Generalised Bianchi permutability for isothermic surfaces

Isothermic surfaces are surfaces which allow a conformal curvature line parametrisation. They form an integrable system, and Darboux transforms of isothermic surfaces obey Bianchi permutability: for two distinct spectral parameters the corresponding Darboux transforms have a common Darboux transform which can be computed algebraically. In this paper, we discuss two-step Darboux transforms with the same spectral parameter and show that these are obtained by a Sym-type construction: All two-step Darboux transforms of an isothermic surface are given, without further integration, by parallel sections of the associated family of the isothermic surface, either algebraically or by differentiation against the spectral parameter.


Introduction
First defined by Bour in [3] as surfaces which admit conformal curvature lines, isothermic surfaces have enjoyed massive interest in the late 19th and early 20th century. Darboux showed in [11] that given an isothermic surface f : M → R 3 from a Riemann surface M into the 3-sphere, one can construct a second isothermic surface via a Ribaucour sphere congruence that depends on a spectral parameter, a transformation which we refer to as Darboux transformation. Then Bianchi, [1], showed that Darboux transformations admit permutability: starting from an isothermic surface f and constructing two Darboux transforms f 1 and f 2 using spectral parameters 1 and 2 , respectively, one can always find a fourth surface f 12 that is both a Darboux transform of f 1 and f 2 with respect to spectral parameters 2 and 1 . Demoulin further showed in [12] that these four surfaces in the permutability enjoy a relationship characterized by cross-ratios: given starting surface f if the spectral parameters are equal. Therefore, permutability gives algebraic methods to find non-trivial successive Darboux transforms as long as the spectral parameters are pairwise distinct.
Note however that one can always integrate twice to find non-trivial two-step Darboux transforms: the condition in Bianchi permutability that the spectral parameters need to be distinct is only essential to obtain non-trivial successive Darboux transforms algebraically.
The aim of this paper is to eliminate the assumption in Bianchi permutability and obtain all successive Darboux transforms without further integration, even in the case when the spectral parameters are equal. Rather than using Bianchi permutability we obtain two-step Darboux transforms with the same spectral parameter by a Sym-type method, [22], that is, by differentiation with respect to the spectral parameter.
The existence of spectral parameters, transformations, and permutability suggested that the class of isothermic surfaces constitutes an integrable system, an approach taken in [10] which renewed modern interest in isothemic surfaces. Various characterisations of Darboux transformations have been obtained since: Darboux transformation can be described in terms of a Riccati type equation [18]; Darboux pairs of isothermic surfaces can be viewed as a curved flat using the Minkowski model [9] or using the quaternionic model [15] of conformal geometry. In fact, isothermic surfaces can be characterised via the existence of a closed 1-form or, equivalently, a one parameter family of flat connections [5,14,19], and one can view Darboux transformations as the parallel sections of the flat connections [16,17]. In addition, many of the aforementioned works have investigated the various transformations of isothermic surfaces and their relationships: for example, the T -transforms, also known as Calapso transforms, can be obtained algebraically from the Darboux transforms, while the Christoffel dual can be obtained via a Sym-type formula from either the T -transforms or the Darboux transforms.
In this paper, we use the quaternionic model and describe Darboux transform by parallel sections of the associated family of flat connections of the isothermic surface. A short review of isothermic surfaces, the associated family d λ and Bianchi permutability in this setting is given in Section 2 to setup the notations and tools for our main result.
Then we tackle the problem to eliminate the need for a second integration for finding two-step Darboux transforms in Section 3. For this, we use the fact that Darboux transforms of isothermic surfaces are indeed given by a simple factor dressing. In particular, the associated family of flat connections d 1 λ of a Darboux transform f 1 with spectral parameter is given by an explicit gauge r λ , which depends smoothly on the spectral parameter and has a simple pole at , of the associated family d λ of f . Although the gauge has a pole, the family d 1 λ = r λ · d λ extends into and we give an explicit form of the associated family.
With this at hand, we obtain the parallel sections ϕ 1 = r λ (ϕ) of f 1 by applying the gauge matrix to parallel sections ϕ given by the isothermic surface f , for spectral parameter away from the pole of r λ . This way, we recover the parallel sections used for Bianchi permutability, the Bianchi-type parallel sections, explicitly as projections of parallel sections ϕ. In the case when the spectral parameter coincide, there is a quaternionic one-dimensional space arising from this construction: to obtain further parallel sections we have to consider limits of parallel sections for spectral parameter λ when λ tends to the pole . We show that these limits, the Sym-type parallel sections, are given by differentiation of a family of d λ -parallel sections with respect to the spectral parameter.
