# Simple factor dressing and the López–Ros deformation of minimal surfaces in Euclidean 3-space

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## Abstract

The aim of this paper is to investigate 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 López–Ros deformation of minimal surfaces is a special case of this transformation. We express the simple factor dressing and the López–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.

## 1 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 [42] 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 \(f: M \rightarrow {\mathbb {R}}^3\) from a Riemann surface *M* into 3-space is a harmonic map, minimal surfaces are exactly the real parts holomorphic curves \(\Phi : M \rightarrow {{\mathfrak {C}}}^3\) into complex 3-space. Due to the conformality of *f*, the holomorphic map \(\Phi \) is a null curve with respect to the standard symmetric bilinear form on \({{\mathfrak {C}}}^3\). A particularly important aspect of this approach is that the *Enneper–Weierstrass representation* formula, see [27, 73], allows to construct all holomorphic null curves, and thus all minimal surfaces, from the *Weierstrass data*\((g, \omega )\) where *g* is a meromorphic function and \(\omega \) 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 [19, 38, 47, 50, 55, 58].

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 \(f: M \rightarrow {\mathbb {R}}^3\) and its holomorphic null curve \(\Phi \): firstly, multiplying \(\Phi \) by \(e^{-i\theta }\), \(\theta \in {\mathbb {R}}\), one obtains the *associated family* of minimal surfaces \(f_{\cos \theta , \sin \theta } ={\mathrm{Re}}\,(e^{-i\theta } \Phi )\) as the real parts of the holomorphic null curves \(e^{-i\theta } \Phi \). The associated family of minimal surfaces was introduced by Bonnet [14], 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 Schwarz [66]: 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* [30], is given by any orthogonal matrix \({\mathcal {A}}\in {{\,\mathrm{O}\,}}(3, {{\mathfrak {C}}})\): since \({\mathcal {A}}\) preserves the standard symmetric bilinear form on \({{\mathfrak {C}}}^3\), the holomorphic map \({\mathcal {A}}\Phi \) is a null curve, and \({\mathrm{Re}}\,({\mathcal {A}}\Phi )\) is a minimal surface in \({\mathbb {R}}^3\). As pointed out by Pérez and Ros [58], 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 [48] 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 [51], the discussion of symmetries of embedded genus *k*-helicoids [1], and in an approach to the Calabi–Yau problem [28].

On the other hand, by the Ruh–Vilms theorem the Gauss map of a minimal surface is a harmonic map \(N: M \rightarrow S^2\) from a Riemann surface *M* into the 2-sphere [63]. 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 non-linear sigma models in the physics of elementary particles. Surveys on the remarkable progress in this topic may be found in [23, 24, 31, 36, 56].

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 [53, 60, 67]: 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 [71] integrable systems methods have been highly successful in the geometric study of harmonic maps from Riemann surfaces into suitable spaces, e.g., [3, 5, 21, 37, 70, 72]. 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 in \({\mathbb {R}}^3\) or \(S^3\), e.g., [10, 32, 59], minimal surfaces in \(S^2\times {\mathbb {R}}\), e.g., [33], Hamiltonian Stationary Lagrangians, e.g., [35, 46], and Willmore surfaces, e.g., [4, 11, 34, 64], and related surface classes such as isothermic surfaces, e.g., [2, 6, 18].

We recall the methods of integrable systems which are relevant for our paper: given a \({\mathbb {C}}_*\)-family of flat connections \(d_\lambda \) of the appropriate form, one can construct a harmonic map from it. In particular, the associated family \(d_\lambda \) of flat connections of a harmonic map gives an element of the *associated family of harmonic maps* by, up to a gauge by a \(d_\mu \)-parallel endomorphism, using the family \(d_{\mu \lambda }\) for some fixed \(\mu \in {\mathbb {C}}_*\). The *dressing operation* was introduced by Uhlenbeck and Terng [70, 71]: as pointed out to us by Burstall, in the case of a harmonic map \(N: M \rightarrow S^2\) the dressing is given by a gauge \({\hat{d}}_\lambda = r_\lambda \cdot d_\lambda \) of \(d_\lambda \) by a \(\lambda \)-dependent dressing matrix \(r_\lambda \) [9]. The *dressing* of *N* is then the harmonic map \({\hat{N}}\) that has \({\hat{d}}_\lambda \) 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 \(r_\lambda \) has a simple pole \(\mu \in {\mathbb {C}}_*\) and is given by a \(d_\mu \)-parallel bundle, then the so-called *simple factor dressing* can be computed explicitly, e.g., [9, 20, 70].

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 [37], and in applications of his methods to constant mean curvature and Willmore tori, e.g., [59, 64]. The holonomy representation of the family \(d_\lambda \) 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*\(\Sigma \), a hyperelliptic curve over \({\mathbb {C}}{\mathbb {P}}^1\) (which is independent of the chosen base point), together with a holomorphic line bundle over \(\Sigma \), 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 \(\Sigma \). This idea can be extended to a more general notion [68] of a spectral curve for conformal tori \(f: T^2 \rightarrow S^4\). 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 [13].

As mentioned above, by the Ruh–Vilms theorem the Gauss map of a minimal immersion \(f: M \rightarrow {\mathbb {R}}^3\) 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 [22], the investigation of minimal surfaces with these dressed harmonic Gauss maps complicates. On the other hand, Meeks, Pérez and Ros [49, 52], use algebro-geometric 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 [12]. 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 \(d_\lambda \) and \(d^S_\lambda \) 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 \(d^S_\lambda \) can be expressed in terms of parallel sections of \(d_\lambda \) and generalisations \(f_{p,q}\), \(p, q\in S^3\), of the associated family of minimal surfaces \(f_{\cos \theta , \sin \theta }\). 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 \(N: M \rightarrow S^2\) one can define the *associated family of harmonic maps* of the harmonic conformal Gauss map of a Willmore surface [8]. 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 [4]. 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 [4, 43].

*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 \((\mu , m, n)\) where \(\mu \in {\mathbb {C}}{\setminus }\{0\}\) is the pole of the simple factor dressing, and \(m, n \in S^3\) determine the \(d^S_\mu \)-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 \(f: M \rightarrow {\mathbb {R}}^3\) 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 \(\mu \in {\mathbb {R}}\) the simple factor dressing is the minimal surface

*f*and a conjugate \(f^*\) of

*f*. In this case, we see immediately that \(f^\mu \) is a Goursat transformation of

*f*with holomorphic null curve \({\mathcal {L}}^\mu \Phi \) where \(\Phi = f+\mathbf{i\,}f^*\) is the holomorphic null curve of

*f*and

*M*then finite total curvature is preserved, too.

In the case when \(m=n\in S^3\), the orthogonal matrix of the Goursat transformation is given as \({\mathcal {R}}_{m,m} {\mathcal {L}}^\mu {\mathcal {R}}_{m,m}^{-1}\in {{\,\mathrm{O}\,}}(3, {{\mathfrak {C}}})\) where the rotation matrix \({\mathcal {R}}_{m,m}\) in 3-space is given by \(m\in S^3\subset {\mathbb {R}}^4\): decomposing \( m = (\cos \theta , q \sin \theta ) \) with \(q\in {\mathbb {R}}^3, ||q||=1\), the matrix \({\mathcal {R}}_{m,m}\) is the rotation along the axis given by *q* about the angle \(2\theta \). In other words, the simple factor dressing in \({\mathbb {R}}^3\) with parameters \((\mu , m, m)\) is obtained from the simple factor dressing (1) with parameter \(\mu \) applied to the (inverse of the) rotation given by *m*.

