Conformal invariant functionals ofimmersions of tori into R3.P.G.Grinevich1M.U.Schmidt21 L.D.Landau Institute for Theoretical Physics, Kosygina 2, Moscow,177940, Russia.e-mail: pgg@landau.ac.ru2 Freie Universit�at, Berlin, Arnimallee 14, D-14195, Berlin, Germany.e-mail: mschmidt@omega.physik.fu-berlin.deAbstractWe show, that higher analogs of the Willmore functional, de�nedon the space of immersions M2 ! R3, where M2 is a two-dimensionaltorus, R3 is the 3-dimensional Euclidean space are invariant underconformal transformations of R3. This hypothesis was formulated re-cently by I. A. Taimanov.Higher analogs of the Willmore functional are de�ned in terms ofthe Modi�ed Novikov-Veselov hierarchy. This soliton hierarchy is asso-ciated with the zero-energy scattering problem for the two-dimensionalDirac operator.1 IntroductionTo start with, we would like to recall the following interesting fact fromthe theory of 2-dimensional surfaces in R3 (see [20], p. 110 and referencestherein). Let X :M2 ! R3 be a smooth immersion of a compact orientablesurface M2 into the Euclidean space R3 (i.e. a smooth map from M2 to R31This work was ful�lled during the author's visit to the Freie Universit�at, Berlin, Ger-many, which was supported by the Humboldt-Foundation. This work was also partiallysupported by Russian Foundation for Basic Research, grant No 95-01-755.2Supported by the DFG, SFB 288 \Di�erentialgeometrie und Quiantenphysik".1

such that its Jacobi matrix is non-degenerate everywhere on M2, but theimage is allowed to self-intersect). Let T be the Willmore functionalT = ZM2 H2dS; (1)whereH denotes the mean curvature, dS is the volume element onM2 gener-ated by the immersion. Then T is invariant under conformal transformationsof R3.It is natural to pose the problem of constructing other conformal invariantfunctional of immersions.It is well-known, that many constructions from the soliton theory havenatural analogs in geometry and vise versa. In particular, important informa-tion how to study immersions of 2-dimensional surfaces into R3 using solitonmethods can be found in the article [1] by A. I. Bobenko. Any immersedsurface possesses (at least locally) a conformal coordinate system (see forexample [3] p. 110), i.e. a coordinate system such that ds2 = f(z; �z)dzd�z.In conformal coordinates this immersion can be locally represented by theGeneralized Weierstrass Formulas (see Section 2 below) and the potentialU(z; �z) is uniquely de�ned. In [17] it was shown, that any analytic immer-sion of a compact orientable 2-dimensional manifold into R3 can be globallyrepresented by these formulas.The GeneralizedWeierstrass Formulas are based on the zero-energy eigen-functions of the two-dimensional Dirac operator with a real potential U(z; �z).The zero-energy spectral problem for this operator arose in the soliton theoryas an auxiliary linear problem for the hierarchy of Modi�ed Novikov-VeselovEquations (MNV) (see [2]). These nonlinear integrable equations with 2spatial variables, introduced by L. V. Bogdanov ,have in�nitely many con-servation laws.In [9] B. G. Konopelchenko and I. A. Taimanov showed that the quadraticMNV conservation lawH1 = 4 Z Z U2(z; �z)dx ^ dy; z = x+ iy (2)coincides with the Willmore functional T . In [18] I. A. Taimanov formulateda hypothesis that all higher MNV conservations laws also generate functionalson immersions of closed orientable 2-dimensional surfaces into R3, invariantunder conformal transformations of R3. He also did some numerical experi-ments with surfaces of revolution, con�rming this assumption.2

During his visit to the Freie Universit�at, Berlin in September{October1996, I. A. Taimanov attracted the authors attention to this problem. In thepresent text we prove this hypothesis for immersions of tori into R3.Of course, it would be natural to extend this result to immersions ofhigher genus surfaces into R3. But here we meet the following problem. Incontrast with H1 the higher conservation laws are non-local in terms of thepotential. For double-periodic potentials they are de�ned in terms of thezero-energy dispersion curve (the Riemann surface of the zero-energy Blochfunction). If the genus is greater than 1, we have to study modular invariantDirac operators in the Lobachevskian plane with a non-abelian group oftranslations. The corresponding Bloch theory has not been constructed untilnow, thus we could not de�ne the higher conservation laws. It would beinteresting to develop the corresponding Bloch theory, but this problem looksrather non-trivial. Thus we restrict ourself to the genus 1 case only, whereall integrals of motion are well-de�ned.The idea of our proof is the following. We show, that in�nitesimal confor-mal transformations of R3 correspond to in�nitesimal Darboux transforma-tions of the Dirac operator (in�nitesimal dressings with degenerate kernels).It is convenient to express these deformations in terms of Cauchy-Baker-Akhiezer kernels, introduced by A. Yu. Orlov and one of the authors in [6](see also the review [7]). From the explicit formulas for such deformationsit follows, that the deformed zero-energy Bloch function is meromorphic onthe same Riemann surface as the original one. Recalling, that this Riemannsurface completely determines all conservation laws, we complete the proof.Remark 1 An alternative proof of the theorem, namely that conformal trans-formations do not change the zero-energy Bloch variety, was obtained byU. Pinkall (private communication). He calculated the action of �nite con-formal transformations on the Bloch function using a quaternionic represen-tation of the Generalized Weierstrass formulas as suggested by G. Kamberov,F. Pedit and U. Pinkall in [8].Remark 2 Finite Darboux transformations for the Dirac operator are dis-cussed in the book by V. B. Matveev and M. A. Salle [13], Laplace transfor-mations for the Dirac operator are discussed in the paper by E. V. Ferapontov[5].The authors are grateful to I. A. Taimanov for interesting discussions.3

2 Generalized Weierstrass constructionLet L be the two-dimensional Dirac operatorL = " @z �U(z; �z)U(z; �z) @�z # (3)with a real potential U(z; �z). Let ~(z; �z) be a zero-energy solution of theDirac equation L~(z; �z) = 0; ~(z; �z) = 1(z; �z)2(z; �z) ! : (4)Then the Generalized Weierstrass Formulas (see [17] and refs. therein)X1(z; �z) + iX2(z; �z) = C1 + iC2 + i zRz0 ��21(z0; �z0)dz0 � �22(z0; �z0)d�z0�X1(z; �z)� iX2(z; �z) = C1 � iC2 + i zRz0 (22(z0; �z0)dz0 �21(z0; �z0)d�z0)X3(z; �z) = C3 � zRz0 �2(z0; �z0) �1(z0; �z0)dz0 +1(z0; �z0) �2(z0; �z0)d�z0� (5)de�nes a map of the plane R2 to the Euclidean space R3. In (5) z0 is a �xedpoint in the z-plane and the integrals are taken over some path connectingthe points z0 and z. From (4) it follows, that the integrands in (5) are closedforms, thus the map does not depend on a speci�c choice of the path. HereC1, C2, C3 are arbitrary real constants.The formulas (5) are equivalent to:d [�2X1 + �1X2 � �3X3] == " �1(z; �z) �2(z; �z)�2(z; �z) 1(z; �z) # " 0 dzd�z 0 # " 1(z; �z) ��2(z; �z)2(z; �z) �1(z; �z) # (6)where �1, �2, �3 are the standard Dirac matrices�1 = " 0 11 0 # ; �2 = " 0 �ii 0 # ; �3 = " 1 00 �1 # : (7)4

