DARBOUX TRANSFORMS OF DUPIN SURFACES

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1 PDES, SUBMANIFOLDS AND AFFINE DIFFERENTIAL GEOMETRY BANACH CENTER PUBLICATIONS, VOLUME 57 INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES WARSZAWA DARBOUX TRANSFORMS OF DUPIN SURFACES EMILIO MUSSO Dipartimento di Matematica Pura ed Applicata, Uniersità di L Aquila Via Vetoio, I-671 L Aquila, Italy musso@uniaq.it LORENZO NICOLODI Dipartimento di Matematica, Uniersità di Parma Via D Azeglio 85/A, I-431 Parma, Italy nicolodi@mat.uniroma1.it Abstract. We present a Möbius inariant construction of the Darboux transformation for isothermic surfaces by the method of moing frames and use it to gie a complete classification of the Darboux transforms of Dupin surfaces. 1. Introduction. The transformation theory of isothermic surfaces was intensiely studied at the turn of the th century and in recent years has receied a renewed attention because of its connection with the theory of integrable systems. The methods of integrable systems theory made their first appearance in the study of isothermic surfaces with Cieśliński Goldstein Sym s zero-curature formulation of the Gauss Codazzi equations of an isothermic surface [13]. This work was taken up by Burstall Hertrich-Jeromin Pedit Pinkall [8], who described the integrable system of isothermic surfaces in the context of Möbius geometry as an example of the cured flat system of Ferus Pedit [16]; an equialent description was gien in [6]. This approach proided a coherent framework for discussing the classical transformations of isothermic surfaces. For recent accounts on the transformation theory of isothermic surfaces, we refer the reader to [18, 19, ], and [7], where the Darboux transformation is described using the loop group formulation according to the general theory of Bäcklund transformations due to Terng Uhlenbeck [6]. Mathematics Subject Classification: Primary 53A3, 53A5; Secondary 35Q51. Key words and phrases: isothermic surfaces, Darboux transformation, Dupin surfaces, T- transformation, Bäcklund transformation. This research was partially supported by the MURST project Proprietà Geometriche delle Varietà Reali e Complesse, by the GNSAGA of the CNR, and by the European Contract Human Potential Programme, Research Training Network HPRN-CT--11 (EDGE). The paper is in final form and no ersion of it will be published elsewhere. [135]

2 136 E. MUSSO AND L. NICOLODI The Darboux transformation of isothermic surfaces is defined by realizing them as the focal surfaces of a -sphere congruence such that the correspondence induced between the two enelopes preseres the curature lines and is conformal. The integrability theorem says that for a gien isothermic surface there exist infinitely many Darboux transforms [15]. Bianchi [3] proed a permutability theorem for this transformation and introduced a related spectral transformation, the T-transformation [4], which was independently discoered also by Calapso [1]. This paper presents a Möbius inariant construction of the Darboux transformation by the method of moing frames. In this approach the T-transform plays a fundamental role in the sense that the Darboux transform is computed algebraically, without quadratures, starting from the T-transform. The latter is computed by soling the Maurer Cartan equations for the associated spectral frame. The general construction is applied to discuss the classical formulae for the Darboux transformation in the context of Möbius geometry and, as a main result, to determine explicit formulae for the Darboux transforms of Dupin cyclides. These together with the isothermic surfaces of spherical type, studied by the authors and U. Hertrich-Jeromin in [], will proide a complete classification of Darboux transforms of Dupin surfaces (see Example 3.). In our discussion, Dupin cyclides arise as T-transforms of a circular cylinder (a constant mean curature cylinder). Finally, as another application of the construction we establish a superposition formula for iterated Darboux transforms and proe Bianchi s permutability theorem. This will be useful for the explicit calculation of multisoliton surfaces (see []). H. Bernstein [], by carrying out a detailed analysis of the Darboux transforms of the standard torus of reolution, has been able to construct new explicit examples of isothermic tori which are not conformally equialent to the known examples, such as tori of reolution and constant mean curature tori in space forms [3]. These tori hae spherical curature lines and can deelop umbilics, either isolated umbilics or cures of umbilics. Much of the background material used by Bernstein in her work is presented in this paper. In [], by the same methods, the authors study the Darboux transforms of constant mean curature surfaces in space forms and their relations with the special isothermic surfaces of Darboux [3]. For the Euclidean and hyperbolic cases see also [19, ]. That there exist constant mean curature Darboux transforms of a constant mean curature surface is a classical result of Bianchi [3], [19]. The particular case of Darboux transforms of the cylinder which hae constant mean curature in Euclidean space has been considered in [19]. These Darboux transforms are related to Sterling Wente s constant mean curature multibubbletons [5] (see also Section 4). Section deals with some basic material. Section 3 presents the construction of the Darboux transformation by the method of moing frames, and Section 4 computes the Darboux transforms of Dupin cyclides. Finally, Section 5 proes a superposition formula and the permutability theorem for iterated Darboux transforms. Acknowledgments. Part of the material presented here has been circulating for some time as informal notes on conformal geometry and isothermic surfaces written by the authors. We thank Holly Bernstein for reading the original notes and making a number of aluable remarks. We also like to thank Fran Burstall and Udo Hertrich-Jeromin for

