Remarks on the Darboux transform of isothermic surfaces

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.


Introduction
Transformations play an important role connecting surface theory with the theory of integrable systems.A well known example is the B acklund transformation of pseudospherical (and spherical 1]) surfaces in Euclidean 3-space which \adds solitons" to a given surface.In case of isothermic surfaces the Darboux transformation takes the role of the B acklund transform for pseudospherical surfaces.Darboux transforms of isothermic surfaces naturally arise in 1-parameter families (\associated families") allowing to rewrite the underlying (system of) partial di erential equation(s) as an (in nite dimensional) integrable system 6], 4].It is mainly for this reason that Darboux transformations provoke new interest among contemporary geometers | even though the subject was well studied around the turn of the century 5], 7] and 2].A key tool in the study of Darboux transforms of an isothermic surface in Euclidean space is a careful analysis of the Christo el transform (or dual isothermic surface) of the surface | which may be considered as a certain limiting case of Darboux transforms.In the present paper, we develop classical results further using quaternionic calculus which makes de nitions elegant and calculations more e cient.Characterizations thus obtained turned out to be necessary in the development of the corresponding discrete theory 10].
In the rst part of the paper, we develop isothermic surface theory in codimension 2 | which is a more appropriate setting when using quaternionic calculus.When restricting to codimension 1, all notions become classical.Here, we rely on the characterizations of Darboux and Christo el pairs in I HP 1 given in 9].The consequent use of the quaternionic setup yields a new and uni ed description for these surface pairs in I R 4 = I H.Even though the quaternionic calculus (as developed in 9]) provides a setting to study the global geometry of surface pairs in M obius geometry (cf.11]) we will restrict to local geometry in this paper, for two reasons: rst, there are a number of possible de nitions of a \globally isothermic surface" whose consequences have not yet been worked out.For example, de nition 1 may well be read as a global de nition but it is far too general to provide any global results.Secondly, Christo el and Darboux transforms of a (compact) surface generally do not exist globally.Moreover, around certain types of umbilics they may not even exist locally.However, up to the problem of closing periods, the results on constant mean curvature surfaces can well be read as global results: here, the Christo el transform can be determined without integration which ensures its global existence (with branch points at the umbilics of the original surface).
A central result is obtained by carefully analyzing the relation between Darboux and Christo el pairs: we derive a Riccati type equation describing all Darboux transforms of a given isothermic surface.This equation is crucial for the explicit calculation of Darboux transforms | in the smooth case (all the pictures shown in this paper are obtained from this equation) as well as in the theory of discrete isothermic nets 10].Moreover, most of our remaining results are di erent applications of the Riccati equation: rst, we extend Bianchi's permutability theorems for Darboux and Christo el transforms for the codimension 2 setup.We then discuss constant mean curvature surfaces in 3-dimensional Euclidean space as \special" isothermic surfaces: they can be characterized by the fact that their Christo el transforms arise as Darboux transforms 3 .Together with the Riccati equation, this provides more detailed knowledge about the 13 constant mean curvature Darboux transforms of a constant mean curvature surface | whose existence is a classical result due to Bianchi 1].Our new proof shows that any such Darboux transform has (pointwise) constant distance to the Christo el transform.This fact provides a geometric de nition for a Figure 1: A Darboux transform of a torus of revolution discrete analog of smooth constant mean curvature surfaces 10].We conclude this paper relating this 3-dimensional family with the Bianchi-B acklund transformation for constant mean curvature surfaces discussed in 12] (cf.1]).
2 Darboux pairs in the conformal 4-sphere In 3-dimensional M obius space (the conformal sphere S 3 ) an isothermic surface may be characterized by the existence of conformal curvature line coordinates around each (nonumbilic) point 4 .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. simultaneously diagonalize all components of a basis representation, the surface's normal bundle has to be at 5 .This is an implicit prerequisite in the following Definition 1 A (2-dimensional) surface in (4-dimensional) M obius 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) M obius space is a 2parameter family of 2-spheres.
A 2-dimensional surface is said to envelope a congruence of 2-spheres if at each point it is tangent6 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 S 4 may have only one envelope | which 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 correspondence between its two envelopes by assigning the point of contact on one surface to the point of contact on the other surface.For a 3-dimensional ambient space it is well known 3] (cf.7]) that two cases can occur if this correspondence preserves curvature lines 7 and is conformal: the congruence consists of planes in a certain space of constant curvature | in which case the two envelopes are M obius equivalent | or, both envelopes are isothermic | in this case one surface is called a \Darboux transform" of the other (see 9], compare 3] or 4]).These remarks may motivate the following 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 correspondence 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