The Generalized Weierstrass Map is conformal, i.e. the metric d~s2 on R2induced by this map is proportional to the standard one: d~s2 = g(z; �z)dzd�z.Assume that we have a map of a 2-dimensional torus into R3. Then thecorresponding potentials U(z; �z) is periodicU(z + �T1; �z + T1) = U(z + T2; �z + �T2) = U(z; �z): (8)Also the eigenfunction ~(z; �z) is periodic or anti-periodic, i.e.~(z+T1; �z+ �T1) =W1~(z; �z); (z+T2; �z+ �T2) =W2~(z; �z); W21 =W22 = 1:(9)The coordinate z is de�ned up to linear transformations z ! az + b,a; b 2 C , a 6= 0. Thus without loss of generality we may assumeT1 = 1; T2 = �; Im � > 0: (10)Conditions (8) and (9) are necessary, but, of course, not su�cient forperiodicity of the Generalized Weierstrass map. Necessary and su�cientconditions for periodicity can be formulated in terms of the Bloch variety.They are obtained in a forthcoming paper by I. A. Taimanov and one of theauthors (M.S.). We do not use these conditions in our text, thus we will notdiscuss them in further details3 Bloch function and Bloch variety.In this Section we assume that U(z; �z) is real, smooth, and double-periodic(8). With any such potential we associate a one-dimensional subvariety � inthe two-dimensional complex space (Cn0)2 .The �rst object we need is the Bloch function. By de�nition, the Blochfunctions ~ (w1; w2; z; �z) are quasiperiodic solutions of the Dirac equation (4)with the following periodicity properties:~ (w1; w2; z + 1; �z + 1) = w1~ (w1; w2; z; �z);~ (w1; w2; z + �; �z + �� ) = w2~ (w1; w2; z; �z): (11)The pairs of multipliers w1, w2 possessing at least one non-zero Blochsolution form a complex one-dimensional subvariety � 2 (Cn0) � (Cn0) (see5

[12]). This variety is called the Bloch variety or the zero-energy disper-sion curve. For a generic potential U(z; �z) the genus of � is in�nite.The Bloch functions form a one-dimensional holomorphic bundle over �(it is shown below, that for generic � 2 � a Bloch solution is unique up tonormalization). It is convenient to �x a section of this bundle ~ (�; z; �z), byassuming 1(�; z; �z) + 2(�; z; �z)jz=z1 = 1; (12)where z1 is an arbitrary �xed point.The logarithms of the multipliers w1(�), w2(�)p1(�) = 1i lnw1(�); p2(�) = 1i j� j lnw2(�); (13)are called quasimomentum functions. Of course, they have non-trivialincrements while going along cycles in �, and they are de�ned up to adding2�n1, 2�n2= j� j respectively, where n1 and n2 are some integers. Thus thefunctions Imp1(�), Imp2(�) are single-valued in �. The di�erentials of thequasimomentum functionsdp1(�) = @@�p1(�)d�; dp2(�) = @@�p2(�)d�; (14)are single-valued and holomorphic on the �nite part of �.In our text the Dirac operator (3) is symmetric and the potential U(z; �z)is real. Let us show, that the corresponding Bloch variety � possesses Z2�Z2as a group of symmetries. An analogous statement for the �xed-energy Blochvariety corresponding to a two-dimensional self-adjoint Schr�odinger operatorwas proved in [10]. The proof from [10] may be applied to (3) after a minimalmodi�cation.The operator (3) with real potential U(z; �z) has the following symmetry.If ~ (w1; w2; z; �z) is a Bloch solution of (4), then the function~ y(w1; w2; z; �z) = � 2(w1; w2; z; �z)� � 1(w1; w2; z; �z) ! (15)is also a Bloch solution of (4) with multipliers �w1 and �w2 respectively. Thusthe surface � possesses an antiholomorphic involution (we denote it by ��for historical reasons)�� : �! �; �� : (w1; w2)! ( �w1; �w2); (16)6

and ~ y(�; z; �z) = n�1(�)~ (��(�); z; �z); (17)where n(�) is a scalar function, meromorphic in � and independent on z, �z.It is less trivial to see that the surface � possesses a holomorphic involu-tion: � : �! �; � : (w1; w2)! (w�11 ; w�12 ): (18)To prove it, let us �x a generic point � 2 �. Denote by Lw1w2 the Banachspace of all locally square-integrable two-component complex-valued vector-functions on R2 with the periodicity properties (11). The space Lw�11 w�12 isnaturally dual to Lw1w2. Namely, if ~f (z; �z) 2 Lw1w2 and ~g(z; �z) 2 Lw�11 w�12 ,then we de�ne a scalar product by< f; g >= 1R0 1R0 dt1 ^ dt2 [f1(t1 + � t2; t1 + ��t2)g1(t1 + � t2; t1 + �� t2)++f2(t1 + � t2; t1 + �� t2)g2(t1 + � t2; t1 + �� t2)] : (19)Let ~f (0) = ~ (w1; w2; z; �z), ~f (1), . . . , ~f (n), . . . be the Jordan basis for the Diracoperator L in the space Lw1w2 . Also let ~g(0), ~g(1), . . . , ~g(n), . . . be the dualbasis in Lw�11 w�12 . The functions ~g(n) form a Jordan basis for the transposedoperator LT . But L is symmetric with respect to this scalar product thus ithas a zero eigenfunction in the space Lw�11 w�12 . Hence if (w1; w2) 2 �, then�(w1; w2) = (w�11 ; w�12 ) 2 �.One of the main properties of the Bloch variety � is the following: � maybe treated as a complete set of integrals of motion for the Modi�edNovikov-Veselov hierarchy.Indeed, consider the space of all real-valued smooth double-periodic func-tions on C 1 = R2 with a �xed pair of periods 1 and � . The Modi�ed Novikov-Veselov hierarchy (MNV) (see Section 4 below) de�nes an in�nite collectionof ows on this space@U(z; �z; t2n+1)@t2n+1 = K2n+1[U ] + �K2n+1[U ]; (20)@U(z; �z; ~t2n+1)@t2n+1 = i � �K2n+1[U ]�K2n+1[U ]� ; (21)where K2n+1[U ] is some integro-di�erential operator in z, �z. Here t2n+1, ~t2n+1are parameters of these ows. 7

Statement 1 Let U(z; �z; t2n+1) be a solution of one of the MNV equations;let L(t2n+1) be the corresponding two-dimensional Dirac operator (3) depend-ing on an extra parameter t2n+1; let �(t2n+1) be the corresponding family ofBloch varieties.Then �(t2n+1) = � does not depend on the MNV time t2n+1.Remark 3 Here and below we use the following notational convention. Ifwe have a complete proof of a mathematical result we call it Theorem orLemma. If we do not have a complete strict proof yet we use the wordStatement.An analogous statement is well-known for soliton systems with one spa-tial variable. In Section 4 we prove this fact at least for algebraic-geometricalpotentials, corresponding to varieties � of �nite genus. (Sometimes such po-tentials are called �nite-gap potentials). It is rather clear, that our proof canbe extended to all smooth potentials, but to do such extension strictly weneed more detailed information about analytic behavior of the Bloch func-tions near in�nity in the momentum space, than we have now. An appropri-ate analytic lemma for the one-dimensional Dirac operator, corresponding tothe surfaces of revolution, was proved by one of the authors (M.S.) in [16].There exists a di�erent way (may be a more natural one) to get a strictproof of Statement 1. It would be interesting to prove the following approx-imation property:Conjecture 1 Any smooth potential can be approximated by the algebraic-geometrical ones with the same periods.From such a result it would follow, that we can restrict ourself to thealgebraic-geometrical potentials in our calculations.We have a map U(z; �z) ! �[U ] from the space of double-periodic realsmooth functions to the space of complex subvarieties in (Cn0)2, which isinvariant under the whole MNV hierarchy. This map generates an in�nitefamily of MNV \normal" conservation laws. Namely, let w1 be a genericpoint in Cn0. Than we have an in�nite collection of numbers w(k)2 [U ] suchthat (w1; w(k)2 [U ]) 2 �[U ]. From Statement 1 it follows that these functionalsw(k)2 [U ] are conservation laws of the whole Modi�ed Novikov-Veselovhierarchy. 8

