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Remarks on the Darboux transform of isothermic surfaces

Udo Hertrich-Jeromin^{⋆} and Franz Pedit^{⋆⋆}

GANG, University of Massachusetts at Amherst, MA 01003 (USA)

August 12, 1997

Summary. We study Darboux and Christoffel transforms of isothermic surfaces in Euclidean space. These transformations play a significant role in relation to integrable system theory. Surfaces of constant mean curvature turn out to be special among all isothermic surfaces: their parallel constant mean curvature surfaces are Christoffel and Darboux transforms at the same time. We prove — as a generalization of Bianchi’s theorem on minimal Darboux transforms of minimal surfaces — that constant mean curvature surfaces in Euclidean space allow ∞^{3} Darboux transforms into surfaces of constant mean curvature. We indicate the connection of these Darboux transforms to Bäcklund transforms of spherical surfaces.

1. Introduction

Transformations play an important role in connecting surface theory in differential geometry with the theory of integrable systems. The best known example might be the Bäcklund transform of pseudospherical (and spherical [1]) surfaces in Euclidean 3-space which “adds solitons” to a given surface. In case of isothermic surfaces the Darboux transform takes the role of the Bäcklund transform for pseudospherical surfaces [6]. Darboux transforms of isothermic surfaces naturally arise in 1-parameter families (“associated families”) allowing to rewrite the underlying (system of) partial differential equation(s) as an (infinite dimensional) integrable system [4], [8]. A key tool in the study of Darboux transforms of an isothermic surface in Euclidean space is a careful analysis of the Christoffel transform (or dual isothermic surface) of the surface. This transformation may

⋆ Partially supported by the Alexander von Humboldt Stiftung as well as by NSF grant DMS 93-12087. ⋆⋆ Partially supported by NSF grant DMS 93-12087.

be considered as a limiting case of Darboux transforms when running through the associated family.

It turns out that isothermic surface theory in codimension 2 is to some extend a more appropriate setting: for example the structure of the Riccati equation for Darboux transforms which we will derive is more clear. For that reason we will study Christoffel and Darboux transforms of isothermic surfaces in 4dimensional Euclidean space. In this case of higher codimension some notions have to be clarified — however, in case of 3-dimensional ambient space all the notions we will introduce become classical. Also, we will give characterizations of Christoffel and Darboux pairs in 4-dimensional ambient space which we will not prove in this paper (therefore, these characterizations are marked as “Propositions”). For more details on these preliminaries the reader is referred to [10] (and [9]). Nevertheless, we will provide some remarks on the proofs in the codimension 1 case. Building on these fundamentals, we discuss relations between Christoffel and Darboux transforms of isothermic surfaces in Euclidean 4-space leading to a Riccati type equation for Darboux transforms.

In the second half of the paper we specialize to 3-dimensional Euclidean ambient space. There, we will give a simple proof of our characterization of Christoffel pairs and find a quite explicit formula relating two surfaces of a Christoffel pair. Using this formula we learn that constant mean curvature surfaces in Euclidean 3-space are characterized by the fact that their Christoffel transforms actually arise as Darboux transforms — giving the constant mean curvature surfaces a special position among all other isothermic surfaces in this context^{1}. After analyzing the geometry of surfaces resulting from Darboux transforms more thoroughly we prove a generalization of Bianchi’s theorem on minimal Darboux transforms of minimal surfaces: a similar theorem holds in the constant mean curvature case. Moreover, in the final section of this paper, we relate these constant mean curvature Darboux transforms to Bianchi-Bäcklund transforms of constant mean curvature surfaces — which are obtained from Bäcklund transforms on the parallel spherical surfaces.

A couple of the results we are going to present in this paper can also be found in [2] — here, we would like to thank Fran Burstall who pointed our attention to Bianchi’s paper after reading a preprint of the present paper. However, at least some of our results seem to be new — certainly those on codimension 2 surfaces will not be found in Bianchi’s work. But it also seemed worth to keep results which might be found in the classical literature since they illustrate the methods we are going to develop. Of special interest might be the Riccati type equation which we derive for Darboux transforms of isothermic surfaces (theorem 2) — this equation leads to a Darboux transform for “discrete isothermic nets” [11] — and the relation of Darboux and Bäcklund transforms for surfaces of constant mean curvature which we establish in theorem 7.

1 While the notion of “Darboux pairs” is naturally a conformal notion (i.e. relates surfaces in Möbius space) the notion of “Christoffel pairs” is a Euclidean one. This might explain the (untypical) fact that constant mean curvature surfaces in Euclidean space have a special position, not constant mean curvature surfaces in any space of constant curvature.

Remarks on the Darboux transform of isothermic surfaces 3 Fig. 1. A Darboux transform of a torus of revolution

2. Darboux pairs in the conformal four sphere

In 3-dimensional Möbius space (the conformal sphere S^{3}) an isothermic surface may be characterized by the existence of conformal curvature line coordinates around each (nonumbilic) point^{2}. Note that the notion of principal curvature directions is conformally invariant — even though the second fundamental form is not. In higher codimensions the second fundamental form (with respect to any metric in the conformal class) takes values in the normal bundle. In order to diagonalize this vector valued second fundamental form, i.e. simultanously diagonalize all components of a basis representation, the surface’s normal bundle has to be flat^{3}. This is an implicit prerequisite in the following

Definition 1. A (2-dimensional) surface in (4-dimensional) Möbius space is called isothermic if around each (nonumbilic) point there exist conformal curvature line coordinates, i.e. conformal coordinates which diagonalize the (vector valued) second fundamental form taken with respect to any conformal metric of the ambient space.

