Finite gap theory of the Clifford torus Iskander A. TAIMANOV 4 ∗ 0 0 2 n a 1 Introduction and main results J 7 In this paper we construct the spectral curve and the Baker–Akhiezer func- 2 tion for the Dirac operator which form the data of the Weierstrass represen- v tation of the Clifford torus. This torus appears in many conjectures from 9 5 differential geometry (see Section 2). 0 By constructing this Baker–Akhiezer function we demonstrate a general 2 procedure for constructing Dirac operators and their Baker–Akhiezer func- 1 3 tions corresponding to singular spectral curves. This procedure is exposed 0 in Section 3. / h The Clifford torus is a torus embedded into R3 which appears in many p important problems of surface theory. The corresponding Dirac operator is - h t 0 ∂ U 0 a = + m D ∂¯ 0 0 U (cid:18) − (cid:19) (cid:18) (cid:19) : v with the potential i X siny r U(z,z¯) = , z = x+iy. (1) a 2√2(siny √2) − We have Theorem 1 The Baker–Akhiezer function of the Dirac operator with the D potential (1) is a vector function ψ(z,z¯,P), where z C and P belongs to ∈ the spectral curve Γ of this operator, such that 1) the spectral curve Γ is a sphere CP1 = C¯ with two marked points + = (λ = ), = (λ = 0) where λ is an affine parameter on C CP1 ∞ ∞ ∞− ⊂ ∗InstituteofMathematics,630090Novosibirsk,Russia;e-mail: [email protected] 1 and with two double points obtained by stacking together the points from the following pairs: 1+i 1+i 1+i 1 i ,− and , − ; 4 4 − 4 4 (cid:18) (cid:19) (cid:18) (cid:19) 2) the function ψ is meromorphic on Γ and has at the marked \{∞±} points (“infinities”) the following asymptotics: ek+z 0 u2 ψ ≈ 0 as k+ = λ → ∞; ψ ≈ ek−z¯ as k− = −|λ| → ∞ (cid:18) (cid:19) (cid:18) (cid:19) where k 1 are local parameters near and u = 1+i; 3) ψ±−has three poles on Γ ∞w±hich are ind4ependent on z and have \{∞±} the form 1+i+√ 2i 4 1+i √ 2i 4 1 p1 = − − − , p2 = − − − − , p3 = . 4√2 4√2 √8 Therewith the geometric genus p (Γ) and the arithmetic genus p (Γ) of g a Γ are as follows: p (Γ) = 0, p (Γ) = 2. g a The Baker–Akhiezer function satisfies the Dirac equation ψ = 0 D at any point of Γ +, ,p1,p2,p3 where the potential U of the Dirac \{∞ ∞− } operator takes the form (1). The Clifford torus is constructed via the Weierstrass representation (2) from the function 1 i ψ = ψ z,z¯, − . 4 (cid:18) (cid:19) Remark that in the proof of this theorem which will be given in Section 4 it is actually will be showed that the Baker–Akhiezer function ψ takes the form ψ1(z,z¯,λ) = eλz−|uλ|2z¯ q1 λ +q2 λ +(1 q1 q2) λ , λ p1 λ p2 − − λ p3 (cid:18) − − − (cid:19) ψ2(z,z¯,λ) = eλz−|uλ|2z¯ t1 p1 +t2 p2 +(1 t1 t2) p3 p1 λ p2 λ − − p3 λ (cid:18) − − − (cid:19) 2 where u = 1+4i and q1,q2,t1,t2 are functions of z,z¯ which are uniquely defined by the conditions 1+i 1+i 1+i 1 i ψ z,z¯, = ψ z,z¯, − , ψ z,z¯, = ψ z,z¯, − . 4 4 − 4 4 (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) It appears that q1,q2,t1,t2 are 2π-periodic functions of y and are indepen- dent on x. We would like to mention one interesting feature: thespectralcurveΓadmitsaholomorphicinvolutionσ(λ) = λwhich • − preserves the infinities . Although both the Jacoby variety J(Γ/σ) ∞± of the quotient space Γ/σ and the Prym variety of this involution are noncomplete Abelian varieties, the potential is a smooth function. This is explained by some effect unfamiliar for other operators. It is as follows. The potential U is written in the terms of Prym functions (as in the case of the two-dimensional Schro¨dinger operator and some other operators, see [3]). Although the Prym variety is isomorphic to C , the potential ∗ dependsononereal-valued variabley inaway thatitisarestriction of some meromorphic function on the Prym variety onto a compact circle S1 C . ∗ ⊂ This circle is a compact subgroup of C . ∗ The correspondencebetween tori in R3 and Dirac operators with real- D valued potentials is established by the Weierstrass representation. It is based on a local representation of any surface immersed into R3 by the formulas xk = xk(0)+ xkdz+xkdz¯ , k = 1,2,3, z z Z (cid:16) (cid:17) (2) i 1 x1z = 2(ψ¯22+ψ12), x2z = 2(ψ¯22 −ψ12), x3z = ψ1ψ¯2, where ψ meets the Dirac equation ψ = 0. D In this event z defines a conformal parameter z = x+iy on the surface, the first fundamental form equals e2αdzdz¯ and Heα U = 2 where H is the mean curvature (see [7, 11]). 