The functionals w(k)2 [U ] are essentially nonlocal. In Section 4 we showthat under the same assumptions as in Statement 1 we can expand thesefunctionals in some asymptotic series near in�nity and the expansion coe�-cients give us the standard \quasi-local" conservation laws.4 Conformal transformations of the Euclideanspace R3, and MNV integrals of motion.In the previous Section we have associated with any double-periodic smoothreal potential a Bloch variety �[U ]. The map is constant on the trajectoriesof the MNV hierarchy. In this Section we associate with any immersion of atorus into R3 a Bloch variety � and show, that � is invariant under conformaltransformations of R3.Let M2 be a torus with a �xed basis of cycles a, b. Let X :M2 ! R3 bea smooth immersion of M2 into the Euclidean space. The standard metricon R3 induces a conformal structure on M2. Let z be a conformal globalcoordinate on the universal covering space ofM2. The coordinate z is de�neduniquely up to a�ne transformations z ! cz + d. If we assume, that theshift of M2 along the a-cycle corresponds to the shift z ! z + 1 then thecoordinate z is de�ned uniquely up to shiftsz ! z + d: (22)The immersion X and coordinate z de�ne a potential U(z; �z) thereforealso a Bloch variety �[U ]. �[U ] is invariant under the shifts (22) thus it iscompletely determined by the immersion X and the cycles a; b, and we maywrite �[X; a; b].Conformal transformations of the Euclidean space R3 do not a�ect theconformal structure of M2, thus they leave the coordinate z invariant up tothe shifts (22). Without loss of generality we shall assume that conformaltransformations of R3 do not change z.Now we are in position to formulate and prove our main result:Theorem 1 Let X :M2 ! R3 be an immersion of a torus with a �xed basisof cycles a, b into the Euclidean space; let �[X; a; b], be the correspondingBloch variety. Then �[X; a; b] is invariant under conformal transformationsof R3. 9

Proof of the Theorem:Step 1: To start with let us recall the well-known facts about the groupof conformal transformations of the standard Euclidean metric on R3 (or onthe sphere S3) (see for example [3]). This group is generated by the followingtransformations:1. Translations Xi ! Xi + (X0)i.2. Rotations ~X ! A ~X, A 2 SO(3).3. Dilations ~X ! k ~X, k 2 R4. Inversions Xi ! Xi � (X0)i< ~X � ~X0; ~X � ~X0 >: (23)5. Re ections ~X ! ~X � 2~v < ~v; ~X >; < ~v;~v >= 1: (24)The connected component of the identity of this group is isomorphic toSO(1; 4) (see [3], page 143). The corresponding Lie algebra is generated bythe following basis of in�nitesimal transformations:1. Translations Pa: �Xi = �ia.2. Rotations ab; a < b: �Xi = �ibXa � �iaXb.3. Dilation D: �Xi = Xi4. Inversions Ka �Xi = 2XiXa � �ia 3Xj=1XjXj : (25)Step 2: Let us calculate deformations of the potential U(z; �z), the eigen-function (z; �z) and the constants Cj in formulas (5) corresponding to allin�nitesimal conformal transformations of R3.1. Translations simply shift the constants Cj and change neither U(z; �z)nor (z; �z). Thus they do not change �[X; a; b], and without loss ofgenerality we can assumeCj = 0; j = 1; 2; 3: (26)10

2. A simple direct calculation based on thr representation (6) (we do notlike to reproduce it here) shows that the rotations in R3 correspond tothe following transformations of the eigenfunction ~(z; �z) 1(z; �z)2(z; �z) !! � 1(z; �z)2(z; �z) !+ � �2(z; �z)��1(z; �z) ! ; j�j2 + j�j2 = 1(27)where � and � are some complex parameters.Both functions ~(z; �z) and~+(z; �z) = �2(z; �z)��1(z; �z) ! ; (28)satisfy the Dirac equation (4) with the same potential U(z; �z). Thusthe rotations does not change the potential and �[X; a; b].3. The dilation is generated by the scaling transform�~(z; �z) = 12 ~(z; �z) (29)and changes neither U(z; �z) nor �[X; a; b].4. The only nontrivial transformations of the Dirac operator correspond tothe inversion generators. Up to conjugations by rotations all generators(25) are equivalent. Thus it is su�cient to prove, that �[X; a; b] isinvariant if we apply the generator �K3:�X1 = �2X1X3; �K2 = �2X2X3; �X3 = �X23 +X21 +X22 : (30)Let us introduce the following notationW (z; �z) = X1(z; �z)� iX2(z; �z): (31)The transformation (30) corresponds to the following transformationof the function ~(z; �z)�1(z; �z) = �X3(z; �z)1(z; �z) + iW (z; �z) �2(z; �z)�2(z; �z) = �X3(z; �z)2(z; �z)� iW (z; �z) �1(z; �z) (32)11

Let us check it.�X3(z; �z) = � zRz0 h��2(z0; �z0) �1(z0; �z0) + 2(z0; �z0)� �1(z0; �z0)� dz0 ++ ��1(z0; �z0) �2(z0; �z0) + 1(z0; �z0)� �2(z0; �z0)� d�z0i == � zRz0 h��2X3(z0; �z0)2(z0; �z0) �1(z0; �z0)��iW (z0; �z0) �21(z0; �z0)� i �W (z0; �z0)22(z0; �z0)� dz0++ ��2X3(z0; �z0)1(z0; �z0) �2(z0; �z0)++iW (z0; �z0) �22(z0; �z0) + i �W (z0; �z0)21(z0; �z0)� d�z0i == � zRz0 h�2X3(z0; �z0)(@z0X3(z0; �z0))��W (z0; �z0)(@z0 �W (z0; �z0))� �W (z0; �z0)(@z0W (z0; �z0))� dz0++ �2X3(z0; �z0)(@�z0X3(z0; �z0))��W (z0; �z0)(@�z0 �W (z0; �z0))� �W (z0; �z0)(@�z0W (z0; �z0))� d�z0i == � zRz0 h�@z0X23 (z0; �z0)� @z0(W (z0; �z0) �W (z0; �z0))� dz0 ++ �@�z0X23 (z0; �z0)� @�z0(W (z0; �z0) �W (z0; �z0))� d�z0i == W (z; �z) �W (z; �z)�X23 (z; �z)�W (z; �z) = i zRz0 2 h�2(z0; �z0)2(z0; �z0)dz0 � �1(z0; �z0)1(z0; �z0)d�z0i == zRz0 2 h��X3(z0; �z0)i22(z0; �z0) +W (z0; �z0) �1(z0; �z0)2(z0; �z0)� dz0 ++ �X3(z0; �z0)i21(z0; �z0) +W (z0; �z0) �2(z0; �z0)1(z0; �z0)� d�z0i == 2 zRz0 h��X3(z0; �z0)(@z0W (z0; �z0))�W (z0; �z0)(@z0X3(z0; �z0))� dz0 ++ ��X3(z0; �z0)(@�z0W (z0; �z0))�W (z0; �z0)(@z0X3(z0; �z0))� d�z0i == �2 zZz0 h�@z0(X3(z0; �z0)W (z0; �z0))� dz0 + �(@�z0(X3(z0; �z0)W (z0; �z0))� d�z0i == �2W (z; �z)X3(z; �z)The corresponding transformation of the potential U(z; �z) reads as�U(z; �z) = 1(z; �z) �1(z; �z)�2(z; �z) �2(z; �z): (33)12

Step 3: Let us calculate the the deformation of the Bloch functionscorresponding to (33).Let ~ (�; z; �z) be the Bloch function of L. For any � such that at leastone of the functions Impx(�), Impy(�) is not equal to zero (\non-physical"�) de�ne the following pair of functions1(�; z; �z) = zR1 2(�; z0; �z0) �1(z0; �z0)dz0 + 1(�; z0; �z0) �2(z0; �z0)d�z02(�; z; �z) = zR1 2(�; z0; �z0)2(z0; �z0)dz0 � 1(�; z0; �z0)1(z0; �z0)d�z0: (34)where the integrals are taken along an arbitrary path in the z-plane, connect-ing the points z and 1 such that the integrand decays exponentially alongthis path. The integrands in (34) are closed 1-forms thus the integrals donot depend on a concrete choice of the path. Using the same arguments asin Section 5.2 below we may easily prove that 1(�; z; �z) and 2(�; z; �z) aremeromorphic in � outside in�nity.The function ~(z; �z) is double-periodic or anti-periodic in z (see (9)), thusthe functions 1(�; z; �z), 2(�; z; �z) have the following periodicity properties:k(�; z + 1; �z + 1) =W1w1(�)k(�; z; �z)k(�; z + �; �z + �� ) =W2w2(�)k(�; z; �z) k = 1; 2: (35)(see (9) for the de�nition of W1, W2.)Lemma 1 The variation of the Bloch function ~ (�; z; �z) corresponding to(33) reads as� 1(�; z; �z) = 1(�; z; �z)1(z; �z)� 2(�; z; �z) �2(z; �z) + �(�) 1(�; z; �z)� 2(�; z; �z) = 1(�; z; �z)2(z; �z) + 2(�; z; �z) �1(z; �z) + �(�) 2(�; z; �z)(36)where �(�) is some meromorphic function, �xed by the normalization condi-tion: � 1(�; z; �z) + � 2(�; z; �z)jz=z1 = 0; (37)Proof of Lemma 1. To start with, let us recall a simple fact from Blochtheory (see for example [10]).Generically, if we calculate variations of the Bloch function, we deform�, and we can not assume both �w1(�) = 0 and �w2(�) = 0 simultaneously.13

