Affine spheres and finite gap solutions of Tzitzèica equation

The purpose of the present paper is to give an explicit form of the finite gap solutions to the Tzitzèica equation (2D Toda equation of type A 2 ( 2 ) ) in terms of Riemann theta function. We give explicit expressions of proper affiene spheres derived from finite gap solutions to the Tzitzèica equation.


Introduction
This paper concerns with the following 2 dimensional Toda lattice (2DTL) of type A 2 2 ( ) : Nowadays, this 2DTL is called the Tzitzèica equation named after a Romanian mathematician G.Tzitzèica.
In addition, Mikhailov [3] pointed out that Tzitzèica equation is obtained as a reduction of periodic 2DTL = -= . This reduction corresponds to the affine root system A 2 2 ( ) (or the reduced root system BC 1 ) [4]. Note that this reduction was already known in projective differential geometry, see [5].
Early year in the twenty century, Tzitzèica [6] studied non-degenerate surfaces, that is, surfaces with nondegenerate second fundamental form in the Euclidean 3-space E 3 with the property that K d 4 is constant over the surface, where K is the Euclidean Gaussian curvature function and d is the Euclidean distance function from the origin of E 3 to each point of the surface. Tzitzèica noticed that K d 4 is invariant under equiaffine transformations of E 3 .
Tzitzèica's observation initiated affine differential geometry of surfaces. The surfaces discovered by Tzitzèica are now refereed as proper affine sphere due to the fact that a proper affine sphere is the set of points where the affine distance from the origin is non-zero constant. In fact the affine distance function d from the origin satisfies K d d 1 4 4 = . On every proper affine sphere, there exists a unique (up to sign) semi-Riemannian metric invariant under equiaffine transformations of E 3 . Such a metric is called the Blashcke metric. Note that the Blaschke metric of a proper affine sphere is conformal to the Euclidean second fundamental form. There are two classes of proper affine spheres and they are said to be indefinite or definite according as the non-degenerate affine metric is indefinite or definite, respectively. In this paper, we study indefinite proper affine spheres.
The Gauss-Codazzi equations for indefinite proper affine spheres are described by the Tzitzèica equation, where h e dx dt u = is Blaschke metric on M and the affine mean curvature is normalized to −1. Tzitzèica equation arises not only in affine differential geometry but also in many other realms of mathematical physics and differential geometry. For instance elliptic versions of Tzitzèica equation are integrability conditions of the following three kinds of surfaces: (1) Lagrangian minimal surfaces in complex projective plane (cf. [7]), (2) Lagrangian minimal surfaces in complex hyperbolic plane [8] and (3) affine spheres with positive definite Blaschke metric [9]. These elliptic versions are closely related to certain Kähler-Einstein metrics [10].
For the geometry of surfaces in 3-dimensional spaces using the soliton theory we refer to [11] and [12]. More generally, for the close relation between minimal surfaces (nonlinear sigma models) in symmetric spaces and 2DTL, for example, see [13,14] and [15].
Some explicit solutions of Tzitzèica equation (1.1) are known. For example, the hexenhut and the Jonas Kelch are realized using the solution of u 0 = and u x t log 1 3 2 cosh 3 2 2 = -+ ( ( ( ( ) ) ), respectively. Moreover, the finite gap solutions of (1.1) may be described by Cherdantsev and Sharipov in [16] as follows : where θ is the Prym-theta function for some compact Riemann surface of genus g 2 and the constant C is determined by some spetral data. They also gave the Baker-Akhiezer function in terms of the spectral data (see [16]). On the other hand, we may expect the existence of the solution of (1.1) in terms of the Jacobi elliptic functions because the elliptic version of (1.1) is solved in terms of the Jacobi elliptic function by Castro-Urbano (see [17]).
In this paper, first of all, we construct the solution of (1.1) in terms of the Jacobi elliptic functions. However, it is not clear what kind of the spectral data produces the solution in terms of the Jacobi elliptic functions as a special case of the finite gap solutions. Therefore, we explicitly give the spectral data for the solution in terms of the Jacobi elliptic functions. Moreover, we can describe the finite gap solutions more explicitly in terms of Riemann theta function. Precisely speaking, we can give the constant C in (1.2) explicitly (see (5.8), (5.9) and theorem 5.5). Moreover, we can give Abelian differentials of second kind on the spectral curve explicitly (see (5.7)) and consequently we obtain a real frame for indefinite proper affine spheres in terms of the Baker-Akhiezer function.
For our purpose stated above, first of all, in section 4, we deduce the solution of the Tzitzèica equation and gives a Blaschke immersion of indefinite proper affine spheres in terms of the Jacobi elliptic functions. Therefore, the well known elliptic function theory hides in the background. We employ the elliptic function theory and reconstruct the whole theory for solutions of the Tzitzèica equation in terms of the Riemann theta function (see (4.17)), and for Blaschke immersions of indefinite proper affine spheres in terms of the meromorphic function Ŷ which is a solution of the Schrödinger equation e x t u ¶ ¶ Y = Ŷˆ(see theorem 4.9).
These reconstructions give a nice story for a general case of the corresponding problems. Since the elliptic curve used in section 4 is the Prym variety of the spectral curve of genus 2, we must consider the spectral curve of genus g 2 and the g-dimensional Prym variety for the general case. The standard argument in the integrable systems and the result in section 4 help us to reconstruct the solutions of the problems for the general case. In section 5, we get the solutions of the Tzitzèica equation and give expressions of the immersions of indefinite proper affine spheres of finite type from the points of views of the algebro-geometric approach to the integrable systems.