Indeed, we can conclude that all parallel sections of the associated family of a Darboux transform are either Bianchi-or Sym-type. In particular, we obtain all non-trivial two-step Darboux transforms with same spectral parameter without need for a second integration, a principle we call generalised Bianchi permutability.
Given an isothermic surface f : M → R 3 , the Darboux transformation is initially a local construction: the used parallel sections exist globally only on the universal cover of the Riemann surface M . Since all two-step parallel sections are given algebraically or by a Sym-type method, we discuss closing conditions for one-and two-step Darboux transforms by investigating the holonomy of the family of flat connections d λ of f only.
We conclude the paper by demonstrating our construction in the explicit example of the round cylinder. In particular, we give explicit formulae for all parallel sections and obtain a complete description of the set of all closed Darboux transforms of a cylinder. Depending on the spectral parameter, four cases can occur: there is exactly one closed Darboux transform, which is the cylinder, there are two distinct Darboux transforms, which are again cylinders, there is a CP 1 -worth of Darboux transforms which are rotation surfaces, or there is a HP 1 -worth of (possibly singular) Darboux transforms which are rotation surfaces or isothermic bubbletons. We then use the parallel sections to give explicit formulae for Sym-type Darboux transforms, including two-step bubbletons. 1 Since the main ingredients for our construction are the associated family and the simple factor dressing, we expect our results to be templates for similar results for other surface classes allowing simple factor dressing, such as CMC surfaces in space forms, and completely integrable differential equations. This should allow to construct new surfaces and, more generally, new solutions to differential equations given by complete integrability.

Background
In this section we will give a short summary of results and methods used in this paper. For details on the quaternionic formalism and isothermic surfaces we refer to [4,8,13,16,18].
2.1. Conformal immersions and quaternions. In this paper we will identify 4-space by the quaternions R 4 = H, and 3-space by the imaginary quaternions R 3 = Im H where H = span R {1, i, j, k} and i 2 = j 2 = k 2 = ijk = −1. For imaginary quaternions the product in the quaternions links to the inner product ·, · and the cross product in R 3 by Here we identify H = Re H ⊕ Im H = R ⊕ R 3 . In particular, we see Thus, if f : M → R 3 is an immersion then its Gauss map N : where * denotes the negative Hodge star operator, that is, * ω(X) = ω(J T M X) for X ∈ T M , ω ∈ Ω 1 (M ). More generally, if f : M → R 4 is a conformal immersion from a Riemann surface into 4-space, the Gauss map is given by a pair of complex structures Note that N = R in the case when f is a surface in 3-space.
Since the theory of isothermic surfaces is conformal, it is useful to also consider conformal immersions into the 4-sphere by identifying S 4 = HP 1 . Then a map f : Therefore, the group of oriented Möbius transformations is in this setup given by GL (2, H). The derivative of L is given by δ = π L d where π L : H 2 → H 2 /L denotes the canonical projection. Then an immersion f is conformal if and only if there are complex structures J L on L and J V /L on H 2 /L such that * δ = J V /L δ = δJ .
In particular, if f : M → R k , k = 3, 4, is an immersion from a Riemann surface into 3-or 4-space we will consider f as a map into the 4-sphere by setting We will identify eH = H 2 /L via the isomorphism π L | eH : eH → H 2 /L where eH = ∞ is the point at infinity with e = 1 0 .
Then N, R : M → S 2 induce the complex structures J L on the line bundles L and J V /L on H 2 /L by setting J L ψ = −ψR and J V /L e = eN : since δψ = edf we obtain indeed * δψ = J V /L δψ = δJ L ψ . This definition links with the Christoffel transformation of an isothermic surface when f is a surface in 3-or 4-space: since im η ⊂ L ⊂ ker η we can write for a 1-form ω with values in H. But then dη = 0 shows that dω = 0, so locally there exists a (possibly branched) immersion f d with df d = ω. Additionally we see from isothermic coordinate (and f does not map into the round sphere) then up to scaling df d = f −1 x dx − f −1 y dy. Conversely, away from umbilics the isothermic coordinate can be constructed from η (see [4, p. 28]).