To compare functions on di�erent subvarieties in C 2 we have to �x someconnection. The simplest way to do this is to assume �w1(�) = 0.Then the variation of the Bloch function can be found as the uniquesolution of the linearized Dirac equation�L~ (�; z; �z) + L� ~ (�; z; �z) = 0 (38)satisfying (37) such that� ~ (�; t1 + � t2; t1 + �� t2) = O �[1 + jt2j] eip1(�)t1+ip2(�)t2� : (39)A simple direct calculation shows that (36) solves (38). From (35) and(9) it follows, that� ~ (w1; w2; z + 1; �z + 1) = w1� ~ (w1; w2; z; �z);� ~ (w1; w2; z + �; �z + �� ) = w2� ~ (w1; w2; z; �z); (40)thus variations of the type (36) satisfy (39). This completes the proof.The function � (�; z; �z) de�ned by (36) has the same periodicity prop-erties as the original Bloch function (�; z; �z) (see formulas (11) and (40)respectively). Thus our special variations satisfy �w1(�) = 0 and �w2(�) = 0simultaneously.Step 4: From (36) it follows, that if we apply in�nitesimal conformaltransformation �K3 to our immersion, the Bloch functions of the deformedDirac operator are meromorphic on the same variety �[X; a; b] as the originalBloch functions and have the same multipliers w1, w2. Thus our deformationdoes not change �[X; a; b]. This completes the proof of Theorem 1.Example 1 Surfaces of revolution. Let be a closed non-sel�ntersectingcurve in the half-plane X2 = 0, X1 > 0 in R3. Rotating about the axesX1 = X2 = 0 we get a surface of revolution. It is always a torus with a �xedpair of periods.Such surfaces are essentially simpler from the soliton point of view. Po-tentials U(z; �z) corresponding to such surfaces depend only on one real vari-able x = Re z. Instead of the �xed energy spectral transform for the 2-dimensional Dirac operator we have the spectral transform for the 2�2 �rst-order matrix di�erential operator in one variable. Periodic direct spectral14

transform for such operators (for both �nite-gap and in�nite-gap potentials)was developed by one of the authors (M.S.) in [16].The MNV equations for surfaces of revolution are reduced to the well-studied Modi�ed Korteweg-de Vries equations (MKdV). In contrast with MNVall higher MKdV conservation laws are local in terms of the potential. Equa-tion (33) in this situation was integrated by V. K. Melnikov in [14] in theclass of potentials su�ciently fast decaying at in�nity. Our theorem for thesurfaces of revolution does not follow formally from [14] because the periodicMKdV theory and the decaying at in�nity one use di�erent technical tools.Nevertheless it is possible to simplify essentially our proof in this speci�ccase.5 Appendix. Modi�ed Novikov-Veselov equa-tions with periodic boundary conditions.In this Appendix we discuss the zero-energy spectral transform for the double-periodic Dirac operator (and the Generalized Weierstrass transform respec-tively) from the soliton point of view. This transform is naturally connectedwith a completely integrable hierarchy of integro-di�erential equations with2 spatial variables known as Modi�ed Novikov-Veselov hierarchy (MNV).Formally, the results of this Section are not used in our proof of Theo-rem 1, but we hope they allow the reader to gain a better understanding ofthe problem.There is a rather big amount of papers dedicated to the periodic prob-lem for soliton equations (see for example the textbook [21]). Nevertheless,the direct spectral transform for two-dimensional Dirac operator was neverstudied in such context in the literature known to us. Important propertiesof Bloch varieties for multidimensional Dirac operators were proved in [12],but they are not su�cient for the purpos of integrating the periodic MNVequations.From the point of view of the Bloch theory the two-dimensional double-periodic Dirac operator is rather similar to the two-dimensional double-periodic Schr�odinger operator. The �xed-energy direct spectral transformfor last one was constructed by soliton methods by I. M. Krichever [10]. Thisproblem was studied more detail by J. Feldman, H. Kn�orrer and E. Trubowitz15

(see [4]). In Section 5.1 we describe the structure of the Dirac Bloch varietyby methods analogous to [10]. It is important to remark, that in spite of thesimilarity between these two problems we have to overcome some additionaltechnical di�culties on this way. The asymptotic expansion of the Blochvariety gives us \quasi-local" MNV conservation laws.The zero-energy scattering problem for the two-dimensional Dirac oper-ator possesses an in�nite-dimensional algebra of symmetries, generated bythe MNV equations. They were constructed by L. V. Bogdanov in [2]. In [2]a generalization of the Miura transform was de�ned, and it was shown, thatthis transform maps the MNV equations to the Novikov-Veselov hierarchy(see [19]), associated with the �xed-energy two-dimensional Schr�odinger op-erator. MNV equations in the space of functions, decaying at in�nity wereintegrated by the so-called method of �@-problem in [2]. Periodic MNV theoryis discussed in Section 5.3.In contrast with the one-dimensional soliton systems, the two-dimensionalones essentially depend on the boundary conditions. To de�ne the periodicMNV hierarchy uniquely we have to �x some constants of integration. Oneof the simplest way to do it is to de�ne the MNV hierarchy in terms of theso-called Cauchy-Baker-Akhiezer (CBA) kernel. This kernel was introducedfor the Kadomtsev-Petviashvily hierarchy by A. Yu.Orlov and one of theauthors (P.G.) in [6] (see also [7]). MNV hierarchy in terms of the CBAkernel is discussed in Section 5.2.In this Section we always assume that U(z; �z) is real, smooth and double-periodic (8).5.1 Asymptotic structure of the Bloch variety.For large Imp1, Imp2 the structure of the Bloch variety can be studied bymethods of perturbation theory. Following [10] we start from the Dirac op-erator with zero potential U(z; �z) � 0. The corresponding Bloch variety isa union of two Riemann spheres �(0) = �(0)1 [ �(0)2 , �(0)1 = �(0)2 = CP1 with acoordinate � and~ (�; z; �z) = 10 ! e��z; � 2 �(0)1 ; ~ (�; z; �z) = 01 ! e�z; � 2 �(0)2 : (41)16