*f*with parameter \(\sigma \in {\mathbb {R}}, \sigma >0,\) is indeed given by

*f*with parameters \((\mu , m, m)\).

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 \((\mu , m, n)\) for a simple factor dressing to be single-valued. 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.

We conclude the paper by demonstrating our results for various well-known minimal surfaces, including the catenoid, Richmond surfaces 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 \(\mu =-\frac{1}{\sqrt{q}}\) 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.

## 2 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 \({\mathbb {R}}^3\), some of our transforms will be surfaces in \({\mathbb {R}}^4\). Therefore, we will study more generally minimal immersions in \({\mathbb {R}}^4\) and specialise to the case of minimal surfaces in 3-space when appropriate.

### 2.1 Minimal surfaces in \({\mathbb {R}}^4\)

*M*into 4-space. If

*f*is minimal, then

*f*is harmonic, i.e.,

*M*, thus, \(*\) is the negative Hodge star operator. In particular, \(*df\) is closed if

*f*is harmonic and there exists a

*conjugate surface*\(f^*\) on the universal cover \({\tilde{M}}\) of

*M*, given up to translation by

*associated family*(when lifting

*f*to the universal cover \({{\tilde{M}}}\)), e.g. [25],

*M*if \(f\circ \pi ^{-1}: M \rightarrow {\mathbb {R}}^4\) is well-defined where \(\pi : {{\tilde{M}}} \rightarrow M\) is the canonical projection of the universal cover \({{\tilde{M}}}\) to

*M*. In this case, we will identify

*f*and \(f\circ \pi ^{-1}\) and write, in abuse of notation, from now on \(f: M \rightarrow {\mathbb {R}}^4\).

*left*and

*right normal*\(N, R: M \rightarrow S^2=\{n\in {\mathrm{Im}}\,{\mathbb {H}}\mid n^2=-1\}\) of

*f*by

*mean curvature vector*\({\mathcal {H}}\) of \(f: M \rightarrow {\mathbb {R}}^4\) satisfies [7, p. 39]

*dR*with respect to the complex structure

*R*. Then the equation of the mean curvature vector becomes

*f*is minimal then

*f*. Thus, both

*N*and

*R*are quaternionic holomorphic sections [29] with respect to the induced quaternionic holomorphic structures on the trivial \({\mathbb {H}}\) bundle \({\underline{{\mathbb {H}}}}^{} = M \times {\mathbb {H}}\). Note also that a map \(R: M\rightarrow S^2\) is harmonic if and only if

*N*and

*R*of a minimal surface are conformal and harmonic.

*f*has the same left and right normal as

*f*since

*f*is harmonic and \(*df =-df^*\), the map

*i*. If \(f: M \rightarrow {\mathbb {R}}^4\) is a minimal conformal immersion, then the holomorphic curve

*f*is given by \(\Phi _{\cos \theta , \sin \theta } = e^{-\mathbf{i\,}\theta } \Phi \) where \(\Phi \) is the holomorphic null curve of

*f*.

*Weierstrass data*of

*f*is given by the meromorphic functions

*m*is the maximum order of poles of \(g_1, g_2,\) and \(g_1^2+g_2^2\) at \(p\in M\) then \(\omega \) has a zero at

*p*of order at least

*m*. Then \((g_1, g_2, \omega )\) gives rise [39] to a holomorphic null curve \(\Phi \), and thus a minimal surface \(f={\mathrm{Re}}\,(\Phi ): {{\tilde{M}}} \rightarrow {\mathbb {R}}^4\), via

*f*to be branched which happens when the order of \(\omega \) at

*p*is bigger than the maximum order of poles of \(g_1, g_2,\) and \(g_1^2+g_2^2\) at \(p\in M\). Note that

*f*is in general only defined on the universal cover \({{\tilde{M}}}\) of

*M*.

*f*is a map into the Grassmannian of oriented two planes in \({\mathbb {R}}^4\). In our case, it is locally given by \(\varphi dz=d\Phi \) where the two-plane

*G*(

*p*) in \({\mathbb {R}}^4\) is spanned by \({\mathrm{Re}}\,\varphi , {\mathrm{Im}}\,\varphi \) at

*p*since \(d\Phi = df +\mathbf{i\,}df^* = df - \mathbf{i\,}*df\). But \( {{\,\mathrm{Gr}\,}}_2({\mathbb {R}}^4) = {{\mathfrak {C}}}{\mathbb {P}}^1\times {{\mathfrak {C}}}{\mathbb {P}}^1\) and the Gauss map

*G*can be identified [57] with the two meromorphic functions \(G_1, G_2: M \rightarrow {{\mathfrak {C}}}\):

*N*and

*R*by

*f*.

Finally we recall the following result due to Chern and Osserman:

### Theorem 2.1

([17, 54]) Let \(f: M \rightarrow {\mathbb {R}}^4\) be a complete (branched) minimal immersion with holomorphic null curve \(\Phi : M \rightarrow {{\mathfrak {C}}}^4\).

Then *f* has finite total curvature if and only if *M* is conformally equivalent to a compact Riemann surface \({\bar{M}}\) punctured at finitely many points \(p_1, \ldots , p_r\) such that \(d\Phi \) extends meromorphically into punctures \(p_i\).

### 2.2 Minimal surfaces in \({\mathbb {R}}^3\)

*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 \({\mathbb {R}}^3\), the Gauss map is thus both harmonic and conformal, that is,

*dN*with respect to the complex structure

*N*.

*f*as

*g*and a holomorphic 1-form \(\omega \), such that if

*g*has a pole of order

*m*at

*p*then \(\omega \) has a zero of order at least 2

*m*, give a minimal surface \(f: {{\tilde{M}}} \rightarrow {\mathbb {R}}^3\) as \(f = {\mathrm{Re}}\,(\Phi )\) where \(\Phi \) is given by the Enneper–Weierstrass representation [27, 73],

*g*,

*dh*) given in terms of the height differential \(dh = g \omega \). Written in terms of (

*g*,

*dh*) the Enneper–Weierstrass representation becomes

*f*is given in terms of the Weierstrass data [41] as

*g*via stereographic projection

As before, complete minimal immersions of finite total curvature can be characterised by the holomorphic null curve \(\Phi \):

### Theorem 2.2

([57]) Let \(f: M \rightarrow {\mathbb {R}}^3\) be a complete (branched) minimal immersion with holomorphic null curve \(\Phi : M \rightarrow {{\mathfrak {C}}}^3\).

Then *f* has finite total curvature if and only if *M* is conformally equivalent to a compact Riemann surface \({\bar{M}}\) punctured at finitely many points \(p_1, \ldots , p_r\) such that \(d\Phi \) 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 \(f: M \rightarrow {\mathbb {R}}^3\) 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 \(D_* = D{\setminus }\{0\}\) and is centered at *p*.

- (i)
*f*has an embedded finite total curvature end at*p*. - (ii)
\(d\Phi \) has order \(-2\) at \(z=0\) and \({{\,\mathrm{res}\,}}_{z=0}d \Phi \) is real.

*planar*, otherwise, it is

*catenoidal*.