Indefinite proper affine spheres 2.1. Fundamental equations
We start with description of affine spheres in equiaffine differential geometry. For more details we refer to the textbook [18] by Nomizu and Sasaki. Let R 3 be a Cartesian 3-space. We denote by R R R R det: 3 3 3´⟶ the determinant function, which defines a volume element of R 3 . Then the triplet D A R, , det 3 3 = ( )is an equiaffine 3-space, i.e., it satisfies D det 0 = , where D is the canonical covariant differentiation for R 3 . An immersion M A : ⟶ of an oriented 2-manifold is said to be an affine immersion if M, y ( )admits a complementary subbundle  of the tangent bundle TM of M in the pull-back bundle In the following, we assume that there is a global transversal vector field ξ to M, y . We then call a triplet M, , y x ( ) an affine immersion. For any vector fields X and Y on M we have the Gauss formula We easily verify that  defines a torsion free linear connection on M and h is a symmetric tensor field on M. The symmetric tensor field h is called the affine fundamental form derived from ξ. We also have the Weingarten formula The endomorphism field S of TM is called the affine shape operator with respect to ξ. The 1-form  is called the transversal connection form. For example, an immersion M A : 0 3 y ⟶ ⧹{ }with the property that the position vector field ψ is transversal to M is an affine immersion and in particular it is called centro-affine immersion. For an affine immersion M, , y x ( ), we may define a volume element ϑ on M by X Y X Y , det , , J x = ( ) ( )for any vector fields X and Y on M. We address here some known facts about affine immersions.
For an affine immersion, the rank of the affine fundamental form is independent of the choice of the transversal vector field.
Ricci equation Here R is the curvature tensor field of the connection . Let M, , y x ( ) be a proper affine sphere. Then all the Blaschke normals meet in one point.

Tzitz`eica equation
In the following, we consider proper affine spheres with indefinite Blaschke metric h. In particular, for such proper affine spheres we have is the affine mean curvature and it is a negative constant. By scaling the affine metric, without loss of generality, we may assume that H 1 = -, hence we may The compatibility condition of the system (2.2) is given by The first equation is the Gauss equation and the last two equations are Coddazi equations. Throughout this paper we assume that indefinite proper affine spheres are weakly regular, that is, AB 0 ¹ [20]. Then the Coddazi equation means that A A x = ( ) and B B t = ( ). Therefore, choosing the coordinate system x t , ( )we may assume that the cubic form is represented as C dx dt 3 3 = + from the first step. In this case, we have A B 1 = = and (2.3) is called the Tzitzèica equation [6].
respectively. We denote by  n ( ) the solution to the system  Before closing this section, we state the assumption on u throughout the paper. We assume that u x t u t x , , = ( ) ( )with respect to the coordinate system x t , ( )so that the cubic form is given by C dx dt 3 3 = + . Of course, this assumption is independent of introducing the non-zero parameter ν.

Twisted loop algebras and twisted loop groups
)be a real special linear group of degree 3 and R 3, = ( ) g sl its Lie algebra. We denote by C 3, C = ( ) g sl the complexification of g. Define two automorphisms σ and γ of C g by Ad ) . We then see that t sg gs = = is an automorphism of order 6 [20]. Let which is a Lie subalgebra of g and is isomorphic to 1, 1

Twisted loop algebras
Let ŝ be an involution on C g defined by f o r a l l a n d , . Then, C L t g is a Banach Lie algebra (see [21,22]). Set We may write ξ as We now define the following Banach Lie subalgebras of L t g : , It then follows from (3.1) that which is a vector space decomposition into Banach Lie subalgebras. For a positive integer d 1 mod 6 º ( ), we define the vector subspace d L ŝ of L s t g by

