Zhicheng LIN（林至誠）

Wuhan Institute of Physics and Mathematics，Chinese Academy of Sciences，Wuhan 430071，ChinaE-mail:flyriverms@qq.com

Erxiao WANG（王二?。?

Department of Mathematics，Hong Kong University of Science and Technology，Clear Water Bay，Kowloon，Hong KongE-mail:maexwang@ust.hk

THE ASSOCIATED FAMILIES OF SEMI-HOMOGENEOUS COMPLETE HYPERBOLIC AFFINE SPHERES?

Zhicheng LIN（林至誠）

Wuhan Institute of Physics and Mathematics，Chinese Academy of Sciences，Wuhan 430071，China
E-mail:flyriverms@qq.com

Erxiao WANG（王二小）?

Department of Mathematics，Hong Kong University of Science and Technology，Clear Water Bay，Kowloon，Hong Kong
E-mail:maexwang@ust.hk

Hildebrand classified all semi-homogeneous cones in R3and computed their corresponding complete hyperbolic affine spheres.We compute isothermal parametrizations for Hildebrand’s new examples.After giving their affine metrics and affine cubic forms，we construct the whole associated family for each of Hildebrand’s examples.The generic member of these affine spheres is given by Weierstrass P，ζ and σ functions.In general any regular convex cone in R3has a natural associated S1-family of such cones，which deserves further studies.

hyperbolic affine spheres;isothermal coordinates;Weierstrass elliptic functions; Monge-Amp`ere equation;Tzitz′eica equation

2010 MR Subject Classification37K25;53A15

1　Introduction

Classical equiaffine differential geometry investigates the properties of hypersurface r（x1，x2，···，xn）in Rn+1invariant under the equiaffine transformations r→ Ar+v，where A∈SLn+1（R）and v∈Rn+1.Although the Euclidean angle is no longer invariant，there exists an affine invariant transversal vector field along r，called the affine normal.The affine metric and the affine cubic form give a set of complete affine invariants.The fundamental theorem tells us that any given affine metric and affine cubic form satisfying certain compatibility conditions（Gauss-Codazzi type）will determine an affine hypersurface uniquely up to equiaffine transformations（see［20］）.The affine metric（conformal to the Euclidean second fundamental form）is definite if and only if the hypersurface is locally strictly convex.The simplest interestingclass of hypersurfaces is the affine spheres，which are defined by the condition that all affine normal lines meet in a point.Specially，definite affine spheres（with a mean curvature H and with center at the origin or infinity）can be represented as a graph of a locally strictly convex function f if and only if the Legendre transform u of f solves a Monge-Amp`ere equation（see Calabi［1］）:

Cheng and Yau［5］showed that on a bounded convex domain there is for H ＜0 a unique negative convex solution of（1.1）extending continuously to be 0 on the boundary.This leads to a beautiful geometric picture of any complete hyperbolic（H＜0）affine sphere r（conjectured by Calabi［1］）:it is always asymptotic to the boundary of the cone given by the convex hull of r and its center;and conversely the interior of any regular convex cone is foliated by complete hyperbolic affine spheres asymptotic to it with all H＜0 and with centers at the vertex.

However，explicit representations are rarely known except for homogeneous cones.Hildebrand classified three-dimensional regular convex cone with an automorphism group of dimension at least 2，which he also called“semi-homogeneous”.In this case，he reduced the Monge-Amp`ere equation to some ODE and was able to solve it with elliptic integrals in［15］.

It is known that 2 dimensional affine spheres have additional features than higher dimensional ones.For example，the affine metric can be expressed simply in some natural isothermal coordinates（definite case）or asymptotic coordinates（indefinite case）.In addition，each 2 dimensional affine sphere has a natural associated family of the same type with the same affine metric but different affine cubic forms.In particular，this implies that the structure equations actually form an integrable system，called Tzitz′eica equation or the affine-Toda field equation（see［7，8，20］）.It is then natural to ask for the isothermal parametrizations and the associated families of Hildebrand’s new examples.Specially，one wish to see the natural associated family of cones.This paper will answer these questions.

