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Moebius geometry of surfaces of constant mean curvature 1 in hyperbolic space

Various transformations of isothermic surfaces are discussed and their interrelations are analyzed. Applications to cmc-1 surfaces in hyperbolic space and their minimal cousins in Euclidean space are presented: the Umehara-Yamada perturbation, the classical and Bryant's Weierstrass type representations, and the duality for cmc-1 surfaces are interpreted in terms of transformations of isothermic surfaces. A new Weierstrass type representation is introduced and a Moebius geometric characterization of cmc-1 surfaces in hyperbolic space and minimal surfaces in Euclidean space is given.


Introduction
This paper aims to establish a relation between two fields of current interest.
On one side, isothermic surfaces and their various transformations gained new interest when an integrable system approach was developed [11]: "Darboux pairs" of isothermic surfaces correspond to "curved flats" in a suitable Grassmannian [6] (cf. [15]), a particularly simple type of integrable system. The "spectral parameter" occuring in the loop group description for curved flats [13] gives rise to a "spectral transformation" for isothermic surfaces. Classically, this transformation was independently discovered by P. Calapso [7], [8] who considered it the most fundamental transformation, and by L. Bianchi [2] who introduced it as the "T -transformation". Later, isothermic surfaces were identified as the only surfaces in conformal geometry allowing a second order deformation [9] that turns out to be exactly the T -transformation [19]. Another central role is played by the "Darboux transformation" for isothermic surfaces, first introduced by G. Darboux [12]: a corresponding permutability theorem and a quaternionic calculus [15] are the keys to the theory of "discrete isothermic nets" [4], and their transformations [14], [17].
On the other side, surfaces of constant mean curvature 1 in hyperbolic space became a field of interest, complementing research on minimal surfaces in Euclidean space, after R. Bryant [5] developed a Weierstrass type representation for them. Subsequently, M. Umehara and K. Yamada described these surfaces (and the corresponding Weierstrass representation) as a deformation of minimal surfaces in Euclidean space (and their Weierstrass representation) [23], and introduced a notion of duality [24] for cmc-1 surfaces in H 3 .
In section 3 of the present paper, we describe the various transformations of isothermic surfaces in terms of a quaternionic calculus that we shortly present in section 2 (for more information see [15]). Here, we emphasize the geometric context of the transformation, e.g. the Christoffel transformation depends on the Euclidean geometry of the ambient space while the Goursat transformation uses the interplay of the ambient space's Euclidean geometry and the conformal invariance of the notion of isothermic surfaces; the Darboux transformation is a transformation for surfaces in Möbius space while the T -transformation is only well defined for Möbius equivalence classes of isothermic surfaces. Then, we discuss the interrelations of these transformations in terms of several "permutability theorems" -some of the proofs become very compact in the quaternionic calculus.
In section 4, we interpret certain results on cmc-1 surfaces in hyperbolic space in terms of transformations of isothermic surfaces: the Umehara-Yamada deformation of minimal into cmc-1 surfaces is identified as the T -transformation and a unified version of the Weierstrass type representations for both surface classes is given in terms of quaternions. In this context, we give a Möbius geometric characterization for cmc-1 and minimal surfaces -in fact, the Umehara-Yamada perturbation families of cmc-1 surfaces appear as spectral families of isothermic surfaces. In terms of Möbius geometry, this means that the induced metric of the central sphere congruence has constant Gauss curvature 1. Finally, we introduce a second Weierstrass type representation for cmc-1 surfaces in H 3 in terms of the Darboux transformation, and discuss its relation with the notion of duality for cmc-1 surfaces.
As all the transformations under discussion require an integration they generally do not preserve periods, and may develop singularities, too. In this respect, our results have to be understood as local results. However, we formulate most of them in an invariant way in order to make the theory we present more accessible for further research on the global geometry of isothermic surfaces and their transformations. For example, it should be an interesting task to examine to what extent the Darboux and T -transformations respect extrinsic symmetries of surfaces, or how they behave in the neighbourhood of an isolated umbilic.