3 DARBOUX TRANSFORMS OF DUPIN SURFACES 137 seeral interesting and pleasant conersations on the subject oer the past years. Finally, we take this opportunity to thank the referee for his useful comments and suggestions.. Isothermic immersions. Consider Minkowski 5-space R 5 1 with linear coordinates x,..., x 4 and Lorentz scalar product gien by the quadratic form x, x = x x 4 + (x 1 ) + (x ) + (x 3 ) = η ij x i x j ; (1) an orientation for which dx dx 4 > ; and a time-orientation defined by the positie light-cone L + = {x R 5 1 : x, x =, x + x 4 > }. Classically, the Möbius space S 3 (conformal 3-sphere) is realized as the projectie quadric {[x] RP 4 : x, x = }. Accordingly, S 3 inherits a natural conformal structure and the identity component G of the pseudo-orthogonal group of (1) acts transitiely on S 3 as group of orientationpresering, conformal transformations (see [1, 9]). In this model for Möbius geometry, the de Sitter space S 4 1 = {y R 5 1 : y, y = 1} parametrizes oriented -spheres in S 3 by y y S 3, and is acted on by G. A Möbius frame is a basis (g,..., g 4 ) of R 5 1 such that g i = gɛ i, for g G, ɛ,..., ɛ 4 standard basis of R 5 1. Geometrically, the unit space-like ectors g 1, g, g 3 represent -spheres which intersect orthogonally, and [g ], [g 4 ] S 3 their intersection points. Regarding g,..., g 4 as R 5 -alued functions defined on G there exist unique 1-forms {ω i j } i,j 4 such that and satisfying the structure equations dg i = ω j i g j, ω k i η kj + ω k j η ki = dω i j = ω i k ω k j. ω = (ω i j ) = g 1 dg is the Maurer Cartan form of G. Definition.1. Consider S 3 endowed with the standard round metric g, and let U R a simply connected domain with preferred coordinates (x, y). A smooth immersion f : U S 3 with no umbilic points is called isothermic if (x, y) are conformal curature line coordinates for the induced Riemannian metric ds f. The notion of isothermic immersion is conformally inariant, that is, it only depends on the conformal class [g ] of the metric on S 3. If a and c denote the principal curatures along the x and y-directions, respectiely, then there exists a function Φ such that Φ (dx + dy ) = 1 4 (a c) ds f. Φ is called the Calapso potential of the isothermic immersion and is a conformal inariant of the immersion. It is uniquely defined up to the choice of conformal curature line coordinates. Geometrically, Φ is the conformal factor relating the metric induced by the central sphere congruence (Gauss conformal map) of f to the metric dx + dy (see for example [8])..1. Principal frames. A Möbius frame field along a conformal immersion f : U S 3 is a smooth map a = (a,..., a 4 ) : U G such that f(p) = [a ](p), for all p U. For any such a field, let α = (α i j ) = a ω. We say that a is a principal frame field along f if α 3 =,

4 138 E. MUSSO AND L. NICOLODI α 3 1 α 1 = α 3 α =, α 1 α. If f is isothermic, then α 1 dx = α dy =, () and there exist smooth functions l 1, l such that α 1 = l α 1 l 1 α. It is not difficult to see that d(α l 1 α 1 l α ) =. (3) Remark.. Principal frames may be defined along any smooth immersion f : U S 3 with no umbilic points. (3) is a necessary and sufficient condition for the existence of a reparametrization h : U U of U such that f h is an isothermic immersion. The totality of principal frames along an isothermic immersion f is a smooth bundle π f : P(f) U whose structure group is diffeomorphic to the group of all elements ( 4 ) 1 ( 4 ) 1t g + (V ) = I 3, = t ( 1,, 3 ), 4 V L +. If we think of P(f) as a submanifold of G, then the Maurer Cartan form ω of G restricted to P(f) satisfies ω 3 =, ω1 = eu dx, ω = eu dy, ω 3 1 = H 1 e u dx, ω 3 = H e u dy, where u, H 1, H are smooth real-alued functions defined on the total space P(f). A straightforward calculation yields e u(b g+ (V )) = 1 4 eu(b), H j (b g + (V )) = 4 H j (b) 3, j = 1,, for each b P(f), and for each g + (V ). This tells us that, whereer H 1 (b) H (b) (that is away from umbilic points), the real-alued function Φ = 1 (H 1 H )e u : P(f) R is constant along the fibers of π f. The function Φ : U R defined by Φ = Φ π f coincides with the Calapso potential of the isothermic immersion f. Example.3 (Euclidean frame). The Euclidean group of rigid motions embeds canonically in the Möbius group G and E 3 is iewed as S 3 minus the point at infinity [ɛ 4 ] = [ t (,,,, 1)]. Let f : U E 3 be an isothermic immersion and consider the principal Euclidean framing e : U E(3) along f gien by e = ( f; f x f x, f y f y, f x f y f x f y Set f x = f y = e ϕ and let a and c be the principal curature of f with respect to e. Considering e as a map into E(3) G, its Maurer Cartan form η = (η i j ) becomes ).