In particular, the definition we are using immediately allows to introduce a spectral parameter ∈ R, see e.g. [5,Theorem 15.4], [6,Proposition 3.6], and we obtain an associated family of flat connections: since d λ = d + λη, λ ∈ R, has curvature R λ = R + λdη + λ 2 η ∧ η = 0 we see that the associated family d λ of f is flat for all λ ∈ R. The converse holds as well: Theorem 2.3. If η ∈ Ω 1 (End(H 2 )) is non-trivial with η 2 = 0 and is flat for all λ ∈ R then ker η can be extended to a quaternionic line bundle L and L is isothermic with retraction form η.
Proof. We follow the arguments in [7, Theorem 3.1], and only give a short outline how the argument there can be adapted to our situation. Let I be the complex structure on H 2 which is given by right multiplication by the quaternion i. Let η 1,0 be the (1, 0)-part of η and E = ker η 1,0 . Since η is quaternionic, ker η = E ⊕ Ej. In [7, Theorem 3.1] it is shown that d induces a holomorphic structure on Γ(K End(C 4 )) when identifying sections in Γ(KK) with 2-rforms in Ω 2 (M ). Since d λ is flat we see that dη = 0, so that also dη 1,0 = 0. Thus, η 1,0 is holomorphic and E = ker η 1,0 extends holomorphically across the zeros of η 1,0 , and so does ker η = EH.
Recall that an isothermic surface f : M → R 3 can be locally characterised as a surface which allows a sphere congruence that conformally envelops f and a second surfacef where f (p) =f (p) for all p. Thenf is called a Darboux transform of f .
In the framework we set up, the Darboux transformation can be formulated in terms of parallel sections of H 2 of the associated family of flat connections, see e.g. [16, §5.4.8]. Here H 2 denotes the trivial H 2 bundle H 2 =M × H 2 over the universal coverM of M . In this situation, the resulting Darboux transform is in general an isothermic surface in the 4-sphere and is defined on the universal cover of M , and is a surface in the 3-sphere only for suitable initial conditions. We will identify, in abuse of notation, a surface f : M → S 4 with the canonical lift f :M → S 4 . Remark 2.5. In the case when the assumption L(p) =L(p) is not satisfied for all p ∈ M , the surfacef is called a singular Darboux transform of f , see [2]. If f,f : M → R 3 are surfaces in 3-space this means that the enveloping sphere congruence degenerates to a point for p ∈ M withf (p) = f (p) andf becomes a branched conformal immersion.
To simplify notations we will abbreviate ϕ = ϕ if it is clear from the context that ϕ is a d -parallel section, and use the superscript only if we want to emphasise the parameter in the family of flat connections that we use. Similarly, we will call the associated surface a Darboux transform, and only refer to it as -Darboux transform or Darboux transform with respect to the parameter for emphasis of a specific spectral parameter.
We now investigate the closing conditions for Darboux transforms, see [2]. Let us recall the notion of sections with multiplier. Definition 2.6. Given a parallel section ϕ ∈ Γ( H 2 ) a multiplier is a group homomorphism h : π 1 (M ) → H * such that where γ is the deck transformation ofM associated to γ. A section with multiplier is a parallel section for which multipliers exist. A spectral parameter ∈ R * is called a resonance point if every d -parallel section is a section with multiplier.
Since a Darboux transform of an isothermic surface f : M → S 4 is given byf = ϕH where ϕ = ϕ is a parallel section of d for some ∈ R * , we see thatf is closed if and only if ϕ is a section with multiplier. In this paper, we consider the "closure condition" to mean that the Darboux transform is defined on the same Riemann surface of the original immersion.
Since for h ∈ H * there exists m ∈ H * with m −1 hm ∈ C * we can assume without loss of generality that h γ ∈ C * by changing ϕ to ϕm in case of an abelian fundamental group. Note that since d is quaternionic, we see that if ϕ is d -parallel with multiplier h then ϕj is d -parallel with multiplierh, so that multipliers come in pairs (h,h) which give both rise to the same surfacef . In particular, in the case when h is real, the corresponding space of parallel sections with multiplier h is at least quaternionic 1-dimensional, whereas in the case of h ∈ R, the space of parallel sections with multiplier h is not quaternionic.