3 Afteraglobalization weobtainforaclosedsurfaceinR3 arepresentation by these formulas where ψ is a solution of the Dirac equation and the Dirac operator acts on smooth sections of some spinor bundles over a conformally equivalent surface of constant curvature [11, 12]. It appears that the spectral curve of on the zero energy level defined D initiallyforthetwo-dimensionalSchro¨dingeroperatorin[2]hastohavesome geometric meaning. Let us briefly recall the origin of the spectral curves in theory of differ- ential operators with double-periodic coefficients. The spectral curve Γ is a complex curve which parameterizes the Flo- quet functions on the zero energy level, i.e. joint eigenfunctions of and D translations T by periods γ T f(z,z¯)= f(z+γ,z¯+γ¯), U(z+γ,z¯+γ¯) = U(z,z¯). γ Here Dψ = 0 (the “eigenvalue” equals zero) and ψ is considered as a formal analytic solution to this equation not necessary belonging to some fixed functional space. Floquet functions are glued together into a meromorphic function ψ(z,z¯,P) on the spectral curve. If Γ is of finite genus, it is also completed by two “infinities” at which ψ has exponential asymptotics. This would be the Baker–Akhiezer function. Given a pair of generators γ1,γ2 of the period lattice, to every Floquet function ψ(z,z¯,P) where corresponds two functions on the spectral curve holomorphic outside the “infinities”, the multipliers µ1(P) and µ2(P), P Γ, such that ∈ T ψ(P) = µ ψ(P). γj j We correspond to a torus in R3 an operator via the Weierstrass rep- D resentation of the torus and the spectral curve of this operator and define the spectral genus of a torus as the geometric genus of the normalization of the spectral curve (see [12]). We conjectured that thespectralcurveΓ correspondingtoatorus inR3 andthemultipliers • µ1,µ2 : Γ C are invariant under conformal transformations of the ambient sp→ace R3 (that was confirmed in [5]); given a conformal class of a torus, the Willmore functional attains its • minima on tori with the minimal value of the spectral genus. In geometric problems the spectral curves related to integrable surfaces are not always smooth. This was discussed for minimal tori in S3 in [6]. For 4 general tori in R3 we have to define the spectral curves via Baker–Akhiezer functions ψ. We show how this is done for the Clifford torus in Theorem 1. The Willmore conjecture reads that the global minimum is attained on the Clifford torus for which the spectral genus vanishes (as it is showed by Theorem 1). We think that given a conformal class of a torus, the Willmore functional attains its • minima on tori with the minimal value of the spectral genus and the minimal value of the arithmetic genus of the spectral curve Γ defined ψ via the Baker–Akhiezer function. We shall discuss this conjecture elsewhere together with the expected relation between the spectral curves of tori in S3 (as they are defined in [14]) and the spectral curves of their stereographic images in R3. For the Clifford torus the mapping 1+i 1+i 1+i 1 i C∗ C∗/ − , − → 4 ∼ 4 − 4 ∼ 4 (cid:26) (cid:27) establishes an isomorphism of the corresponding spectral curves and the equivalence of the multipliers µ1,µ2. This was the basic idea of the compu- tations in Section 4, i.e. in the proof of Theorem 1. We summarize the contents of the paper: in Section 2 we explain the geometry of Clifford torus in R3; in Section 3 we expose the procedure for constructiing Dirac opera- tors and their Baker–Akhiezer functions corresponding to singular spectral curves; in Section 4 we prove Theorem 1. Thisworkwas supportedbyRFBR (grant03-01-00403), theProgramme “LeadingScientificSchools”(NS-2185.2003.1), andMax-Planck-Institute on Mathematics in Bonn. 2 The Clifford torus In fact, in