5 η = DARBOUX TRANSFORMS OF DUPIN SURFACES 139 e ϕ dx ϕ y dx ϕ x dy ae ϕ dx e ϕ dy ϕ y dx + ϕ x dy ce ϕ dy ae ϕ dx ce ϕ dy e ϕ dx e ϕ dy In terms of the Euclidean data, the Calapso potential is gien by Φ = 1 (a c)eϕ. Example.4 (Central frame). Let f : U R S 3 be an immersion with no umbilic points. Then there exists a frame b : U G along f (cf. [9]), the central frame, with connection form β gien by. q β 1 + q 1 β p 1 β 1 + p β p β 1 + p 3 β β 1 q 1 β 1 q β β 1 p 1 β 1 + p β β q 1 β 1 + q β β p β 1 + p 3 β β 1 β β 1 β q β 1 q 1 β, (4) such that β 1 dx = β dy =, β 1 β. The smooth functions p 1, p, p 3,q 1, q form a complete system of conformal inariants for f and satisfy the following structure equations: dβ 1 = q 1 β 1 β, dβ = q β 1 β, (5) dq 1 β 1 + dq β = (1 + p 1 + p 3 + q 1 + q )β 1 β, (6) dq β 1 dq 1 β = p β 1 β, (7) dp 1 β 1 + dp β = (4q p + q 1 (3p 1 + p 3 ))β 1 β, (8) dp β 1 dp 3 β = (4q 1 p q (p 1 + 3p 3 ))β 1 β. (9) If b = (b i ) 4 i= is a central frame, any other central frame is gien by (b, b 1, b, b 3, b 4 ). The mapping b 3 : U S 4 1 coincides with the central sphere congruence (Gauss conformal map) of f. Its second enelope [b 4 ] : U S 3 is called the conformal transform of f. Let e = (e,..., e 4 ) : U E(3) G be a Euclidean principal framing along f and denote by η = (ηj i) its Maurer Cartan form. Setting H = 1 (a + c), R = 1 (a c), and dh = R(uη 1 + η), we may adapt e to a central framing b = (b,..., b 4 ), where A direct calculation yields b = Re, b 1 = e 1 ue, b = e + e, b 3 = e 3 + He, b 4 = 1 { 1 ( H + u + ) } e ue 1 + e + He 3 + e 4. R β 1 = Rη, 1 β = Rη, β 3 =, β1 3 = β, 1 β 3 = β, β3 = (1) β = 1 [ a1 β 1 c β R ], β 1 = 1 [ c β 1 + a 1 β ], (11) R β1 = 1 {[ ( ) H H R R a u 1 a R u ] [ u β 1 + R R u c ] } 1 β, (1) R β = 1 {[ u R R + 1 ua ] [ ( ) H H β 1 + R R c + uc 1 R + u ] } β, (13) R

6 14 E. MUSSO AND L. NICOLODI where for any function g, g 1 and g denote the directional deriaties defined by dg = g 1 β 1 + g β. This allows us to express the set of inariant functions p 1, p, p 3, q 1, q in Euclidean terms. If the immersion f is isothermic, then p anishes, and conersely [1]. In this case, there exists a unique central frame such that β 1 = Φdx, and Φ =: e ψ is the Calapso potential of f. β = Φdy.. The Calapso equation and the T-transformation. Let f : U R S 3 be an isothermic immersion and let us retain the notations of the preceding example. From (5) it follows that q 1 = e ψ ψ y and q = e ψ ψ x and (4) becomes β = dψ p 1 e ψ dx p 3 e ψ dy e ψ dx ψ y dx + ψ x dy e ψ dx p 1 e ψ dx e ψ dy ψ y dx ψ x dy e ψ dy p 3 e ψ dy e ψ dx e ψ dy e ψ dx e ψ dy dψ Moreoer, from equation (6), we get Substituting into equations (8) and (9) yields, p 1 + p 3 = 1 e ψ ψ. (14) d ( e ψ (p 1 p 3 ) ) = e ψ { (e ψ ψ) x + 4ψ x (1 + e ψ ψ) } dx + e ψ { (e ψ ψ) y 4ψ y (1 + e ψ ψ) } dy. (15) The integrability condition of this equation is precisely the Calapso equation 1 (e ψ (e ψ ) xy ) + (e ψ ) xy =. (16) Conersely, if Φ = e ψ is a solution of (16), the right-hand-side of equation (15) is a closed 1-form, say γ Φ. Thus γ Φ = dh for some function H determined up to an additie constant. Define e ψ (p 1 p 3 ) = H + m, m any real constant, and set (cf. 14) p 1 = p 1 + me ψ, p 3 = p 3 me ψ. Accordingly, consider the G-alued 1-forms β(m) defined by dψ (p 1 e ψ + me ψ )dx (p 3 e ψ me ψ )dy e ψ dx ψ y dx + ψ x dy e ψ dx (p 1 e ψ + me ψ )dx e ψ dy ψ y dx ψ x dy e ψ dy (p 3 e ψ me ψ )dy e ψ dx e ψ dy. (17) e ψ dx e ψ dy dψ The forms β(m) satisfies the Maurer Cartan integrability condition dβ(m) = β(m) β(m), 1 The deduction of Calapso s equation follows an argument similar to that in [8]; there a slightly different Möbius inariant frame is considered. The equation named after Calapso was obtained independently by Rothe [4] and Calapso [1] as defining equation for isothermic surfaces.