Example 2.7. In the case of a surface of revolution f : M → R 3 , the holonomy of d is for all spectral parameter ∈ R \ {0, 0 } diagonalisable and has at most two distinct multipliers, h and h −1 , see [20] and Proposition 4.4 in the case of a round cylinder. The spectral parameter 0 ∈ R * is determined by the choice of dual surface: scaling of f d by some factor will result in a scale of 0 . In the case when f (x, y) = ip(x) + jq(x)e −iy with smooth real-valued functions p, q satisfying p 2 + q 2 = q 2 is a conformally parameterised surface of revolution in the conformal With such choices, for the unique spectral parameter = − 1 4 with non-diagonalisable holonomy there is exactly one parallel section with multiplier h (up to quaternionic scaling), which indeed is h = −1, and the corresponding Darboux transform is a rotation of f , see Theorem 4.5 in the case when f is a round cylinder and Remark 4.6 for the general case. 1 4 there are exactly two distinct real multipliers h, h −1 ∈ R, and two H-linearly independent d -parallel sections ϕ 1 , ϕ 2 with multiplier h and h −1 respectively. These give two distinct Darboux transforms of f which are both rotations of f . Since ϕ 1 j, ϕ 2 j have the same real multipliers as ϕ 1 and ϕ 2 respectively, there are no further Darboux transforms, see Theorem 4.5 and Remark 4.6.
there are exactly two complex multipliers h ∈ S 1 \ {±1}, and two H-linearly independent d -parallel sections ϕ 1 , ϕ 2 with multiplier h. Since any complex linear combination ϕ ϕ = ϕ 1 m 1 +ϕ 2 m 2 , m 1 , m 2 ∈ C, is a d -parallel section with multiplier h, we obtain a CP 1 family of closed (possibly singular) Darboux transforms, giving in case of the round cylinder general rotation surfaces, see Theorem 4.5 and Remark 4.6. Since ϕ j has multiplierh = h −1 and ϕ H = ϕ jH we obtain no further Darboux transforms in this case.
In the case of a surface of revolution, the only other case which can occur is that the spectral parameter is a resonance point: every d r -parallel section ϕ r is a section with multiplier, that is, every Darboux transform with parameter r is a closed Darboux transform.
Put differently, given a basis {ϕ r 1 , ϕ r 2 } of d r -parallel sections at a resonance point r every d r -parallel section, and thus, every (possibly singular) r -Darboux transform, is given by Note that this shows that all d -parallel sections at a resonance point ∈ R * have the same multiplier h, and since multipliers appear as pairs (h,h) we also see that h ∈ R.
The corresponding Darboux transforms in case of a surface of revolution are rotation surfaces or isothermic bubbletons: in this case resonance points r = k 2 −1 4 are parametrised by positive integers k ∈ Z, k > 1, such that the corresponding Darboux transforms have k lobes. Special initial conditions give, in the case of a Delaunay surface, again Delaunay surfaces and CMC bubbletons, see Proposition 4.4 for the case of a round cylinder. Given two Darboux transforms f 1 , f 2 of f with respect to parameter 1 , 2 ∈ R, there is a common Darboux transform of both f 1 , f 2 which can be computed from the parallel sections without further integration.
In particular, the Darboux transform f 12 is f : in contrast to the case when 1 = 2 we do not get all Darboux transforms of f 1 with parameter 1 = 2 by this construction. We will discuss how to obtain all Darboux transforms by a Sym-type argument in the next section.
We also know [20] that ϕ 12 = ϕ 2 12 is a parallel section of the family of flat connections of f 1 for spectral parameter 2 , and ϕ 21 = ϕ 1 21 := ϕ 12 χ −1 is a parallel section of the family of flat connections of f 2 at 1 . In particular, f 12 = ϕ 12 H = ϕ 21 H = f 21 : Given an isothermic surface f with associated family d λ and a Darboux transform f 1 given by spectral parameter 1 ∈ R and d 1 -parallel section ϕ 1 = ϕ 1 1 , Bianchi permutability allows to compute Darboux transforms of f 1 for all spectral parameter 2 = 1 by solely knowing the parallel sections of the family of flat connections of f and performing an algebraic operation. However, in the case when := 1 = 2 we only obtain one Darboux transform of f 1 via Bianchi permutability, namely f 12 = f . In this section we show that we still obtain all Darboux transforms of f 1 without integration by the parallel sections of the associated family of f . The Darboux transform in this case is not given algebraically but by a Sym-type argument: we will differentiate parallel sections with respect to the spectral parameter.