Twisted loop groups
Let G L t be a twisted loop group defined by G g S G g g g g S g : , f o r a l l a n d whose Lie algebra is L t g . Here we denote the Lie group automorphism corresponding to τ by the same letter.
. We extend the involution ŝ to the above map by the rule , 3.4. Affines spheres of finite type Now we return to indefinite proper affine spheres. Let D R A : 2 3 y Ì  be an indefinite proper affine sphere parametrized by global asymptotic coordinates x t , ( )as before. The coordinate extended framing F l ( )can be extended analytically on C*. The (extended) map F l ( )is uniquely determined by the values on S C 1 * Ì (cf. [20], p. 234). Hence F l ( ) is regarded as a map into G L t . In addition, since we assumed that u x t u t x , , ( ). Therefore, the coordinate extended framing F l ( )can be considered as a map F G D * Î L s t ( ) . Moreover, since ⟶ . Here we give the following definition (compare with [20,23] * a s a = -ˆ( ) is said to be an extended framing for indefinite proper affine sphere. Next we introduce the following notion for affine spheres.
Definition 3.4. An indefinite proper affine sphere is said to be of finite type if its extended framing a is obtained from the following differential equation under certain initial condition:  Take a matrix A Hence the resulting surface 0 y is affine congruent to the hexenhut Z X Y 1 Figure 1). The hexenhut 0 y corresponds to the trivial solution u 0 = to the Tzitzèica equation [25].
Remark 3.6. One can establish the Symes method (also called AKS-scheme) for constructing indefinite proper affine sphere of finite type. This scheme was established in a separate publication [26].

Blaschke immersions in terms of elliptic functions
In this section, we give a solution of Tzitzèica equation (1.1) in terms of elliptic functions and represent affine spheres in terms of elliptic functions. Although these formulas have been known in some literature, our purpose here is to represent them in terms of Riemann theta function. This achievement gives a nice story for the description of the solutions in terms of the spectral curves of higher genus and Prym-theta functions.
Introducing new coordinates x, yˆby We now assume that a solution u depends only on the parameter yˆand write u u y = (ˆ). We suppose that an initial condition e u , 0 0 (ˆ) is the Jacobi sn-function with modulus p. Therefore, we obtain a solution of (1.1) as follows.
The function Y y (ˆ) can be extended as a function of complex variables. Extending yˆas a complex valued function, we see that Y y (ˆ) is a doubly-periodic function with the periods 2 , 2

The solution of this equation is given by
Weierstrass Ã-function z Ã( ), which is related to the Jacobi sn-function by z z k sn , , Since we know that k p = ¢ and k p ¢ = , the above equation yields the following.
is also a solution of (1.1). The Weierstrass z Ã( )-function is defined by The sum of the right hand side is absolutely and uniformly convergent on a compact subset of C. It is an elliptic function.
On the other hand, rewriting where c 1 , c 2 and c 3 are non-zero real numbers with c c c 2 .
This is a rotational surface as an orbit of the group SO 1, 1 ( )of hyperbolic rotations (Lorentz boosts) up to affine transformations. When α tends to 2, the solution converges to a trivial solution u 0 = .
Remark 4.2. Analogously, we may get a solution of (1.1) which depends only on the variable x as follows. The corresponding affine sphere is obtained as follows.
herefore, this is a rotational surface as a orbit of the rotational group SO(2) of elliptic type up to affine transformations. When α tends to 2, the above solution converges to the 1-soliton solution of Tzitzèica equation. The rotational surface corresponding to the 1-soliton is called the Jonas Kelch ( Figure 2, see also [25]).

4.2.
In the rest of the section 4, we represent the formulas (4.1) and (4.6) in terms of the Riemann theta function.
We consider an elliptic curve  defined by B 4 . We may choose a cycle a b , 1 1 { } of  so that the following holds : , then we easily see that Then, the elliptic curve has a Riemann period matrix = -, so that the following relations hold: Thus, the curve 0  has a Riemann period matrix It is a meromorphic function on C and a odd function, but not an elliptic function. In fact, we have the following properties : In the sequel, we keep on using this notation when there is no confusion. The last equation in (4.10) follows from the 2nd and 3rd equations in (4.10) and from the fact . Therefore, 0 W ¥ is the normalized Abelian differential of 2nd kind.
which absolutely and uniformly converges on a compact subset of C. The Jacobi theta functions are described in terms of the Riemann theta function as follows (see [27] and [28] [28]). Although the Jacobi theta functions are not elliptic functions, there are some relations between the Jacobi theta functions and the Jacobi elliptic functions as follows.
Moreover, the following beautiful formula is known is the complete elliptic integral of 2nd kind. We now calculate where we have used (4.2) and p 2 which is a solution of (1.1) (see (4.1)). On the other hand, for e 1 p = we see that Thus, we obtain the following formula.

