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Dressing Actions on Proper Definite Affine Spheres

We will first clarify the loop group formulations for both hyperbolic and elliptic definite affine spheres in R^3. Then we classify the rational elements with 3 poles or 6 poles in a real twisted loop group, and compute dressing actions of them on such surfaces. Some new examples with pictures will be produced at last.


Introduction
All intuitive geometrical notions about position, size and shape developed in school geometry are invariant under rigid motions comprised of translations, rotations and reflections. Following Klein's famous Erlangen Program (1872), affine geometry studies the constructs invariant under the affine group A(n) = GL(n)⋉R n comprised of nondegenerate linear maps and translations. While two fundamental measurements angle and distance in Euclidean geometry are no longer invariant under affine motions, notions for some relative positions such as parallels and midpoints (more generally center of mass) still make sense. As the standard directional derivative operator D is preserved by A(n), differential geometry of curves and surfaces can also be generalized. Similarly, the symmetry group for equiaffine geometry is SA(n) = SL(n) ⋉ R n preserving in addition the standard volume form on R n .
In 1907 Gheorghe Tzitzéica discovered a particular class of hyperbolic surfaces in R 3 whose Gauss curvature at any point p is proportional to the fourth power of the distance from a fixed point to the tangent plane at p. He used the structure equation ω xy = e ω − e −2ω (1.1) to describe the (indefenite) affine spheres. The reader may refer to [20] for a concise survey on affine spheres and to [31] for an extensive survey on their relations to real Monge-Ampère equations, projective structures on manifolds, and Calabi-Yau manifolds. Here we would like to emphasize the Monge-Amprère equation (see Calabi [8]). The graph of a locally strictly convex function f is a mean curvature H affine sphere centered at the origin or infinity if and only if the Legendre transform u of f solves: Cheng & Yau [10] showed that on a bounded convex domain there is for H < 0 a unique negative convex solution of (1.2) extending continuously to be 0 on the boundary. This eventually leads to a beautiful geometric picture of complete hyperbolic (H < 0) affine spheres (conjectured by Calabi): such a hypersurface is always asymptotic to a convex cone with its vertex at the center and the interior of any regular convex cone is a disjoint union of complete hyperbolic affine spheres asymptotic to it with all negative mean curvature values and with centers at the vertex. Even in R 3 , it is highly nontrivial 1 to see the above equation is equivalent to the usual surface structure equations (see Simon & Wang [38]): ψ zz + H 2 e ψ + |U | 2 e −2ψ = 0, Uz = 0, (1.3) where e ψ |dz| 2 is the affine metric and U dz 3 is equivalent to the affine cubic form.
The equation (1.3) for nonzero U is an elliptic version of the classical Tzitzéica equation, or the Bullough-Dodd-Jiber-Shabat equation, or the affine a (2) 2 -Toda field equation, or the first SL(3, R)/ SO(2, R) elliptic system (see Terng [40]). Dunajski & Plansangkate [19] have given gauge invariant characterization of (1.3) in terms of SU(2, 1) Hitchin equations or reduction of Anti-Self-Dual-Yang-Mills equations, and linked the local equation around a double pole of U to Painlevé III. Zhou [46] have given Darboux transformations for two dimensional elliptic affine Toda equations. Hildebrand [24] have given analytic formulas for complete hyperbolic affine spheres asymptotic to semi-homogenous convex cones.
In the background of mirror symmetry, Loftin, Yau & Zaslow [30] constructed some global hyperbolic/elliptic affine sphere immersions from Riemann sphere minus 3 points, also called "trinoids". This has motivated our extensive studies of such surfaces from integrable system or soliton theory. In this paper we compute the dressing actions and soliton examples. The Permutability Theorem and group structure of dressing actions will be presented in a subsequent paper [44]. Their Weierstrass or DPW representation has been studied in [15], using an Iwasawa decomposition of certain twisted loop group. The equivariant solutions have been constructed in [16]. The conformal parametrization of Hildebrand's complete affine spheres will be presented in [29].
The rest of the paper is organized as follows. In section 2, we will recall the fundamental notions and theorems of equiaffine differential geometry, and the definitions of affine spheres. We specialize to two dimensional proper definite affine spheres in section 3 and give the loop group descriptions of them. We then classify the simple rational elements in certain twisted loop group in section 4 and compute the dressing actions of them on such surfaces in section 5. Some examples will be presented in the last section.
