4 Surfaces in the four-space and the 0 0 Davey–Stewartson equations 2 n a Iskander A. TAIMANOV ∗ J 9 2 ] G 1 Introduction D . h The Weierstrass representation for surfaces in R3 [8, 13] was generalized for at surfaces in R4 in [12] (see also [4]). This paper uses the quaternion lan- m guage and the explicit formulas for such a representation were written by [ Konopelchenko in [9] for constructing surfaces which admit soliton defor- 1 mation governed by the Davey–Stewartson equations. This generalizes his v results from [8] where he introduced the formulas for inducing surfaces in 2 the three-space which involve a Dirac type equations and defined for such 1 4 surfaces a deformation governed by the modified Novikov–Veselov (mNV) 1 equations. 0 It was shown in [13] that the formulas for inducing surfaces in R3 [8] 4 0 describe all surfaces and that the modified Novikov–Veselov equation de- / h formstoriintotoripreservingtheWillmorefunctionalwhichnaturallyarises t and plays an important role in this representation. The spectral curve of a m the corresponding Dirac operator is invariant under this deformation. The : global Weierstrass representation at least for real analytic surfaces could be v obtained by an analytic continuation from a local representation. Thus a i X moduli space of immersed tori is embedded into the phase space of an in- r tegrable system with the Willmore functional and, moreover, the spectral a curve as conservation quantities. Looking forward to understand the spectral curves for tori in R4 we consider in this paper the analogous problems for surfaces in R4 and show thatthiscaseisverydifferentfromthethree-dimensionalcase, inparticular, by the following features which were overlooked until recently: ∗ InstituteofMathematics,630090Novosibirsk,Russia;e-mail: [email protected] 1 • for tori in R4 every equation from the Davey–Stewartson (DS) hi- erarchy describes not one but infinitely many geometrically different soliton deformations; • the multipliers on the spectral curve for a torus in R4 are not uniquely defined and different complex curves in C2 (the spectral curves im- mersedviathemultipliers)areinvariantsofdifferentDSdeformations. Thereasonforthatisquiteclearandconsistsbasically inthenonunique- ness of a Weierstrass representation. A surface in R3 is constructed in terms of one vector function ψ (spinor) which is a lift of the Gauss mapping into non-vanishing spinors. Such a lift is defined up to a sign by fixing a conformal parameter on the surface. This function ψ satisfies a Dirac equation. A surface in R4 is constructed in terms of two vector functions ψ and ϕ which formagain aliftof theGauss mapping. However inthis case byfixing a conformal parameter one defines a lift only up to a gauge transformation given byef wheref is any smooth function. Moreover notevery lift satisfies the Dirac equations and the lifts meeting these equations are defined up to gauge transformations eh where h is a any holomorphic function. In particular, given aWeierstrass representation of asurfaceΣ ⊂ R4 and a domain W ⊂ Σ we can replace a representation of the domain by gauge- equivalent using a transformation eh where h is a holomorphic function on W which is not analytically continued onto the surface. Thus we obtain a representation of a domain which is not continued (i.e., expanded to a representation of a surface). This also makes a difference with the case of surfaces in R3. Another importantpointis that theDS equations contain theadditional potentials which are defined by resolving the constraint equations. Such resolutions are not unique and we have to choose the potentials carefully to make the DS deformations geometric: for some special choices of the additional potentials the corresponding DS deformations map tori into tori preserving the Willmore functional. However in general this is not the case and we show how to achieve that in §4. TheworkwassupportedbyRFBR(grant03-01-00403) andMax-Planck- Institute on Mathematics in Bonn. We thank U. Abresch for comments and discussions. 