differential geometry two objects are called the Clifford torus: 1) (a torus in S3) the product of circles of the same radii which lies in the unit 3-sphere S3 R4 and therewith is defined by the equations: ⊂ 1 2 2 2 2 x +x = x +x = 1 2 3 4 2 where (x1,...,x4) are Euclidean coordinates in R4; 5 2) (a torus in R3) the following torus of revolution: take in the x1x3 plane a circle γ of radius r = 1 such that the distance between its center and the x1 axis equals R = √2 and obtain a torus of revolution in R3 by rotating this circle γ around the x1 axis. The torus in S3 is considered up to isometries of S3 and in symplectic geometry it is considered as a torus in a symplectic 4-space R4 and is also called by this name. The torus in R3 is considered up to conformal transformations of R3. In thiseventitisdistinguishedamongtoriofrevolutionbytheratioR/r = √2. These tori are related to many conjectures in geometry: the Willmore functional defined on immersed surfaces in R3 by the • formula 2 (M) = (H K)dµ, W − ZM wheredµistheareaformintheinducedmetric,H andK arethemean curvature and the Gaussian curvature, respectively, for tori is not less than2π2 andattainsitsminimumontheCliffordtorus(anditsimages under conformal transformations of R3) (the Willmore conjecture); the Clifford torus in S3 is the unique (up to isometries) embedded • minimal torus in S3 (the Lawson conjecture); the area of any minimal torus in S3 is not less than 2π2 which is the • area of the Clifford torus; the Clifford torus in S3 R4 minimizes the Willmore functional • ⊂ 1 2 (M) = H dµ, W 4 | | ZM where H is the mean curvature vector, for tori in R4, or at least for Lagrangian tori in R4 with the standard symplectic structure; the Clifford torus in S3 R4 minimizes the area in its Hamiltonian • isotopy class of Lagrangi⊂an tori in R4, i.e. among tori obtained from the Clifford torus by Hamiltonian deformations (the Oh conjecture). The relation between two different notions of the Clifford torus and the corresponding conjectures is supplied by the stereographic projection F of S3 onto R3 = R¯3. For that project S3 (0,0,0,1) onto R3 = x4 = 0 from the∪{n∞ort}h pole N = (0,0,0,1) S\3{ mapping}N into { R¯3. } ∈ ∞ ∈ 6 This mapping establishes the conformal equivalence of S3 and R¯3 and in coordinates takes the form x1 x2 x3 2 2 2 2 F(x1,x2,x3,x4) = , , , x1+x2+x3+x4 = 1. 1 x4 1 x4 1 x4 (cid:18) − − − (cid:19) It is known that for a minimal torus T2 S3 its area coincides with the ⊂ value of the Willmore functional on its stereographic image: 2 2 Area(T ) = (F(T )). W Therefore the third conjecture follows from the Willmore conjecture al- though The opposite is not true (these conjectures are not equivalent). Moreover the Lawson conjecture also implies the third conjecture (for expo- sitions of the conjectures, some related results and their relations for tori in R3 see, for instance, [1, 11] and for tori in R4 see [9]). Let us parameterize the Clifford torus in S3 as follows: cosx sinx cosy siny x1 = , x2 = , x3 = , x4 = √2 √2 √2 √2 where 0 ,x,y 2π. Then the Clifford torus in R3 is given by the formulas ≤ ≤ cosx sinx cosy r(x,y) = , , . (3) √2 siny √2 siny √2 siny (cid:18) − − − (cid:19) It is easy to compute that x,y define a conformal parameter z = x+iy on the torus in which the induced metric takes the form dzdz¯ 2 ds = , (√2 siny)2 − the normal vector (assuming that the orientation on the surface is defined by the positively oriented frame (r ,r )) equals x y cosx(1 √2siny) sinx(1 √2siny) cosy N = − , − , − , √2 siny √2 siny √2 siny! − − − the second fundamental form is (√2siny 1) 1 2 2 − dx + dy , (√2 siny)2 (√2 siny)2 − − the principal curvatures take very simple forms: κ1 = 1, κ2 = √2siny 1, − 7 and therefore the mean curvature equals siny H = . √2 The potential of the Weierstrass representation is siny U = . (4) 2√2(√2 siny) − Other important functions related to this representation are ∂ (x1+ix2)= iψ¯22, ∂x3 = ψ1ψ¯2 ∂z ∂z and for the Clifford torus they are equal to ∂ ∂ cosx+isinx i √2 cosy siny (x1+ix2)= = eix − − , ∂z ∂z √2 siny 2 (√2 siny)2 − − (5) ∂x3 ∂ cosy i √2siny 1 = = − . ∂z ∂z√2 siny 2(√2 siny)2 − − 3 Baker–Akhiezer functions and Dirac operators Let us recall the definition of the Baker–Akhiezer (vector) function ψ corre- sponding to the Dirac operator 0 ∂ U 0 = + . D ∂¯ 0 0 V (cid:18) − (cid:19) (cid:18) (cid:19) By definition it depends on a complex variable z C and on a parameter ∈ P on a complex curve Γ of finite arithmetic genus g = p (Γ) and meets the a following conditions: 1) ψ is meromorphicin P outsidea coupleof markedpoints Γ and ∞± ∈ has poles at g+1 points P1+ +Pg+1 (the points +, ,P1,...,Pg+1 ··· ∞ ∞− are nonsingular); 2) ψ has the following asymptotics near : ∞± + 1 ξ ψ ≈ ek+z 0 + ξ1+ k+−1+O(k+−2) as P → ∞+, 2 (cid:20)(cid:18) (cid:19) (cid:18) (cid:19) (cid:21) 0 ξ ψ ek−z¯ + 1− k 1+O(k 2) as P , − − ≈ 1 ξ2− − − → ∞− (cid:20)(cid:18) (cid:19) (cid:18) (cid:19) (cid:21) 8 1 where k are local parameters near infinities such that − ± 1 k ( ) = 0. − ± ∞± It follows from the general theory of Baker–Akhiezer functions ([8, 3]) that the such function is unique (for a generic divisor D = P1 + +Pg+1) and ··· therefore we can find an operator with potentials U and V such that ψ D D is meromorphic with analogous singularities but with the asymptotics ψ = ek+zO(k+−1) as P +, ψ = ek−z¯O(k−1) as P . D → ∞ D − → ∞− The uniqueness of ψ implies that ψ =0. We have D Theorem 2 ([12, 13]) The Baker–Akhiezer function ψ satisfies the Dirac equation + ψ = 0 with U = ξ , V = ξ . 2 1− D − IfthecurveΓissingularweassumethatithassomespecialform. Letus now define what it is this form and recall some algebro-geometric properties of such singular curves following [10]. Let Γnm be a nonsingular complex curve. Take on Γnm effective divisors D1,...,Dn, i.e. for any k = 1,...,k such a divisor Dk is a formal sum of finitely many points on Γnm with positive coefficients: Dk = ak1Qk1+···+akmkQkmk, Qkj Γnm,akj > 0,akj Z,j =1,...,mk,k = 1,...,n. ∈ ∈ We assume that the supports suppDk = Qk1+···+Qkmk, k = 1,...,n, of these divisors are pairwise nonintersecting. Then we denote by Γ D1,...,Dn the singular curve obtained by contracting all points from each divisor D k into one singular point S Γ. This is done with respect to the multi- k ∈ plicities. This means that a function f which is meromorphic on Γ is, by definition, a meromorphic function f : Γnm C such that it has no poles → at suppD , and for each divisor k k ∪ Dk = ak1Qk1+···+akmkQkmk we have f(Qk1)= ··· = f(Qkmk), ∂mf(Qkl)= 0 for l = 1,...,(akl−1), 9 where k = 1,....n. Here ∂ is the Cauchy derivation with respect to a complex parameter on Γ. The natural projection π : Γnm Γ → is the normalization mapping. If Γ is singular we assume that it takes the form Γ = Γ . D1,...,Dn Let T Γ and T is empty or contains only nonsingular points. ⊂ D1,...,Dn Recall that a one-form ω is called a regular form on Γ T if ω is a meromorphic form on Γnm π−1(T) which may haveDp1,o..l.,eDsno\nly at the \ supports of D1,...,Dk and at each point Qkl the degree of the pole is not greater than a and moreover for every k the inequality kl Resfω(Q ) = 0 kl l X which holds for all meromorphic functions on Γ. For a nonsingular curve Γ the notions of regular and holomorphic forms coincide. For nonsingular curves the values of the arithmetic genus p and the a geometric genus p coincide. For a singular curve Γ = Γ we have g D1,...,Dk pg(Γ) = pg(Γnm), pa(Γ)= pg(Γ)+ (degDk 1) − k X where the degree of the divisor D equals to k degDk = deg(ak1Qk1+···+akmkQkmk)= ak1+···+akmk. We say that σ : Γ Γ is an involution of Γ if σ1 = 1 and its pull-back → defines an involution on Γnm such that the singularity divisors D1,...,Dn falls into two groups: 1) divisors which are preserved: D = σ(D ); j j 2) divisors which are interchanges with others: D = σ(D ),j =k. j k 6 The first group corresponds to fixed points S = π(D ) of an involution j j on Γ although the points from D could be permuted. j Theorem 3 ([12, 13]) 1) Let σ be a holomorphic involution σ : Γ Γ → such that σ( ) = , σ(k ) = k ∞± ∞± ± − ± 10