Our extensive studies of definite affine spheres were motivated by Loftin，Yau and Zaslow’s‘trinoid’construction（see［22］）with applications in mirror symmetry.The dressing actions on proper definite affine spheres and soliton examples were presented in［20］.The Permutability Theorem and group structure of dressing actions will be presented in a subsequent paper［29］. Their Weierstrass or DPW representations were studied in［8］，using an Iwasawa decomposition of certain twisted loop group.The equivariant solutions were constructed in［9］.

The rest of the paper is organized as follows.In Section 2，we introduce the fundamental concepts of affine sphere and review the main results in［15］.In Section 3，we compute the isothermal parametrizations and the affine invariants of Hildebrand’s examples case by case. We also get the whole associated family for each case by solving their structure equations.In general any regular convex cone in R3has a natural associated S1-family of such cones，which deserves further studies.

2　Affine Spheres and Semi-homogeneous Cones

Let

be an immersion with a non-degenerate second fundamental form.Introduce

is equiaffine invariant，and it is called the affine（or Blaschke）metric of the immersion.The surface is said to be definite or indefinite if this metric g is so.The surface is definite if and only if it is locally strictly convex.A transversal vector field ξ on a surface r（M）is called affine normal if it satisfies

where Δgis the Laplace-Beltrami operator of the affine metric.The connection?induced by the affine normal is called the Blaschke connection.The affine metric and the affine normal are then uniquely determined（up to a sign）by requiring the following decomposition into tangential and transverse components:

where D is the canonical flat connection on R3and X，Y are any tangent vector fields on the surface.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.

If all affine normals of r meet in one point（the center），then the surface is called affine sphere.If this point is not infinite it may be chosen as the origin of R3so that

H is called the affine mean curvature，and such surface is called proper affine sphere.For a convex proper affine sphere，if the center is outside the surface，it is called hyperbolic.

In the sequel we consider the cone K ?R3，with the vertex at the origin.The complete hyperbolic affine spheres foliating K are the level sets of the solution F:K?→ R of the following Monge-Amp`ere equation（see［3，4，11，21］for detail）:

Let AutK denote the automorphism group of the cone K.The equiaffine invariance of（2.1）implies F is invariant under unimodular automorphisms g∈AutK（see［11］or［14］）.Then the number of variables of F can be effectively reduced by the generic dimension of the orbits of AutK.If there are orbits of dimension 2，the PDE（2.1）will reduce to an ODE.This kindof cone is also called semi-homogeneous cone.Hildebrand provided an important classification theorem of such cones:

Fig.1　Semi-homogeneous cone

Theorem 2.1（［15］，Theorem 3.2）Let K?R3be a regular convex cone such that dim AutK≥2.Then K is isomorphic to exactly one of the following cones:

1.the cone obtained by the homogenization of the epigraph of the exponential function;

We choose p=5 for each case and give the pictures of these five cases of semi-homogeneous cones in Fig.1.It could be useful for knowing the shape of semi-homogeneous cones.

3　The Associated Families of Affine Spheres

For convex affine spheres，the affine metric is positive-definite，and thus it provides a conformal structure in dimension two.There exists local complex coordinate z=x+iy so thatThe affine cubic formThe affine normal satisfies:

Simon-Wang derived in［23］the following equations for the frame

and showed that the compatibility conditions（Tzitz′eica equation）are:

The above system is invariant underInserting the parameter λ=e3itinto the frame equations（3.1），we obtain the following Lax representation of system（3.2）:

In this section we compute the isothermal parametrizations of Hildebrand’s examples，then obtain the affine invariants.Substituting these affine invariants into the frame equations（3.3）and solving them，we get the whole associated family for each examples.

3.1The Isothermal Parametrization and the Associated Family of Case 1

In this subsection we consider the complete hyperbolic affine sphere asymptotic to the boundary of the cone obtained by the homogenization of the epigraph of the exponential function（case 1 in Theorem 2.1）.The corresponding example in［15］is given by:

Let us briefly review the classical results of isothermal coordinates in［24］.If the metric is given locally as

then in the complex coordinate z=x+iy，it takes the form:

In isothermal coordinates（u，v）the metric should take the form

The complex coordinate ω=u+iv satisfies

so that the coordinates（u，v）will be isothermal if the Beltrami equation

has a diffeomorphic solution.