3.1. Simple factor dressing. Let f : M → S 4 be an isothermic surface with associated family d λ and letf = f 1 be a Darboux transform given by a d -parallel section ϕ. To find all parallel sections of the associated familyd λ = d 1 λ off at λ = in terms of parallel sections of d λ we need to understandd λ at . To this end, we recall the so-called simple factor dressing: it is known that a suitable λ-dependent gauge matrix r λ with a simple pole given by gives via gauging the associated familyd λ = r λ · d λ of a -Darboux transform.
be a d -parallel section with corresponding Darboux transformf :M → H given bŷ L = ϕH. Denote byπ and π the projections ontoL and L respectively along the splitting H 2 =L ⊕ L and define Thend λ = r(λ) · d λ is the family of flat connections of the Darboux transformf .
In particular, the family of flat connectionsd λ = r λ · d λ extends into the pole of r λ . We will now investigate parallel sections ofd λ at λ = and their corresponding Darboux transforms in terms of parallel sections of d λ .

3.2.
Bianchi-type and Sym-type parallel sections. Let f 1 be the Darboux transform of an isothermic surface f : M → S 4 which is given by ∈ R * and a d -parallel section ϕ 1 = ϕ 1 , and d 1 λ its associated family of flat connections. For λ = all parallel sections of d 1 λ are given by Bianchi permutability. We are now investigating parallel sections of d 1 λ at λ = . Proposition 3.2. Assume that ϕ 2 = ϕ 2 is d -parallel and independent of ϕ 1 over H. Then ϕ 12 = πϕ 2 is a parallel section of the flat connection Here π and π 1 are the projections onto L and L 1 respectively along the splitting H 2 = L 1 ⊕ L. We call ϕ 12 a Bianchi-type section. The associated Darboux transform of f 1 is f 12 = f .
Since all d 1 -parallel sections arising from Bianchi permutability are sections in L and therefore quaternionic multiples of ϕ 12 we know that there exist d 1 -parallel sections on the universal coverM of M which do not arise from Bianchi permutability since d 1 is a flat connection on H 2 . We now investigate these.
Now letφ ∈ Γ( H 2 ) be an arbitrary d 1 -parallel section. We first show thatφ is a Bianchi-type parallel section if π 1φ = 0. In this case,φ ∈ Γ(L) and since bothφ and ϕ 12 are non-vanishing d 1 -parallel sections of the line bundle L we havê ϕ = ϕ 12 m, m ∈ H * , But thenφ = π(ϕ 2 m) is a Bianchi-type parallel section. We can therefore now assume that π 1φ = 0 so that We aim to show thatφ is a Sym-type Darboux transform of f . Therefore, we have to find a smooth extensionφ λ 1 near λ = so thatφ is its associated Sym-type parallel section, that is,φ Since ϕ 11 , ϕ 12 are linearly independent over H we can writê ϕ = ϕ 11 m 1 + ϕ 12 m 2 , m 1 , m 2 ∈ H .
This immediately gives a generalisation of Bianchi permutability, Theorem 2.8: Denoting by f 11 the Sym-Darboux transform given by a Sym-type parallel section ϕ 11 and by f 12 a Darboux transform given by Bianchi permutability by a Bianchitype parallel section ϕ 12 we see the following picture: For i ∈ R, i = 1, 2, let ϕ i = ϕ i i be d i -parallel sections of the associated family of flat connections d λ of f . Assume that ϕ i have multipliers h i ∈ C, that is, γ * ϕ i = ϕ i h i (γ) for all γ ∈ π 1 (M ). Then both associated Darboux transforms f i : M → S 4 are closed surfaces. The function χ defined by dϕ 2 = dϕ 1 χ satisfies χ * = h −1 1 χh 2 so that ϕ 12 = ϕ 2 − ϕ 1 χ has multiplier h 2 . In particular, we see that the two-step Darboux transforms, which are obtained by Bianchi permutability from closed Darboux transforms, are closed too: Consider now the remaining case when := 1 = 2 and the Darboux transform f 11 of f 1 is given by a Sym-type parallel section, that is, it is given by ϕ 11 = ϕ 1 − π( ∂ ∂λ ϕ λ 1 )| λ= where ϕ λ 1 is d λ -parallel near λ = and ϕ λ= 1 = ϕ 1 . If ϕ λ 1 is a section with multiplier h λ 1 for all λ near then and thus ϕ 11 = ϕ 1 − π( ∂ ∂λ ϕ λ 1 )| λ= has the same multiplier h 1 as ϕ 1 . In particular, the resulting Darboux transform f 11 of f 1 is closed.