Here, \(\Phi =(\Phi _1, \Phi _2, \Phi _3)\) is the holomorphic null curve of *f* and \({{\,\mathrm{ord}\,}}_{z=0}d\Phi \) is the minimum of \({{\,\mathrm{ord}\,}}_{z=0} d\Phi _i\) for \(i=1,2,3\).

### Proof

We first assume that the end is vertical. By [38] an embedded complete finite total curvature vertical end has logarithmic growth \(\alpha \in {\mathbb {R}}\) satisfying \({{\,\mathrm{res}\,}}_{z=0} d\Phi = -(0, 0, 2\pi \alpha )\). The end is planar if \(\alpha =0\) 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 \({{\,\mathrm{ord}\,}}_{z=0} d\Phi =-2\).

If the end is not vertical, we can apply a rotation \({\mathcal {R}}\in {{\,\mathrm{SO}\,}}(3, {\mathbb {R}})\) on *f* to obtain a minimal surface \({{\tilde{f}}} = {\mathcal {R}} f\) with a vertical end. Then the holomorphic null curve of \({{\tilde{f}}}\) is \({\tilde{\Phi }} ={\mathcal {R}} \Phi \) and thus, \({{\,\mathrm{ord}\,}}_{z=0} d\Phi ={{\,\mathrm{ord}\,}}_{z=0} d{\tilde{\Phi }} =-2\). Moreover, the residues at \(z=0\) vanish at a planar end, whereas \({{\,\mathrm{res}\,}}_{z=0}d\Phi = {\mathcal {R}}^{-1}{{\,\mathrm{res}\,}}_{z=0}d\tilde{\Phi }\) is real for a catenoidal end.

Conversely, we will show that if \({{\,\mathrm{ord}\,}}_{z=0} d\Phi =-2\) and \({{\,\mathrm{res}\,}}_{z=0}d\Phi \) 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 \(m\ge 1\) and \({{\,\mathrm{ord}\,}}_{z=0} dh = m-2\) holds for the height differential. From this we conclude that the end has finite total curvature.

*g*has a pole or zero at

*p*of order \(m\ge 1\). Since the residue of \(d\Phi \) at \(z=0\) vanishes, we see that

*dh*cannot have a simple pole. But then \({{\,\mathrm{ord}\,}}_{z=0} d\Phi =-2\) implies that

*dh*is holomorphic with \({{\,\mathrm{ord}\,}}_{z=0} dh=m-2\), \(m\ge 2\), so that with the Enneper–Weierstrass representation (6)

*p*is a planar end.

If the order of \(d\Phi =-2\) and \({{\,\mathrm{res}\,}}_{z=0} d\Phi \not =0\) is real then we can rotate the end so that \({{\,\mathrm{res}\,}}_{z=0} d\Phi =-(0,0,2\pi \alpha )\) for some \(\alpha \in {\mathbb {R}}_*={\mathbb {R}}{\setminus }\{0\}\). Then the order of *dh* is either \(-1\) or \(-2\).

*dh*at the end is \(-2\). We write

*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 \(\Phi \) can be written as

*z*| we see that in the limit \(r\rightarrow \infty \) the multiplicity is again 1, and the end is embedded by [40]. Following the arguments in the proof of [38, Prop. 2.1],

*f*is a graph over (the exterior of a bounded domain of) the \((f_1, f_2)\) plane with asymptotic behaviour

The *López–Ros deformation* of a minimal surface \(f: M \rightarrow {\mathbb {R}}^3\) with Weierstrass data \((g, \omega )\) is [48] the minimal surface \(f_r: {{\tilde{M}}} \rightarrow {\mathbb {R}}^3\) given by the new Weierstrass data \((r g, \frac{\omega }{r})\) with \(r\in {\mathbb {R}}_* \). Obviously this can be extended to a deformation \(f_\sigma \) with complex parameter \(\sigma \in {{\mathfrak {C}}}_*={{\mathfrak {C}}}{\setminus }\{0\}\) by using the Weierstrass data \((\sigma g, \frac{\omega }{\sigma })\).

Since any matrix in \({{\,\mathrm{O}\,}}(3,{{\mathfrak {C}}})=\{ {\mathcal {A}} \in \mathrm {GL}(3, {{\mathfrak {C}}}) \mid {\mathcal {A}}^t = {\mathcal {A}}^{-1}\}\) preserves the standard symmetric bilinear form on \({{\mathfrak {C}}}^3\), we obtain new minimal surfaces via the Goursat transformation [30]: if \(f: M\rightarrow {\mathbb {R}}^3\) is minimal with holomorphic null curve \(\Phi = f+\mathbf{i\,}f^*\) then \({\mathcal {A}}\Phi \) is again a holomorphic null curve and \({\mathrm{Re}}\,({\mathcal {A}} \Phi )\) is a minimal surface in \({\mathbb {R}}^3\).

As pointed out by Pérez and Ros [58], the López–Ros deformation is a special case of the Goursat transformation:

### Theorem 2.4

### Proof

*f*. Putting

*f*. Indeed, if \((g, \omega )\) denotes the Weierstrass data of

*f*then the Weierstrass data of \({{\tilde{f}}}\) computes to

*f*with parameter \(\sigma \in {{\mathfrak {C}}}_*\). \(\square \)

### 2.3 Willmore surfaces

*M*whose fibers at \(p\in M\) are given by

*L*is given by

*L*given by an immersion \(f: M \rightarrow {\mathbb {R}}^4\) via \(L \mapsto B L, B\in {{\,\mathrm{GL}\,}}(2,{\mathbb {H}})\). Every pair of unit quaternions \(m, n\in S^3\) gives an element \({\mathcal {R}}_{m,n} \in {{\,\mathrm{SO}\,}}(4,{\mathbb {R}})\) by

*L*of an immersion \(f: M \rightarrow {\mathbb {R}}^4\) is given by

### Definition 2.5

([7, p. 27]). The *conformal Gauss map* of a conformal immersion \(f: M \rightarrow S^4\) is the unique complex structure *S* on \({\underline{{\mathbb {H}}}}^{2}\) such that *S* and *dS* stabilise the line bundle *L* of *f* and its Hopf field *A* is a 1-form with values in *L*.

*Hopf field*

*A*of

*S*is the 1-form given by

*S*with respect to the complex structure

*S*.

*N*,

*R*are the left and right normal of

*f*and \(H = -R{\bar{{\mathcal {H}}}}\) with mean curvature vector \({\mathcal {H}}\). Thus, we see that \(S\psi = -\psi R\) and

*S*and

*dS*indeed stabilise the line bundle \(L=\psi {\mathbb {H}}\) of

*f*. In affine coordinates, the Hopf field computes [7, Prop 12, p. 42] to

*A*is indeed a 1-form with values in

*L*. Note that the Hopf field

*A*is holomorphic [7, p. 68] with \({{\,\mathrm{im}\,}}A \subset L\). In particular, if \(A \not \equiv 0\), the Hopf field

*A*gives the line bundle

*L*by holomorphically extending \({{\,\mathrm{im}\,}}A\) into the isolated zeros of

*A*. Therefore, when fixing the point at \(\infty \), the immersion \(f: M \rightarrow {\mathbb {R}}^4\) is uniquely determined by

*A*.

### Corollary 2.6

*A*of the conformal Gauss map

*S*of

*f*satisfies

In particular, if \(f: M\rightarrow {\mathbb {R}}^4\) is minimal then \(*dR = - RdR\) and thus \(df\wedge dR =0\) by type arguments. Therefore, the Hopf field is harmonic, that is, \(d*A =0\).