7 DARBOUX TRANSFORMS OF DUPIN SURFACES 141 and then integrates to a map b(m) : U G, b(m) 1 db(m) = β(m), which is the central frame of the isothermic immersion f m = [b(m) ]. This 1-parameter family of isothermic immersions amounts to the second order Möbius deformations of f [17, 1, 1]. So, a solution to Calapso s equation introduces an auxiliary parameter which describes the conformal deformation through isothermic surfaces. All deformations hae the same Calapso potential Φ. Definition.5 (T-transformation). Two isothermic immersions f, ˆf : U S 3 are said to be T-transforms (spectral deformations) of each other if they hae the same Calapso potential. The T-transformation for isothermic surfaces was originally introduced by Calapso [1] and Bianchi [4]. Remark.6. The spectral family f m constructed aboe describes all T-transforms of f = f. In fact, any umbilic free T-transform of f is Möbius equialent to f m for some m R ([1, 11]). Example.7 (Dupin surfaces). The Calapso potential of an isothermic immersion f is constant if and only if the inariant functions q 1 and q anish. The anishing of q 1 and q is a necessary and sufficient condition for f being a cyclide of Dupin (cf. Section 4). We recall that an immersion in S 3 is a Dupin surface if each principal curature is constant along any line of curature tangent to its principal direction. It is known that the only Dupin surfaces in R 3 are spheres, planes and the cyclides of Dupin. Examples of Dupin cyclides include tori of reolution, circular cylinders and cones. It turns out that all Dupin cyclides are conformally equialent to either one of these examples [5, 7]. Section 4 will describe Dupin cyclides as T-transforms of a circular cylinder. Example.8 (Willmore isothermic surfaces). In terms of the conformal inariants, Willmore isothermic surfaces are characterized by the equations 3 (cf. [9, 1]) By (8) and (9), and then for a constant c. But, by (14), p 1 = p 3, p =. dp 1 = 4 ( q β 1 q 1 β ) = β p, 1 p 1 = ce 4ψ, Then, p 1 = 1 ( e ψ ψ + 1 ). ψ = ce ψ e ψ, (18) Two immersions f, f : U S 3 are (second order) G-deformations of each other if there is a smooth map b : U G with the property that b(p) f and f agree up to order two at p, for each p U. f is said deformable if it admits a non-triial deformation, i.e., a deformation f that is not G-equialent to f. The non-triial deformations of f arise in a 1-parameter family. 3 The equation p 1 = p 3 is equialent to the Euler Lagrange equation H + H(H K) = for the Willmore functional. In fact, using the expressions for p 1 and p 3 gien by (1) and (13), one computes p 3 p 1 = 1 R 3 ( H + H(H K)).

8 14 E. MUSSO AND L. NICOLODI c a constant. It is easily erified that any solution of (18) satisfies the Calapso equation. Therefore, (18) defines Willmore isothermic surfaces. According as c is positie, negatie or zero, the equation (18) accounts, up to conformal equialence, for the minimal surfaces in 3-dimensional space forms H 3, S 3, R 3, respectiely [5]. In [, 11], it has been proed that the T-transforms of minimal immersions in 3-dimensional space forms are locally conformally equialent to constant mean curature surfaces in space forms and that all constant mean curature surfaces in space forms arise in this way. 3. The Darboux transformation by moing frames Definition 3.1 (Darboux transformation). A Darboux isothermic frame or a cured flat frame 4 is a principal frame a : U G along an isothermic immersion f : U S 3 such that ( α 1 + iα ) (α 1 + iα ) =, α 3 =, α 1 α. If a is a cured flat frame, then (a 4, a 1, a, a 3, a ) defines a principal frame along ˆf := [a 4 ] and (x, y) are conformal curature line coordinates for ˆf. Thus ˆf is another isothermic immersion. Moreoer, the mapping σ := a 3 : U S 4 1 defines a sphere congruence whose enelopes are f and ˆf. The correspondence induced by σ between f(u) and ˆf(U) preseres curature lines (α 1 α 1 = α α = ) and is conformal. In the classical terminology [5], this is expressed by saying that σ is a conformal Ribaucour congruence, i.e., a Darboux congruence. Accordingly, ˆf is called a Darboux transform of f. Example 3. (Isothermic surfaces of spherical type). In general, the second enelope of the central sphere congruence (the conformal transform) of an isothermic immersion need not be isothermic, see for example [8]. An isothermic immersion is said of spherical type if its central sphere congruence is a Darboux congruence 5. This is equialent to p 1 + p 3 =, p =, and hence, by the structure equations, to ψ + e ψ =. (19) As proed in [], non-minimal isothermic surfaces of spherical type are Darboux transforms of sphere pieces. Moreoer, the congruence realizing the Darboux transform is the central sphere congruence. This will complement the discussion in Section 4 to gie a complete classification of Darboux transforms of Dupin surfaces. Let a : U G be a principal frame along an isothermic immersion f : U S 3 with connection form α, and let ρ be a smooth function such that dρ ρ = (α l 1 α 1 l α ). 4 For the related notion of cured flats in symmetric spaces we refer to [16, 8]. 5 Equialently, its central framing is a cured flat framing.