As mentioned above, many of the results of this paper can be formulated in a very elegant way using 2. The Quaternionic formalism for Möbius differential geometry, as introduced in [15] (cf. [22], [25]) -some results even depend on that approach. On the other hand, the classical approach of considering Möbius geometry as a subgeometry of real projective geometry is the better known formalism and allows a more direct treatment of certain aspects in local differential geometry. In this section, we intend to discuss the relation between the two approaches briefly, focussing on 3-dimensional geometry 1) .
Classically, the conformal n-sphere S n is embedded into the real projective (n + 1)-space IRP n+1 . Then, the Möbius transformations of S n become the projective transformations of IRP n+1 that fix S n ⊂ IRP n+1 as an absolute quadric. Up to scaling, there is a unique Lorentz scalar product on the space IR n+2 1 of homogeneous coordinates such that the points of S n are exactly the isotropic lines of that Lorentz product. Via orthogonality (polarity), hyperspheres can be identified with spacelike lines in IR n+2 1 , and the Möbius transformations appear as pseudo orthogonal transformations (cf. [3]).
On the other hand, the orientation preserving Möbius transformations of the conformal 4-sphere IH ∪ {∞} can be identified with the projective transformations of the quaternionic projective line IHP 1 -just as the (orientation preserving) Möbius transformations of the conformal 2-sphere / C ∪{∞} can be identified with the projective transformations of / CP 1 . Thus, in homogeneous coordinates 2) v ∈ IH 2 for points vIH ∈ IHP 1 ∼ = S 4 , the orientation preserving Möbius transformations appear as linear transformations v → Av with A ∈ Gl(2, IH). Or, in affine coordinates x ≃ (x, 1) t , the Möbius transformations become fractional linear transformations The relation between the two models can be established using the space of quaternionic hermitian forms s : IH 2 × IH 2 → IH on IH 2 , s(u, vλ + wµ) = s(u, v)λ + s(u, w)µ and s(w, v) = s(v, w).
Fixing a basis (e 1 , e 2 ) of the quaternionic plane IH 2 , any quaternionic hermitian form is determined by its values s(e 1 , e 1 ), s(e 2 , e 2 ) ∈ IR and s(e 1 , e 2 ) ∈ IH. Thus, the space of quaternionic hermitian forms is a 6-dimensional real vector space. Endowed with the determinant as a quadratic form 3) , s, s := − det s(e 1 , e 1 ) s(e 1 , e 2 ) s(e 2 , e 1 ) s(e 2 , e 2 ) = |s(e 1 , e 2 )| 2 − s(e 1 , e 1 ) · s(e 2 , e 2 ), this space becomes the 6-dimensional Minkowski space IR 6 1 of the classical model for 4-dimensional Möbius geometry: space-and lightlike lines of quaternionic 1) For a more comprehensive treatment -particularly for the 4-dimensional setting that also appears natural when using quaternions -the reader is referred to [15].
2) We consider IH 2 as a right vector space over the quaternions.
3) Indeed, the Lorentz product introduced this way is only well defined up to a scaling: a change of basis (e 1 , e 2 ) → (ẽ 1 ,ẽ 2 ) results in a rescaling of s, s → λ 2 s, s (cf. [15]). This is the same scaling ambiguity that arises in the classical construction, above. hermitian forms can be identified with their null cones in IH 2 to encode hyperspheres s ⊂ IHP 1 and points p ∈ IHP 1 (cf. [15], [22]). Moreover, using the Study determinant D (cf. [1]) for quaternionic 2 × 2-matrices, the special linear group As hyperspheres s ⊂ IHP 1 are encoded as spacelike lines of quaternionic hermitian forms, the Möbius group of the conformal 3-sphere can be obtained as the subgroup of Sl(2, IH) fixing a given quaternionic hermitian form, say 4) In affine coordinates, this choice corresponds to the imaginary quaternions as Euclidean 3-space, ImIH ∼ = IR 3 : the null cone of the above form is In this setting, the Lie algebra of the Möbius group of S 3 becomes −r + a | r ∈ IR; a, b, c ∈ ImIH} ⊂ sl(2, IH).
Space-and lightlike quaternionic hermitian forms that are perpendicular to the above fixed form representing S 3 can now be identified with 2-spheres and points, respectively, in the conformal 3-sphere S 3 .