A basic characterization for Darboux pairs
In order to discuss (Darboux) pairs of surfaces in 4-(or 3-) dimensional M obius geometry we consider the conformal 4-sphere as the quaternionic projective line 9]: Note that we consider the space I H 2 of homogeneous coordinates of the quaternionic projective line as a right vector space over the quaternions I H. Now, let (f; f) : M 2 !P be an immersion into the (symmetric) space of point pairs8 in S 4 , P := f(x; y) 2 S 4 S 4 j x 6 = yg: (2) We may write the derivatives of f and f as9 where '; !; '; !: T M !I H denote suitable quaternionic valued 1-forms.Then, the integrability conditions d 2 f = d 2 f = 0 for f and f | the Maurer Cartan equations | read Since the quaternions are not commutative ' ^' 6 = 0 in general.Before continuing, let us list some useful identities for quaternionic 1-forms: let ; : T M !I H be quaternionic valued 1-forms and g : M !I H be a quaternionic valued function; then ^g In this framework we are now able to state a basic characterization for Darboux pairs of isothermic surfaces (for more details 10 including a proof see 9]): where !; !: T M !I H are de ned by 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; f) 7 !(fa; fâ), a; â : M !I H, we have 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 pairs11 : writing df r = f r ' + fr (r 2 !); d fr = f r (r 2 !) + fr ' (9)   with a parameter r 2 I R the Gau equations for f r and fr become 0 while the Codazzi equations remain unchanged.This shows that if there exist surface pairs | not necessarily Darboux | (f r ; fr ) 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 ; fr ) of Darboux pairs a special situation will occur when r !0. To discuss this, we assume ' = ' = 0 without loss of generality: we have 0 = d' + ' ^' and 0 = d ' + ' ^' and thus at least locally ' = da a 1 and ' = dâ â 1 with suitable functions a; â : M !I H. Rescaling by those and applying (8) gives ' = ' = 0. Thus, df r = fr (r 2 !); d fr = f r (r 2 !); (11)   and after the rescaling (f; f) 7 !(f 1 r ; fr) (or (f; f) 7 !(fr; f 1 r ), respectively) we see that f (or f) becomes a xed point in the conformal 4-sphere | which should be interpreted as a point at in nity.Thus, the other limit surfaces, f 0 and f0 , naturally lie in (di erent) Euclidean spaces.Identifying these two Euclidean spaces \correctly" we obtain df 0 = !and d f0 = !9].
These two limit surfaces fc 0 := f 0 and f c 0 := f0 usually do not form a Darboux pair | in general they do not even envelope a congruence of 2-spheres 12  As for the characterization of Darboux pairs (page 317) a proof may be found in 9].However, in case of 3-dimensional ambient space we will present an easy proof later (page 323) using some of the calculus we are going to develop.Now we are prepared to study (15) As a rst consequence of these equations we derive the equations for any Darboux pair (f; f).Since (15) also implies 0 (17) we conclude that the Christo el transforms f c and fc of f and f are given by Finally, if we x the translations of f c and fc such that So far we learned how to derive the Christo el transforms f c and fc of two surfaces f and f forming a Darboux pair.But usually it will be much easier to determine an isothermic surface's Christo el transform than a Darboux transform.In the next section we will see that deriving Darboux transforms f and fc of two surfaces f and f c forming a Christo el pair15 comes down to solving (20) Using our characterization (12) of Christo el pairs it is easily seen that this equation is \completely" (Frobenius) integrable.Note that | in agreement with our previous results | the common transform g c = g 1 for Riccati equations yields showing that fc = f c + g c 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 f of an isothermic surface f provides a Christo el transform f c of f via (18) every Darboux transform comes from a solution of (20) | if we do not x the scaling of the Christo el transform f c .On the other hand every solution g of (20) de nes a Darboux transform f = f +g of f since df ^g 1 d(f +g) = d(f + g) ^g 1 df = 0.This seems to be worth formulating as a where r 6 = 0 is a real parameter.For the derivatives of f and a Darboux transform Interpreting f; f : M 2 !I H = I H f1g as homogeneous coordinates of the point pair map (f; f) : M 2 !P we may choose new homogeneous coordinates by performing a rescaling (f; f) 7 !(fr; f(rg) 1 ) to obtain17 Even though this system resembles very much our original system (9) which describes the associated family of Darboux pairs, there is an essential di erence: in (9) the forms ', !, ' and ! 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 ; fr ) of Darboux pairs obtained from (9) both surfaces, f r as well as fr , change with the parameter r whereas the parameter contained in the Riccati equation just e ects the Darboux transform f = fr while the original surface f remains unchanged.However, the original system (9) appears in the linearization of our Riccati equation18 which indicates a close relation of these two parameters.As a rst application of this parameter which occurs from rescalings of the Christo el transform f c in our Riccati equation we may prove an extension of Bianchi's permutability theorem 2] for Darboux transforms: (26) where we xed any scaling for the Christo el transform f c of f.Then, there exists an isothermic surface f : M 2 !I H which is an r 1 -Darboux transform of f2 and an r 2 -Darboux transform of f1 at the same time 19 : Moreover, the points of f lie on the circles determined by the corresponding points of f, f1 and f2 , the four surfaces having a constant (real) cross ratio 20 r 2 r 1 To prove this theorem we simply de ne the surface f : M 2 !I H by solving the cross ratio equation 21 (28) for f: 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 3-dimensional Euclidean space I R 3 = ImI H: 7 Christoffel pairs in I R 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 Christo el pairs and to write down the Christo el transform of an isothermic surface quite explicitly.First we note that our characterizations (15) and ( 12) of Darboux and Christo el pairs of isothermic surfaces reduce to just one equation: if f; f : M 2 !ImI H both take values in the imaginary quaternions, In order to continue we will collect some identities present in the codimension 1 case.
