boh10], [LR10], [BQ14] seemed to indicate that the Darboux transformation preserves a surface class which is given by a harmonicity condition as long as the Darboux transform is given by a parallel section of the associated family of the harmonic map. These Darboux transforms are the so-called µ-Darboux transform. We will show that this does not hold for minimal surfaces: a µ-Darboux transform of a minimal surface is a (non-minimal) Willmore surface in R 4 . However, the Darboux transforms are still closely related to the simple factor dressing of the minimal surface, and in particular, a µ-Darboux transform is given by complex holomorphic data.
Let us recall that two isothermic immersions f, f : M → R 4 form a classical Darboux pair [Dar99] if there exists a sphere congruence enveloping both f and f . In particular, a minimal surface f : M → R 3 in R 3 is isothermic, and a classical Darboux transform f = f + T of f is given [HJP97] by a solution T of the Riccati equation where N is the Gauss map of f and r ∈ R * .
By weakening the enveloping condition the notion of a classical Darboux transformation has been extended in [BLPP12] to any conformal immersion f : M → S 4 . In case of a conformal torus f : T 2 → S 4 , there exists at least a Riemann surface worth of Darboux transforms f : T 2 → S 4 of f . This way, one obtains a geometric interpretation of the spectral curve Σ of the conformal torus f as the normalisation of the set of closed Darboux transforms of f .
In this paper however, we are only interested in local theory, so we will assume from now on that M is simply connected. Denoting as before by Here we identify H 2 /L = eH via (π L )| eH : eH → H 2 /L where π L : H 2 → H 2 /L is the canonical projection.
If f : M → R 4 is minimal we have the associated family (20) of flat connections 1) , λ ∈ C * of the harmonic conformal Gauss map S of f , where A is the Hopf field of S . Since im A ⊂ L we see that for fixed µ ∈ C * every d S µ -parallel section ϕ ∈ Γ(H 2 ) has dϕ ∈ Ω 1 (L), and thus L = ϕH is a Darboux transform of f , a so-called µ-Darboux transform of f .
If f has constant right normal R then by (10) the Hopf field A vanishes and d µ = d for all µ ∈ C \ {0, 1}. That is, all µ-Darboux transforms of f are in this case the constant sections Γ(H 2 ). Therefore, from now on we will assume that f is not the twistor projection of a holomorphic curve in CP 3 .
With Proposition 3.2 at hand, we can again discuss all µ-Darboux transforms of f . If ϕ = en, n ∈ H * then the corresponding µ-Darboux transform is the constant point ∞ = eH. On the other hand, every non-constant d S µ -parallel section ϕ ∈ Γ(H 2 ), µ ∈ C \ {0, 1}, is given by Here f * is a conjugate surface of f and a = µ+µ −1 2 , b = i µ −1 −µ 2 and m ∈ H * . The µ-Darboux transform is in this case given by L = (eα + ψβ)H , where β is nowhere vanishing. Therefore, the µ-Darboux transform is given by the affine coordi- We summarise: Theorem 7.1. Every non-constant µ-Darboux transform of a minimal surface f : M → R 4 , M simply connected, is given by where f * is a conjugate surface of f and ρ = m i(1+µ) Note that a µ-Darboux transform depends non-trivially on the choice of the conjugate surface f * .
Moreover, we see with T = αβ −1 and dα = −d f β that Since dβ = dRm by (16) this is a generalised Riccati equation (away from the zeros of α) In particular, if f : M → R 3 is minimal in R 3 then the Gauss map N is the right normal of f , and the above equation generalizes (39). Note however that non-constant µ-Darboux transforms of a minimal surface f : M → R 3 are never classical; for f = f + T to be classical, mα −1 ∈ R * has to hold but α is not constant. However, we will show that the µ-Darboux transformation still preserves geometric information of the minimal surface: it is a Willmore surface which is given by a minimal surface in the associated family of f : of the minimal surface h, the associated Willmore surface g of − h 2 , see Theorem 2.11, is given by Here we used again thatâ 2 +b 2 = 1. In other words, g is by (41) the µ-Darboux transform f of f . By Theorem 7.1 every µ-Darboux transform arises this way.
By Theorem 2.11 every twistor projection of a holomorphic curve in CP 3 which is not minimal in R 4 is the associated Willmore surface of a minimal surface. In particular, there is an induced transformation on Willmore surfaces which are given by complex holomorphic curves in CP 3 : Corollary 7.3. Let g : M → R 4 be the twistor projection of a holomorphic curve in CP 3 and R its right normal. Then In particular, g is the twistor projection of a holomorphic curve in CP 3 .
By Theorem 7.2 and Theorem 2.11 the right normal of a Darboux transform is given by the negative of the right normal of a minimal surface in the right associated family: Corollary 7.4. Let f : M → R 4 be a minimal surface with right normal R.
Then the right normal R of a µ-Darboux transform f of f is given by a simple factor dressing of −R, and vice versa.