In order to understand the notion of a “Darboux pair of isothermic surfaces” we also have to learn what a “sphere congruence” is and what we will mean by “envelope of a sphere congruence”:

Definition 2. A congruence of 2-spheres in (4-dimensional) Möbius space is a 2-parameter family of 2-spheres.

A 2-dimensional surface is said to envelope a congruence of 2-spheres if at each point it is tangent^{4} to a corresponding 2-sphere.

Note that the requirements on a congruence of 2-spheres in 4-space to be enveloped by two surfaces are much more restrictive than on a hypersphere congruence [9]. Also, a congruence of 2-spheres in may have one envelope — which case generically does not occur in the hypersphere case. In the second half of the paper we will concentrate on the more familiar situation in 3-space.

If, however, we have two surfaces which envelope a congruence of 2-spheres the congruence will establish a point to point correspondance between its two envelopes by assigning the point of contact on one surface to the point of contact on the other surface. In case of 3-dimensional ambient space it is well known [3] (cf. [7]) that if this correspondance preserves curvature lines^{5} and is conformal two cases can occur: the congruence consists of planes in a certain space of constant curvature — in which case the two envelopes are Möbius equivalent — or both envelopes are isothermic — in this case one surface is called a “Darboux transform” of the other (see [10], compare [3] or [4]). These remarks may motivate the following

2 There are several ways of defining isothermic surfaces — which are all equivalent away from umbilics. Since in this paper we intend to demonstrate some new techniques rather than struggeling with exceptional points we will always assume the absence of umbilics which is no restriction as long as we do local geometry of surfaces.

3 Since the principal directions of the (scalar) second fundamental forms with respect to any normal vector are conformally invariant, as in the codimension 1 case, the flatness of the normal bundle is a conformal invariant, too.

4 As usually done in the 3-dimensional case, we also want to allow the surface to degenerate. 5 This is what is called a “Ribaucour sphere congruence”.

Definition 3. If a congruence of 2-spheres (which is not a plane congruence in a certain space of constant curvature) is enveloped by two isothermic surfaces, the correspondance between its two envelopes being conformal and curvature line preserving, the surfaces are said to form a Darboux pair. Each of the two surfaces is called a Darboux transform of the other.

Before studying Darboux pairs in Euclidean space we will recall

3. A basic characterization for Darboux pairs

In order to discuss (Darboux) pairs of surfaces in four (or three) dimensional Möbius geometry we consider the conformal 4-sphere as the quaternionic projective line [9]:

S^{4} ≅ IHP^{1} = {x · IH | x ∈ IH^{2}}.  (1)

Note that we consider the space IH^{2} of homogeneous coordinates of the quaternionic projective line as a right vector space over the quaternions IH.

Now, let (f, f̂) : M^{2} → P be an immersion into the (symmetric) space of point pairs^{6} in S^{4} where

P := {(x, y) ∈ S^{4} × S^{4} | x ≠ y}.  (2)

We may write the derivatives of f and f̂ as^{7}

df = fφ + f̂ω, df̂ = fω̂ + f̂φ̂  (3)

where φ, ω, φ̂, ω̂ : T M → IH denote suitable quaternionic valued 1-forms. Then, the integrability conditions d^{2}f = d^{2}f̂ = 0 for f and f̂ — the Maurer Cartan equations — read

0 = dφ + φ ∧ φ + ω̂ ∧ ω  (Gauß equation for f),
0 = dω + ω ∧ φ + φ̂ ∧ ω  (Codazzi equation for f),
0 = dω̂ + ω̂ ∧ φ̂ + φ ∧ ω̂  (Codazzi equation for f̂),
0 = dφ̂ + φ̂ ∧ φ̂ + ω ∧ ω̂  (Gauß equation for f̂).

(4)

Since the quaternions are not commutative φ ∧ φ ≠ 0 in general. Before continuing, let us list some useful identities for quaternionic 1-forms: let α, β : T M → IH be quaternionic valued 1-forms and g : M → IH be a quaternionic valued function; then

\begin{aligned}α ∧ gβ & = & αg ∧ β, \\ \overline{α ∧ β} & = & −β̄ ∧ ᾱ, \\ d(gα) & = & dg ∧ α + g · dα, \\ d(αg) & = & −α ∧ dg + dα · g, \end{aligned}  (5)

6 The homogeneous coordinates of a pair of (different) points in IHP^{1} form a basis of IH^{2}. Thus, P can be identified with the symmetric space \frac{Gl(2,IH)}{IH_{∗}×IH_{∗}}. Sometimes it is more convenient to use suitably normalized coordinates: the group Gl(2, IH) may be replaced by a 15-dimensional subgroup Sl(2, IH) which is isomorphic to the group of orientation preserving Möbius transformations of S^{4} [9].

7 We will use “f” and “f̂” for the point maps into S^{4} as well as for their homogeneous coordinates.

where (α ∧ β)(x, y) := α(x)β(y) − α(y)β(x).