A pair �1 2 �(0)1 , �2 2 �(0)2 is called resonant ife�1��2 = 1; e�1����2� = 1; (42)and non-resonant otherwise. All resonant pairs are given by the followingformulas: �(m;n)1 = �mRe � � �nIm � + i�m; �(m;n)2 = ��(m;n)1 ; m; n 2 Z: (43)Let us call a point � 2 �(0)1 non-resonant if the pair � 2 �(0)1 and �� 2 �(0)2 isnon-resonant. The antiholomorphic involution �� maps �(0)2 to �(0)1 thus it issu�cient to develop a perturbation theory only on �(0)1 .Let ", R be some positive constants. Denote by �(0)";R the domain, obtainedfrom CP1 by removing " neighbourhoods of all resonant points �(m;n)1 and thedisk j�j � R.Lemma 2 For any " > 0 there exists a constant R(") such that in the do-main �(0)";R(") there exists an unique solution of the Dirac equation (4) withnormalization (12) such that~ (�; z + 1; �z + 1) = e�+h(�) ~ (�; z; �z);~ (�; z + �; �z + ��) = e���+h(�)� ~ (�; z; �z); (44)where h(�) is uniquely de�ned under the condition that h(�)! 0 as �!1.The functions ~ (�; z; �z), and h(�) are holomorphic in � in the domain �(0)";R(").A proof of this statement is to appear in a forthcomming paper by I. A. Taimanovand one of the authors (M.S.). It is rather long and rather technical. We donot want to present it in our text.Statement 2 The functions ~ (�; z; �z), h(�) de�ned in Lemma 2 posses thefollowing asymptotic expansions as �!1~ (�; z; �z) = e�(�z��z1)+h(�)(z�z1) 1 + �1(z;�z)� + �2(z;�z)�2 + �3(z;�z)�3 + �4(z;�z)�4 + : : :�1(z;�z)� + �2(z;�z)�2 + �3(z;�z)�3 + �4(z;�z)�4 + : : : !(45)h(�) = h1� + h3�3 + h5�5 + : : : (46)(the Bloch variety � has a symmetry � : (w1; w2)! (w�11 ; w�12 ), �(�) = ��,thus all even coe�cients in (46) are identically zero.)17

Unfortunately, at this moment we do not have any complete proof of thisStatement. It is rather clear how to do it, but this proof needs a rather bigpiece of asymptotic analysis, and we are not in a position to do it presently.But we know, that it is ful�lled at least in two important speci�c situations:1. U(z; �z) is an algebraic-geometrical (or, equivalently, �nite-gap) poten-tial. It means, that the normalization of the zero-energy Bloch varietyis algebraic (has �nite genus).2. U(z; �z) depend only on one real variable x = Re z. This fact was provedby one of the authors (M.S.) in [16]. Such potentials corresponds tosurfaces of revolution.We would like to remark, that in [4] a class of Riemann surfaces was intro-duced, which are in some sense similar to compact Riemann surfaces, andthe zero-energy level of a two-dimensional Schr�odinger operator belongs tothis class. Therefore it is natural to expect that the zero-energy level of thetwo-dimensional Dirac operator also belongs to this class.To de�ne the Modi�ed Novikov-Veselov equations (MNV) and their con-servation laws it is su�cient to have a formal solution of the Dirac equationin the form (45), (46). Let us show that such solution exists and is unique ifwe assume that all �k(z; �z), �k(z; �z) are bounded in the whole z-plane.Inserting (45) and h(�) = h1� + h2�2 + h3�3 + : : : (47)in (4) we get the following system of equations�1(z; �z) = �U(z; �z);�k(z; �z) = �@�z�k�1(z; �z)� U(z; �z)�k�1(z; �z); k > 1;@z�k(z; �z) = U(z; �z)�k(z; �z)� hk �Pk�1j=1 hj�k�j(z; �z): (48)We solve this system by induction. First we �nd �1(z; �z), then �1(z; �z), then�2(z; �z), then �2(z; �z) and so on. To �nd �k(z; �z) at each step we di�erentiatesome double-periodic functions, obtained at previous steps. Thus they arede�ned uniquely and are automatically double-periodic. To �nd �k(z; �z)we have to invert the operator @z in the space of functions, bounded in18

the whole z-plane (any bounded solution �k(z; �z) is automatically double-periodic). This is possible if and only if the mean value of the right-handside is equal to zero:<< U(z; �z)�k(z; �z) � hk � k�1Xj=1 hj�k�j(z; �z) >>= 0: (49)where: << F (z; �z) >>= Z 10 Z 10 dt1dt2F (t1 + � t2; t1 + �� t2): (50)Thus at each step we �nd hk from (49) and then calculate �k(z; �z). Thefunction �k(z; �z) is determined by the system (48) uniquely up to addingan arbitrary constant. This constant is �xed by the normalization condition(12).Let us check that the function h(�) does not depend on the normalizationpoint z1. If we change the point z1, then we change the integration constants.But these constants can be arbitrary shifted bymultiplying the whole solutionto a formal series in �" 1 + �1(z;�z)� + �2(z;�z)�2 + : : :�1(z;�z)� + �2(z;�z)�2 + : : : #!�1+�1� +�2�2 +: : :� " 1 + �1(z;�z)� + �2(z;�z)�2 + : : :�1(z;�z)� + �2(z;�z)�2 + : : : # :(51)This multiplication does not a�ect h(�).We have proved, that the constants h1, h3, h5, . . . are completely deter-mined by the potential U(z; �z) and all h2k = 0. Thus we have constructedan in�nite sequence of functionals h2k+1[U ]. The formulas for the �rst twoof them are: h1 = � << U2(z; �z) >> (52)h3 = � << U(z; �z)U�z�z(z; �z)� (U2(z; �z) + h1)V1�z(z; �z) >>; (53)where V1(z; �z) = @�1z (U2(z; �z) + h1): (54)Let us point out that adding an arbitrary constant to V1(z; �z) does not a�ecth3. We have de�ned an in�nite collection of functionals h2k+1[U ], k = 0; 1; 2; : : :.The following Statement explains why these functionals are so important.19

Statement 3 The quantities h2k+1[U ] are conservation laws for the wholehierarchy of the Modi�ed Novikov-Veselov equations (MNV).Assuming that Statement 2 is ful�lled we the prove this Statement at theend of the Section.It is well-known in the soliton theory that integrable systems with onespatial variable usually have in�nitely many local conservation laws. Formultidimensional soliton systems we normally have the opposite situation:almost all conservation laws are nonlocal. Let us brie y discuss the case ofthe MNV hierarchy.A functional Q[U ] is called local if it possesses the following representa-tion: Q[U ] =<< q(U;Uz; U�z; Uzz; Uz�z ; U�z�z; : : :) >> (55)where the density q(: : :) depend only on U(z; �z) and �nite number of itsderivatives. Of course h1[U ] is local. The next conservation law h3[U ] isnonlocal because the corresponding density depends on an auxiliary func-tion V1�z(z; �z), and to calculate V1�z(z; �z) we have to know U(z; �z) on thewhole z-plane. It it rather evident that higher functionals h2k+1[U ] are alsononlocal. This nonlocality creates no serious problems if the potential isdouble-periodic, but it is very di�cult to extend existing de�nitions to widerclasses of boundary conditions.In [10] perturbation theory was developed also in neighbourhoods of res-onant pairs. It can be shown that for su�ciently large � the surface � isobtained from the �(0) by attaching small handles to the resonant pairs. Wedo not use this fact, thus we do not want to discuss it now. However we usethe following property of �:Corollary 1 The Bloch variety � has two in�nite points, corresponding tothe points � =1 in �(0)1 and �(0)2 . We denote them by 1+ and 1� respec-tively.This statement follows immediately from Lemma 2.Further we shall use the following notation: �!1+, where � is a pointof �, always means that � tends to 1+ in the domain �(0)";R("); the notation�!1� always means that �� 2 �(0)";R(").20

5.2 Cauchy-Baker-Akhiezer kernel.To study symmetries of the soliton equation it is convenient to use theCauchy-Baker-Akhiezer kernel (CBA) (see [6]). In this Section we de�ne theCBA kernel on the zero-energy Bloch variety of the two-dimensional Diracoperator.To start with, suppose (�; �) is a pair of points in � such that1. ~ (�; z; �z) is non-singular at the points � = � and � = ��. (Let usrecall, that outside � = 1 the poles of ~ (�; z; �z) arose due to thenormalization (12) and do not depend on z and �z.)2. At least one of the following conditions is ful�lled:Imp1(�) � Imp1(�) 6= 0 or Imp2(�)� Imp2(�) 6= 0: (56)Then we may de�ne ~!(�; �; z; �z) by:~!(�; �; z; �z) = Z z1 d~!(�; �; z0; �z0); (57)whered~!(�; �; z0; �z0) = 2(�; z0; �z0) 2(��; z0; �z0)dz0 � 1(�; z0; �z0) 1(��; z0; �z0)d�z0(58)The integral in (57) is taken along some path in the z-plane such that1. connects the point z with 1.2. The form d~!(�; �; z0; �z0) decays exponentially as z0!1 along .Condition (56) guarantees the existence of such path.From the Dirac equation (4) if follows that the form d~!(�; �; z0; �z0) isclosed that the integral (57) is well-de�ned.The next step is to show, that for a �xed � our function ~!(�; �; z; �z) ismeromorphic in � on � outside the points 1+ and 1�.Suppose, that a pair (�; �) satis�es the following condition, which is muchweaker than (56).20. At least one of the following combinations12� (p1(�)� p1(�)) or 12�j� j (p2(�)� p2(�)) is non-integer, (59)21