We may orient an immersion f : M 2 !I R 3 = ImI H by choosing a unit normal eld n : M 2 !S 2 .This de nes 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 = hn; dfi + n df = n df = df n: (32) The Hodge operator is then given as the dual of this complex structure: = J (33) 19 Note, that this claim makes no sense before we x a scaling for the Christo el transforms fc 1;2 of f1;2 .But, according to our \permutability theorem" for Christo el and Darboux transforms (theorem 1) there is a canonical scaling for fc 1;2 after we xed the scaling of f c . 20For a comprehensive discussion of the (complex) cross ratio in IR 4 = IH see 10].The idea for the proof given in this paper actually originated from the discrete version of this theorem. 21Note that the denominator does not vanish as long as f1 6 = f2 .For r 1 = r 2 we get f = f.
for any 1-form on M.
With this notation we are now able to give a useful reformulation22 of the equation arising in our characterizations of Darboux pairs and Christo el pairs: if : T M !I H is any quaternionic valued 1-form we have for any x 2 T M. Consequently, df ^ = 0 if and only if This criterium shows that the space of imaginary solutions : with suitable functions a; b : M !I R.But one (imaginary) solution to the equation 0 = df ^ is easily found: it is well known that where H is the mean curvature of f.Thus df ^(dn which gives an injective solution = dn + H df away from umbilics of f.At this point, we are ready to give the announced proof of our characterization of Christo el pairs (12) in the 3-dimensional case: Theorem 4 Two surfaces f; f c : M 2 !I R 3 = ImI H form a Christo el pair if and only if df ^df c = 0: (39) Generically, the Christo el transform f c of f is uniquely determined by f up to homotheties and translations of I R 3 .
The fact that both surfaces of a Christo el pair in 3-space are isothermic is classical (see for example 5]) | and thus we omit this calculation.Now, in order to prove this theorem we note that from the above we know that This equation shows that whenever one of the surfaces of a Christo el pair is a minimal surface the other is totally umbilic (namely, a scaling of its Gau map) and vice versa.This brings us back to our previous problem of the uniqueness of Christo el transforms: assume we have a Christo el pair (f; n) consisting of a minimal surface f and its Gau map n.Then all the pairs with real constants a and t will also form Christo el pairs.Up to homotheties (given by a) this will run us through the associated family of minimal surfaces (given by t) re ecting the fact that associated minimal surfaces have the same Gau map 25 .Another fact that can be derived from (46) is that the (correctly scaled and positioned) Christo el transform of a surface of constant mean curvature H 6 = 0 is its 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 !ImI H 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) di erential equation.Thus our Riccati type equation (20) will produce a Darboux transform f = f + g of constant mean curvature Ĥ = H out of a surface of constant mean curvature (H 6 = 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 f(p) of f = f +g always lie in distance 1 2 H c o the tangent planes f(p) + d p f(T p M) of f.Since we also have the freedom of rescaling the Christo el transform f c of f we end up with a 3-parameter family of minimal Darboux transforms of a minimal surface (cf.2]).A minimal Darboux transform of the Catenoid is shown in gure 3. Sending H c ! 1 | note that in case of surfaces of constant mean curvature the associated family of Darboux pairs may be parameterized 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 Christo el transform and a minimal surface (Fig. 4).
In case of a surface of constant mean curvature H 6 = 0 we may reformulate the condition h H (g) = 0 as jH g nj 2 = 1 H c H (61)  showing that the points f(p) lie on spheres centered on the parallel surface f + 1 H n and with constant radius 1 H p 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 1 H containing 0 (without loss of generality we assume Fixing the scaling of the Christo el transform f c of f such that H c = H = 1 2 , i.e. f c = f + 2n, it is an unpleasant but straightforward calculation to see that our Riccati type equation dg = g( sinh 2 ( ) 4 df c )g df ( 65) is equivalent to the above linear system (64) de ning the function .Thus we have: Theorem 7 Any Bianchi-B acklund transform of a surface of constant mean curvature is a Darboux transform.
Analyzing the e ect of the three parameters ( and initial values for ' and ) contained in the Bianchi-B acklund transform on the function g : M !I R 3 at an initial point we nd that any solution of our Riccati equation (20) with a positive multiple of the parallel constant mean curvature surface f + 2n as Christo el transform f c can be obtained via a Bianchi-B acklund transform29 .Those constant mean curvature Darboux transforms of a constant mean curvature surface where the Christo el transform is taken a negative multiple of the parallel constant mean curvature surface (see Fig. 6) seem not to occur as Bianchi-B acklund transforms.
. But they do form what is called a Christo el pair: Definition 4 Two surfaces f 0 ; f0 : M 2 !I R 4 = I H in Euclidean 4-space are said to form a Christo el pair if they induce conformally equivalent metrics on M and have parallel tangent planes with opposite orientations.Each of the surfaces of a Christo el pair is called a Christo el transform or dual of the other.Note that the two surfaces of a Christo el pair are automatically isothermic; in fact, isothermic surfaces can be characterized by the (local) existence of a Christo el transform 9].The Christo el transform of an isothermic surface is unique 13 up to a homothety and a translation | so that in the sequel we will denote the Christo el transform of an isothermic surface f by f c .Finally, let us state a characterization of Christo el pairs similar to that for Darboux pairs: Proposition 2 Two surfaces f 0 ; f0 : M 2 !I R 4 = I H form a Christo el pair if and only if d f 0 ^d f0 = d f0 ^d f 0 = 0: (12) Both surfaces of a Christo el pair are isothermic.