In this framework we are now able to state a basic characterization for Darboux pairs of isothermic surfaces (for more details^{8} including a proof see [10]):

Proposition 1. A pair of surfaces (f, f̂) : M^{2} → P is a Darboux pair if and only if

ω ∧ ω̂ = ω̂ ∧ ω = 0  (6)

where ω, ω̂ : T M → IH are defined by

df = fφ + f̂ω, df̂ = fω̂ + f̂φ̂.  (7)

It is easy to see that this characterization does not depend upon the choice of homogeneous coordinates for the two surfaces: given a change of homogeneous coordinates (f, f̂) ⇝ (fa, f̂â), a, â : M → IH, we have

\begin{aligned}d(fa) & = & (fa) · (a^{−1}φa + a^{−1}da) + (f̂â) · (â^{−1}ωa), \\ d(f̂â) & = & (fa) · (a^{−1}ω̂â) + (f̂â) · (â^{−1}φ̂â + â^{−1}dâ). \end{aligned}  (8)

4. Christoffel pairs of isothermic surfaces in Euclidean space

Another observation is that introducing a real parameter into the Maurer Cartan equations (4) we can obtain the Darboux pair equations (6) together with the original integrability conditions as integrability conditions of a 1-parameter family of Darboux pairs — the “associated family” of Darboux pairs^{9}: writing

df_{r} = f_{r}φ + f̂_{r}(r^{2}ω), df̂_{r} = f_{r}(r^{2}ω̂) + f̂_{r}φ̂  (9)

with a parameter r ∈ IR the Gauß equations for f_{r} and f̂_{r} become

\begin{aligned}0 & = & dφ + φ ∧ φ + r^{4} · ω̂ ∧ ω, \\ 0 & = & dφ̂ + φ̂ ∧ φ̂ + r^{4} · ω ∧ ω̂ \end{aligned}  (10)

while the Codazzi equations remain unchanged. This shows that if there exist surface pairs — not necessarily Darboux — (f_{r}, f̂_{r}) for more than one value of r > 0, then, we have a whole 1-parameter family of Darboux pairs.

Assuming we have such a 1-parameter family (f_{r}, f̂_{r}) of Darboux pairs a special situation will occur when r → 0. To discuss this, we assume φ = φ̂ = 0 without loss of generality: we have 0 = dφ + φ ∧ φ and 0 = dφ̂ + φ̂ ∧ φ̂ and thus at least locally φ = −da a^{−1} and φ̂ = −dâ â^{−1} with suitable functions a, â : M → IH. Rescaling by those and applying (8) gives φ = φ̂ = 0. Thus,

df_{r} = f̂_{r}(r^{2}ω), df̂_{r} = f_{r}(r^{2}ω̂),  (11)

and after the rescaling (f, f̂) ⇝ (f \frac{1}{r}, f̂r) (or (f, f̂) ⇝ (fr, f̂\frac{1}{r}), respectively) we see that f̂ (or f) becomes a fixed point in the conformal 4-sphere — which

8 In fact, this proposition states the connection between Darboux pairs and “curved flats”

[8] in the symmetric space of point pairs.

9 As we mentioned in a previous footnote (8) Darboux pairs are actually curved flats in the symmetric space of point pairs — and curved flats arise in associated families.

should be interpreted as the point at infinity of Euclidean space — and df_{0} = ω (or df̂_{0} = ω̂).

These two limit surfaces^{10} f̂_{0}^{c} := f_{0} and f_{0}^{c} := f̂_{0} usually do not form a Darboux pair — in general they do not even envelope a congruence of 2-spheres^{11}. But they do form what is called a Christoffel pair [10]:

Definition 4. Two surfaces f_{0}, f̂_{0} : M^{2} → IR^{4} ≅ IH in Euclidean 4-space are said to form a Christoffel pair if they induce conformally equivalent metrics on M and −f̄_{0} and f̂_{0} have parallel tangent planes with opposite orientations. Each of the surfaces of a Christoffel pair is called a Christoffel tranform or dual of the other.

Note that the two surfaces of a Christoffel pair are automatically isothermic; in fact, isothermic surfaces can be characterized by the (local) existence of a Christoffel transform [10]. Also, the Christoffel transform of an isothermic surface can be shown to be unique^{12} up to a homothety and a translation — so that in the sequel we will denote the Christoffel transform of an isothermic surface f by f^{c}.

Finally, let us state a characterization of Christoffel pairs similar to that for Darboux pairs:

Proposition 2. Two surfaces f_{0}, f̂_{0} : M^{2} → IR^{4} ≅ IH form a Christoffel pair if and only if

df_{0} ∧ df̂_{0} = df̂_{0} ∧ df_{0} = 0.  (12)

Both surfaces of a Christoffel pair are isothermic.

As for the characterization of Darboux pairs (page 6) a proof may be found in [10]. However, in case of 3-dimensional ambient space we will present an easy proof later (page 12) using some of the calculus we are going to develop.

Now we are prepared to study

5. Darboux pairs in IR^{4}

Let (f, f̂) : M^{2} → P denote a pair of surfaces with

df = fφ + f̂ω, df̂ = fω̂ + f̂φ̂,  (13)

as before. Assuming that f, f̂ : M → IH × {1} ≅ IH take values in Euclidean 4-space we see that φ = −ω and φ̂ = −ω̂, and hence

df = (f̂− f) · ω, df̂ = (f − f̂) · ω̂.  (14)

10 Actually, it would have been more precise if we had defined f_{0} := lim_{r→0}(f_{r} \frac{1}{r}) and f̂_{0} := lim_{r→0}(f̂_{r} \frac{1}{r}). 11 This might seem more natural if we remember that f_{0} and f̂_{0} take values in “different” Euclidean spaces (cf. [4]). — However, one of these surfaces and the point at infinity (which are the remains of the other surface) do form a (degenerate) Darboux pair. 12 Except in one case: Christoffel transforms of the 2-sphere appear in 1-parameter families. We will discuss this case later (see page 13).