2. Fundamental theorem of affine differential geometry and affine spheres Classical affine differential geometry studies the properties of an immersed hypersurface r : M n → R n+1 invariant under the equiaffine transformations r → Ar + b, where A ∈ SL n+1 (R) and b ∈ R n+1 . Here R n+1 is viewed as an affine space equipped with standard equiaffine structure: the canonical connection D and the parallel volume form given by the standard determinant. The following form in local coordinates (u 1 , · · · , u n ) is naturally an equiaffine invariant: which is independent of choice of local coordinates and thus defines a global quadratic form with values in the line bundle of top forms on M . Denote the determinant coefficients in (2.1) by Λ ij and their determinant by det Λ. Assuming det Λ = 0 everywhere, a pseudo-Riemannian metric g is induced uniquely up to sign (depending on the orientation of M ) by the equation or defined explicitly by called the affine metric. The hypersurface is said to be definite or indefinite if this metric g is so. A smooth hypersurface is definite if and only if it is locally strictly convex. Although the Euclidean angle is not invariant under affine transformations, there exists an invariant transversal vector field ξ along r(M ) defined by △ g r / n, called the affine normal. Here △ g is the Laplace-Beltrami operator of g. Another way to find the affine normal up to sign is by modifying the scale and direction of any transversal vector field (such as the Euclidean normal) to meet two natural characterizing conditions: (i) D X ξ or dξ(X) is tangent to the hypersurface for any X ∈ T p M , (ii) ξ and g induce the same volume measure on M : det (r * X 1 , · · · , r * X n , ξ) 2 = | det(g(X i , X j ))| for any X i ∈ T p M .
The formula of Gauss gives the following decomposition into tangential and transverse components: which induces a torsion-free affine connection ∇ on M , called Blaschke connection. LetĈ denote the difference tensor between the induced Blaschke connection ∇ and g's Levi-Civita connection ∇ g . The affine cubic form measures the difference between the induced Blaschke connection ∇ and g's Levi-Civita connection ∇ g : It is actually symmetric in all 3 arguments and is a 3rd order invariant. The Pick invariant J is simply C 2 g /(n 2 − n). Similar to the Euclidean case, the affine shape operator S defined by the formula of Weingarten: is again self-adjoint with respect to g. The affine mean curvature H and the affine Gauss curvature K are defined as H = TrS / n, K = det S.
Note that the affine normal is also a 3rd order invariant. Hence S is a 4th order invariant, which can actually be computed from g and C: where R g = scalar curvature of g.
The readers may refer to several textbooks [5,28,34] for more details of the above basic notions.
Theorem 2.1 (Dillen-Nomizu-Vrancken type fundamental theorem [17]). Given a nondegenerate symmetric 2-form g and a symmetric 3-form C on simply connected M satisfying two compatibility conditions: the apolarity condition Tr g C = 0, and the conjugate connection ∇ := ∇ g −Ĉ is projectively flat, there exists a global immersion into R n+1 unique up to equiaffine motions such that (g, C) are the induced affine metric and cubic form respectively. Remark 2.2. When we scale the immersion f to ρf for some positive constant ρ, the above invariants changes by: Historically the first important class of affine surfaces is the affine spheres studied by Tzitzéica in a series of papers from 1908 to 1910. A proper affine sphere is a surface whose affine normal lines all meet in a point (the center), such as all ellipsoids and hyperboloids. An improper affine sphere is a surface whose affine normals are all parallel, such as all paraboloids and all ruled surfaces of the form x 3 = x 1 x 2 +f (x 1 ). An equivalent definition is S = H· Id, which implies from the compatibility conditions that the affine mean curvature H must be constant. When S ≡ 0, ξ is a constant vector and it is improper. So improper affine spheres are special affine maximal surfaces (H = 0) and have been integrated explicitly for dimension 2 in [5]. When S = H· Id = 0, it is proper and ξ = −H(x − x 0 ) with some x 0 being the center. For simplicity, we will always make x 0 = 0 by translating the surface. In the indefinite case, we can scale the surface and change the sign of ξ if necessary to normalize H = −1. However, in the definite case we have a preferred choice of the signs of g and ξ by requiring g to be positive definite, or equivalently by choosing ξ to point to the inside of the convex surface. So we need to distinguish two cases according to the sign of H: (1) the elliptic affine spheres whose center is inside the convex surface (H > 0), such as all ellipsoids; (2) the hyperbolic affine spheres whose center is outside the convex surface (H < 0), such as all hyperboloids of two sheets and x 1 x 2 x 3 = 1.