Given a surface f : M 2 → S 3 , a congruence of 2-spheres s : M 2 → IR 6 1 in S 3 is said to be enveloped by f if each point f (p) lies on the corresponding sphere s(p), and if the tangent planes d p f (T p M ) = T f (p) s(p) coincide. As discussed, incidence is encoded by s(p)(f (p), f (p)) = 0 in terms of homogeneous coordinates 5) f : M 2 → IH 2 . A simple calculation (cf. [15]) shows that Lemma. An immersion f : M 2 → S 3 envelopes a congruence s : Fixing a basis (e 1 , e 2 ) of IH 2 , a map F : M 2 → Sl(2, IH) will be called an adapted frame of an immersion f : And, F is an adapted frame for a surface pair f,f : M 2 → S 3 if f = F e 1 andf = F e 2 . Note, that for a 4) Fixing one of the two length 1 representatives of the spacelike line amounts to fixing an orientation on the conformal 3-sphere S 3 . 5) We will use the same letter f for the immersion into S 3 ⊂ IHP 1 and for its homogeneous coordinates into IH 2 .
surface pair (f,f ) ≃ F the fact that F is invertible is equivalent to f (p) =f (p) at all points p ∈ M 2 . Denoting s f , sf ∈ IR 6 1 two generators of the lightlike lines of quaternionic hermitian forms corresponding to f andf , respectively, s f (f, f ) = 0 together with |s f | 2 = 0 implies that also s f (f,f ) = 0. Thus, choosing a suitable scaling for s f and sf , we have If s : M 2 → IR 6 1 is a sphere congruence in S 3 containing the points of both surfaces of a surface pair (f,f ), then the first condition in (1) yields with a function c : M 2 → ImIH into the imaginary quaternions. Given three such sphere congruences, say s i , s j , s k : a classical frame for Möbius geometry in S 3 (cf. [3]) is obtained: it consists of three congruences of orthogonally intersecting spheres s i , s j , and s k , and the maps s f ∼ = f and sf ∼ =f describing their intersection points. Since d(F −1 ·s c ) = 0 for all the above defined frame vectors s c , one calculates ds c = −s c (., Φ.) − s c (Φ., .) with the connection form Φ = F −1 dF . Using this equation allows to write the structure equations of the classical frame in terms of the components of Φ -thus obtaining a "translation table" between the classical and the quaternionic setting (in the codimension 1 case): For example, the surface f = F e 1 , orf = F e 2 , will envelope the sphere congruence s i if and only if the i-component df, s i = 0, or df , s i = 0 respectively, in the corresponding off-diagonal form of Φ vanishes (cf. (1)). At this point, we are prepared to collect (and reformulate) some results on

Transformations of Isothermic surfaces
Classically, an immersed surface f : where n : M 2 → S 2 is the normal field of f and u : M 2 → IR is a suitable real valued function. This definition obviously runs into problems when umbilics are present -and, since we are planning to consider sphere pieces as isothermic surfaces, we look for a more appropriate definition. On the tangent bundle of an immersed surface f : M 2 → ImIH, (quaternionic) left multiplication with the unit normal field n : M 2 → S 2 yields a 90 0 -rotation; the pull back of this structure provides a complex structure J on M 2 : df (Jx) = n·df (x). Clearly, the immersion f is conformal with respect to this complex structure. It is well known that, away from umbilics, the existence of conformal curvature line coordinates is equivalent to the fact that the Hopf differential η of a conformal immersion f is a real multiple of a holomorphic quadratic differential q on M 2 : η = ̺q with ̺ : M 2 → IR. Thus, we give the definition of an isothermic immersion of a Riemann surface with "prescribed" principal curvature lines: Definition (isothermic immersion). A Riemann surface M 2 equipped with a holomorphic quadratic differential 6) q is called a polarized surface; a conformal immersion f : (M 2 , q) → ImIH of a polarized surface is called isothermic if its Hopf differential η = ̺q is a real multiple, ̺ : M 2 → IR, of q.