This allows us to rewrite condition (6) on f and f̂ to form a Darboux pair^{13} as

0 = df ∧ (f − f̂)^{−1}df̂ = df̂∧ (f̂− f)^{−1}df.  (15)

As a first consequence of these equations we derive the equations

\begin{aligned}0 & = & df ∧ (f̂− f)^{−1}df̂(f̂− f)^{−1} & = & (f̂− f)^{−1}df̂(f̂− f)^{−1} ∧ df, \\ 0 & = & df̂∧ (f − f̂)^{−1}df(f − f̂)^{−1} & = & (f − f̂)^{−1}df(f − f̂)^{−1} ∧ df̂ \end{aligned} (16)

for any Darboux pair (f, f̂). Since (15) also implies

0 = d[(f̂− f)^{−1}df̂(f̂− f)^{−1}] = d[(f − f̂)^{−1}df(f − f̂)^{−1}]  (17)

we conclude that the Christoffel transforms f^{c} and f̂^{c} of f and f̂ are given by

\begin{aligned}df^{c} & = & (f̂− f)^{−1}df̂(f̂− f)^{−1}, \\ df̂^{c} & = & (f − f̂)^{−1}df(f − f̂)^{−1}. \end{aligned}  (18)

Finally, if we fix the translations of f^{c} and f̂^{c} such that

(f^{c} − f̂^{c}) = (f − f̂)^{−1}  (19)

— note that we have d(f − f̂)^{−1} = d(f^{c} − f̂^{c}) — we learn from the above characterization (15) of Darboux pairs that f^{c} and f̂^{c} also form a Darboux pair (cf. [2]):

Theorem 1. If f, f̂ : M^{2} → IR^{4} form a Darboux pair, then, their Christoffel transforms f^{c}, f̂^{c} : M^{2} → IR^{4} (if correctly scaled and positioned) form a Darboux pair, too.

So far we learned how to derive the Christoffel transforms f^{c} and f̂^{c} of two surfaces f and f̂ forming a Darboux pair. But usually it will be much easier to determine an isothermic surface’s Christoffel transform than a Darboux transform. In the next section we will see that deriving Darboux transforms f̂ and f̂^{c} of two surfaces f and f^{c} forming a Christoffel pair^{14} comes down to solving

6. A Riccati type equation

Solving (18) for df̂ we obtain df̂ = (f̂− f)df^{c}(f̂− f). This yields the following Riccati type partial differential equation^{15} for g := (f̂− f):

dg = g df^{c} g − df.  (20)

13 Hopefully, the reader will forgive our context dependent notation: f and f̂ denote points in IHP^{1} ≅ S^{4}, vectors in IH^{2} or numbers in IH ≅ IR^{4}. 14 Note that the notation f̂^{c} for a Darboux transform of f^{c} makes sense because of our previous theorem: we have _{f^{c}} = f̂^{c}. 15 The pictures in this paper were produced using Mathematica to numerically integrate this Riccati type equation.

Using our characterization (12) of Christoffel pairs it is easily seen that this equation is completely integrable. Note that — in agreement with our previous results — the common transform g^{c} = \frac{1}{g} for Riccati equations yields

dg^{c} = g^{c} df g^{c} − df^{c},  (21)

showing that f̂^{c} = f^{c} + \frac{1}{g} will provide a Darboux transform of f^{c} whenever f +g is a Darboux transform of f coming from a solution g of (20).

Since every Darboux transform f̂ of an isothermic surface f provides a Christoffel transform f^{c} of f via (18) every Darboux transform comes from a solution of (20) — if we do not fix the scaling of the Christoffel transform f^{c}. On the other hand every solution g of (20) defines a Darboux transform f̂ = f + g of f since df ∧ g^{−1}d(f + g) = d(f + g) ∧ g^{−1}df = 0. This seems to be worth formulating a

Theorem 2. If f, f^{c} : M^{2} → IR^{4} form a Christoffel pair of isothermic surfaces every solution of the completely integrable Riccati type partial differential equation

dg = g df^{c} g − df  (22)

provides a Darboux transform f̂ = f + g of f. On the other hand, every Darboux transform f̂ of f is obtained this way — if we do not fix the scaling of f^{c}.

Fig. 2. Darboux transforms of the Catenoid when H^{c} → ∞

At this point, we should discuss the effect of a rescaling of the Christoffel transform f^{c} in the equation (20). To this extend we examine the equations

dg = g (±r^{4}df^{c}) g − df  (23)

where r ≠ 0 is a real parameter. For the derivatives of f and a Darboux transform f̂ = f + g of f this yields

\begin{aligned}df & = & f · [−g^{−1}df] & + & f̂· [g^{−1}df], \\ df̂ & = & f · [∓r^{4}df^{c} g] & + & f̂· [±r^{4}df^{c} g]. \end{aligned}  (24)

Interpreting f, f̂ : M^{2} → IH ≅ IH × {1} as homogeneous coordinates of the point pair map (f, f̂) : M^{2} → P we may choose new homogeneous coordinates by performing a rescaling (f, f̂) ⇝ (fr, f̂(rg)^{−1}) to obtain^{16}

16 Note that this rescaling provides an Sl(2, IH) framing of the point pair map (f, f̂) [9].

\begin{aligned}d[fr] & = & [fr] · [−g^{−1}df] & + & [f̂(rg)^{−1}] · [r^{2}df], \\ d[f̂(rg)^{−1}] & = & [fr] · [∓r^{2}df^{c}] & + & [f̂(rg)^{−1}] · [df g^{−1}]. \end{aligned}  (25)

Even though this system ressembles very much our original system (9) which describes the associated family of Darboux pairs, there is an essential difference: in (9) the forms φ, ω, φ̂ and ω̂ are independent of the parameter r whereas the forms g^{−1}df and df g^{−1} in the system we just derived do depend on r. In fact, in the associated family (f_{r}, f̂_{r}) of Darboux pairs obtained from (9) both surfaces, f_{r} as well as f̂_{r}, change with the parameter r whereas the parameter contained in the Riccati equation just effects the Darboux transform f̂ = f̂_{r} while the original surface f remains unchanged. However, the original system (9) appears in the linearization of our Riccati equation^{17} which indicates a close relation of these two parameters — a more comprehensive discussion will be given in [10].