Improper definite affine spheres (H = 0), are naturally called parabolic affine spheres, such as all elliptic paraboloids. While quadrics are the only global examples of elliptic and parabolic affine spheres (by Calabi, Cheng-Yau, Jörgens, Pogorelov), there are many global hyperbolic affine spheres asymptotic to each sharp (or regular) convex cone (by Cheng-Yau). One of the main goals of this paper is to construct families of local examples of both elliptic and hyperbolic affine spheres in R 3 using integrable system techniques.

Loop group description of definite affine spheres
From now on we consider only the surface case in R 3 where its affine metric g is positive definite: this means that r(M ) is locally strongly convex and oriented so that Λ is positive valued. In particular its Euclidean Gauss curvature is positive and the affine normal ξ is chosen to point to the concave side of the surface. This essentially induces a unique Riemann surface (or complex) structure on M in whose coordinates g = e ψ |dz| 2 and the orientation given by idz ∧ dz is consistent with the orientation induced by ξ. Note that −r would have the same fundamental invariants as r but reverse the orientation. So it is not considered to be equi-affinely equivalent to r. Indeed 3 × 3 negative identity matrix has determinant −1 and is not in SL(3, C). Alternatively we are studying "affine-conformal" immersions of any Riemann surface M into R 3 : | r z rz r zz | = 0 affine-conformal condition. (3.1) Then the following two determinants completely determine the fundamental invariants: with g = e ψ |dz| 2 and C = U dz 3 +Ū dz 3 . This can also be illustrated by computing the evolution equations for the positively oriented frameF = (r z , rz, ξ = 2e −ψ r zz ): The compatibility condition (F zz =Fz z ) or the flatness ofF −1 dF determines every entry in the above matrix one-form satisfying two structure equations (first derived by Radon): Conversely, given a positive definite metric g and fully symmetric cubic form C as above satisfying (3.3) and (3.4), there exits a definite affine surface unique up to equiaffine motions such that (g, C) are the induced affine metric and cubic form respectively. This is the fundamental theorem for definite affine surface. Here H is indeed the affine mean curvature function since the affine shape operator S takes the following matrix form in terms of the basis (∂ z , ∂z): In summary we obtain the governing equations for definite affine spheres in R 3 (see also Simon-Wang [38]): whose solution determines a unique definite affine sphere (up to equi-affine motions) with constant affine mean curvature H and affine metric e ψ |dz| 2 , by integrating the frame equation:α It is now clear that U dz 3 is a globally defined holomorphic cubic differential (i.e. in H 0 (M, K 3 ) where K is the canonical bundle of M ). Recall Pick's Theorem: C ≡ 0 if and only if r(M ) is part of a quadric surface. So U is nonzero except for the quadrics. Away from its isolated zeroes one could make a holomorphic coordinate change to normalize U to a nonzero constant but we will not do that now. These zeroes of U will be called "planar" points of the affine surface.
We would like to emphasize that the immersion is analytic for any definite affine sphere, since the defining equation is elliptic (cf. [5] §76 ).
The following observation is crucial for the integrability of definite affine spheres: the system (3.5) is invariant under U → e iθ U with any constant θ ∈ R/2πZ. Thus an S 1 -family of definite affine spheres r θ can be associated to any given one, with the same affine metric but different affine cubic forms C = e iθ U dz 3 + e −iθŪ dz 3 .
Note that τ is a conjugate linear involution of sl(3, C) whose fixed point set is isomorphic to sl(3, R); and σ is an order 6 automorphism of sl(3, C) giving the following eigenspace decomposition or Z 6 -gradation: We verify that στ = τ σ and they define a 6-symmetric space "SL(3, R)"/ SO(2, R).
The induced automorphisms on SL(3, C) (still denoted by τ and σ) are: If we solve F −1 λ dF λ = α λ uniquely with certain initial condition F λ (p 0 ) at any base point p 0 of M , it is easy to show that F λ (p 0 ) −1 F λ satisfies the reality conditions (3.8) and therefore lies in the corresponding twisted loop group.
Remark 3.1. Hereafter we will always choose the initial loop F λ (p 0 ) = I. Then we may conjugate the complex frame to a real SL(3, C)-frame: In fact F R = (e 1 , e 2 , ξ) with {e 1 , e 2 } being simply an orthonormal tangent frame w.r.t. the affine metric. So we obtain an affine sphere immersion r = −H −1 ξ = ∓ξ. It is clear now that we may also simply take the real part of the last column of F λ to get an equivalent affine sphere modulo affine motions.