9 DARBOUX TRANSFORMS OF DUPIN SURFACES 143 Consider the deformed form α α1 + ρα1 α ρα α 3 α 1 α1 α1 3 α1 + ρα 1 α ρ := α α1 α 3 α ρα α1 3 α 3 α3. () α 1 α α Since α ρ satisfies the Maurer Cartan integrability condition dα ρ = α ρ α ρ, α ρ = a ρ 1 da ρ for some map a ρ : U G, unique up to left multiplication by a constant element of G. Let V R 5 1 be any constant ector. Then is a solution of V = a ρ 1 V : U R 5 1 dv = α ρ V. (1) Any solution of (1) is obtained in this way. In particular, a solution V : U L + of (1) is completely determined by a solution a ρ : U G of the equation da ρ = a ρ α ρ and a constant ector V L +. Let V = t (,..., 4 ) : U L + be a solution of (1) for a fixed ρ. It is not difficult to see that, generically, the zero locus of 4 is discrete 6. Away from this discrete set, we consider the change of frame gien by g + (V ) = ( 4 ) 1 ( 4 ) 1t I 3 4, = t ( 1,, 3 ). Lemma 3.3. The frame ag + (V ) is a cured flat framing and the mapping av from U into the light-cone L + represents a Darboux transform of f. Proof. The proof follows immediately by looking at the corresponding Maurer Cartan form α + = (g + ) 1 dg + + (g + ) 1 αg +, which takes the form 4 ρα 1 4 ρα α 1 α α α 4 α α ρα 1 α α 4 1 α α 4 α α 4 4 ρα. α1 3 3 α 1 4 α 3 3 α 4 α 1 α 4 4 We now apply the aboe construction to the Euclidean and central frames. 6 Clearly, the locus { 4 = } is contained in { 1 = = 3 = }. From (1), it follows that d 1 d { 1 = = 3 =}. By the inerse function theorem, we then conclude that { 1 = = 3 = }, if not empty, is made of isolated points.

10 144 E. MUSSO AND L. NICOLODI Example 3.4 (Darboux transforms ia Euclidean frames). Let f be an isothermic immersion and let e = (e,..., e 4 ) denote the Euclidean principal frame along f with connection form η. In this case the first order system (1) coincides with the Darboux fundamental differential system 7 ([3], p. 99) d = me ϕ dx me ϕ dy e ϕ dx η1 k 1 e ϕ dx me ϕ dx e ϕ dy η1 k e ϕ dy me ϕ dy k 1 e ϕ dx k e ϕ dy e ϕ dx e ϕ dy for a constant m. Then, eg + (V ) is a cured flat framing and the stereographic projection from [ɛ 4 ] of its second enelope eg + (V )(ɛ 4 ) gies the classical Darboux transform D m of f [3]. By construction D m (f) is represented by the mapping ev = e + j e j + 4 ɛ 4 into the light-cone. Since e is Euclidean, away from the set of isolated points where anishes, D m (f) = f + j a j(f), j = 1,, 3, where (a 1 (f), a (f), a 3 (f)) is the principal framing of f (cf. [3], p. 1, eq. (7)). Example 3.5 (Darboux transforms ia central frames). Let b be the central frame along an isothermic immersion f : U S 3. It is easily checked that ρ = me ψ for a real constant m, and that the deformed connection form β ρ =: β(m) is gien by (17). Let V : U L + be a solution of dv = β(m)v, and consider the cured flat framing (cf. Lemma 1) ˆb := bg + (V ) : U R G. The first enelope ˆb(ɛ ) represents the original immersion f. Let d m (f) := [ˆb(ɛ 4 )] denote the Darboux transform of f. The Calapso potential of d m (f) can be computed as follows. Suppose m > and consider b := (ˆb 4, ˆb 1, ˆb, ˆb 3, ˆb ). b is a principal frame along d m (f) whose Maurer Cartan form β is of the form eψ e dx ψ 4 dy 4 4 me ψ dx β1 e ψ dx+ 1 4 e ψ dy (1 3 4 )e ψ dx eψ 4 dx 4 4 me ψ dy β1 + e ψ dx 1 4 e ψ dy ( )e ψ e dy ψ 4 dy. 4 (1 3 )e ψ dx ( )e ψ dy 4 4 me ψ dx 4 me ψ dy Following the procedure illustrated in Example.4, we may adapt the principal frame b further to a central frame. From this we will learn the expression for the Calapso potential 1 3 4, 7 In Bianchi s notations we hae = mσ, 1 = λ, = µ, 3 = w, 4 = ϕ, where λ, µ, w, ϕ, σ are the fie transforming functions (funzioni trasformatrici) from f to D m (f).