As a first application of this parameter which occurs from rescalings of the Christoffel transform f^{c} in our Riccati equation we may prove an extension of Bianchi’s permutability theorem [2] for Darboux transforms:

Theorem 3. Let f̂_{1,2} : M^{2} → IH be two Darboux transforms of an isothermic surface f : M^{2} → IH,

df̂_{1,2} = r_{1,2}(f̂_{1,2} − f) df^{c}(f̂_{1,2} − f),  (26)

where we fixed any scaling for the Christoffel transform f^{c} of f. Then, there exists an isothermic surface f̂: M^{2} → IH which is an r_{1}-Darboux transform of f̂_{2} and an r_{2}-Darboux transform of f̂_{1} at the same time^{18}:

df̂ = r_{2,1}(f̂− f̂_{1,2}) df̂_{1,2}^{c}(f̂− f̂_{1,2}).  (27)

Moreover, the points of f̂ lie on the circles determined by the corresponding points of f, f̂_{1} and f̂_{2}, the four surfaces having a constant (real) cross ratio^{19}

\frac{r_{2}}{r_{1}} ≡ (f − f̂_{1})(f̂_{1} − f̂)^{−1}(f̂− f̂_{2})(f̂_{2} − f)^{−1}.  (28)

To prove this theorem we simply define the surface f̂: M^{2} → IH by solving the cross ratio equation^{20} (28) for f̂:

f̂:= [r_{2}f̂_{1}(f̂_{1} − f)^{−1} − r_{1}f̂_{2}(f̂_{2} − f)^{−1}] · [r_{2}(f̂_{1} − f)^{−1} − r_{1}(f̂_{2} − f)^{−1}]^{−1}. (29)

Using this ansatz, it is a straightforward calculation to verify the Riccati equations (27) which proves the theorem.

As indicated earlier, from now on we will concentrate on surfaces in 3dimensional Euclidean space IR^{3} ≅ ImIH:

17 Here, we would like to thank Fran Burstall for helpful discussions. 18 Note, that this claim makes no sense before we fix a scaling for the Christoffel transforms f̂_{1,2}^{c} of f̂_{1,2}. But, according to our “permutability theorem” for Christoffel and Darboux transforms (theorem 1) there is a canonical scaling for f̂_{1,2}^{c} after we fixed the scaling of f^{c}. 19 For a comprehensive discussion of the (complex) cross ratio in IR^{4} ≅ IH see [11]. The idea for the proof given in this paper actually originated from the discrete version of this theorem. 20 Note that the denominator does not vanish as long as f̂_{1} ≠ f̂_{2}. For r_{1} = r_{2} we get f̂ = f.

7. Christoffel pairs in IR^{3}

In this situation, much of our previously developed calculus will simplify considerably. For example, we will be able to give an easy proof of our characterization of Christoffel pairs and also write down the Christoffel transform of an isothermic surface quite explicitely. First we note that our characterizations (15) and (12) of Darboux and Christoffel pairs of isothermic surfaces reduce to just one equation: if f, f̂: M^{2} → ImIH both take values in the imaginary quaternions we have

\begin{aligned}df̂∧ df & = & −\overline{df ∧ df̂}, \\ df̂∧ (f̂− f)^{−1}df & = & −\overline{df ∧ (f − f̂)^{−1}df̂}. \end{aligned}  (30)

In order to continue we will collect some identities present in the codimension 1 case. We may orient an immersion f : M^{2} → IR^{3} ≅ ImIH by choosing a unit normal field n : M^{2} → S^{2}. This defines the complex structure J on M via

df ∘ J = n df  (31)

— note that since f and n take values in the imaginary quaternions

n df = −⟨n, df⟩ + n × df = n × df = −df n.  (32)

The Hodge operator is then given as the dual of this complex structure:

∗ η = −η ∘ J  (33)

for any 1-form η on M.

With this notation we are able to give a useful reformulation^{21} of the equation arising in our characterizations of Darboux pairs and Christoffel pairs: if η : T M → IH is any quaternionic valued 1-form we have

(df ∧ η)(x, Jx) = df(x) · (− ∗ η(x) + n η(x))  (34)

for any x ∈ T M. Consequently, df ∧ η = 0 if and only if

∗ η = n η.  (35)

This criterium immediately shows that the space of imaginary solutions η : T M → ImIH of the equation 0 = df ∧ η is pointwise 2-dimensional^{22} — if η is an (injective) solution, then, every other solution η̃ is of the form

η̃ = (a + b n) · η  (36)

with suitable functions a, b : M → IR. But one (imaginary) solution to the equation 0 = df ∧ η is easily found: it is well known that

d ∗ df = −dn ∧ df = H df ∧ df  (37)

21 At this point we would like to thank Ulrich Pinkall for many helpful discussions — this criterium is actually due to him. 22 The space of solutions with values in the full quaternions is 4-dimensional as is easily seen:

(36) becomes

η̃ = (a + b n) · η + (∗α + nα)

with an arbitrary real 1-form α : T M → IR.

where H is the mean curvature of f. Thus

df ∧ (dn + H df) = 0  (38)

which gives an injective solution η = dn + H df away from umbilics of f.