### Theorem 2.7

From this characterisation of Willmore surfaces and the fact that the conformal Gauss map of a minimal surface is harmonic we see:

### Corollary 2.8

Every minimal immersion in \({\mathbb {R}}^4\) is a Willmore surface in \({\mathbb {R}}^4\).

## 3 Harmonic maps and their associated families of flat connections

It is well-known [71] 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 [70] and Darboux transforms [9, 15]. In this paper, we investigate the links between the first two transformations, when applied to the left and right normals and to the conformal Gauss map of a minimal surface \(f: M \rightarrow {\mathbb {R}}^4\). For the resulting Darboux transforms and their relation to the dressing we refer to [44].

### 3.1 The harmonic right normal and its associated family

We equip \({\mathbb {H}}\) with the complex structure *I* which is given by the right multiplication by the unit quaternion *i*. This way, we identify \({\mathbb {C}}^2=({\mathbb {H}}, I)\). It is worthwhile to point out that this complex structure *I* differs from the complex structure \(\mathbf{i\,}\) we used before. We will use the symbol \({\mathbb {C}}={{\,\mathrm{span}\,}}_{\mathbb {R}}\{1, I\}\) to indicate that we use the complex structure *I* whereas \({{\mathfrak {C}}}= {{\,\mathrm{span}\,}}_{\mathbb {R}}\{1, \mathbf{i\,}\}\).

*M*can be written as, see for example [9, 15],

*d*is the trivial connection on \({\underline{{\mathbb {H}}}}^{}\), \(\lambda \in {\mathbb {C}}_*={\mathbb {C}}{\setminus }\{0\}\), and \(Q=-\frac{1}{2}(*dR)'' = \frac{1}{4}(RdR -*dR)\) is the Hopf field of

*R*. Moreover,

*Q*with respect to the complex structure

*I*.

*R*is the right normal of a minimal surface \(f: M\rightarrow {\mathbb {R}}^4\) the map

*R*is conformal with \(*dR=-RdR\), so that \( 2*Q = dR\) and, using \(RdR =-dRR\),

*I*operates from the right we obtain \( d_\lambda = d+ \alpha _\lambda \) with

*R*is obtained from (5), that is, from \(d*Q =0\).

*m*is constant: since \(a^2+b^2=1\), \(RdR =-dRR\) and \(d\beta = dRm \) by (13) we have

### Lemma 3.1

*R*of

*f*. For \(\mu \in {\mathbb {C}}{\setminus }\{0,1\}\) every (non-trivial) \(d_\mu \)-parallel section \(\beta \in \Gamma ({\underline{{\mathbb {H}}}}^{})\) is given by

In a similar way, the associated family of the left normal of a minimal surface can be discussed.

### Remark 3.2

### 3.2 The conformal Gauss map and its associated family

*I*is given by right multiplication by the unit quaternion

*i*. If the conformal Gauss map of a conformal immersion \(f: M \rightarrow S^4\) is harmonic, that is \(d*A=0\), by the same arguments as in the case of harmonic maps into the 2-sphere, the \({\mathbb {C}}_*\)-family of connections

*M*where as before

*A*with respect to

*I*.

*S*is the conformal Gauss map of a minimal immersion in \({\mathbb {R}}^4\). We fix \(\mu \in {\mathbb {C}}_*\) and compute all parallel sections of \(d^S_\mu \). If \(\mu =1\) then \(d^S_\mu =d\) is trivial, and every constant section is parallel. Assume from now on that \(\mu \not =1\), and let

*f*. We denote by \( \widetilde{{\underline{{\mathbb {H}}}}}^{2}\) the pull back of the trivial \({\mathbb {H}}^2\) bundle under the canonical projection \(\pi : {{\tilde{M}}} \rightarrow M\) of the universal cover \({{\tilde{M}}}\) to

*M*. Since \(e{\mathbb {H}}\oplus L ={\underline{{\mathbb {H}}}}^{2}\) every \(d^S_\mu \)-parallel section \(\varphi \in \Gamma (\widetilde{{\underline{{\mathbb {H}}}}}^{2})\) can be written as

*f*and thus from (11) we see

*f*, that is,

### Proposition 3.3

*S*of

*f*. For \(\mu \in {\mathbb {C}}{\setminus }\{0,1\}\) every \(d^S_\mu \)-parallel section is either a constant section \(\varphi =e n, n\in {\mathbb {H}}\), or is given by \(\varphi = e\alpha + \psi \beta \in \Gamma (\widetilde{{\underline{{\mathbb {H}}}}}^{2})\) with

## 4 The associated family of a minimal surface

Motivated by our observation (22) that parallel sections of the associated family of flat connections of a minimal surface \(f: M \rightarrow {\mathbb {R}}^4\) are given by a quaternionic linear combination of *f* and its conjugate surface \(f^*\), we now discuss a generalisation of the associated family of a minimal surface. This associated family is indeed given by the associated family of the harmonic conformal Gauss map.

### 4.1 The generalised associated family

Given a minimal surface \(f: M \rightarrow {\mathbb {R}}^4\) with left and right normal *N* and *R* respectively, we have seen that its conjugate surface \(f^*: {{\tilde{M}}} \rightarrow {\mathbb {R}}^4\) has the same left and right normals *N* and *R* respectively since \(df^* =-*df\).

*R*of

*f*gives, away from the isolated zeros of \(p+Rq\), via

*N*of

*f*is the left normal

### Definition 4.1

*f*, is called the

*right associated family*of

*f*. The family of (branched) minimal immersion

*left associated family*of

*f*.

Note that for \(p,q\in {\mathbb {R}}, (p,q)\not =(0,0),\) 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 \({\mathbb {R}}^4\) for \(n\in {\mathbb {H}}_*\).

### Theorem 4.2

The right (left) associated family is a \(S^3\)-family of minimal surfaces in \({\mathbb {R}}^4\) which contains the classical associated family \(f_{\cos \theta , \sin \theta }, \theta \in {\mathbb {R}},\) 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 \({\mathbb {R}}^4\).

### Proof

*f*if and only if \(f_{p, q} = f_{n\cos \theta , n\sin \theta }\) for some \(\theta \in {\mathbb {R}}\) and \(n\in S^3\). Assume first that \(f_{p,q}\) is isometric to

*f*. By multiplying with a unit quaternion from the right we may assume that \(p\in {\mathbb {R}}\). Then \(|df_{p,q}| = |df|\) implies by (23)

*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 \({\mathrm{Re}}\,(Rq) = {\mathrm{Re}}\,(R\, {\mathrm{Im}}\,q) = -< R, {\mathrm{Im}}\,q>\) is constant only if \({\mathrm{Im}}\,q=0\). But then the above equation gives \(p^2+q^2=1\) with \(q\in {\mathbb {R}}\) and \((p,q) = (\cos \theta ,\sin \theta )\) for some \(\theta \in {\mathbb {R}}\). If

*R*is constant then \(df^* = - *df = df R\) gives \(f_{p,q} = f(p+Rq) = f_{p+Rq, 0}\) with constant \(p+Rq\in S^3\).

Conversely, \(f_{p,q} = f_{n\cos \theta , n\sin \theta }, n\in S^3\), then \(|p+Rq|=1\) and thus \(|df_{p,q}| = |df|\) by (23).