The Christoffel transformation
If two non homothetic immersions f, f * : M 2 → ImIH induce the same conformal structure and have parallel normal fields then [10] they are either associated minimal surfaces (n * = n), or n * = −n and both surfaces are isothermic. The latter case can be characterized (cf. [15]) by the equation On the other hand, given an isothermic surface in terms of isothermic curvature line coordinates, f : ( / C, dz 2 ) → ImIH, df * := f −1 x dx − f −1 y dy defines, at least locally, a second surface satisfying df ∧df * = 0. Obviously, the surface f * is only determined up to homothety (and translation) by (4). Canonically trivializing the complex line bundle span{1, n} ∼ = M 2 × / C and using the above scaling of f * , one obtains the polarization q back: Lemma. Let f : M 2 → ImIH be a conformal immersion of a Riemann surface and τ : T M → ImIH a 1-form with values in the imaginary quaternions. Then df ·τ : T M → span{1, n} defines a quadratic differential if and only if df ∧τ = 0. 6) In order to establish a theory of "globally isothermic surfaces" it seems too restrictive to ask q to be holomorphic -for example, with this assumption the ellipsoid is no longer isothermic [18]. In order to give a satisfactory global definition, the consequences of various possible assumptions on q have still to be worked out.
This lemma allows us (cf. [18]) to define the Christoffel transform of a polarized isothermic surface (up to translation), i.e. to fix the scaling of f * : The Christoffel transform Cf of an isothermic immersion f is unique up to translations and it is isothermic with respect to the polarizationq = dCf · df . The Christoffel transform of Cf is the original surface, f = C 2 f .
As an example, we consider a minimal surface f : M 2 → ImIH in Euclidean 3-space: since df ∧dn = −Hdf ∧df , minimal surfaces are exactly those isothermic surfaces whose Christoffel transforms are totally umbilic (cf. [16]). As all the surfaces f t in the associated family, df t = (cos t+sin t·n)df , have the same Gauss map the additional information provided by the (holomorphic) Hopf differential df · dn as polarization is required to reconstruct the original surface from its Christoffel transform (Gauss map). The Goursat transformations form a group acting on isothermic immersions of a polarized Riemann surface since C 2 = id. If the Möbius transformation M is a similarity, then the corresponding Goursat transform Gf of an isothermic immersion is clearly similar to the original immersion f . If the Möbius transformation M is "essential", however, Gf will generally be not Möbius equivalent to the original surface. In this way, the Goursat transforms of an isothermic immersion f provide a 3-parameter family of non Möbius equivalent isothermic immersions. Thus, we may restrict the attention to essential Möbius transformations -that are (up to similarity) of the form x → (x − m) −1 with m ∈ ImIH:

The Darboux transformation
A sphere congruence enveloped by two surfaces induces a point-to-point correspondence between the two surfaces. If, under this correspondence, the induced latter situation can be characterized by the equation Comparison with (4) leads (cf. [16]) to the following Lemma. Let f : M 2 → ImIH be an immersion. If there is a second immersion f : M 2 → ImIH satisfying (7) then f is isothermic with with some λ ∈ IR \ {0} defines the Christoffel transform of f .
The linear version 0

8)
Choosing an initial condition v(p 0 ) ∈ S 3 the solution will stay in S 3 , thus D λ f : M 2 → ImIH.
. Consequently,F λ e 1 and F λ e 1 are just different homogeneous coordinates for the same surface in S 3 : the T λ -transformation is a well defined transformation for isothermic surfaces in Möbius geometry. In fact, the T λ -transform of an isothermic immersion can be defined using any adapted frame: observing the effect of a Gauge transformation of adapted frames, F 0 →F 0 = F 0 A with Ae 1 = e 1 a and a : M 2 → IH, on the corresponding connection forms shows that a suitable definition forΦ λ provides a family of integrable connection forms corresponding to (10). Given an adapted frameF 0 : M 2 → Sl(2, IH) of an isothermic immersion f : (M 2 , q) → ImIH, is integrable withF λ e 1 ≃ T λ f . Herein, ψ * is defined -analogously to dCfby the equation 9) ψ · ψ * ≃ q.