Expression of Blaschke immersions in terms of Riemann theta function
We gave a formula of Blaschke immersion in terms of elliptic functions in (4.6). In this section, we rewrite the formula (4.6) in terms of Riemann theta function. For this purpose, we consider the elliptic curve 0  as a Prym variety of a compact Riemann surface  of genus 2. When u u y = (ˆ) is given by (4.1), the spectral curve is defined by the equation , where U l ( ) and V l ( ) are as in (2.5). Since the spectral curve is independent of the parameter yˆ, if we take y 0 = then the initial condition of u u y = (ˆ) implies that the defining equation of the spectral curve is , , , old (see Figure 3). Let w w , It is well known that the Jacobian variety J  (ˆ) of  is a complex for j 1 = , 2. The Riemann bilinear relation means that C 0 11 ¹ . Thus, we find w w , 1 2 : We have the relations as follows.
We may consider the function f on  defined by f P P e  q = -(ˆ) ( (ˆ) ) for e C Î , where θ is the Riemann theta   [29]). If f P P e 0  q = -(ˆ) ( (ˆ) ) ≢ then the zeros of f is a degree 2 divisor  . Moreover, we have K e 2 mod   º + G (ˆ) ( ) , where K is given by and b 0 ĵ ( ) is the initial point of the path b ĵ in the boundary 0  ¶ˆ.
This is, of course, a special one of the higher genus case, which is stated in the next section. Fay states the property of the divisor  in terms of the Abelian map J :    ⟶ (ˆ). Therefore, we here address the outline of the proof. When P a b resp. j j Îˆ(ˆ), we denote by P the corresponding point of a b resp. j j If f is not identically zero, then the number n of the zeros of f is given by Using the same calculation as in above, we obtain (1) Ŷ is a meromorphic function on P P , 0  ¥ ⧹{ˆˆ} and the divisor of the poles is nonspecial and given by p p , {ˆˆ} which is independent of the parameters x and t, (2) Ŷ has the following asymptotic expansions. n n -ˆ, respectively. Since the residue of w Ŷ is zero, it follows from the condition (1) that there are two points of zeros, which are denoted by q x t , 1 ( )and q x t , 2 ( ). Therefore, w Ŷ may be described as follows.
which is nothing but the consequence of the reciprocity law for differentials of 1st and 2nd kinds.
The integration of (4.22) over the cycle b ĵ and the reciprocity law gives . We then assume that z z z , , . We denote by P 1 , P 2 , P 3 the points on  corresponding to 1 h , 2 h , 3 h , respectively. Precisely, P , 1 ). We choose a path γ from P 0 to P ¥ as in Figure 3 and we fix a path from P 0 to P 1 in the following. Since we have w w We put here e C Î is chosen so that f P 0 (ˆ) ≢ and the divisor of the poles of F q is p p , is as that in (4.26), and we fix the path from P 0 to P 1 .
Moreover, using the expression of Ŷ we see that the following reality condition holds: x t P t x  P  x t  P  e  e  , , , , , , , , As we see later, we may prove that Ŷ satisfies the Schrödinger equation ) then we write down the system of differential equations which we must solve as follows.
{ˆˆ} is a solution of the system of differential equations (4.29) above.
Set e e , , It follows from (4.29) that Ŵ is independent of the parameters x and t. Thus, we may write W x t P W P , , = (ˆ)ˆ(ˆ). We will arrive at the following conclusion.
which is a solution of (2.1), where c c c Proof. Proof. Since Ŷ is single-valued, we carry out the calculations of integrations using the paths b a , , 1 1 g + + +b y the parts of the disjoint unions b b b a a a , , {ˆˆˆ} be the points of  as above, which are also points of 1 n = . It follows from (4.10) and (4.11) that