A similar argument shows the statement for the left associated family. \(\square \)

### Remark 4.3

For any immersion \(f: M \rightarrow {\mathbb {R}}^3 ={\mathrm{Im}}\,{\mathbb {H}}\) in Euclidean 3-space, the left and right normal coincide. A surface in the right associated family \( f_{p,q} \) of a minimal surface \(f: M \rightarrow {\mathbb {R}}^3\) has left normal \(N_{p,q} = N\) and right normal \(R_{p,q} = (p+Nq)^{-1}N(p+Nq)\). In particular, we have \(N_{p,q} \not = R_{p,q}\) in general and thus, elements of the right associated families of a minimal surface \(f: M \rightarrow {\mathbb {R}}^3\) are not necessarily minimal in 3-space but are minimal surfaces in \({\mathbb {R}}^4\).

### 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 \(f:M \rightarrow {\mathbb {R}}^3\) the Gauss map \(N: M \rightarrow S^2\) of *f* is by the Ruh–Vilms theorem [63] harmonic and its associated family \({\hat{d}}_\lambda \) of flat connections (18) gives rise to the associated family of constant mean curvature surfaces: for \(\mu \in {\mathbb {C}}_*\) the map *N* is harmonic with respect to the quaternionic connection \({\hat{d}}_\mu \) for all \(\mu \in S^1\), see e.g. [9] for details. Thus, for any \({\hat{d}}_\mu \)-parallel section \(\varphi \in \Gamma ({\underline{{\mathbb {H}}}}^{})\) the map \(N_\mu = \varphi ^{-1}N\varphi \) is harmonic with respect to *d*. Note that \(\varphi \) is unique up to a right multiplication by a constant quaternion. This family \(N_\lambda \), \(\lambda \in S^1\), is called the *associated family* of *N*. The Sym–Bobenko formula [10] relies on the link (4) between the differentials of *f* and *N* to obtain the constant mean curvature surface *f* by differentiating a family \(\varphi _\lambda \) of parallel sections of \(d_\lambda \) with respect to the parameter \(\lambda \). This way, one can obtain from the associated family \(N_\lambda \) of *N* a family of constant mean curvature surfaces, the *associated family* of *f*.

*R*, we have seen in Lemma 3.1 that every parallel section \(\beta \in \Gamma ({\underline{{\mathbb {H}}}}^{})\) of the associated family of flat connections of

*R*is, using (16), given by

To obtain a non-trivial family of minimal surfaces with right normal \(R_\mu \) for \(\mu \in {\mathbb {C}}_*\), we consider instead the harmonic conformal Gauss map *S* of *f*:

### Theorem 4.4

*S*. Then, up to Möbius transformation, the line bundle of a minimal surface \(f_{\cos \theta , \sin \theta }, \theta \in {\mathbb {R}},\) in the associated family of

*f*is given by

### Proof

*B*constant so that \(L_{{\tilde{\phi }}} = B^{-1}L_\phi \) is given by a Möbius transformation. Thus, we can assume without loss of generality that for \(\mu \not =1\)

By definition \(S_\phi \) leaves \(L_\phi \) invariant. The final statement follows by a similar argument as in [9]. In fact, this is a special case of a corresponding statement for (constrained) Willmore surfaces \(f: M \rightarrow {\mathbb {R}}^4\), see [4, 8]. \(\square \)

## 5 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 \(d_\lambda \) is the associated family of flat connections of a harmonic map, one can gauge \(d_\lambda \) with a \(\lambda \)-dependent dressing matrix \(r_\lambda \) to obtain a new family of flat connections \(\breve{d}_\lambda = r_\lambda \cdot d_\lambda \). If \(r_\lambda \) satisfies appropriate reality and holomorphicity conditions, then \(\breve{d}_\lambda \) is the associated family of a new harmonic map, a so-called *dressing* of the original harmonic map, see [70, 71].

### 5.1 Simple factor dressing of the left and right normals

*R*then the simple factor dressing matrix \(r_\lambda \) is obtained by choosing \(\mu \in {\mathbb {C}}{\setminus }\{0,1\}\) and a parallel section \(\beta \) of the flat connection \(d_\mu \) of the associated family as

*simple factor dressing*of

*R*, and

*R*is

*R*so that also

### Theorem 5.1

Let \(f: M \rightarrow {\mathbb {R}}^4\) 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*.

### Proof

Let \(\mu \in {\mathbb {C}}{\setminus }\{0,1\}\) be the pole of the simple factor dressing and put, as usual, \(a= \frac{\mu +\mu ^{-1}}{2}, b=\frac{i(\mu ^{-1}-\mu )}{2}\). Write \({\hat{c}} = m c m^{-1}\) for \(c\in {\mathbb {C}}\) where \(m\in {\mathbb {H}}_*\) determines the \(d_\mu \)-parallel bundle of the simple factor dressing.

*f*. From (24) we see that the right normal of \(h= f_{{\hat{b}}, {\hat{a}}-1}\) is given by

*R*so that also

*h*is (26) the dressing of

*R*. \(\square \)

### 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 \(f_{\hat{b}, {\hat{a}}-1}\) is one example of such a minimal surface but so is \(pf_{{\hat{b}}, {\hat{a}}-1} + qf^*_{\hat{b}, {\hat{a}}-1}\) for any \(p, q \in {\mathbb {H}}_*. \)

An analogue theorem holds for the left normal *N* of a minimal surface \(f: M \rightarrow {\mathbb {R}}^4\):

### 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 \(f: M \rightarrow {\mathbb {R}}^3\) is a minimal surface in \({\mathbb {R}}^3\) then the left and right associated families give in general minimal surfaces in \({\mathbb {R}}^4\). In particular:

### Corollary 5.4

Let \(f: M \rightarrow {\mathbb {R}}^3\) be a minimal surface with Gauss map *N* and assume that *f* is not a plane. Let \(f_{{\hat{b}}, {\hat{a}}-1}\) be the minimal surface in the associated family of *f* whose right normal \(R_{{\hat{b}}, {\hat{a}}-1}\) is the simple factor dressing of *N* given by \(\mu \in {\mathbb {C}}{\setminus }\{0,1\}\), \(m\in {\mathbb {H}}_*\), where \({\hat{a}} = m \frac{\mu +\mu ^{-1}}{2}m^{-1}, {\hat{b}} = m i\frac{\mu ^{-1}-\mu }{2}m^{-1}\).

Then \(f_{\hat{b}, {\hat{a}}-1}\) is a minimal surface in \({\mathbb {R}}^3\) if and only if \(\mu \in S^1, \mu \not =1\).

### Proof

Conversely, for \(\mu \in S^1\) we know \({\hat{a}}, {\hat{b}}\in {\mathbb {R}}\) so that \(f_{{\hat{b}}, {\hat{a}}-1} = f {\hat{b}} + f^*({\hat{a}}-1)\) takes values in \({\mathbb {R}}^3\). \(\square \)

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 [4, 61]. 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).