11 DARBOUX TRANSFORMS OF DUPIN SURFACES 145 of d m (f). The Calapso potential of the Darboux transform d m (f) is computed to be Φ = 3 4 Φ. Note that the zero locus of 3 is the umbilic locus of d m (f). We are then in a position to state: Lemma 3.6 (Bäcklund s theorem). With the notations introduced aboe, let Φ be the Calapso potential of an isothermic immersion f and let V : U L + be a solution of the completely integrable linear system: Then dv = β m (Φ)V. Φ m = 3 4 Φ is the Calapso potential of the Darboux transform d m (f). The set { 3 = } is the umbilic locus of d m (f). 4. Darboux transforms of Dupin cyclides. Let f : U S 3 be an umbilic free immersion and let a, c denote its principal curatures. Let e be a Euclidean principal frame along f and a be the central frame obtained from e as indicated in Example.4. From (4) and (11), we get q 1 = c R, q = a 1 R. Thus f is a Dupin cyclide, that is, equations a x = c y = hold true, if and only if q 1 = q =. Using the structure equations, we also get that f is isothermic (p = ), and that the Calapso potential is a constant function. Moreoer, equations (8), (9) and (14) imply that p 1 and p 3 are constant functions such that 1 + p 1 + p 3 =. Without loss of generality, we may assume Φ = 1 and set p 1 = k = const., p 3 = (1 + k). So, the central frame a along a Dupin cyclide f has Maurer Cartan form kdx (1 + k)dy dx dx kdx α = dy dy (1 + k)dy dx dy. dx dy 4.1. The central frame of Dupin cyclides. In order to compute the central frame a = (a,..., a 4 ) of a cyclide f, we need to sole the following linear system da = a 1 dx + a dy da 1 = (ka + a 3 + a 4 )dx da = [ (1 + k)a a 3 + a 4 ]dy da 3 = a 1 dx + a dy da 4 = ka 1 dx (1 + k)a dy. ()

12 146 E. MUSSO AND L. NICOLODI Note that a 1 = a 1 (x), a = a (y), and set From (), it follows that a 1 = db 1 dx, a = db dy. a = b 1 + b + A a 3 = b 1 + b + A 3 a 4 = kb 1 (1 + k)b + A 4, (3) and d b 1 dx = (k 1)b 1 + ka + A 3 + A 4 d b dy = (k + 3)b (1 + k)a A 3 + A 4, where A, A 3, A 4 are constant ectors. The discussion can be reduced to the study of the following three cases: I : 1 < k < 1, II : k > 1, III : k = 1. Case I: 1 < k < 1 b 1 = C 1 cos 1 kx + C sin 1 kx + 1, b = C 3 cos k + 3y + C 4 sin k + 3y +, (4) C i, α constant ectors. Thus a 1 = 1 k( C 1 sin 1 kx + C cos 1 kx) a = k + 3( C 3 sin k + 3y + C 4 cos k + 3y) (5) Since a 1 = a = 1 and a 1, a =, it follows that C 1 = C = 1 1 k, C 3 = C 4 = 1 k + 3, C i, C j =, i j. We may assume that C 1 = ɛ 1 1 k, C = ɛ ɛ 4 (1 k), C 3 = From (3), (5), (6), and (4), it follows that a = + k + 3 sin 1 kx (1 k)(k + 3) ɛ k + 3, C 4 = ɛ + cos 1 kx ɛ 1 + cos k + 3y ɛ 1 k k sin k + 3y ɛ 3 + k + 3 sin 1 kx ɛ 4, k + 3 (1 k)(k + 3) ɛ 3 k + 3. (6)

13 DARBOUX TRANSFORMS OF DUPIN SURFACES 147 a 1 = cos 1 kx ɛ sin 1 kxɛ 1 cos 1 kx ɛ 4 a = sin k + 3yɛ + cos k + 3yɛ 3, a 3 = (k + 1) k + 3 sin 1 kx (1 k)(k + 3) ɛ cos 1 kx ɛ 1 + cos k + 3y ɛ 1 k k sin k + 3y ɛ 3 + (k + 1) + k + 3 sin 1 kx ɛ 4, k + 3 (1 k)(k + 3) a 4 = 1 + k k + 3 sin 1 kx (1 k)(k + 3) ɛ + k cos 1 kx ɛ 1 (1 + k) cos k + 3y ɛ 1 k k + 3 (1 + k) sin k + 3y ɛ k k + 3 sin 1 kx ɛ 4. k + 3 (1 k)(k + 3) This yields Lemma 4.1. For 1 < k < 1, the Dupin cyclide gien by the stereographic projection of f(u) is Möbius equialent to a torus of reolution in E 3. Case II: k > 1 b 1 = Γ 1 cosh k 1x + Γ sinh k 1x + ka + A 3 + A 4, 1 k b = L 1 cos k + 3y + L sin k + 3y + (1 + k)a A 3 + A 4, k + 3 Γ α, L α constant ectors. Then a 1 = k 1(Γ 1 sinh k 1x + Γ cosh k 1x) a = k + 3(L 1 sin k + 3y L cos k + 3y) Since a 1 = a = 1 and a 1, a =, it follows that Γ 1 = 1 k 1, Γ = 1 k 1, Γ 1, Γ =, L 1 = L = 1 k + 3, L 1, L =, L i, Γ j =, and we may assume Γ 1 = ɛ + ɛ 4, Γ = (k 1) By imposing the other conditions ɛ 1 k 1, L 1 = ɛ k + 3, L = ɛ 3 k + 3. a =, a 3 = 1, a 4 = a i, a j =, i j, i, j =, 3, 4, we obtain the following expressions for a, a 1, a, a 3, a 4 : k + 3 cosh k 1x + a = ɛ + sinh k 1x ɛ 1 + cos k + 3y ɛ (k 1)(k + 3) k 1 k sin k + 3y k + 3 cosh k 1x ɛ 3 + ɛ 4, k + 3 (k 1)(k + 3)