As in [14] and [43], we write It follows that where Let C × := C \ {0}. We adopt the following notations for loop groups: , σ(g(λ)) = g(ǫλ)}, where 0 < r < 1 is sufficiently small and Similar notations can apply to their Lie algebras. Then α λ in (3.6) is a Λ σ,τ C × sl(3, C)valued flat connection, and the corresponding frame F λ lies in Λ σ,τ C × SL(3, C) for any 0 < r < 1. It is remarkable that a simple "algebraic" condition characterize such extended frames of proper definite affine spheres: The reality conditions (3.8) guarantee that F −1 F z must be linear in λ. So we have with A ∈ g −1 , B ∈ g 0 , C = τ (A), and D = τ (B). The fixed points of both σ and τ are of the form diag(e iβ , e −iβ , 1). Gauging by them respect the reality conditions. Let e iβ = ± A 13 |A 13 | . Use it to gauge A 13 to a real positive function which is then set to e ψ/2 . The rest follows from the equations of flatness. 4. Simple elements in Λ σ,τ I GL(3, C) A rational element in any loop group is usually called a simple element if it has the least number of simple poles. To construct the simple elements in the twisted loop group Λ σ,τ I GL(3, C), we will handle order 3 σ 2 -twisting first: has three simple poles and always take the following form: where A is diagonal and To construct simple element in Λ σ I GL(3, C), plug (4.1) into σ 1 -reality condition in (3.12), we obtain that g ∈ Λ σ I GL(3, C) iff g(−λ) P 12 g t (λ) = P 12 , where P 12 is given in (3.11), i.e.
Compute the LHS of (4.2) at λ = ∞ to get , and we have Compute the residue of the LHS of (4.2) at λ = α to get i.e.
Substituting (4.6) into (4.5) and computing directly, we have Residue of LHS of (4.2) at λ = ǫ 2 α gives This is equivalent to (4.4). Residue at λ = ǫ 4 α is also equivalent to (4.4). So the loop group element of rank 1 type is as follows: By explicit computation, if rank(R) = 2, we have with the corresponding rank 2 type loop group element We also compute that the determinant of g(λ) for the both cases are [(λ 3 + α 3 )/(λ 3 − α 3 )] rank(R) , i.e. only depending on the poles and the rank of the residues. Let l := (b, c, 1), ℓ be the line C · l, and introduce the following 'cone': , or z 3 = 0}. Then 2bc = 1 is equivalent to ℓ ∆. We observe that Image(Res α g t ) for rank 1 type is and Kernel(Res α g P 12 ) for rank 2 type are both ℓ t := C · l t .
Notation. Since a line ℓ not in ∆, a complex number α and a diagonal matrix A (4.3) determine g(λ) uniquely in both types, we always use Ag α,ℓ (λ) to denote the rank 1 type element (4.8) and use Am α,ℓ (λ) to denote the rank 2 type element (4.9).
We conclude the above computations into the following theorem. (1)Take any α ∈ S 1 and b ∈ C so that |b| 2 > 1 2 . Then c =b and Then h(λ) := Ag α,ℓ 1 (λ)gᾱ−1 ,ℓ 2 (λ) ∈ Λ σ,τ I GL(3, C) . Remark 4.4. It would be interesting to consider the converse question: whether any rational element with 6 poles must take the form of h(λ)? More generally, it would be interesting to know whether any rational element can be factorized into product of simple ones: Am α,ℓ (λ)(or Ag α,ℓ (λ)) and h(λ)? Such factorization problem has been treated by Uhlenbeck [42] in the case of unitary group. We will answer these questions in a subsequent paper.

Dressing actions of simple elements
In this section we use the simple element to give the dressing actions on definite affine spheres.
Let us review the technique of dressing action (the idea of dressing was dated from [45] but see [23] or [40] for an elementary introduction). Let G = SL(3, C), g(λ) ∈ Λ τ,σ I G, and E(z,z, λ) ∈ Λ σ.τ C × G is the frame of a definite affine sphere. Assume we can do the following factorization for each fixed (z,z): ThenẼ(z,z, λ) will also be the frame of a new definite affine sphere. We sketch the proof here. By Theorem 3.2, it suffices to prove thatẼ −1Ẽ z andẼ −1Ẽz are linear in λ and λ −1 respectively. But On the left hand sideẼ −1Ẽ z is holomorphic in λ ∈ C × ; on the right it has a simple pole at ∞. SoẼ −1Ẽ z =ũ 0 + λũ 1 , whereũ 0 ,ũ 1 are independent of λ. Similarly,Ẽ −1Ẽz is linear in 1 λ . This completes the proof.