2 2 Explicit formulas for a representation and so- liton deformations Let us recall the explicit formulas for inducing a surface and its soliton deformation via the Davey–Stewartson equation. The following proposition is derived by straightforward computations. Proposition 1 ([9]) Let vector functions ψ and ϕ be defined in a simply- connected domain W ⊂ C (with a complex parameter z) and meet the Dirac equations ∨ Dψ = 0, D ϕ = 0 where 0 ∂ U 0 ∨ 0 ∂ U¯ 0 D = + , D = + . −∂¯ 0 0 U¯ −∂¯ 0 0 U (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) Then the 1-forms η = f dz+f¯dz¯, k = 1,2,3,4, k k k with i 1 f = (ϕ¯ ψ¯ +ϕ ψ ), f = (ϕ¯ ψ¯ −ϕ ψ ), 1 2 2 1 1 2 2 2 1 1 2 2 (1) 1 i f = (ϕ¯ ψ +ϕ ψ¯ ), f = (ϕ¯ ψ −ϕ ψ¯ ) 3 2 1 1 2 4 2 1 1 2 2 2 are closed and the formulas xk = xk(0)+ η , k = 1,2,3,4, (2) k Z define a surface in R4 (here the integral is taken over any path in W and by the Stokes theorem does not depend on a choice of path). The induced metric equals e2αdzdz¯= (|ψ |2+|ψ |2)(|ϕ |2+|ϕ |2)dzdz¯ (3) 1 2 1 2 and the norm of the mean curvature vector H = 2xzz¯ meets the equality e2α |H|eα |U| = . (4) 2 3 For U = U¯ and ψ = ±ϕ these formulas reduce to the Weierstrass repre- sentation for surfaces in R3. The existence of a local representation of any surface in R4 by these for- mulas is not proved in [9] although it was indicated in [12] that the Weier- strass representation for surfaces in R3 is generalized for surfaces in R4 and involves in this case two vector functions ψ and ϕ and a complex valued potential U. We expose such a derivation in the next section revealing some features not taking place in thethree-dimensional case. Remark that for Lagrangean surfaces in R4 this representation was discovered in other terms by Helein and Romon [7]. Let 0 ∂ −p 0 L = + . −∂¯ 0 0 q (cid:18) (cid:19) (cid:18) (cid:19) Let us consider deformations of this operator which take the form of Man- akov’s L,A,B-triple: L +[L,A ]−B L = 0 (5) t n n or [L,∂ −A ]+B L = 0. t n n Notice that if L meets (5), then the solution of the equation Lψ = 0 is evolved as follows: ψ = A ψ. t n The following two propositions are proved by straightforward computa- tions. Proposition 2 For −∂2−v q∂¯−q A = 1 z¯ , 2 −p∂+p ∂¯2+v z 2 (cid:18) (cid:19) ∂2+∂¯2+(v +v ) −(p+q)∂¯+q −2p B = 1 2 z¯ z¯ , 2 (p+q)∂ −p +2q −(∂2+∂¯2)−(v +v ) z z 1 2 (cid:18) (cid:19) where v = −2(pq) , v = −2(pq) , 1z¯ z 2z z¯ 4 the equations (5) takes the form p = p +p +(v +v )p, t zz z¯z¯ 1 2 (6) q = −q −q −(v +v )q. t zz z¯z¯ 1 2 Proposition 3 For ∂3+ 3v ∂ −3w q∂¯2−q ∂¯+q + 3v q A = 2 1 1 z¯ z¯z¯ 2 2 , 3 p∂2−p ∂ +p + 3v p ∂¯3+ 3v ∂¯−3w (cid:18) z zz 2 1 2 2 2 (cid:19) b b B = 11 12 , 3 b b 21 22 (cid:18) (cid:19) where 3 b = −b = ∂¯3−∂3− (v ∂ −v ∂¯)+3(w −w ), 11 22 1 2 1 2 2 3 b = −(p+q)∂¯2− (p+q)v −(3p −q )∂¯−(3p +q ), 12 2 z¯ z¯ z¯z¯ z¯z¯ 2 3 b = −(p+q)∂2 − (p+q)v −(3q −p )∂ −(3q +p ) 21 1 z z zz zz 2 and v = −2(pq) , v = −2(pq) , 1z¯ z 2z z¯ w = (pq ) , w = (qp ) , 1z¯ z z 2z z¯ z¯ the equations (5) takes the form 1 3 p = p +p + (v p +v p )−3(∂−1[(qp ) ]+∂¯−1[(qp ) ])p, t zzz z¯z¯z¯ 1 z 2 z¯ z¯ z¯ z z 2 (7) 3 q = q +q + (v q +v q )−3(∂−1[(pq ) ]+∂¯−1[(pq ) ])q. t zzz z¯z¯z¯ 1 z 2 z¯ z¯ z¯ z z 2 The equations (6) and (7) are called the Davey–Stewartson equations. In fact these are the equations DSII and DSII from the DSII hierarchy. 