At this point, we are ready to give the announced proof of our characterization of Christoffel pairs (12) in the 3-dimensional case:

Theorem 4. Two surfaces f, f^{c} : M^{2} → IR^{3} ≅ ImIH form a Christoffel pair if and only if

df ∧ df^{c} = 0.  (39)

Generically, the Christoffel transform f^{c} of f is uniquely determined by f up to homotheties and translations of IR^{3}.

The fact that both surfaces of a Christoffel pair in 3-space are isothermic is classical (see for example [5]) — and thus we omit this calculation.

In order to prove this theorem we note that from the above we know that f^{c} : M^{2} → ImIH satisfies (39) if and only if

∗ df^{c} = n df^{c}.  (40)

But this equation means that in corresponding points f and f^{c} have parallel tangent planes and that the almost complex structure induced by f^{c} with respect to n^{c} := −n is just J — the same as that induced by f with respect to n. Thus,

df ∧ df^{c} = 0  (41)

if and only if f, f^{c} : M^{2} → IR^{3} have parallel tangent planes with opposite orientations and they induce conformally equivalent metrics, i.e. they form a Christoffel pair.

Now assume we have not just one but two Christoffel transforms f^{c} and f̃^{c} of an isothermic surface f : M^{2} → IR^{3}. Then we know from (36) that

df̃^{c} = (a + b n) · df^{c}.  (42)

The integrability condition for f̃^{c} reads

0 = da ∧ df^{c} + db ∧ ∗df^{c} + b H^{c}df^{c} ∧ df^{c}  (43)

showing that a = const and b = 0 since df^{c}∧ df^{c} takes values in normal direction while all other components are tangential — provided that f^{c} is not a minimal surface^{23}. This concludes the proof.

With (38) it also follows that

dn + H df = (a + b n)df^{c}  (44)

for suitable functions a, b : M → IR. Similarly, we obtain

− dn + H^{c} df^{c} = (a^{c} + b^{c}n)df  (45)

by interchanging the roles of f and f^{c}. Adding these two equations yields a = H^{c}, a^{c} = H and b = b^{c} = 0 since the forms df, n df, df^{c} and n df^{c} are linearly

23 The case of minimal Christoffel transforms will be discussed below.

independent (over the reals). As a consequence, we have a quite explicit formula relating the two surfaces of a Christoffel pair:

H^{c}df^{c} = dn + H df.  (46)

This equation shows that whenever one of the surfaces of a Christoffel pair is a minimal surface the other is totally umbilic (namely, a scaling of its Gauß map) and vice versa. This brings us back to our previous problem of the uniqueness of Christoffel transforms: assume we have a Christoffel pair (f, n) consisting of a minimal surface f and its Gauß map n. Then all the pairs

( a ∫ (cos(t) + sin(t) n) · df , n)  (47)

with real constants a and t will also form Christoffel pairs. Up to homotheties (given by a) this will run us through the associated family of minimal surfaces (given by t) reflecting the fact that associated minimal surfaces have the same Gauß map^{24}.

Fig. 3. A Darboux transform of the Catenoid

As another interesting consequence of the equation (46) we see that the (correctly scaled and positioned) Christoffel transform of a surface of constant mean curvature H ≠ 0 is its parallel surface f + \frac{1}{H} n of the same constant mean curvature H^{c} = H. Note that this parallel surface induces a conformally equivalent metric on the underlying manifold M^{2} and consequently it is also a

24 However, choosing “curvature lines” for the Gauß map will fix the minimal surface.

Darboux transform of the original surface^{25} — the enveloped sphere congruence consisting of spheres with constant radius \frac{1}{2H}. Later, we will see that constant mean curvature surfaces in Euclidean space can be characterized by the fact that their Christoffel transforms are Darboux transforms too. Thus, in the remaining part of this paper we will study constant mean curvature (H ≠ 0 or H = 0) Darboux transforms of

8. Surfaces of constant mean curvature

Using the reformulation (35) of our characterizing equation (15) of Darboux pairs we conclude that for any Darboux transform f̂ = f + g of f : M^{2} → IR^{3}

∗ g df̂ = n g df̂  (48)

where we used the fact that g^{−1} = −\frac{1}{|g|^{2}} g for g ∈ ImIH. Consequently, the

normal field n̂ of f̂ is given by^{26}

n̂ = \frac{gng}{|g|^{2}} = \frac{1}{|g|^{2}} (|g|^{2}n − 2⟨n, g⟩g)  (49)

since we must have ∗df̂ = −n̂df̂.

Thus, if the normal field of a Darboux transform f̂ of an isothermic surface f : M^{2} → ImIH equals that of its Christoffel transform,

n̂ = n^{c} = −n,  (50)

then g = a n for a suitable constant a ∈ IR (remark that a has to be constant in order to obtain parallel tangent planes of f̂ and f). With (46) we conclude

Hdf + dn = H^{c}df^{c} = H^{c}(df + dg) = H^{c}df + H^{c}a dn  (51)

which implies that either one of the surfaces is minimal and the other is totally umbilic, or, H = H^{c} = \frac{1}{a} which means that f and f̂ = f^{c} form a pair of parallel constant mean curvature surfaces.