We summarise: We now investigate cases where we can guarantee existence of closed two-step Darboux transforms in terms of the behaviour of the holonomy of d λ .
Corollary 3.11. Let f : M → S 4 be isothermic and d λ its associated family of flat connections. If ∈ R * is a spectral parameter such that there are four distinct complex multipliers of the holonomy of d , then every closed Darboux transform f 1 has exactly two closed Darboux transforms with parameter .
Remark 3.12. Homogeneous tori are examples of isothermic surfaces which have exactly four distinct complex multipliers: we will return to this topic in a future paper.
Proof. If one of the multipliers is real then there exist two complex independent parallel sections ϕ, ϕ j with the same multiplier which contradicts the assumption that the holonomy has 4 distinct eigenvalues with complex one-dimensional eigenspaces.
Since complex multipliers appear as pairs of conjugate complex multipliers we have exactly two d -parallel sections ϕ 1 , ϕ 2 with complex multiplier h 1 and h 2 , h 1 = h 2 , respectively which are H-independent. Thus, all multipliers are given by Since f 1 is closed, it is given by one of these parallel sections, say ϕ 1 . The multipliers depend smoothly on the spectral parameter and since there are four distinct multipliers for λ near , we can extend ϕ 1 around to a smooth family of d λ -parallel sections ϕ λ 1 with multipliers h λ 1 . Then the Sym-type formula shows that ϕ 11 is a section with multiplier h 1 and f 11 is closed. Since f 11 = f we obtain the second closed Darboux transform from Bianchi permutability and the parallel section ϕ 2 . Since h 1 = h 2 we cannot have further closed Darboux transforms of f 1 . Proof. Since h = h λ= ∈ R we see that is not a resonance point. Let ϕ 1 be a d -parallel section with multiplier and f 1 the Darboux transform given by ϕ 1 . Since multipliers come in pairs of complex conjugates, we know that the holonomy of d λ is diagonalisable with complex 2-dimensional, d λ -stable eigenspaces E λ = span C {ϕ λ 1 , ϕ λ 2 } and E λ j with multipliers h andh. Therefore, we can assume without loss of generality that the d -parallel section ϕ 1 has multiplier h by replacing ϕ 1 by ϕ 1 j if necessary. Moreover, we can write ϕ 1 = ϕ λ= 1 m 1 + ϕ λ= 2 m 2 , m 1 , m 2 ∈ C, and thus can also assume without loss of generality that ϕ 1 = ϕ λ= 1 by replacing The Sym-type parallel section Here π is the projection onto L along the splitting H 2 = L ⊕ L 1 .
On the other hand, the Bianchi-type Darboux transform f 12 of f 1 is given Example 3.14. This case appears for surfaces of revolution in 3-space: If > − 1 4 , = 0, is not a resonance point then a closed Darboux transform f 1 with parameter in 3-space is a surface of revolution and so is every Darboux transform with parameter of f 1 in 3-space.
At resonance points r it is possible that a Darboux transform f 1 has r as a resonance point.  Proof. By Theorem 3.5 every parallel section of the family of flat connections of f 1 is either a Sym-type or a Bianchi-type parallel section. Every Bianchi-type parallel section ϕ 12 gives rise to the Darboux transform f 12 = f and is given by a parallel section ϕ 2 with real multiplier h 2 = h 1 since r is a resonance point.
By Theorem 3.10 we know that a Sym-type Darboux transform is closed if ϕ 1 can be extended by a d λ -parallel sections ϕ λ 1 with multiplier h λ 1 . In this case, ϕ 11 has multiplier h 1 and ϕ 12 and ϕ 11 have the same real multiplier. Since any parallel sectionφ is a linear combination ϕ = ϕ 11 m 1 + ϕ 12 m 2 with m 1 , m 2 ∈ H we see that every parallel section has multiplier h 1 . Thus, r is a resonance point of f 1 .

Sym-type Darboux transforms of the round cylinder
In this section we will demonstrate explicitly the construction of Sym-type Darboux transforms in the example of a conformally parametrised round cylinder (referred to simply as cylinder, hereafter). We will first show that the Darboux transform of a real-analytic surface of revolution, which does not have constant mean curvature, has constant mean curvature if and only if the Darboux transform is again a surface of revolution. This will allow to rule out later that closed surfaces obtained by Sym-type Darboux transforms are constant mean curvature surfaces.