5
Darboux pairs in I R 4 Let (f; f) : M 2 !P denote a pair of surfaces with df = f' + f!; d f = f !+ f '; (13) as before.Assuming that f; f : M !I H f1g = I H take values in Euclidean 4-space we see that ' = !and ' = !, and hence df = ( f f) !; d f = (f f) !: (14) This allows us to rewrite condition (6) on f and f to form a Darboux pair 14 as 0 = df ^(f f) 1 d f = d f ^( f f) 1 df:

|
note that d(f f) 1 = d(f c fc ) | we learn from the above characterization (15) of Darboux pairs that f c and fc also form a Darboux pair (cf.2]): Theorem 1 If f; f : M 2 !I R 4 form a Darboux pair, then, their Christo el transforms f c ; fc : M 2 !I R 4 (if correctly scaled and positioned) form a Darboux pair, too.

6 A
Riccati type equation Solving (18) for d f we obtain d f = ( f f)d f c ( f f).This yields the following Riccati type partial di erential equation 16 for g := ( f f): dg = g d f c g df:

f c : M 2 !
ImI H satis es (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 !I R 3 have parallel tangent planes with opposite orientations and they induce conformally equivalent metrics, i.e. they form a Christo el pair.Now assume we have not just one but two Christo el transforms f c and fc of an isothermic surface f : M 2 !I R 3 .Then we know from (36) that d fc = (a + b n) df c :(42)The integrability condition for fc reads0 = 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 24 .This concludes the proof.With (38) it also follows thatdn + H df = (a + b n)df c(44)for suitable functions a; b : M !I R. 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 independent (over the reals).As a consequence, we have a quite explicit formula relating the two surfaces of a Christo el pair: H c df c = dn + H df:

Figure 5 :
Figure 5: A Darboux transform of the cylinder

Figure 6 :
Figure 6: Another Darboux transform of the cylinder