Together with our previous remark (page 13) this leaves us with the following characterization of constant mean curvature surfaces:

25 Note that in order to obtain g = \frac{n}{H} as a solution of our Riccati type equation (20) the Christoffel transform df^{c} of f has to be scaled such that H^{c} = \frac{1}{H} — then, the Riccati equation is equivalent to (46). This means that the parallel constant mean curvature surface appears at a well defined location in the associated family. 26 Note that with this formula we easily see that f̂ is the second envelope of a sphere congruence enveloped by f:

2⟨g, n⟩f + |g|^{2}n = 2⟨g, n⟩f̂+ |g|^{2}n̂.

The second fundamental form of f̂ is quite complicated, but at least, when introducing frames it can be seen that it has the same principal directions as the seond fundamental form of f^{c}. Since f̂ also induces the conformally equivalent metric |df̂|^{2} = |g|^{4}|df^{c}|^{2} we get half of a proof for our characterization (15) of Darboux pairs in the case of 3-dimensional ambient space.

Theorem 5. The (correctly scaled and positioned) Christoffel transform f^{c} of an isothermic surface f : M^{2} → IR^{3} is also a Darboux transform f̂ of f if and only if f is a surface of constant mean curvature H ≠ 0. In this case f̂ = f^{c} is the parallel surface of constant mean curvature.

In order to study constant mean curvature Darboux transforms of constant mean curvature surfaces in general we have to calculate the mean curvature of a Darboux transform f̂ of an isothermic surface. There are several ways to do so: we could calculate the second fundamental form of f̂ — which is not convenient because this second fundamental form looks quite difficult — or, we could use (37) to directly calculate Ĥ with the help of our Riccati type equation (20). This second way is quite straightforward but not very interesting. So, we will present another way which grew out of discussions with Ulrich Pinkall^{27}: observing that df̂ = −ḡdf^{c}g the integrability condition for f̂ becomes

0 = ḡ(\overline{dg g^{−1}} ∧ df^{c} − df^{c} ∧ dg g^{−1})g,  (52)

i.e. the reality of the form df^{c} ∧ dg g^{−1}. Since the volume form \frac{1}{2}df^{c} ∧ ∗ df^{c} induced by f^{c} is a basis of the real 2-forms on M this may be reformulated as

0 = df^{c} ∧ (dg g^{−1} − \frac{1}{2}U ∗ df^{c})  (53)

with a suitable function U : M → IR. With (35) we obtain the equivalent equation

n^{c}dg − ∗dg = U df^{c}g  (54)

— the “Dirac equation” with reference immersion f^{c}.

Using this equation we may calculate the mean curvature Ĥ of f̂ in terms of the function U via

d ∗ df̂ = \frac{1}{|g|^{2}} (U − H^{c}) df̂∧ df̂  (55)

since

∗ α ∧ ∗β = α ∧ β  (56)

for any two 1-forms α, β : T M → IH on a Riemann surface and hence

∗ df^{c} ∧ dg = \frac{1}{2}(∗df^{c} ∧ dg − ∗df^{c} ∧ ∗ ∗ dg) = \frac{1}{2}df^{c} ∧ (n^{c}dg − ∗dg).  (57)

Substituting our Riccati type equation (20) into the Dirac equation yields U = 2⟨n, g⟩ and consequently

Ĥ = \frac{1}{|g|^{2}} (2⟨n, g⟩ − H^{c}).  (58)

Now we assume the mean curvature H of our original surface f to be constant — and consequently H^{c} is constant too — and rewrite this equation as

27 The Dirac equation (54) which we will discover on our way can be considered as a replacement for the Cauchy Riemann equations in a generalized “Weierstraß representation” for surfaces in IR^{3}. Given an immersion f : M^{2} → IR^{3} this generalized “Weierstraß representation” will provide us with any immersion f̂ which induces the same complex structure on M.

0 = hĤ(g) := Ĥ |g|^{2} − 2⟨n, g⟩ + H^{c}.  (59)

Taking the derivative of this function h_{C} where C denotes any constant and assuming H^{c} to be constant yields

dh_{C}(g) = −2⟨df^{c}, g⟩ · h_{C}(g) − 2⟨df, g⟩ · (C − H)  (60)

where we got rid of dn by using (46). This shows that whenever we choose an initial value g(p_{0}) = g_{0} for a function g : M^{2} → ImIH such that h_{H}(g_{0}) = 0 the trivial solution h_{H} ≡ 0 will be the unique solution to the above (linear and homogeneous: C = H) differential equation. Thus our Riccati type equation (20) will produce a Darboux transform f̂ = f +g of constant mean curvature Ĥ = H out of a surface of constant mean curvature (H ≠ 0 or H = 0).

To conclude let us study the geometry of the condition h_{H}(g) = 0: for a minimal surface this simply says that the points f̂(p) of f̂ = f + g always lie in distance \frac{1}{2}H^{c} off the tangent planes f(p) + d_{p}f(T_{p}M) of f. Since we also have the freedom of rescaling the Christoffel transform f^{c} of f we end up with a 3parameter family of minimal Darboux transforms of a minimal surface (cf. [2]). A minimal Darboux transform of the Catenoid is shown in figure 3. Sending H^{c} → ±∞ — note that in case of surfaces of constant mean curvature the associated family of Darboux pairs may be parametrized by H^{c} — the Darboux transforms look more and more like the original surface (Fig. 2) while sending H^{c} → 0 the Darboux transforms approach a planar surface patch — the best compromise between the Catenoid’s Christoffel transform and a minimal surface (Fig. 4).