or, equivalently, at least one of the following inequalities is ful�lledw1(�)w�11 (�) 6= 1; or w2(�)w�12 (�) 6= 1: (60)Assume that w1(�)w�11 (�) 6= 1. Then we have either~!(�; �; z; �z) = Z 0�1 ~!x(�; �; z + t; �z + t)dt (61)or ~!(�; �; z; �z) = � Z 10 ~!x(�; �; z + t; �z + t)dt; (62)where~!x(�; �; z0; �z0) = 2(�; z0; �z0) 2(��; z0; �z0)� 1(�; z0; �z0) 1(��; z0; �z0); (63)and t 2 R.If ���w1(�)w�11 (�)��� � 1 from (61) we get:~!(�; �; z; �z) == 0R�1 + �1R�2 + �2R�3 + : : :! ~!x(�; �; z + t; �z + t)dt == �w1(�)w1(�) + w21(�)w21(�) + w31(�)w31(�) + : : :� 1R0 ~!x(�; �; z + t; �z + t)dt == w1(�)w1(�)�w1(�) 1R0 ~!x(�; �; z + t; �z + t)dt (64)Similarly if ���w1(�)w�11 (�)��� � 1 from (62) we get:~!(�; �; z; �z) = w1(�)w1(�)�w1(�) 1R0 ~!x(�; �; z + t; �z + t)dt (65)The formulas (64) and (65) de�ne meromorphic continuations of the function~!(�; �; z; �z) to the whole surface �. To conclude the proof, it remains to notethat formulas (64) and (65) coincide.If w2(�)w�12 (�) 6= 1 then the integration path can be chosen along theline z0 = z + � t, and we obtain a new representation for the same function~!(�; �; z; �z): ~!(�; �; z; �z) = w2(�)w2(�)�w2(�) 1R0 ~!2(�; �; z + � t; �z + �� t)dt; (66)22

where~!2(�; �; z0; �z0) = � 2(�; z0; �z0) 2(��; z0; �z0)� �� 1(�; z0; �z0) 1(��; z0; �z0): (67)Now we are in position to de�ne the Cauchy-Baker-Akhiezer kernel!(�; �; z; �z): !(�; �; z; �z) = � 12� dp1(�)< ~!x(�; �; z; �z) >x ~!(�; �; z; �z) (68)where < ~!x(�; �; z; �z) >x= Z 10 ~!x(�; �; z + t; �z + t)dt: (69)It is a simple exercise to check that the function < ~!x(�; �; z; �z) >x does notdepend on z and is even in �, i.e.< ~!x(�; �; z; �z) >x=< ~!x(��; ��; z; �z) >x : (70)< ~!x(�; �; z; �z) >x= �1 + c(�)2�2 + c(�)4�4 + : : : ; �!1� (71)Lemma 3 The Cauchy-Baker-Akhiezer kernel !(�; �; z; �z) de�ned above hasthe following properties:1. For any �xed z, !(�; �; z; �z) is a meromorphic function of � and ameromorphic 1-form in � (both on the �nite part of �).2. For generic �, !(�; �; z; �z) has poles at the poles of ~ (�; z; �z) and at thepoint � = �. It is holomorphic outside these points at the �nite part of�.3. For a generic � and �! �!(�; �; z; �z) = 12�i d�� � � + regular terms; (72)or, equivalently, I�2S !(�; �; z; �z) = 1; (73)where S is a small contour, surrounding the point �.23

4. Let �!1+ and let � be a �xed point in �. Then we have the followingformal expansion:!(�; �; z; �z) = d�2�i 24 1(�; z; �z)� + 1Xk=2 R(+)k [@�z] 1(�; z; �z)�k 35 e���z�h(�)z;(74)where R(+)k [@�z] = @k�1�z + k�2Xl=0 v(+)kl (z; �z)@l�z (75)are di�erential operators in �z of order k� 1. The coe�cients v(+)kl (z; �z)do not depend on � and �, are di�erential polynomials of the asymptoticexpansion coe�cients �j(z; �z), �j(z; �z), j < k. Each function v(+)kl (z; �z)depends also on a �nite number of constants h2j+1, c(�)2j .Similarly for �!1�, we have:!(�; �; z; �z) = d�2�i 24 2(�; z; �z)� + 1Xk=2 R(�)k [@z] 2(�; z; �z)�k 35 e��z�h(�)�z :(76)where R(�)k [@z] = @k�1z + k�2Xl=0 v(�)kl (z; �z)@lz: (77)5. !(�; �; z + 1; �z + 1) = w1(�)w�11 (�)!(�; �; z; �z);!(�; �; z + �; �z + �� ) = w2(�)w�12 (�)!(�; �; z; �z): (78)Statement 4 If Statement 2 is ful�lled, then the CBA kernel has the fol-lowing additional properties:1. The formal expansions (74), (76) are asymptotic.2. Let f (+)(�; z; �z) be the following formal series in �f (+)(�; z; �z) = 8<: 1Xk=�N f (+)k (z; �z)�k 9=; e��z: (79)24

Then2�i res�����=1+!(�; �; z; �z)f (+)(�; z; �z) = 8<: 0Xk=�N f (+)k (z; �z)�k +O �1��9=; e��z;(80)as �!1+ and2�i res�����=1+!(�; �; z; �z)f (+)(�; z; �z) = O �1�� e�z; (81)as �!1�.Similarly if f (�)(�; z; �z) is the following formal series in �f (�)(�; z; �z) = 8<: 1Xk=�N f (�)k (z; �z)�k 9=; e�z; (82)then2�i res�����=1�!(�; �; z; �z)f (�)(�; z; �z) = 8<: 0Xk=�N f (�)k (z; �z)�k +O �1��9=; e�z;(83)as �!1� and2�i res�����=1�!(�; �; z; �z)f (�)(�; z; �z) = O �1�� e��z; (84)as �!1+.Remark 4 The formulas (80), (81), (83), (84) require some comments. Onthe left-hand side we have formal series in �, thus the analytic de�nition ofthe residue as an integral does not work. Fortunately the terms, containingexponents in � annihilate each other and we can de�ne the residue as thecoe�cient of 1=�: res�����=1 1Xk=�N ak�k d� = a1 (85)Proof of Lemma 3. 25

1. We have constructed ~!(�; �; z; �z) as a meromorphic function. The func-tion < ~!x(�; �; z; �z) >x is meromorphic in � on the whole � and dp1(�)is a meromorphic 1-form on �. Thus (68) gives us a meromorphicfunction with appropriate tensor properties.2. From Lemma 2 it follows that for generic �(a) ~ (�; z; �z) is nonsingular at the points � = �, � = ��,(b) < !x(�; �; z; �z) >x 6= 0(c) w1(�)w�11 (�) = w2(�)w�12 (�) = 1 i� � = �.If � ful�ll these conditions, if � 6= � and if ~ (�; z; �z) is nonsingular,then formulas (65), (66), (68), de�ne a nonsingular function of z, �z.3. Let �! �. To calculate the asymptotics of the CBA kernel we expandthe function w1(�) in (64) in a series in ��� using (13), (14). We see,that we have a �rst-order pole and the normalization in (68) is chosenso that the residue is exactly 1.4. To calculate the asymptotics of ~!(�; �; z; �z) as � ! 1 we substitutethe asymptotic expansion (45) to (57) and use the following asymptoticformula:Z z e��zf(�; z0; �z0)dz0 + e��zg(�; z0; �z0)d�z0 = e��z 1Xk=1(�1)k�1@k�1�z g(�; z; �z)�k ;(86)where the functions f(�; z0; �z0), g(�; z0; �z0) are some asymptotic seriesin � f(�; z0; �z0) = 1Xk=0 fk(z; �z)�k ; g(�; z0; �z0) = 1Xk=0 gk(z; �z)�k (87)such that @�z �e��zf(�; z0; �z0)� = @z �e��zg(�; z0; �z0)� ; (88)and all expansion coe�cients fk(z; �z), gk(z; �z) are smooth functions, allderivatives of these functions are bounded in the z-plane.The proof of the formula (86) is rather standard and we do not repro-duce it here. 26