14 148 E. MUSSO AND L. NICOLODI a 1 = sinh k 1x ɛ + cosh k 1xɛ 1 + sinh k 1x ɛ 4 a = sin k + 3yɛ + cos k + 3yɛ 3 k + 3 cosh k 1x + (k + 1) a 3 = ɛ sinh k 1x ɛ 1 + cos k + 3y ɛ (k 1)(k + 3) k 1 k sin k + 3y k + 3 cosh k 1x (k + 1) ɛ 3 ɛ 4, k + 3 (k 1)(k + 3) a 4 = k k + 3 cosh k 1x + 1 (k 1)(k + 3) ɛ + k sinh k 1x ɛ 1 (k + 1) cos k + 3y ɛ k 1 k + 3 (k + 1) sin k + 3y k + 3 ɛ 3 + k k + 3 cosh k 1x 1 (k 1)(k + 3) ɛ 4. From this we hae Lemma 4.. For k > 1, the Dupin cyclide gien by the stereographic projection of f(u) is a Möbius equialent to a circular cone in E 3. Case III: k = 1. In this case b 1 = 1 ( ) 1 A + A 3 + A 4 x + Γ 1 x + Γ b = L 1 cos y + L sin y + 1 ( 3 ) 4 A A 3 + A 4. According to (3) and the orthogonality conditions we may choose the constant ectors of integration so that a = 1 ( x + 1 ) ɛ + xɛ (cos yɛ + sin yɛ 3 ) + ɛ 4, In this case we hae a 1 = xɛ + ɛ 1, a = sin yɛ + cos yɛ 3, a 3 = 1 ( x 3 ) ɛ xɛ (cos yɛ + sin yɛ 3 ) ɛ 4, a 4 = 1 ( x + 9 ) ɛ xɛ (cos yɛ + sin yɛ 3 ) + 1 ɛ 4 Lemma 4.3. For k = 1, the Dupin cyclide gien by the stereographic projection of f(u) is Möbius equialent to a circular cylinder in E 3. Remark 4.4. As already obsered, from the preious formulae we can conclude that Dupin cyclides arise as T-transforms of the circular cylinder.

15 DARBOUX TRANSFORMS OF DUPIN SURFACES The Darboux transformation of Dupin cyclides. The linear system (1) becomes where k = k + m. Note that d = k 1 dx + (1 + k) dy d 1 = dx + 3 dx k 4 dx d = dy 3 dy + (1 + k) 4 dy d 3 = 1 dx + dy d 4 = 1 dx dy, 1 = 1 (x) = (y). Let u 1, u be functions such that du 1 = 1 dx, du = dy. Then that is d = kdu 1 + (1 + k)du d 3 = du 1 + du d 4 = du 1 du, = ku 1 + (1 + k)u + α 3 = u 1 + u + α 3 4 = u 1 u + α 4, α, α 3, α 4 constants. Combining (7) and (8) yields ODEs for u 1 and u : d u 1 dx + (1 k)u 1 = α + α 3 kα 4 d u dy + (3 + k)u = α α 3 + (1 + k)α 4. Fie cases may occur: 1 I : k >, II : 1 k =, III : 3 < k < 1, IV : 3 k =, V : 3 k <. Case I. Case II. u 1 = c 1 cosh k 1x + c sinh k 1x + ( α + α 3 kα 4 ) 1 k u = l 1 cos k + 3y + l sin k + 3y + ( α α 3 + (1 + k)α 4 ) 3 + k u 1 = 1 ) ( α + α 3 α4 x + c 1 x + c ( α α ) α4 u = l 1 cos y + l sin y Case III. u 1 = c 1 cos 1 kx + c sin 1 kx + ( α + α 3 kα 4 ) 1 k u = l 1 cos k + 3y + l sin k + 3y + ( α α 3 + (1 + k)α 4 ) 3 + k (7) (8)

16 15 E. MUSSO AND L. NICOLODI Case IV. ( α + α α4 ) u 1 = c 1 cos x + c sin x u = 1 ) ( α α 3 α4 y + l 1 x + l Case V. u 1 = c 1 cos 1 kx + c sin 1 kx + ( α + α 3 kα 4 ) 1 k u = l 1 cosh k 3y + l sinh k 3y + ( α α 3 + (1 + k)α 4 ) 3 + k Remark 4.5. Starting from the solution Φ 1 of Calapso s equation, according to Bäcklund s theorem (Lemma 3.6), we obtain new solutions of the Calapso equation. Φ = u1 + u + α 3 u 1 + u α 4 If V = t (,..., 4 ) is a solution of (7), then the Darboux transform of f associated with V is represented by the mapping av : U { 4 = } L + gien by ( ku 1 + (1 + k)u + α )a + du1 dx a 1 + du dy a + ( u 1 + u + α 3 )a 3 + ( u 1 u + α 4 )a 4. The constants of integration must be chosen so that av =. Remark 4.6. Bernstein [] constructs explicit examples of isothermic tori which are neither conformally equialent to tori of reolution nor to constant mean curature tori in space forms. These tori are obtained by carefully analyzing the Darboux transforms of the torus of reolution computed aboe. They hae spherical curature lines and, in contrast to the original surface, can hae umbilics; either isolated umbilics or lines of umbilics can occur. The proof uses the expression for the Calapso potential of the Darboux transforms, computed according to Bäcklund s theorem (Lemma 3.6), in combination with equation (18) whose solutions correspond to constant mean curature surfaces in 3-dimensional space forms (cf. Example.8). The existence of constant mean curature Darboux transforms of a constant mean curature surface is a classical result of Bianchi [3], [19]. In particular, the constant mean curature Darboux transforms of the cylinder hae been considered in [19]. They are related to Sterling Wente s constant mean curature multibubbletons which in turn are obtained from the cylinder ia the Bianchi Bäcklund transformation [5]. In fact, Hertrich-Jeromin Pedit [19] proe that any Bianchi Bäcklund transform of a constant mean curature surface is a Darboux transform. For recent studies on constant mean curature Darboux transforms of constant mean curature surfaces in space forms and their relations with the classical special isothermic surfaces of Darboux [3] we refer the reader to [].