2 3 The equation DSII takes the form (5) where A equals n n (−1)n+1∂n 0 A = +... n 0 ∂¯n (cid:18) (cid:19) 1Theequation (5) after a formal substitution of A and B reduces to thesystem 3 1 pt =pzzz+pz¯z¯z¯+ (v1pz+v2pz¯)+3(w1−w2+ v1z)p, 2 2 3 1 qt =qzzz+qz¯z¯z¯+ (v1qz+v2qz¯)−3(w1−w2− v2z¯)q, 2 2 5 (here by ... we denote terms of lower order). For n= 1 we have ∂ q ∂¯−∂ −(p+q) A = , B = , 1 p ∂¯ 1 −(p+q) ∂−∂¯ (cid:18) (cid:19) (cid:18) (cid:19) and the DSII equations are 1 p = p +p , q = q +q . t z z¯ t z z¯ Remark that the DSI hierarchy is a hierarchy of nonlinear equations obtained from the DSII hierarchy by replacing the variables z,z¯ by real- valued variables x,y. Let us consider the reduction of the DSII hierarchy for the case p = −u, q = u¯. (8) The equation (6) is not compatible however the substitution A → iA , B → iB 2 2 2 2 into (5) gives a reduction of (6) compatible with (8): u = i(u +u +2(v+v¯)u), t zz z¯z¯ (9) v = (|u|2) . z¯ z The substitution of (8) into (7) gives ′ u = u +u +3(vu +v¯u )+3(w+w )u, t zzz z¯z¯z¯ z z¯ (10) v = (|u|2) , w = (u¯u ) , w′ = (u¯u ) . z¯ z z¯ z z z z¯ z¯ For brevity we shall call the equations (9) and (10) by the DS and DS 2 3 equations respectively. In differencewith(9)theDS equation is compatible with theconstraint 3 u = u¯ and for real-valued potentials it reduces to the modified Novikov– Veselov equation: 3 u = u +u +3(vu +v¯u )+ (v +v¯ )u, t zzz z¯z¯z¯ z z¯ z z¯ 2 (11) v = (u2) . z¯ z Notice that A depends on two functional parameters which are p and n q and put A+ = A for p = −u,q = u¯, A− = A for p = −u¯,q = u. n n Now let us recall the definition of the DS deformations of a surface introduced in [9]. 6 Proposition 4 ([9]) Let a surface Σ be defined by the formulas (1) and (2) for some ψ0,ϕ0 and let U(z,z¯,t) be a deformation of the potential described by the equation (9) or (10). Then the formulas (1) and (2) and the equations ψ = iA+ψ, ϕ = −iA−ϕ, t 2 t 2 (12) ψ = A+ψ, ϕ = A−ϕ t 3 t 3 with ψ = ψ0,ϕ = ϕ0, define deformations of the surface governed by t=0 t=0 the equations (9) and (10) respectively. The proof of this proposition is as follows. Since the deformation of U = uisdescribedbytheequation(5),thevectorfunctionsψ andϕmeeting ∨ Dψ = 0 and D ϕ = 0 are deformed via equations (12) and for any t they meet again the Dirac equations. Therefore by Proposition 1 they define a surface Σ via the Weierstrass formulas (1) and (2). Thus we have a t deformation Σ such that Σ = Σ. t 0 Any equation of the DSII hierarchy defines such a deformation (for n even we have to substitute A → iA to preserve the reduction p = −q¯). n n We write down only two equations, DS and DS , because they resemble 2 3 the main properties of others and do not involve very large expressions. For u = u¯ such a deformation reduces to the mNV deformation defined in [8] and studied in [13, 14, 5, 10, 2]. 3 The Weierstrass representation An oriented two-plane in R4 is defined by a positively-oriented orthonormal basis e = (e ,...,e ), e = (e ,...,e ) 1 1,1 1,4 2 2,1 2,4 which is defined up to rotations. There is a one-to-one correspondence {(e ,e )} ↔ (y :y : y : y ), y = e +ie , k = 1,2,3,4, 1 2 1 2 3 4 k 1,k 2,k between the moduli space of such planes (which is the Grassmannian G ) 4,2 and points of the quadric Q ⊂ CP3 defined by the equations e y2+y2+y2+y2 = 0. 1 2 3 4 ′ ′ In terms of another homogeneous coordinates y ,...,y such that 1 4 i 1 ′ ′ ′ ′ y = (y +y ), y = (y −y ), 1 2 1 2 2 2 1 2 7 1 i ′ ′ ′ ′ y = (y +y ), y = (y −y ) 3 2 3 4 4 2 3 4 this quadric is written as ′ ′ ′ ′ y y = y y . 