Furthermore, g * E :=Ẽ defines a group action of Λ τ,σ I G on the frames of the definite affine spheres, which is called the dressing action.
When we impose the τ -reality condition in the following, the aboved will be uniquely determined by requiring positiveness (see Remark 5.2). Now we begin to compute the dressing actions of two types of elements in Theorem 4.3.

4)
where φ := (b,b, 1)r −α is a scale solution of (3.7) with parameter −α. This new definite affine sphere has new affine metricd 2 · e ψ and the same affine cubic form (and the same affine mean curvature). For the elliptic affine sphere, we have the similar results. Only need to change −α to α and a scaling for (5.4) by λ 3 −α 3 λ 3 +α 3 .
Proof. Since ℓE(z,z, −α) (b,b, 1), we get Recall the discussion in the beginning of this section,Ẽ −1Ẽ z be expressed as: We can getũ 1 byũ Compareũ 1 with the coefficient matrix of λ in (3.7), it is easy to see that this new definite affine sphere has new affine metricd 2 · e ψ and the same affine cubic form (and the same affine mean curvature).

Basic examples
In this section, we will apply our results to construct some basic examples. 72 . One can integrate (3.7) to obtain the whole family of frames. Note that the surface is independent of the parameter λ. So the associated family is really a family of parameterizations of the same affine sphere.
We may choose the following conformal parametrization of the vacuum definite affine sphere after certain affine transformation: where ǫ = e πi 3 , R(λ) = exp(λz + 1 λz ). Then Normalize this frame in a neighborhood of 0, we get where Example 6.2 (The one 'soliton' solution). Choose simple element Ag α,ℓ with α = i, ℓ = C(− 1 2 + i, − 1 2 − i, 1), and E(z,z, λ) is the normalized frame of vacuum definite affine sphere in (6.1). Then we have the factorization: By Theorem 5.4, we can get the solution of Tzitzéica equation (3.5) (new affine metric): Recall the one soliton type solution of Tzitzéica equation in [26]: Here we adopt the usual notation τ for τ -function, not the same as the reality condition before. After some computation, we can see our solution (6.2) is coincide with (6.3) with k = √ 3 and s = ln( 2 . We give the picture new solution in Figure 1. Recall a complete hyperbolic affine sphere in [24]: which is asymptotic to the boundary of the cone obtained by the homogenization of the epigraph of the exponential function. To compare with this hyperbolic affine sphere, we use the conformal parametrization in [29]: with the affine metric, affine mean curvature and affine cubic form are: We only need to make a motion of variable x by : x → x − s 2 √ 3 and choose λ = i in our case. Then we get an definite affine sphere with the same invariants with (6.4), i.e. one of the family of definite affine sphere (5.4) is a complete hyperbolic affine sphere. Since the specific formula is too long to place in this paper, we give the picture of this affine sphere instead. (1) Choose simple element Am α,ℓ , where α = i, ℓ = C(1 + i, 1 − i, 1), and E(z,z, λ) is (6.1). Then we have the factorization +s 2 + (k 1 − k 2 ) 2 (k 2 1 − k 1 k 2 + k 2 2 ) (k 1 + k 2 ) 2 (k 2 1 + k 1 k 2 + k 2 2 ) e (k 1 +k 2 )z+( 3 )z+s 1 +s 2 (6.6) Then we get the τ -function of the new solution h: It is coincide with the τ -functions (6.6) with parameters k 1 = 3 ). We give the picture of this solution in Figure 3. Notice in Figure 3, the solution goes below the red plane (z=0) only in the very small region, which means the metric become negative in a curve. It is easy to see that Tzitzéica equation (3.5) equals to h zz h − h z hz − h 3 + 1 = 0. (6.7) and the negative parts are also the solution of (6.7). From geometric point of view, if we fix the affine mean curvature H = −2, h is positive means the affine metric is positive definite, then it gives a local hyperbolic affine sphere; h is negative means the affine metric is negative definite, it gives a local elliptic affine sphere.
So the solution h = eψ can be extended to be a global solution of Tzitzéica equation (6.2) and we get a global mixed definite affine sphere. Some parts of it are hyperbolic type, the other are elliptic.
We also use the formula (5.4) to get the new definite affine sphere: Since the specific formula of new definite affine sphere is too long to place in this paper, we give the picture of it in Figure 4 with λ = e 2 3 πi instead. Remark 6.1. By computation, we find out that when Re(b) = − 1 2 the simple element with 3 simple poles will give the one soliton type solution, and other cases it gives two soliton type solution.