17 DARBOUX TRANSFORMS OF DUPIN SURFACES Superposition and permutability. As another application of our construction we proide a superposition formula for iterated Darboux transforms of a gien isothermic surface and gie a proof of Bianchi s permutability theorem [3]. This formula is particularly useful for the explicit calculation of multisoliton surfaces (see figures and []). Let a(1), a() : U G be two cured flat frames haing the same isothermic immersion (1) () f = [a ] = [a ] : U S3 as first enelope and with second enelopes f (1) and f (), respectiely. Let α(1) = (αji ), α() = (α ji ) be the corresponding connection forms 8. Fig. 1. Darboux transforms of the torus 8 (αij ) can be described by α = α3 = α3 = α1 = eu dx, α = eu dy, α31 = H1 eu dx, α1 = me u dx, α3 = H eu dy, α = me u dy for functions H1, H, u : U R, and some constant m (see [8]).

18 15 E. MUSSO AND L. NICOLODI Fig.. Darboux transforms of the cylinder The deformed forms associated with α(1) are gien by λα1 λα 3 α1 α α 1 1 (1) 3 αλ := α α1 α α13 α3 α1 α λα1 λα for some constant λ. Similarly for α(). According to Section 3, there is a solution V = t (,..., 4 ) : U L+ of the integrable linear system (1) dv = αλ V, (9) for some λ, such that, on the complement of the discrete zero locus of 4, the frames a(1), a() are related by the gauge change a() = a(1) g + (V ). Next, away from the set of isolated points where anishes, define the mapping gλ (V ) : U G by 1 (λ) I3 gλ (V ) =. 1 1 λ 4 ( ) λt λ( ) We hae

19 DARBOUX TRANSFORMS OF DUPIN SURFACES 153 Lemma 5.1. ā (1) := a (1) g λ (V ) is a cured flat framing whose second enelope [ā(1) 4 ] = f (1). Its connection form is gien by λ α λ 1 α ( 1 λ 1)α1 α1 +λ α 1 λ 1 α α1 3 +λ 3 α λ 1 α 1 ( 1 λ 1)α α1 λ α 1 +λ 1 α α 3 +λ 3 α λ α. α1 3 λ 3 α 1 α 3 λ 3 α ( 1 λ 1)α1 ( 1 λ 1)α Definition 5.. Let f = [ā (1) ] be the first enelope of ā(1). We say that the isothermic map f is the superposition of the two Darboux transforms f (1) and f () of f. We write f = f (1) f f (). Proposition 5.3 (Permutability Theorem). If an isothermic immersion f has two Darboux transforms f (1) and f (), then there is another isothermic immersion f which is a Darboux transform of f (1) and f () and is such that Proof. We can write f = f (1) f f () = f () f f (1). a (1) = a () g + (V ) 1 = a () g + ( ˆV ), where ˆV = t ( t, 4, 1 ). (3) 4 By a direct calculation it is easily erified that, if V : U L + is a solution of (9), then ˆV is a solution of d ˆV = α µ () ˆV, where µ is gien by µ = λ(µ 1). (31) Thus, f () f f (1) is the isothermic map represented by the first column of the framing ā () = a () gµ ( ˆV ), where gµ ( ˆV ) is defined in analogy with (5). It is now easily seen that, if ˆV and µ are related to V and λ as in (3) and (31), then [ā () ] = [ā(1) ]. References [1] M. Babich and A. Bobenko, Willmore tori with umbilic lines and minimal surfaces in hyperbolic space, Duke Math. J. 7 (1993), [] H. E. Bernstein, Non-special, non-canal isothermic tori with spherical lines of curature, Trans. Amer. Math. Soc. 353 (1), 45 74; Isothermic tori with spherical lines of curature, thesis, Washington Uniersity, St. Louis, [3] L. Bianchi, Ricerche sulle superficie isoterme e sulla deformazione delle quadriche, Ann. Mat. Pura Appl. (3) 11 (195), [4] L. Bianchi, Complementi alle ricerche sulle superficie isoterme, Ann. Mat. Pura Appl. (3) 1 (195), 54.

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