4.32
) by (4.2) and (4.12). We now set u y 2 2 1 z z = -ˆ. It then follows from (4.13), (4.14), (4.31) and (4.32) that x t x t Take a function C P (ˆ) on  and define Ỹ by where C P C P C P P 1, , , . Let P  be the point near P 2 along a 1+ .
We may write P a We define W using Ỹ as well as the way we defined Ŵ using Ŷ . We then see that Here, we note that We set e u t 1 x = ¶ˆ. We then see that e 0 as 0.
x t u 1 ince the poles of Ŷ are independent of the parameters x t , , we see that the poles of e , , if 0 Ŷ ≢ then it follows from lemma 4.5 that the zeros of Ŷ are the degree 2 divisor q x t q x t , , , {ˆ( )ˆ( )}. We then have q x t q x t H , , , {ˆ( )ˆ( )}is a non-special divisor at the origin x t , 0,0 = ( ) ( ), it remains non-special near the origin. Thus, F must be a constant on  . Evaluating it at Finally, we show that the e u above coincides with one in (4.17). Near P 0 , we may write 1 2 (4.18). Differentiating log Ŷ by n and setting 0 n = , we obtain from lemma 4.7.
where C x 0 ( ) is a function of the parameter x only. It then follows from e u t 1 . It follows from the reciprocity law for 1st and 2nd kinds that

Blaschke immersions of finite type in terms of Prym-theta functions
Let d 1 mod 6 º and x t D , , ].
For λ, C m Î , the spectral curve is defined by the equation . However, since a Ad 1 x x l = -( ) ( )for some initial data d x l Î L s ( )ˆ, we see that the spectral curve is independent of the parameters x, t and given by the equation , which becomes trace det where we have expressed d as d k 6 1 = + . Therefore, we see that the spectral curve has d 12 branch points. Projectivizing this affine plane curve, we have a compact Riemann surface , whose genus is given by d ). A three-fold covering map given by , , , , . The curve  also admits the involutions σ and ρ. We assume that  has no branch points over S 1 n Î . We denote by P n ( ) and Q n ( ) the corresponding Laurent polynomials obtained by setting We find the properties u u u u , We then see that w w , , ˆ. Moreover, define two matrices ij P = P ( )and T T ij = ( ) where e e , , d 1  is a standard basis of C d and z z z , , ) for e C d Î . The method similar to those in section 4.4 yield the following.
Lemma 5.1 (cf. [29]). If f P P e 0  q = -(ˆ) ( (ˆ) ) ≢ for some e C d Î then the zeros of f is a degree g ) and each K i is given by ) then we write down the system of differential equations which we must solve as follows. (1) Ŷ is e meromorphic function on P P , 0  ¥ ⧹{ˆˆ} and the divisor of the poles is given by which is independent of the parameters x and t, (2) Ŷ has the following asymptotic expansions. , ) . We choose three points P P P , , {ˆˆˆ} on  , which are expressed as P , 0 , here e C Î is chosen so that f P 0 (ˆ) ≢ and the divisor of the poles of F q is p p p , , , g where x t P , , e F (ˆ) is as that in (5.10), and we fix the path from P 0 to P 1 .
For our chosen canonical homology basis of  in Figure 4, we decompose a ĵ and b ĵ into the disjoint unions a a a j j j where a j+ is the part of a ĵ lying on Im 0  m ( ) and b j+ is the part of b ĵ lying on the upper sheet over 0 n =  . We also decompose γ, which is the fixed path from P 0 to P ¥ , into the disjoint union È g g g = + -, where g + is the part of γ lying on the upper sheet over 0 n =  and it is nothing but the fixed path from P 0 to P 1 .
We here prepare the following lemma. where σ is the involution of  . We may verify that the following reality condition of Ŷ holds : x t P x t P e e , , , , , , .
We define W x t P , , (ˆ) as in (4.30). We then see from (5.4) that W x t P , , (ˆ) is independent of the parameters x and t. Therefore, we may write it as W P (ˆ). We now obtain the following. where C is as in (5.9). A Blaschke immersion y of an indefinite proper affine sphere with e det u  = may be described as . Since  is non-special at the origin x t , 0,0 = ( ) ( ), it remains non-special near the origin. Thus, F must be a constant on  . Evaluating it at 0 n = we obtain 0 F ô , which implies that x t ¶ ¶ Y - On the other hand, we find that W P (ˆ) may be expressed as W P e P P P P P P . Calculating V x t P , , x t j ¶ ¶ (ˆ) and using e x t u ¶ ¶ Y = Ŷˆwe have W P V x t P e V x t P 3 , , , , , which implies that each W P ĵ (ˆ) is real. We now assume that P j 1, 2, 3 ) is a zero of f P P e  q = -(ˆ) ( (ˆ) ). Then P ĵ is also a zero of f P s ( (ˆ)) by (5.2) and the properties of the theta function. Therefore, P ĵ is a pole of W P |ˆ(ˆ)| . Thus, the pole of x t P e , , , Ŷ (ˆ) cancel with the pole of W P |ˆ(ˆ)| each other. Therefore, as in the proof of theorem 4.9, introducing x t P , , Ỹ(ˆ) and W P(ˆ) with C P C P 1 1 2 = = (ˆ) (ˆ) and