To understand the geometry of T λ f , we first note that Φ λ defined by (10) takes values in the Lie algebra of the Möbius group of S 3 ⊂ IHP 1 . Consequently, T λ f can be assumed (after a suitable Möbius transformation) to take values in S 3 , T λ f : M 2 → S 3 = ImIH ∪{∞}. Using affine coordinates, F λ = gbĝb bb , yields 9) Note that ψ ⊥ (similar to df ⊥ ) defines a complex line bundle over M 2 as (F 0 e 1 )IH : According to the above lemma, g ≃ T λ f is isothermic with Christoffel transform Cg given by µdCg = (ĝ − g) −1 dĝ(ĝ − g) −1 . Calculating the corresponding quadratic differential, shows that the curvature lines of f and T λ f correspond 10) . Moreover, the Christoffel transform Cg of g can be scaled, µ = −λ, such that dg · dCg ≃ q. Then,ĝ = D −λ g; on the other hand,ĝ = T λ Cf as the Gauge transformation (e 1 , e 2 ) → ( 1 λ |λ|e 2 , |λ|e 1 ) interchanges the roles of df and dCf in (10), i.e.ĝ ≃ T λ Cf . This way, we obtain the first 3.5 Permutability theorem (cf. [2]), T λ Cf = D −λ T λ f . As the T λ -transformation can be defined using any adapted frame, it is obvious that the 1-parameter family λ → T λ f of Ttransforms of an isothermic immersion f is a 1-parameter group: Given an isothermic immersion f : (M 2 , q) → ImIH, and a Darboux trans- Hence, Cf and CD λ f can be positioned such that [2], [16]) that CD λ f = D λ Cf . Combining this permutability theorem with the first, we obtain the following scheme: Fig. 3. A permutability theorem Finally, consider two D λ -transforms (D λ f ) 1 and (D λ f ) 2 of an isothermic immersion f : (M 2 , q) → ImIH. The T λ -transforms of these three surfaces are only well 10) In fact, the family λ → T λ f is the "conformal deformation" [19] of f (cf. [9]): isothermic surfaces are the only non rigid surfaces in Möbius geometry with prescribed curvature lines and conformal metric (Calapso potential), resp. with prescribed "conformal Hopf differential" [21]. defined up to Möbius transformation. But, after suitable Möbius transformations, T λ (D λ f ) 1 is the Christoffel transform of T λ f , as well as T λ (D λ f ) 2 is -after possibly other Möbius transformations. Thus, T λ (D λ f ) 2 = GT λ (D λ f ) 1 : after a T λ -transformation, the "difference" between two D λ -transforms of an isothermic surface is given by a Goursat transformation.

Surfaces of constant mean curvature 1 in H 3
Thus far, we have not yet given any examples of Darboux or T -transforms. To do so, it turns out to be convenient to use a suitably adapted frame for the two envelopes f,f : M 2 → S 3 ≃ s 1 of a sphere congruence s : M 2 → IR 6 1 : given that the curvature lines on both envelopes do correspond (s is "Ribaucour",f is a "Ribaucour transformation" of f ), there are two congruences of 2-spheres in S 3 such that each one intersects both surfaces orthogonally to one family of curvature lines. Moreover, homogeneous coordinates f,f : M 2 → IH 2 can be chosen such that these sphere congruences become s j and s k with respect to the frame F = (f,f ) : M 2 → Sl(2, IH), s = s i (cf.(3)), and that df,f = 0. This way, the homogeneous coordinates of the two envelopes are fixed up to constant rescaling. We will refer to such a frame as a "Ribaucour frame" for f andf .

The D-transformation and cmc-1 surfaces in H 3
Now, f andf induce conformally equivalent metrics on M 2 if either λ ≡ 0 or λ ≡ 0 while the other is constant by (14). In the former case, the two surfaces, f andf , are Möbius equivalent; in the latterf is a Darboux transform of f (and vice versa), and F is a "curved flat frame" (cf. [15]).