1 2 3 4 Therefore the correspondence ′ ′ ′ ′ y = a b , y = a b , y = a b , y = a b 1 2 2 2 1 1 3 2 1 4 1 2 establishes a biholomorphic equivalence G = CP1×CP1 4,2 where (a1 : a2) and (b1 : b2)eare homogeneous coordinates on the copies of CP1. This mapping CP1×CP1 → Q ⊂ CP3 is called the Segr´e mapping. Letr : W → R4 beanimmersionofasurfacewithaconformalparameter z ∈ W ⊂ C. The conformality condition reads 4 hr ,r i= (xk)2 = 0 z z z k=1 X where xk = ∂xk, k = 1,2,3,4. The Gauss map takes the form z ∂z G : W → G , Q ∈ W → (x1(Q),: ··· :x4(Q)). 4,2 z z By using the equivalencee G4,2 = CP1×CP1, decompose G into two maps e G = (Gψ,Gϕ) where G = (ψ : ψ¯ )∈ CP1, G = (ϕ :ϕ¯ ) ∈ CP1 ψ 1 2 ϕ 1 2 and rewrite the formulas for xk in terms of these maps as follows z i 1 x1 = (ϕ¯ ψ¯ +ϕ ψ ), x2 = (ϕ¯ ψ¯ −ϕ ψ ), z 2 2 2 1 1 z 2 2 2 1 1 (13) 1 i x3 = (ϕ¯ ψ +ϕ ψ¯ ), x4 = (ϕ¯ ψ −ϕ ψ¯ ). z 2 2 1 1 2 z 2 2 1 1 2 We have dxk = η , k = 1,2,3,4, k where the forms η take the same shapes as in Proposition 1. k 8 This decomposition is not unique and functions ψ and ϕ are defined up to gauge transformations ψ efψ ϕ e−fϕ 1 → 1 , 1 → 1 (14) ψ ef¯ψ ϕ e−f¯ϕ (cid:18) 2 (cid:19) (cid:18) 2 (cid:19) (cid:18) 2 (cid:19) (cid:18) 2 (cid:19) By choosing a representative ψ for G we fix a function ϕ. ψ The formula (1) gives exactly a gauge transformation between different lifts of the Gauss mapping G = (G ,G ) to non-vanishing spinors, i.e. to ψ ϕ (C2\{0})×(C2\{0}), which we mention in the introduction. The closedness conditions for the forms η are k (ϕ¯ ψ ) = (ϕ¯ ψ ) , ϕ¯ ψ¯ = − ϕ¯ ψ¯ (15) 2 1 z¯ 1 2 z 2 2 z¯ 1 1 z (cid:0) (cid:1) (cid:0) (cid:1) ∨ and they do not have the form of Dirac equations Dψ = 0 and D ϕ = 0 for arbitrary representatives (ψ ,ψ ) and (ϕ ,ϕ ) of the mappings G and G . 1 2 1 2 ψ ϕ Suchrepresentatives havetobefoundbysolvingsomedifferentialequations. Let us start with the following lift for G = (ψ :ψ¯ ) which is correctly ψ 1 2 defined up to a ±1 multiple: s = eiθcosη, s = sinη. 1 2 We look for a pair of functions ψ ,ψ meeting two conditions: 1 2 1) G = (ψ : ψ¯ ) = (s :s¯ ); ψ 1 2 1 2 2) Dψ = 0 for some potential U. By the first condition, ψ has the form ψ = egs , ψ = eg¯s . 1 1 2 2 The second condition is written as ∂(eg¯sinη)+Ueg+iθcosη = 0, ∂¯(eg+iθcosη) = U¯eg¯sinη. These equations are rewritten as follows: eg¯ U = − (g¯ sinη+η cosη), eg+iθcosη z z eg¯ U = (g¯ cosη−iθ cosη−η sinη), eg+iθsinη z z z which imply g +iθ cos2η = 0, z¯ z¯ 9 U = −eg¯−g−iθ(iθ sinηcosη+η ). z z The ∂¯-problem for g is solved by the well-known means and its solution is defined up to holomorphic functions. Therefore the potential U is defined uptoamultiplication byeh¯−h wherehis anarbitraryholomorphicfunction. It is derived by straightforward computations that Dψ = 0 implies that ∨ the condition (15) takes the form of the Dirac equation D ϕ= 0. Thus the following theorem is derived. Theorem 1 Let r : W → R4 be an immersed surface with a conformal parameter z and let G = (eiθcosη : sinη) be one of the components of its ψ Gauss map. There exists another representative ψ of this mapping G = (ψ : ψ¯ ) ψ 1 2 such that it meets the Dirac equation Dψ = 0 with some potential U. A vector function ψ = (eg+iθcosη,eg¯sinη) is defined from the equation g = −iθ cos2η, (16) z¯ z¯ up to holomorphic functions and the corresponding potential U is defined up by the formula U = −eg¯−g−iθ(iθ sinηcosη+η ) z z up to multiplications by eh¯−h where h is an arbitrary holomorphic function. Given the function ψ, a function ϕ which represents another component G of the Gauss map meets the equation ϕ ∨ D ϕ = 0. Different representations (lifts) of the Gauss mapping G of the surface W are related by gauge transformations of the form ψ1 → ψ′ = ehψ1 , ϕ1 → ϕ′ = e−hϕ1 , (cid:18) ψ2 (cid:19) (cid:18) eh¯ψ2 (cid:19) (cid:18) ϕ2 (cid:19) (cid:18) e−h¯ϕ2 (cid:19) (17) U → U′ = eh¯−hU, where h is an arbitrary holomorphic function on W. 10