5. The last statement of the Lemma follows directly from the transforma-tion rules for the Bloch functions.Proof of Statement 4. Assume for de�niteness, that � ! 1+. If � ! 1+then: !(�; �; z; �z) = 24 12�i(�� �) + Xi;j>0 !(++)ij (z; �z)�i�j 35 e(���)�zd�: (89)From (89) it follows, that2�i res�����=1+!(�; �; z; �z)f (+)(�; z; �z) == res�����=1+ d�� � �e(���)�zf (+)(�; z; �z) +O �1�� e��z: (90)Applying the well-known formula:res�����=1 d��� ��n = ( �n n � 00 n < 0 (91)we get the right-hand side of (80).If �!1�, then using (74) we get:!(�; �; z; �z) = 24Xi;j>0 !(+�)ij (z; �z)�i�j 35 e��z��zd�: (92)Formula (84) follows automatically from from (92).Formulas (83), (81) are proved exactly in the same way.Remark 5 A formula, representing the Cauchy kernel on a Riemann surfaceas a semi-in�nite sum of quadratic combinations of eigenfunctions �rst arosein the article [11] by I. M. Krichever and S. P. Novikov. In [11] the spatialvariable x was discrete. In [6] the spatial variable was continuous, and theCBA kernel was de�ned as an integral similar to (57).Remark 6 A form, similar to ~!(�; �; z; �z), but with integration path, start-ing from a �nite point Z in the z-plane, arises in the theory of �nite Darbouxtransformations (see [13], formula (6.1.18)). In fact, the same object arosein the Generalized Weierstrass map. We do not discuss this analogy in ourtext, but it seems possible that this analogy has some deep roots.27

5.3 Modi�ed Novikov-Veselov equations.Lemma 4 Let � be a generic point in �. De�ne a deformation �(�) of thefunction ~ (�; z; �z) associated with this point by:�(�) 1(�; z; �z) = ~!(�; �; z; �z) 1(�; z; �z) + �(�; �) 1(�; z; �z)�(�) 2(�; z; �z) = ~!(�; �; z; �z) 2(�; z; �z) + �(�; �) 2(�; z; �z); (93)where �(�; �) is a meromorphic function in �, uniquely �xed by the followingrequirement �(�) 1(�; z; �z) + �(�) 2(�; z; �z)���z=z1 = 0: (94)Then (93) is a deformation of the Bloch function, corresponding to the fol-lowing deformation of the Dirac operator�(�)L = " 0 ��(�)1 U(z; �z)�(�)2 U(z; �z) 0 # ; (95)where �(�)1 U(z; �z) = 1(�; z; �z) 2(��; z; �z)�(�)2 U(z; �z) = 2(�; z; �z) 1(��; z; �z): (96)For generic � these deformations result in non-self-adjoint Dirac opera-tors (�(�)1 U(z; �z) 6= �(�)2 U(z; �z)) with complex-valued potentials. But the theBloch function and the Bloch variety are well de�ned for such Dirac opera-tors.Proof of Lemma 4. A simple direct calculation shows, that�L + ��(�)L� �~ (�; z; �z) + ��(�)~ (�; z; �z)� = O(�2); (97)Thus (93) de�nes deformations of the eigenfunctions, corresponding to (95).From (78) it follows that the new eigenfunctions satisfy (11) with the samemultipliers w1(�), w2(�) as the old ones. Thus the new eigenfunctions arede�ned on the same curve � and have the same periodicity properties. Thelast property plays the key role, when we prove below that the functionalsh2k+1[U ] are invariant under some deformations.The deformations of the Dirac operator, generated by all �(�) form a linearspace. Deformations preserving the class of self-adjoint Dirac operators withreal potentials are the most interesting ones. Let us check that the subspaceof such deformations is su�ciently large.28

Lemma 5 Denote by �(�) the following linear combination of deformations�(�): �(�) = �(�) + �(��)< ~!x(�; �; z; �z) >x + �(��) + �(���)< ~!x(��; ��; z; �z) >x (98)(recall that < ~!x(�; �; z; �z) >x is an even function in � (see (70)) and doesnot depend on z, �z.)Then �(�) acts on the space of self-adjoint Dirac operators with real po-tentials, i.e.�(�)1 U(z; �z) = �(�)2 U(z; �z) = �(�)1 U(z; �z) = �(�)2 U(z; �z) == 1(�;z;�z) 2(��;z;�z)+ 1(��;z;�z) 2(�;z;�z)<~!x(�;�;z;�z)>x + 1(��;z;�z) 2(���;z;�z)+ 1(���;z;�z) 2(��;z;�z)<~!x(��;��;z;�z)>x (99)and the Bloch variety �[U ] de�ned in Section 3 is invariant under thesedeformations.Proof of Lemma 5. Formula (99) follows directly from (96) and (98). TheBloch function ~ (�; z; �z) for real U(z; �z) has symmetry property (17), thusthe right-hand side of (99) is real. As we pointed out in the previous Lemma,all deformations generated by �(�) do not change the Bloch variety �[U ] andthe multipliers w1, w2. This completes the proof. If Statement 2 is ful�lled,then these ows do not change h2k+1[U ].Equations (99) are essentially nonlocal and rather complicated. Namelythe right-hand side is expressed in terms of the Bloch function, and it isdi�cult to calculate Bloch solutions either analytically or numerically. For-tunately the space of deformations generated by all �(�) contains simplerequations such that the right-hand side can be expressed via U(z; �z) in termsof quadratures.Let � ! 1. If Statement 2 is ful�lled we may expand �(�) to thefollowing asymptotic series�(�)U(z; �z) = �2 d�idp(�) 1Xk=0 K2k+1[U ]�2k+2 + K2k+1[U ]��2k+2 : (100)(We write the term d�=dp(�) to gain some standard normalization of theMNV hierarchy. If we omit this multiplier, our expansion coe�cients will be29

linear combinations of Novikov-Veselov generators with constant coe�cients,which in most situations is not essential).Any K2k+1[U ] is a quadratic polynomial of �l(z; �z), �l(z; �z), l = 1; : : : ; 2k,with coe�cients depending on the h2l�1 and c2l, l = 1; : : : ; k, where the c2lare coe�cients of the asymptotic expansion (71).For any odd integer 2k + 1, k � 0 we have the following pair of ows onthe space of real double-periodic functions:@U(z; �z; t2k+1)@t2k+1 = 2ReK2k+1[U ]: (101)@U(z; �z; ~t2k+1)@~t2k+1 = 2 ImK2k+1[U ]: (102)De�nition 1 The equations (101), (102) are called theModi�ed Novikov-Veselov equations (MNV).Remark 7 To de�ne the MNV ows associated to (101), (102) it is suf-�cient to have a formal expansion for the Bloch function and the functionh(�). Thus these ows are well-de�ned for any smooth potential. But withoutStatement 2 we can not prove that they conserve the functional h2k+1[U ].Using this de�nition of the MNV hierarchy we may immediately proveStatement 3. All functionals h2k+1[U ] de�ned in Section 3 are integrals ofmotion for all ows �(�), thus they are conservation laws for their expansioncoe�cients.The representation (101) may look rather unusual. To check that ourde�nition of the MNV equations coincides with the standard one, let uscalculate the deformation of the Bloch function, corresponding to the ows(101). To gain a standard answer we shall use a normalization of the Blochfunction di�erent from th eone used earlier. Instead of (94) we assume thatthe function �(�; �) in (93) is identically zero.Statement 5 Let the Statement 2 be valid. Then the deformation of theBloch function corresponding to the ows (101) has the following represen-tations: @ ~ (�; z; �z; t2k+1)@t2k+1 =30