Examples
We conclude this paper by demonstrating some of our results for well-known examples of minimal surfaces, including surfaces with one planar end, the first Scherk surface and a punctured torus. In particular, as we can control the periods and the end behaviour at punctures of simple factor dressings by choosing appropriate parameters, we obtain simple factor dressings which are minimal surfaces with one planar end, doubly-periodic surfaces and minimal puncture tori respectively. Our first example is the catenoid for which all computations can be done completely explicitly.
The images were implemented by using the software jReality and the jTEM library of TU Berlin.
8.1. The catenoid. We consider the catenoid f : C → R 3 in the conformal parametrisation where z = x + iy is the standard conformal coordinate on C with * dz = idz. The left and right normal of the catenoid are given by the Gauss map A conjugate surface is the helicoid f * (x, y) = iy + ji sinh x e −iy and, identifying z = x + i y, we obtain the holomorphic null curve and the Weierstrass data g(z) = e z −i e z +i and ω = − i 2 e −z (e z + i ) 2 dz. The right associated family f p,q = f p + f * q is given (26) by f p,q (x, y) = i(px + qy) + je −iy (p cosh x + iq sinh x), p, q ∈ H .  The associated Willmore surface f = f R − f * of f computes to and has right normal R = −R. The associated Willmore surface f is the twistor projection of the holomorphic curve where we used that by Theorem 2.9 and (9) the line subbundle E = F C is given by  The periods of the simple factor dressing with parameter µ are given by Corollary 6.8: Lemma 8.1. The simple factor dressingf of the catenoid with parameters (µ, m, m) has translational periodsf In particular,f (x, y + 2π) =f (x, y) if and only if m ∈ C * or m ∈ C * j or µ ∈ S 1 . In this case,f is a (reparametrised) catenoid.
is the simple factor dressing with parameters (µ, m, m) it is enough to investigate the periods of the simple factor dressingf µ with parameter µ of the minimal surfacef = R −1 m,m f . Sincef (x, y + 2π) =f (x, y) andf * (x, y + 2π) =f * (x, y) + 2πm −1 im we see by Corollary 6.8 that the simple factor dressingf µ has vanishing periods if and only if µ ∈ S 1 or m −1 im = ±i, that is, m ∈ C * or m ∈ C * j. In the former case, the simple factor dressing of f is trivial. In the latter case we see with Lemma 5.5 thatf is the simple factor dressing of f with parameter µ orμ −1 . But the simple factor dressing of f with parameter µ is by (38) given by . Thus, for m ∈ C * we see thatf = f µ is a reparametrisation of the catenoid. Using again Lemma 5.5 we obtain also the case m ∈ C * j.
In the case of general parameters (µ, m, m), m ∈ H * , we obtain with (33) that Thus, in general the simple factor dressing of a catenoid will have translational periods. Although the resulting surfaces resemble Catalan's surface (see Figure 4 and Figure 5) the simple factor dressing of a catenoid has by Corollary 5.9 no branch points.
If we allow the simple factor dressing to be a minimal surface in R 4 we obtain with Corollary 6.5 further closed minimal surfaces: for example, when choosing m = 1+k 2 , n = i− j 2 then the simple factor dressing with parameters (µ, m, n) gives a minimal immersion into R 4 bŷ f (x, y) = ix + k sin y cosh x + (sin y sinh x sinh s + j cos y cosh x cosh s)e − jt , where s = − ln |µ|, t = argμ −1 µ(1−µ) . In particular, we immediately see thatf (x, y + 2π) =f (x, y). Figure 6. Simple factor dressing of the catenoid with parameters (− i 2 , 1+k 2 , i− j 2 ), various orthogonal projections to R 3 .
In particular, the end near the puncture (x, y) = (0, 0) is asymptotic to the plane spanned by (i + k sinh se −it ) and je −it . Figure 12. Simple factor dressing of a minimal surface with one planar end, l = 1, with parameters µ = − i 2 and µ = − 1 2 + i 2 .
By Theorem 2.4 the López-Ros deformation of f with parameter σ = e s+it ∈ C * is x cos t − 1 x 2 +y 2 e −s + 3y 2 −x 2 3 e s − y sin t − 1 x 2 +y 2 e −s + y 2 −3x 2 3 e s x sin t − 1 x 2 +y 2 e −s + 3y 2 −x 2 3 e s + y cos t − 1 x 2 +y 2 e −s + y 2 −3x 2 3 e s x f σ has a vertical planar end at the puncture z = x + iy = 0. From the holomorphic null curves of f and f σ , σ 1, we see that f σ is not a reparametrisation of f . Finally, we include some pictures of the simple factor dressing for more general parameters. Note that the surfaces are single-valued for all parameters (µ, m, m), and have a planar end at z = 0. Figure 14. Simple factor dressing of a minimal surface with one planar end, l = 1, with parameters (µ, m, m) with µ = − i 2 and µ = − 1 2 + i 2 , m = 1 2 − k.