We then will give all Darboux transforms of a cylinder explicitly by computing all parallel sections of the family of flat connections. With this at hand, we will consider the case when the one-step Darboux transform is a surface of revolution but not CMC. In this case, we give two surprisingly explicit examples of Sym-type transforms, one which is a surface of revolution and one which is not.  In this case, the Darboux transform given by ϕ can be written in affine coordinates asf = f + T so that df = T df d T .
Next we recall that for an isothermic surface f : M → R 3 the mean curvature of a Darboux transformf = f + T in 3-space is given in terms of the mean curvature of a dual surface f d of f .
where H d is the mean curvature of the dual surface f d of f .
Similar to the case when f is CMC in [8] one can now derive a necessary condition for a Darboux transform of an isothermic surface to have constant mean curvature: It remains to show that where we used equation (4.2) and N df d = −df d N .
We can now use the previous lemma to discuss the mean curvature of Darboux transforms of surfaces of revolution. Proof. Since f is conformally parametrised we can write f (x, y) = ip(x) + jq(x)e −iy with smooth real-valued functions p, q satisfying p 2 + q 2 = q 2 .
. Therefore, T y = in y + j(m y − im)e −iy is a scale of f y = kqe −iy by a real-valued function, and thus n y = 0. Since f x , T , p , q only depend on x this shows that also m y = 0. Therefore we have shown thatf is a surface of revolution ifĤ = H.

4.2.
Darboux transforms of a cylinder. We will compute all Darboux transforms of a conformally parametrised cylinder, of constant mean curvature H = 1 Consider the dual surface f d given, up to translation, by df d = f −1 x dx − f −1 y dy. We choose f d (x, y) = −2(ix − je −iy ) and observe that the dual surface has constant mean curvature H d = − 1 4 .
To find all d -parallel sections, = 0, we recall (2.1) that From this we observe that ϕ has complex multiplier h if and only if α has also multiplier h.
Differentiating the above equations again we obtain in the isothermic coordinate z = x + iy the differential equation We write c ± = c ± 0 + jc ± 1 with complex valued function c ± 0 , c ± 1 . Then β = −f −1 y α y gives .
It remains to find the complex-valued functions c ± i . Since dα = −df β we see that * dα = N dα where N = −je −iy is the Gauss map of f . Therefore, we can find c ± by solving the differential equation α y = N α x which gives the linear system The solutions of this system are given by Thus, we have now computed all parallel sections of a cylinder explicitly. We summarise: Moreover, every d -parallel section, = − 1 4 , is given by Finally, the resonance points of the cylinder are given by In this case, every d k -parallel section has multiplier h k = (−1) k+1 .
From the explicit form of the parallel sections we have now complete information about the set of closed Darboux transforms: 1+4 has a complex 2-dimensional space E ± of parallel sections with multiplier h ± . Moreover, then there is exactly one closed Darboux transform, which is the rotation of f with angle θ = π in the jk-plane, i.e,f (x, y) = 1 2 (ix − je −iy ) is a cylinder. • if > − 1 4 , = k 2 −1 4 , k ∈ Z, k ≥ 1, then there is a CP 1 -worth of closed Darboux transforms which are rotation surfaces.
• if = k 2 −1 4 , k ∈ Z, k > 1, then is a resonance point. In this case, all Darboux transforms are closed and are either rotation surfaces or isothermic "bubbletons" with k lobes.
In the case when < − 1 4 the two parallel sections ϕ ± have real multipliers h ± ∈ R and the eigenspaces of the multipliers h ± are quaternionic. Therefore, in this case ϕ + H and ϕ − H give two closed Darboux transforms f ± = f + T ± . Our explicit expressions give and both surfaces f ± = f + T ± = 1 2 (ix + je ±iθ e −iy ) are cylinders where e iθ = − 1+t 1−t ∈ S 1 since t ∈ iR.
In the case when = − 1 4 we have real multiplier h + = h − = −1 and ϕ + H = ϕ − H gives one closed Darboux transform. Since there is no other section with multiplier, there are no other closed Darboux transforms in this case. The same computation as in the case < − 1 4 shows that the surface is a cylinder (with t = √ 1 + 4ρ = 0).