= 2�i( res�����=1++res�����=1�)!(�; �; z; �z; t2k+1; )�2k+1 ~ (�; z; �z; t2k+1); (103)= �2k+1 ~ (�; z; �z; t2k+1) +8<: O � 1�� e��z as �!1+O � 1�� e�z as �!1� (104)= (@2k+1z + @2k+1�z + 2kXl=0Wl(z; �z)@lz +Wl(z; �z)@l�z) ~ (�; z; �z; t2k+1); (105)Proof of Statement 5, given the validity of Statement 2. By de�nition�idp(�)2 �(�) ~ (�; z; �z) == �i h!(�; �; z; �z)~ (�; z; �z) + [!(�;��; z; �z)~ (��; z; �z)i�!1+ ++�i h!(�; ��; z; �z)~ (��; z; �z) + [!(�;���; z; �z)~ (���; z; �z)i�!1� (106)(in this formula we use �� = ��, �� = ���).The residues in (103) are exactly the coe�cients of the terms d�=�2k+2and d��=��2k+2 respectively. Thus these coe�cients gives us the action ofK2k+1[U ] and K2k+1[U ] on the Bloch function.Formula (104) follows directly from (103) and formulas (80)-(84).To prove (105) let us substitute the asymptotic expansions (74), (76) in(103). We get: @ 1(�; z; �z; t2k+1)@t2k+1 == (R(+)2k+1[@�z] + 2kPl=0�2k+1�l(z; �z)R(+)l [@�z]) 1(�; z; �z; t2k+1)++( 2kPl=0�(�)2k+1�l(z; �z)R(�)l [@z]) 2(�; z; �z; t2k+1) = (107)= @2k+1�z 1(�; z; �z; t2k+1) + U(z; �z)@2kz 2(�; z; �z; t2k+1) + lower order terms =(108)= n@2k+1�z + @2k+1z o 1(�; z; �z; t2k+1) + lower order terms; (109)(the �(�)l (z; �z) denote the expansion coe�cients of the function 1(�; z; �z) atthe point � =1�). A similar calculation shows the following relation:@ 2(�; z; �z; t2k+1)@t2k+1 = n@2k+1�z + @2k+1z o 2(�; z; �z; t2k+1) + lower order terms:(110)31

Statement 5 is proved.From (105) it follows that the MNV equations are the compatibility con-ditions for a pair of di�erential operators on the space of zero eigenfunctionsof L. It is well-known in soliton theory that such compatibility conditionsare equivalent to the existence of standard L � A � B representations (see[2]).Remark 8 In this article we constructed a soliton hierarchy in terms of theCauchy-Baker-Akhiezer kernel.Using the CBA kernel we construct, in fact, a much wider hierarchy, in-cluding essentially nonlocal equations. All these equations preserve the spec-tral curve. In Section 4 we show that in fact the deformations correspondingto conformal transformations of the Euclidean space, lie in this wider hier-archy.In terms of U(z; �z) this hierarchy looks rather unnatural. But we maytreat it as a system of di�erential equations on a bigger collection of functionssimultaneously; namely we may consider the potential U(z; �z) and the wavefunction in a �nite number of �xed points on the spectral curve as unknownfunctions, connected by the Dirac equations. Similar systems associated withone-dimensional soliton equations were discussed in the literature (see [14]and references therein) from both a mathematical and a physical point of view.In [15] it was shown, that starting from the KP equation we get a hierarchynaturally containing many other well-known soliton systems.References[1] A. I. Bobenko, Surfaces in terms of 2 by 2 matrices. Old and new inte-grable cases. in: Harmonic Maps and integrable systems, ed A. P. Fordy,Vieweg, 1994 83-127.[2] L. V. Bogdanov, Veselov-Novikov equation as a natural two-dimensionalgeneralization of the Korteveg-de Vries equation, Theoret. Math. Phys.70 (1987), 219-223.[3] B. A. Dubrovin, A. T. Fomenko, S. P. Novikov, Modern Geome-try { Methods and Applications. Part 1. The Geometry of Surfaces,32

Transformation Groups and Fields, Springer-Verlag, New York-Berlin-Heidelberg-Tokyo, 1984.[4] J. Feldman, H. Kn�orrer, E. Trubowitz, Riemann surfaces of in�nitegenus. Part III, Preprint ETH, Z�urich.[5] E. V. Ferapontov, Laplace transformations of hydrodynamic type sys-tems in Riemann invariants, Teor and Math. Physika 110 (1997) 86-97.[6] P. G. Grinevich, A. Yu. Orlov, Virasoro action on Riemann sur-faces, Grassmannians, det �@ and Segal-Wilson � - function, in Prob-lems of modern quantum �eld theory, ed. A. A. Belavin, A. U. Klimyk,A. B. Zamolodchikov, Springer-Verlag, 1989, 86-106.[7] P. G. Grinevich, A. Yu. Orlov, E. I. Schulman, On the symmetries ofthe integrable systems. in: Important developments in soliton theory, ed.A. Fokas, V. E. Zakharov, Springer-Verlag, Berlin - Heidelberg - NewYork, 1993, 283-301.[8] G. Kamberov, F. Pedit, U. Pinkall, Bonnet pairs and isothermic surfaces,preprint (dg-ga/9511005), to appear in Duce Math. Journal.[9] B. G. Konopelchenko, I. A. Taimanov, Generalized Weierstrass formu-lae, soliton equations and Willmore surfaces. I. Tori of revolution andthe mKdV equation. Preprint Ruhr-Universit�at, Bochum, 1995.[10] I .M .Krichever, Spectral theory of two-dimensional periodic operatorsand its applications, Russian. Math Surveys, 44:2 (1989), 145-225;I. M. Krichever, Perturbation theory in periodic problems for the two-dimensional integrable systems. Soviet Sci. Rev., Math. Phys. Rev., Gor-don and Breach, Harwood Academic Publishers. 9 (1992).[11] I. M. Krichever, S. P. Novikov, Virasoro-type algebras, Riemann surfacesand strings in Minkovsky space. Funct. Anal. Appl. 21 (1987), 294-307.[12] P. Kuchment. Floquet theory for partial di�erential equations.Birkh�auser 1993.[13] V. B. Matveev, M. A. Salle, Darboux transformations and solitons,Springer-Verlag, 1991. 33

[14] V. K. Melnikov. New method for deriving nonlinear integrable systems.J.Math.Phys. 31, No 5 (1990), 1106-1113.[15] A. Yu. Orlov, Volterra operators algebra for zero curvature represen-tation. Universality of KP, in: Nonlinear (and turbulent) processes inphysics. Editors: A. Fokas, D. Kaup, A. Newell, V. E. Zakharov, 126-131, Springer seria in nonlinear dynamic 1991; A. Yu. Orlov, Sym-metries for unifying di�erent soliton systems into a single hierarchy",Preprint IINS/Oce-04/03 (March 1991); B. Enriquez, A. Yu. Orlov,V. N. Rubtsov, Dispersionful analogues of Benney's equations and N -wave systems, Inverse Problems 12 (1996), 241-250.[16] M. U. Schmidt. Integrable systems and Riemann surfaces of in�nitegenus. Memoirs of the American Mathematical Society, 551 (1996).[17] I. A. Taimanov. Modi�ed Novikov-Veselov equation and di�erential ge-ometry of surfaces, preprint, November 1995 (dg-ga/9511005), to appearin Translations of the American Mathematical Society.[18] I. A. Taimanov. Surfaces of revolution in terms of solitons (dg-ga/9610013).[19] A. P. Veselov, S. P. Novikov, Finite-zone, two-dimensional potentialSchr�odinger operators. Explicit formulas and evolution equations. So-viet Math. Dokl., 30 (1984) 588-591; A. P. Veselov, S. P. Novikov,Finite-zone, two-dimensional potential Schr�odinger operators. Potentialoperators. Soviet Math. Dokl., 30 (1984) 705-708;[20] T. J. Willmore, Riemannian geometry, Claderon Press 1993.[21] V. E. Zakharov, S. V. Manakov, S. P. Novikov, L. P. Pitaevsky, SolitonTheory, New York: Plenum 1984.34