The second surface,f , of such a pair F = (f,f ) of Darboux transforms of each other is totally umbilic if and only if H ≡ 0 in (12). Obviously, this is equivalent to the fact that tr df, ds = 0 with respect to the induced conformal structure df, df of f , characterizing the enveloped sphere congruence s as the "central sphere congruence" of f (cf. [3]). By the Codazzi equations, alsoĤ is constant; consequently,Ĥf + λ s =: s ∞ is constant, ds ∞ =Ĥdf + λ ds ≡ 0, thus describing a constant 2-sphere (as long as λ = 0). As f , s ∞ ≡ 0,f takes values in that 2-sphere s ∞ . IfĤ ≡ 0, also f takes values in s ∞ and a "Darboux pair of meromorphic functions" is obtained as a degenerate case. IfĤ = 0, we can assumeĤ ≡ 1 without loss of generality 11) . Interpreting the two components of as hyperbolic spaces of sectional curvature −4λ 2 with infinity boundary s ∞ , the sphere congruence s becomes a congruence of horospheres in one of these hyperbolic spaces, H 3 ⊂ S 3 . Using the tangent plane congruence t := s + 2λf of f in hyperbolic space 12) , the first and second fundamental form of f become showing that f has constant mean curvature 13) h ≡ −2λ (as a surface in hyperbolic space). Summarizing, we obtain the following To confirm the last statement of the theorem, we just notice that the circles intersecting the central spheres s orthogonally in f andf are the hyperbolic geodesics orthogonal to f . Their intersection points with the infinity boundary s ∞ are given byf andf := f + 1 2λ s + 1 4λ 2f . As f andf induce conformally equivalent metrics on M 2 (while f andf do not) the claim follows (see [5]).

The T-transformation and cmc-1 surfaces in H 3
Having a closer look at the connection form 12) As t, s ∞ ≡ 0, the spheres t intersect the infinity boundary s ∞ orthogonally. 13) Here, we could as well have argued that the mean curvature of f at any point coincides with the mean curvature −2 s, s ∞ of the central sphere s at that point, measured in any space form. of the above Ribaucour frame for the surface f λ := f : M 2 → H 3 4λ 2 of constant mean curvature −2λ and its hyperbolic Gauss mapf λ :=f : M 2 → s ∞ =: s ∞ (λ) shows immediately 14) that f λ = T λ f 0 is a T λ -transform of an isothermic surface f 0 : (M 2 , q) → ImIH: sending λ → 0, the infinity sphere s ∞ (λ) →f 0 ≡ p ∞ degenerates to a point, df 0 ≡ 0, that we interprete as the point at infinity of Euclidean 3-space ImIH. Since the central spheres s 0 of f 0 all contain the point p ∞ at infinity they are planes in ImIH. Consequently, the surface f 0 is minimal in ImIH. On the other hand, away from umbilics, every minimal immersion f 0 : M 2 → ImIH carries conformal curvature line coordinates z : M 2 → / C (minimal immersions are isothermic) such that the Hopf differential q = dz 2 . Thus, every minimal immersion has a frame with connection form Φ 0 , and Theorem. The surfaces of constant mean curvature h = −2λ in hyperbolic space of sectional curvature k = −4λ 2 are in one-to-one correspondence with the minimal surfaces in Euclidean 3-space via the T λ -transformation.
As all surfaces f λ = T λ f 0 : M 2 → H 3 −4λ 2 , λ ∈ IR, are isometric, I λ = I 0 , and their second fundamental forms satisfy II λ = II 0 − 2λI 0 the T λ -transformation yields exactly the "Umehara-Yamada perturbation" [23] of minimal surfaces into constant mean curvature surfaces in hyperbolic space. Consequently, our previously discussed version (6) of the Weierstrass representation for minimal surfaces in Euclidean 3-space "perturbs" into a version of Bryant's Weierstrass type representation [5]: given Weierstrass data (g, ω) and λ ∈ IR the corresponding surface f λ of constant mean curvature −2λ in H 3 −4λ 2 (together with its hyperbolic Gauss mapf λ , if λ = 0) or IR 3 , respectively, is obtained by integrating the system Remark. The Gauss map (i − jg)i(i − jg) −1 of the minimal surface f 0 is called the "secondary Gauss map" of f λ , and f 0 its "minimal cousin" (cf. [5]).

Isothermic surfaces of spherical type
The Gauss equation (13) for the connection form (16) associated with a surface f λ : M 2 → H 3 −4λ 2 of constant mean curvature −2λ (or, its minimal cousin f 0 : M 2 → IR 3 ) reduces to the Liouville equation Consequently, the induced metric ds, ds = e −2u (dx 2 + dy 2 ) of the (central) sphere congruence s has constant Gauss curvature 1: 14) Remember that the T -transformation is frame independent, see (11).
Definition (spherical type). An isothermic immersion is called isothermic of spherical type if its central sphere congruence induces a metric of constant Gauss curvature 1.