*S*of a Willmore surface \(f: M\rightarrow S^4\) is given explicitly by parallel sections of a connection \(d^S_\mu \) of the associated family of flat connections [43]: for \(\mu \in {\mathbb {C}}{\setminus }\{0,1\}\) let \(W_\mu \) be a \(d^S_\mu \)-stable, complex rank 2 bundle over \({{\tilde{M}}}\) with \(W_\mu \oplus W_\mu j =\widetilde{{\underline{{\mathbb {H}}}}}^{2}\). For two \(d^S_\mu \)-parallel sections \(\varphi _1, \varphi _2\in \Gamma (W_\mu )\) with \(\phi =(\varphi _1, \varphi _2)\) regular, define a conformal immersion \({\hat{f}}: {{\tilde{M}}} \rightarrow S^4\) by

*simple factor dressing*of

*S*. In particular, \({\hat{S}}\) is harmonic, and \({\hat{f}}\) is a Willmore surface. It is known that \({\hat{L}}\), and thus \({\hat{f}}\), is independent of the choice of basis for \(W_\mu \) [43]. We call the Willmore surface \({\hat{f}}\) a

*simple factor dressing*of

*f*. Note that \(\frac{b}{a-1}\in {\mathbb {R}}\) for \(\mu \in S^1\) so that \({\hat{f}} = f\) in this case.

Note that this gives a conformal theory. However, in the case of a minimal surface \(f: M \rightarrow {\mathbb {R}}^4\) 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 \(S + \phi \frac{b}{a-1} \phi ^{-1}\) stabilises the point at infinity \(\infty = e{\mathbb {H}}\). Because \(Se = eN\) by (9) where *N* is the left normal of *f*, we have to restrict to \(\phi \) with \(\phi \frac{b}{a-1} \phi ^{-1}\infty = \infty \). Since \(\frac{b}{a-1}\in {\mathbb {R}}\) if and only if \(\mu \in S^1\) we can assume that \(\frac{b}{a-1}\) is not real as otherwise \({\hat{f}} =f\).

*f*, and

*R*. Assume that \(\phi \frac{b}{a-1} \phi ^{-1}\) fixes the point at infinity. Then there exists \(\eta : M \rightarrow {\mathbb {H}}_*\) with

Both \(\beta _1\) and \( \beta _2\) are \(d_\mu \)-parallel, and thus, \(q\in {\mathbb {C}}_*\) is constant.

If *R* is constant, that is, if *f* is the twistor projection of a holomorphic curve in \({\mathbb {C}}{\mathbb {P}}^3\), then \(f^* = fR + c\) with some \(c\in {\mathbb {H}}\), so that with (30) and (31) we obtain \(\varphi _1 + \varphi _2 q= en\) with \(n= -c(m_2 q+m_1)\) and \(q\in {\mathbb {C}}_*\). In other words, *en* is a \(d^S_\mu \)-parallel section of \( W_\mu \) and we can replace \(\phi \) in the definition of \({\hat{S}}\) by \({\tilde{\phi }} = (en, \varphi _2)\).

If *R* is not constant, we use again the explicit forms (31) to obtain from \(\beta _1 + \beta _2q=0\), \(q\in {\mathbb {C}}_*\), that \(R(m_1 + m_2q)\) is constant, which implies that \(m_1 =- m_2q\). But then (30) and (31) show that \(\varphi _1 = -\varphi _2 q\) which contradicts the assumption that \(\phi \) is regular.

### Theorem 5.5

Let \(f: M \rightarrow {\mathbb {R}}^4\) be a minimal surface and \({\hat{f}}: {{\tilde{M}}} \rightarrow {\mathbb {R}}^4\) a simple factor dressing of *f* in 4-space.

Then \({\hat{f}}\) is a minimal surface, and \({\hat{f}}\) is preserved when changing the parameters \((\mu , m, n)\) to \((\mu , mz, nw)\) or \(({\bar{\mu }}^{-1}, mj, nj)\) for \(z, w\in {\mathbb {C}}_*\). For \(\mu \in S^1\) the simple factor dressing \({\hat{f}}= f\) is trivial.

### Proof

*f*. In particular, \({\hat{f}}\) is minimal. From the explicit formula above we see that \({\hat{f}}\) is preserved when changing \((\mu , m, n)\) to \((\mu , mz, nw)\) with \(z, w\in {\mathbb {C}}_*\). Since \(\frac{{\bar{\mu }}^{-1}+ {\bar{\mu }}}{2}={\bar{a}}, i \frac{{\bar{\mu }} - {\bar{\mu }}^{-1}}{2} = {\bar{b}}\) and \((mj) {\bar{z}} (mj)^{-1}= mzm^{-1}\) for all \(z\in {\mathbb {C}}, m\in {\mathbb {H}}_*\), we obtain the same simple factor dressing for the parameters \((\mu , m, n)\) and \(({\bar{\mu }}^{-1}, mj, nj)\). The final statement follows from the fact that \(\frac{b}{a-1} \in {\mathbb {R}}\) for \(\mu \in S^1\). \(\square \)

### 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 \(W_\mu \) and the family of flat connections satisfies a reality condition [9], the surface is preserved under the given changes of parameter. The last statement holds for general simple factor dressings with \(\mu \in S^1\).

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

*simple factor dressing*of a minimal surface \(f: M \rightarrow {\mathbb {R}}^4\) with

*parameters*\((\mu , m, n)\) is the minimal surface \({\hat{f}}: {{\tilde{M}}} \rightarrow {\mathbb {R}}^4\) given by

If \(m=n=1\) then we refer to \(f^\mu = {\hat{f}}\) as the *simple factor dressing* of *f* with *parameter*\(\mu \).

*f*is given by

*f*with parameters \((\mu , m, n)\). Thus, all simple factor dressings are build from rigid motions of the simple factor dressings with parameter \(\mu \):

### Proposition 5.8

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:

### Corollary 5.9

The simple factor dressing of a minimal immersion \(f: M \rightarrow {\mathbb {R}}^4\) with parameters \((\mu , m,n)\) is a minimal immersion \({\hat{f}}: {{\tilde{M}}} \rightarrow {\mathbb {R}}^4\).

The right and left normal of \({\hat{f}}\) are given by simple factor dressings of the right and left normal of *f* respectively. Moreover, \({\hat{f}}\) is complete if and only if *f* is complete.

### Proof

*f*which are given by the pole \(\mu \) and the parallel sections \(Rm + m\frac{b}{a-1}\) and \(Nn + n\frac{b}{a-1}\) respectively.

*p*if and only if \((N(p)+ n \frac{b}{a-1} n^{-1}) =0\) or \((R(p) + m \frac{b}{a-1} m^{-1})=0\). We already have seen that \(\beta = R m + m \frac{b}{a-1}\) is nowhere vanishing if \(m\not =0\). A similar argument, as given before Lemma 3.1, gives the corresponding statement for the expression in

*N*, so that

*f*and \({\hat{f}}\) have the same conformal class, that is,

*f*is complete. \(\square \)

From the explicit form (32) of the simple factor dressing of a minimal surface we immediately see that the simple factor dressing commutes with the conjugation:

### Corollary 5.10

Let \(f: M \rightarrow {\mathbb {R}}^4\) be a minimal surface and \(f^*\) a conjugate surface of *f*. Then a conjugate surface of the simple factor dressing of *f* is given by a simple factor dressing of the conjugate surface \(f^*\).

Moreover, the choice of a different conjugate surface results in a translation of the simple factor dressing in 4-space.