Fig. 4. Darboux transforms of the Catenoid when H^{c} → 0

In case of a surface of constant mean curvature H ≠ 0 we may reformulate the condition h_{H}(g) = 0 as

|H g − n|^{2} = 1 − H^{c}H  (61)

showing that the points f̂(p) lie on spheres centered on the parallel surface f + \frac{1}{H} n and with constant radius \frac{1}{H} \sqrt{1 − H^{c}H}. Since the radius has to be real to provide real Darboux transforms we see that we have to have H^{c}H ≤ 1 which restricts the range of the parameter H^{c} to a ray H^{c} ≤ \frac{1}{H} containing 0 (without loss of generality we assume H ≥ 0). As H^{c} → −∞ and H^{c} → 0 we obtain the original surface and its Christoffel transform, respectively. But now, we obtain the Christoffel transform a second time — as a Darboux transform when H^{c} = \frac{1}{H}, i.e. when the spheres h_{H}(g) = 0 collapse to points. Figures 5 and 6 show constant mean curvature Darboux transforms of the cylinder.

To summarize the results we found in this section we may formulate a theorem generalizing a theorem of Bianchi on minimal Darboux transforms of minimal surfaces [2]:

Theorem 6. Any surface of constant mean curvature (H ≠ 0 or H = 0) in Euclidean 3-space allows a 3-parameter family of Darboux transforms into surfaces of the same constant mean curvature.

Having a second look at the Darboux transform of the cylinder shown in figure 5 we recognize a strong similarity to Ivan Sterling’s “doublebubbleton” [12]. This suggests a relation between our constant mean curvature Darboux transform and

Fig. 5. A Darboux transform of the cylinder

9. The Bianchi-Bäcklund transform of constant mean curvature surfaces

We may supply any surface f : M^{2} → IR^{3} of constant mean curvature H = \frac{1}{2} with conformal coordinates (x, y) : M^{2} → IR^{2} such that

I = e^{2u}(dx^{2} + dy^{2}),
II = e^{u}(sinh(u)dx^{2} + cosh(u)dy^{2})

(62)

— reflecting the fact that every surface of constant mean curvature is isothermic. Then, a new surface of constant mean curvature — a “Bianchi-Bäcklund transform” of the original surface — can be obtained as f̂ = f + g where

g = \frac{2}{sinh(β) cosh(β + φ)} (cosh(β)e^{−u}[cos ψf_{x} − sin ψf_{y}] − sinh φn) , (63)

β denoting a real parameter and φ + iψ = θ being given by the linear system

θ_{x} + iu_{y} = sinh β sinh θ cosh u + cosh β cosh θ sinh u
iθ_{y} + u_{x} = − sinh β cosh θ sinh u − cosh β sinh θ cosh u.

(64)

In fact, this transformation is obtained by applying two successive Bäcklund transforms to the surface of constant Gauß curvature [1] which is parallel to the original surface of constant mean curvature and then, taking the (correct) parallel surface of constant mean curvature [12]. In this construction, the second Bäcklund transform has to be matched to the first one such that the resulting surface of constant Gauß curvature is a real surface again.

Fixing the scaling of the Christoffel transform f^{c} of f such that H^{c} = H = \frac{1}{2}, i.e. f^{c} = f + 2n, it is an unpleasant but straightforward calculation to see that our Riccati type equation

dg = g(\frac{sinh^{2}(β)}{4}df^{c})g − df  (65)

is equivalent to the above linear system (64) defining the function θ. Thus we have:

Theorem 7. Any Bianchi-Bäcklund transform of a surface of constant mean curvature is a Darboux transform.

Analyzing the effect of the three parameters (β and initial values for φ and ψ) contained in the Bianchi-Bäcklund transform on the function g : M → IR^{3} at an initial point we find that any solution of our Riccati equation (20) with a positive multiple of the parallel constant mean curvature surface f + 2n as Christoffel transform f^{c} can be obtained via a Bianchi-Bäcklund transform^{28}. Those constant mean curvature Darboux transforms of a constant mean curvature surface where the Christoffel transform is taken a negative multiple of the parallel constant mean curvature surface (see Fig. 6) seem not to occur as Bianchi-Bäcklund transforms.

28 Hereby, we also have to allow singularities φ → ∞ to obtain vertical values of g too.

Remarks on the Darboux transform of isothermic surfaces 19 Fig. 6. Another Darboux transform of the cylinder

References

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2. L. Bianchi: Ricerche sulle superficie isoterme e sulla deformazione delle quadriche; Annali Mat. 11 (1905) 93-157

3. W. Blaschke: Vorlesungen über Differentialgeometrie III; Springer, Berlin 1929

4. F. Burstall, U. Hertrich-Jeromin, F. Pedit and U. Pinkall: Curved flats and Isothermic surfaces; to appear in Math. Z.

5. E. Christoffel: Ueber einige allgemeine Eigenschaften der Minimumsflächen; Crelle’s J. 67 (1867) 218-228

6. J. Cieśliński: The Darboux-Bianchi transformation for isothermic surfaces; IP-WUD Preprint 15, Warsaw 1994

7. G. Darboux: Sur les surfaces isothermiques; Comptes Rendus 128 (1899) 1299-1305, 1538 8. D. Ferus, F. Pedit: Curved flats in symmetric spaces; to appear in manuscripta math.

9. U. Hertrich-Jeromin: A quaternion calculus for Möbius differential geometry; manuscript (1996)

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12. I. Sterling, H.C. Wente: Existence and Classifiation of Constant Mean Curvature Multibubbletons of Finite and Infinite Type; Indiana Univ. Math. J. 42 (1993) 1239-1266

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