We will see that this is a (Möbius geometric) characterization for surfaces of constant mean curvature h in hyperbolic space of sectional curvature k = −h 2 , and minimal surfaces in Euclidean space: it is well known [3] that a surface f : M 2 → S 3 is isothermic if and only if its central sphere congruence s is Ribaucour. Therefore, we can associate a Ribaucour frame F = (f,f ) to f wherê f is the second envelope of the central sphere congruence. In the connection form (12), the fact that s = s i is the central sphere congruence of f is reflected by H ≡ 0. By (14),Ĥ is constant so that without loss of generalityĤ ≡ 1 (see footnote 11). Now, u satisfies the Liouville equation (18) if and only ifλ ≡ 0, if and only if the connection form (12) takes the form (16). Thus, Theorem. An isothermic immersion is isothermic of spherical type iff the second envelope of its central sphere congruence is a Darboux transform -this Darboux transform is then either totally umbilic, or a pointiff the surface is either a surface of constant mean curvature h in a hyperbolic space of sectional curvature k = −h 2 , or a minimal surface in Euclidean space.
Consequently, every surface f : M 2 → H 3 k of constant mean curvature h in a space of sectional curvature k = −h 2 can be obtained as a D λ -transform, λ = − h 2 , of a totally umbilic immersion n h : M 2 → S 2 ∼ = ∂H 3 k . This gives rise to another Weierstrass type representation for such surfaces: given Weierstrass data (g, ω), integration of Darboux's linear system 15) as a D λ -transform of its hyperbolic Gauss map n h = −jg -as long as an initial condition for v is chosen such that v 2 v −1 1 ∈ / Cj. The secondary Gauss map of f # λ and its minimal cousin are obtained as T λ -transforms of −jg and f # λ , respectively. Thus, comparing this second Weierstrass type representation for surfaces of constant mean curvature h in H 3 −h 2 with Bryant's, the roles of the hyperbolic and secondary Gauss maps of the surface are interchanged.

Duality for cmc-1 surfaces in H 3
In [24] M. Umehara and K. Yamada introduced a notion of "duality" for cmc-1 surfaces in hyperbolic space: given a cmc-1 surface f : M 2 → H 3 with hyperbolic and secondary Gauss maps n h , n s : M 2 → S 2 ∼ = ∂H 3 , the "dual cmc-1 surface" f # : M 2 → H 3 is obtained by interchanging the roles of the two Gauss maps, n # h = n s and n # s = n h . However, the hyperbolic Gauss map n h is only well defined up to Möbius transformation of S 2 since isometries of H 3 extend to Möbius transformations of S 2 ∼ = ∂H 3 . Consequently, the dual cmc-1 surface f # depends on the position of f in hyperbolic space; as there is a 3-parameter family of essential Möbius transformations of the 2-sphere, generally, there is a 3-parameter family of dual cmc-1 surfaces. In the context of transformations of isothermic surfaces, this duality relation occurs as a special case of the permutability theorem sketched in Fig. 3: as discussed above, the hyperbolic Gauss map n h =f of a cmc-1 surface f is a D1  − cosh( √ λz)k][cosh( √ λz)− √ λ sinh( √ λz)k] −1 } that have the above minimal surfaces as their minimal cousins. 16) With respect to the original polarization, f # has constant mean curvature -1; changing the direction of the normal field, q → −q, yields mean curvature 1 (cf. [24]).
In order to determine the Darboux transforms of T λ f too, one has to integrate Darboux's linear system again with a connection form Φ λ,µ = Φ µ (T λ f ). As the Darboux transformation is a Möbius geometric notion any choice of representative for T λ f will lead to the same (in terms of Möbius geometry) result. Thus, with the above choice T λ f = −j tanh( , we obtain F λ,µ e 1 = F λ+µ e1 cosh( √ λz) confirming that T µ T λ f = T λ+µ f . In the special case µ = −λ of our permutability theorem (see Figure 3), F λ,0 F −1 λ,−λ = [F 0 F −1 λ ] −1 so that the Darboux transforms ] −1 } can be determined without further integration. Note that according to the permutability theorem all the surfaces D −λ T λ f have the Enneper surface as their minimal cousin, and they are dual cmc surfaces of D λ f in the sense of [24].