## 6 Simple factor dressing and the López–Ros deformation

Given a minimal surface \(f: M \rightarrow {\mathbb {R}}^4\) in 4-space with Weierstrass data \((g_1, g_2, \omega )\) denote, in analogy to the case of a minimal surface in \({\mathbb {R}}^3\), by \(f^\sigma \) the *López–Ros deformation* of *f* with complex parameter \(\sigma \in {{\mathfrak {C}}}\), that is, the minimal surface given by the Weierstrass data \((\sigma g_1, \sigma g_2, \frac{\omega }{\sigma })\). Similarly, the *Goursat transformation* is defined by \({\mathrm{Re}}\,({\mathcal {A}}(f+\mathbf{i\,}f^*))\) where \({\mathcal {A}}\in {{\,\mathrm{O}\,}}(4,{{\mathfrak {C}}})\) 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 \({\mathbb {R}}^4\)

Since by Proposition 5.8 any simple factor dressing is given in terms of the simple factor dressing with parameter \(\mu \), we will first show that these simple factor dressings are Goursat transformations.

### Theorem 6.1

*f*whose holomorphic null curve is

*f*and

### Proof

*f*with parameter \(\mu \) is given by

*s*and

*t*. Thus, with \(a-1 = \frac{(1-\mu )^2}{2\mu }, b = i \frac{1-\mu ^2}{2\mu }\) we have

*f*with parameter \(\mu \) is then given by

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 \(f: M \rightarrow {\mathbb {R}}^4\) is a Goursat transformation of *f*.

### Proof

*f*then the null curve of \({\mathcal {R}}_{n,m}^{-1}(f)\) is \({\mathcal {R}}_{n,m}^{-1}\Phi \) since \({\mathcal {R}}_{n,m}\) is real. But then the holomorphic null curve of the simple factor dressing of \({\mathcal {R}}_{n,m}^{-1}(f)\) with parameter \(\mu \) is \({\mathcal {L}}^\mu {\mathcal {R}}_{n,m}^{-1}\Phi \) by (35). Thus, the holomorphic null curve of the simple factor dressing \({\hat{f}}\) with parameters \((\mu , m, n)\) is given by

*f*. \(\square \)

Note that the simple factor dressing is a special case of the Goursat transformation: its matrix \({\mathcal {A}}\in {{\,\mathrm{O}\,}}(4, {{\mathfrak {C}}})\) has \(\det {\mathcal {A}} =1\) 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

*f*is given by

In particular, the López–Ros deformation \(f^\sigma \) of *f* with parameter \(\sigma = e^{s+ \mathbf{i\,}t}\in {{\mathfrak {C}}}_*, |\sigma |\not =1,\) is the simple factor dressing \({\hat{f}}\) of *f* with parameters \((\mu , m, m)\) where \(\mu =\frac{1-e^{-(s+i t)}}{1- e^{s-it}} \in {\mathbb {C}}{\setminus }\{0,1\}\) and \(m=\frac{1-i-j-k}{2}\in S^3\).

From this we see again that the López–Ros deformation is a trivial rotation in the *ij*-plane if \(\sigma \in S^1\subset {{\mathfrak {C}}}\). Moreover, if \(\sigma \in {\mathbb {R}}\) then \(\mu = -\frac{1}{\sigma }\).

### Proof

Let \(f^\sigma \) be the Lopez–Ros deformation of *f* with parameter \(\sigma =e^{s+\mathbf{i\,}t}\in {{\mathfrak {C}}}_*, |\sigma |\not =1\). The first equation (36) is an analogue computation as in the proof of Theorem 2.4.

By assumption \(|\sigma | \not =1\) so that \(\mu = \frac{1-e^{-(s+it)}}{1-e^{s-it}} \in {\mathbb {C}}{\setminus }\{0,1\}\) is well defined. Put, as usual, \(a= \frac{\mu + \mu ^{-1}}{2}, b = i \frac{\mu ^{-1}-\mu }{2}\).

*f*with parameters \((\mu , m, m)\)

*f*. \(\square \)

### 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*.

If \(f: M \rightarrow {\mathbb {R}}^4\) is a periodic minimal surface then the periods of the simple factor dressing \(f^\mu \) with parameter \(\mu \) are immediately given by the explicit formulation (34):

### Corollary 6.5

*f*with parameter \(\mu \) is periodic with \(\gamma ^* f^\mu = f^\mu + \tau ^\mu \) where

*f*, that is, \(\gamma ^* f^* = f^* + \tau ^*\), and \(s =- \ln |\mu |, t =\arg \frac{{\bar{\mu }}-1}{{\bar{\mu }}(1-\mu )}\).

From this, we can immediately compute the periods of all simple factor dressings by Proposition 5.8.

*M*, and that there exist \( m, n\in {\mathbb {H}}_*\) such that all periods of the conjugate surface \(f^*\) can be rotated simultaneously into the 1,

*i*-plane, that is,

*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 \(f: M \rightarrow {\mathbb {R}}^4\) has finite total curvature and if the simple factor dressing \({\hat{f}}: M \rightarrow {\mathbb {R}}^4\) of *f* with parameters \((\mu , m, n)\) is single-valued on *M* then \({\hat{f}}\) has finite total curvature.

### Proof

Since *f* has finite total curvature, we can assume by Theorem 2.1 that \(M ={\bar{M}} {\setminus }\{p_1, \ldots , p_r\}\) where \({\bar{M}}\) is a Riemann surface punctured at finitely many \(p_i\). Moreover, if \(\Phi \) denotes the holomorphic null curve of *f* then we can assume that \(d\Phi \) extends meromorphically into the \(p_i\). Since the simple factor dressing is a Goursat transformation, the holomorphic null curve \({\hat{\Phi }}\) of \({\hat{f}}\) is given by \(\hat{\Phi }= {\mathcal {A}} \Phi \) with \({\mathcal {A}}\in {{\,\mathrm{O}\,}}(4,{{\mathfrak {C}}})\). Thus, \(d{\hat{\Phi }}\) extends meromorphically into the punctures \(p_i\). \(\square \)

### 6.2 Simple factor dressing in \({\mathbb {R}}^3\)

*f*with parameter \(\mu \in {\mathbb {C}}{\setminus }\{0,1\}\) gives a minimal surface

*f*with parameters \((\mu , m, n)\) is given by a simple factor dressing with parameter \(\mu \) and an operation of \({\mathcal {R}}_{n,m}\in {{\,\mathrm{SO}\,}}(4, {\mathbb {R}})\), we see from (34) that \({\hat{f}}\) is in 3-space if \({\mathcal {R}}_{n,m}\) stabilises \({\mathbb {C}}={{\,\mathrm{span}\,}}_{\mathbb {R}}\{1 , i\}\). In particular:

### Theorem 6.7

Let \(f: M \rightarrow {\mathbb {R}}^3\) be minimal. The simple factor dressing \({\hat{f}}\) with parameters \((\mu , m, n)\) with \(m=n\lambda \), \(\lambda \in {\mathbb {C}}_*\), is a minimal surface \({\hat{f}}:M \rightarrow {\mathbb {R}}^3\) in 3-space.

As before, we also obtain the periods of the simple factor dressing:

### Corollary 6.8

*f*with parameter \(\mu \) is periodic with \(\gamma ^* f^\mu = f^\mu + \tau ^\mu \) where

*f*, that is, \(\gamma ^* f^* = f^* + \tau ^*\), and \(s =- \ln |\mu |, t =\arg \frac{{\bar{\mu }}-1}{{\bar{\mu }}(1-\mu )}\).