Finally, if = k 2 −1 4 , k ∈ Z, k > 1, is a resonance point then h + = h − ∈ R and every parallel section is a section with multiplier. The closed Darboux transforms given by L ± = ϕ ± H are non-singular and give rotation surfaces. The closed Darboux transforms with ϕ = ϕ + + ϕ − , ϕ ± = 0, give isothermic bubbletons which may be singular Darboux transforms.  Remark 4.6. We should note that similar arguments as in Proposition 4.4 and Theorem 4.5 allow to investigate parallel sections with multiplier and Darboux transforms of surfaces of revolution, see [20]. Although in general, the differential equations for c ± cannot be solved explicitly, the corresponding shape of the functions α, β is still enough to find all possible multipliers and to conclude that all Darboux transforms are surfaces of revolution. We will consider the case when the one-step Darboux transform of the cylinder is a surface of revolution but not CMC. Otherwise, the Darboux transform is again a cylinder, and all of its Darboux transforms are already known, or an (isothermic) bubbleton which has the original cylinder f as its only closed Darboux transform.
We will fix our spectral parameter as the resonance point = 3 4 and choose, according to Proposition 4.4, the parameter m + 0 = m + 1 = 1 and m − 0 = m − 1 = 0. Then the d -parallel section is given by ϕ = eα + ψβ with The resulting Darboux transform is a surface of revolution in 3-space wherê In particular,f is real-analytic and we see by Theorem 4.3 that a Darboux transform f off can only have constant mean curvature iff is a surface of revolution.
We now demonstrate in two examples how to explicitly construct Sym-type Darboux transforms of f . The first one is obtained by extending ϕ near λ = to d λ -parallel sections ϕ λ . Here ϕ is the section which gives the above Darboux transformf . To obtain the Sym-type parallel section we then computê where π is the projection along the splitting H 2 = L ⊕L ,L = ϕH. with t = √ 1 + 4λ and so that indeed ϕ λ= = ϕ. Abbreviating the λ-derivative evaluated at by a dot, we haveφ = ( ∂ ∂λ ϕ λ )| λ= = eα + ψβ .
Thus, using (1 + m) −1 = 1 − m(1 + m) −1 we obtain which givesf =f +T witĥ In particular,f is again a surface of revolution in 3-space. Sincef is not a Delaunay surface, we see thatf is isothermic but not CMC.
We now compute another Sym-type Darboux transform of the cylinder by using Theorem 3.5: all Darboux transformsf off are given by parallel sections which are quaternionic linear combinations ofφ and ofφ 2 = πϕ 2 , where π is the projection to L along the splitting L ⊕L,L = ϕH, and ϕ 2 is a d -parallel section ϕ 2 which is H-independent of ϕ.
Note that for the resonance point = 3 4 all Darboux transforms obtained this way are closed surfaces. Moreover, iff = f thenf is a Sym-type Darboux transform of f : recall that by Theorem 3.6 a two-step Darboux transform is either Sym-type or Bianchi type; in the latter case, it is the original cylinderf = f whereas in the formerf = f . Let c 2 0 = c + 0 (x, i, −i) and consider the corresponding parallel sectionφ which is quaternionic independent of ϕ by construction. To obtain a CMC bubbleton, see [20], we put ϕ 2 = ϕ +φj = eα 2 + ψβ 2 with The resulting Darboux transform f 2 of f can be explicitly computed as   To obtain a surface in 3-space from linear combinations of the two parallel sectionŝ ϕ andφ 2 = πϕ 2 , we need to satisfy an initial condition: if we usê ϕ +φ 2 ir where r ∈ R is a free parameter, the resulting Darboux transformsf : M → R 3 of f : M → R 3 are surfaces in 3-space and Sym-type Darboux transforms of f sincê π(φ +φ 2 ir) =φ = 0 , that isf = f . The resulting Darboux transforms off can be computed explicitly. For example, for r = 50 we obtainf =f +T withT = (T 1 ,T 2 ,T 3 ) wherê  Similarly, one can obtain other Sym-type Darboux transforms explicitly where k gives the number of lobes: Figure 13. Sym-type Darboux transforms of a cylinder at resonance points k = k 2 −1 4 , k = 2, 3.
To conclude this section we observe that we also obtain all closed Darboux transform of higher order of the cylinder f by information on the multipliers of parallel sections of the associated family d λ of f , without further integration. Figure 14. Triple Darboux transforms at a resonance point: the first one is obtained as a Darboux transform of a Sym-type twostep transform surface of revolution at the resonance point 2 , whereas the second one is obtained by Bianchi permutability from a non-rotational Sym-type Darboux transform, using the two different resonance points 2 , 3 .
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