In particular, \( f^\mu \) is closed along \(\gamma \) if and only if \( \tau _1 =0\) and \(\begin{pmatrix} \tau _2 \\ \tau _3 \end{pmatrix} = \begin{pmatrix} \tau _3^* \\ -\tau _2^* \end{pmatrix} \tanh s. \)

We can also investigate the behaviour of simple factor dressings in \({\mathbb {R}}^3\) at ends:

### Theorem 6.9

Let \(f: M\rightarrow {\mathbb {R}}^3\) be a minimal surface on a punctured disc \(M=D{\setminus }\{p\}\) and \({\hat{f}}: {{\tilde{M}}} \rightarrow {\mathbb {R}}^3\) its simple factor dressing with parameter \((\mu , m, m)\), \(m\in S^3\).

- (i)
If

*f*has a planar end at*p*then \({\hat{f}}: M \rightarrow {\mathbb {R}}^3\) is single-valued on*M*and \({\hat{f}}\) has a planar end at*p*. - (ii)
If

*f*has a catenoidal end at*p*and \({\hat{f}}: M \rightarrow {\mathbb {R}}^3\) is single-valued on*M*then \({\hat{f}}\) has a catenoidal end at*p*.

### 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 \(n\in {\mathbb {H}}_*\) such that \({{\tilde{f}}} = {\mathcal {R}}_{n, n}^{-1}f\) has vertical normal at *p*. Since \({\mathcal {R}}_{m, m} = {\mathcal {R}}_{n, n} \circ {\mathcal {R}}_{n^{-1}m,n^{-1}m}\) and \({\hat{f}} = {\mathcal {R}}_{m,m}(({\mathcal {R}}_{m,m}^{-1}(f))^\mu )\) by Proposition 5.8, the simple factor dressing of *f* is up to rotation given by the simple factor dressing of \({{\tilde{f}}}\) with parameters \((\mu , n^{-1}m, n^{-1}m)\).

In a conformal coordinate *z* on the punctured disk \(D{\setminus }\{0\}\), we know from [38], see also Theorem 2.3, that the holomorphic null curve \(\Phi \) of *f* has \({{\,\mathrm{ord}\,}}_{z=0}d\Phi =-2\) and \({{\,\mathrm{res}\,}}_{z=0} d\Phi =-(0, 0, 2\pi \alpha )\) where \(\alpha \in {\mathbb {R}}\) is the logarithmic growth.

By [38] the periods of the conjugate surface \(f^*\) around the end are given by \({{\,\mathrm{res}\,}}_{z=0} d\Phi \). 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 \(-2\pi \alpha k\). 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 \((\mu , m, m)\) such that \(m^{-1}k m =\pm i\), that is, \(m=(1\mp j)\lambda \) with \(\lambda \in {\mathbb {C}}\).

We know from Corollary 5.9 that the simple factor dressing preserves completeness, that is, the end of \({\hat{f}}\) at *p* is complete. Since the simple factor dressing \({\hat{f}}\) is a Goursat transformation, the holomorphic null curve of \({\hat{f}}\) is given by \({\hat{\Phi }} = {\mathcal {A}} \Phi \) with \({\mathcal {A}}\in {{\,\mathrm{O}\,}}(3, {{\mathfrak {C}}})\) and thus, \({{\,\mathrm{ord}\,}}_{z=0} d{\hat{\Phi }} =-2\). At a planar end we have \({{\,\mathrm{res}\,}}_{z=0} d{\hat{\Phi }} = {{\,\mathrm{res}\,}}_{z=0} d\Phi = (0,0,0)\). Therefore, \({\hat{f}}\) has a planar end by Theorem 2.3.

Again, we obtain from Theorem 6.3 the link to the López–Ros deformation:

### Theorem 6.10

Let \(f: M\rightarrow {\mathbb {R}}^3\) be a minimal surface in \({\mathbb {R}}^3\) with conjugate surface \(f^*\). The López–Ros deformation with complex parameter \(\sigma =e^{s+\mathbf{i\,}t}\in {{\mathfrak {C}}}_*, |\sigma | \not =1,\) is the simple factor dressing of *f* with parameters \((\mu , m, m)\) where \(\mu = \frac{1-e^{-(s+it)}}{1-e^{s-it}}\in {\mathbb {C}}\) and \(m=\frac{1-i-j-k}{2}\).

We obtain as a consequence of the last two theorems the following well-known result [48]:

### Corollary 6.11

Let \(f: M\rightarrow {\mathbb {R}}^3\) be a minimal surface on a punctured disk \(M = D{\setminus }\{p\} \) and \(f^\sigma : {{\tilde{M}}} \rightarrow {\mathbb {R}}^3\) a López–Ros deformation with parameter \(\sigma \).

- (i)
If

*f*has a planar end at*p*then \(f^\sigma : M \rightarrow {\mathbb {R}}^3\) is single-valued on*M*and \(f^\sigma \) has a planar end at*p*. - (ii)
If

*f*has a catenoidal end at*p*and \( f^\sigma : M \rightarrow {\mathbb {R}}^3\) is single-valued on*M*then \(f^\sigma \) has a catenoidal end at*p*.

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 \(2m=1-i-j-k =(1-j)(1-i)\). In particular, the López–Ros deformation of *f* has a catenoidal end at *p*, too.

## 7 Examples

We conclude this paper by demonstrating some of our results for well-known examples of minimal surfaces, including Richmond surfaces and the first Scherk surface. Further examples, such as the Riemann minimal examples and the Costa surface can be found in [45]. 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 and doubly-periodic surfaces 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.

### 7.1 The catenoid

The periods of the simple factor dressing with parameter \(\mu \) are given by Corollary 6.8:

### Lemma 7.1

In particular, \({\hat{f}}(x, y+2\pi ) = {\hat{f}}(x,y)\) if and only if \(m\in {\mathbb {C}}_*\) or \(m\in {\mathbb {C}}_*j\) or \(\mu \in S^1\). In this case, \({\hat{f}}\) is a (reparametrised) catenoid.

### Proof

*f*is trivial. In the latter case we see with Theorem 5.5 that \({\hat{f}}\) is the simple factor dressing of

*f*with parameter \(\mu \) or \(\bar{\mu }^{-1}\). But the simple factor dressing of

*f*with parameter \(\mu \) is by (37) given by

Thus, in general the simple factor dressing of a catenoid will have translational periods. Although the resulting surfaces resemble Catalan’s surface see (Fig. 4) the simple factor dressing of a catenoid has by Corollary 5.9 no branch points.

### 7.2 Richmond surfaces

*f*has a planar end at the puncture \(z=0\) since \({{\,\mathrm{ord}\,}}_{z=0} d\Phi = -2\) and the residue of \(d\Phi \) at \(z=0\) vanishes (Fig. 6).

*f*with parameter \(\mu \) in more detail for the case \(l=1\), that is,

*f*with parameter \(\sigma =e^{s+it}\in {{\mathfrak {C}}}_*\) is

*f*and \(f_\sigma \), \(\sigma \not =1\), we see that \(f_\sigma \) is not a reparametrisation of

*f*(Fig. 9).

Finally, we include some pictures of the simple factor dressing for more general parameters. Note that the surfaces are single-valued for all parameters \((\mu , m, m)\), and have a planar end at \(z=0\) (Fig. 10).

### 7.3 Scherk surfaces

*f*with parameter \(\sigma \) is doubly-periodic (Figs. 12, 13) with

## Notes

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