To compute the isothermal parametrization of the affine sphere，we should compute the affine metric.Then we get

The Beltrami equation（3.5）can be written as

Then we divide our computation into the following steps.

and

Summarize the above computations，we use the coordinate transformation

By a direct computation，we can get the affine metric，the affine mean curvature and the affine cubic form:

It is easy to see that g gives an explicit solution of Tzitz′eica equation（3.2）.Substituting g，H，U into frame equations（3.3）and solving them，then we can conclude the above computations into the following theorem.

Theorem 3.1The associated family of the complete hyperbolic affine sphere（3.4）in［15］is given by:

here

with the affine metric，the affine mean curvature and the affine cubic form are as follows:

Remark 3.2Expression（3.10）is gauged by the initial condition

With this initial condition，one can checkby direct computation.

3.2The Isothermal Parametrizations of Remaining Cases

The complete hyperbolic affine spheres asymptotic to the boundary of the remaining cones can be treated in a common framework.In this subsection we compute the common isothermal coordinates transformation for these affine spheres by Weierstrass elliptic functions.Let us review the results in［15］.

Now we begin to compute the transformation.We consider Blaschke metric of the immersion

Similar to case 1，we divide the computation into the following 3 steps:

Summarizing the above computation，we obtain the transformation:

The above integral can be expressed by Weierstrass elliptic function（see［13］or［17］）:

By this expression and a gauge transformation:

we can simplify（3.18）-（3.20）as

Then we obtain the affine metric，the affine mean curvature and the affine cubic form:

Invariants（3.25）-（3.26）imply that the two affine spheres coresponding to s=1 and s=-1 are in the same associated family.The coefficients U of these two affine spheres can be expressed asrespectively，in other words

4　The Associated Families of Remaining Cases

With the generic formulas（3.22）-（3.24），we can construct the complete hyperbolic affine spheres and the associated families case by case.Some parts of construction have referred to the formulas of the affine spheres in［15］.We consider the following cases.

Since φ is analytic in t，it must be an even function of t.For each negative t，s=-1，we have

This affine sphere is asymptotic to the cone given in case 4 of Theorem 2.1，for α 6=1.The affine invariants are given in（3.25）-（3.26）.When computing the associated family，we can consider the coefficient U of this affine sphere isThen，for convenience，substitute g，H，into frame equations（3.3）and solve them.We conclude the above discussions into the following theorem.

Theorem 4.1The associated family the complete hyperbolic affine sphere（3.13）-（3.15） within［15］is given by

What’s more，the affine metric，the affine mean curvature and the affine cubic form are given as

Case（ii）c=0.In this case s=1，β=1 and（3.22）-（3.24）become

for each positive t.Then for each negative t，we have

This affine sphere is asymptotic to the cone given in case 4 of Theorem 2.1，with α=1.The affine metric，the affine mean curvature and the affine cubic form of this affine sphere are

Remark 4.2By direct computation，it is easy check the expression of affine spheres（4.4）-（4.5）can get from（4.1）-（4.2）when c→0.Henceforth，the associated families in this case can also get from（4.3）by c→0.

Case（iii）c=-2（p+q），β=1.In this case s=1，and（3.22）-（3.24）become

This affine sphere is asymptotic to the cone given in case 5 of Theorem 2.1.The affine metric，the affine mean curvature and the affine cubic form of this affine sphere are

Theorem 4.3The associated family of the complete hyperbolic affine sphere in［15］:

with the affine metric，the affine mean curvature and the affine cubic form are as follows:

This affine sphere is asymptotic to the cone given in case 3 of Theorem 2.1.Since this affine sphere is in the same associated family with the previous case，their affine metrics and affine mean curvatures are the same，and the cubic form of this case is

At last，we explain that any regular convex cone in R3should have a natural associated family of such cones.

If we use the transformationsand

it is easy to see that the associated family in Theorem 3.3 is the same with it in Theorem 3.1，i.e.，case 1，case 3 and case 5 are all in the same associated family.Conversely，any member in this family can be found in these three cases.The cones which these members are asymptotic to also can be found in the corresponding cases of the semi-homogeneous cones，which form a natural associated family of cones.

There are similar results for case 4.To show this，we should use the following relationship between Weierstrass P-function and Jacobi SN-function:

In conclusion，in general any regular convex cone in R3has a natural associated S1-family of such cones，which deserves further studies.Given any natural associated family of some semi-homogeneous cone，it may contain case 1，case 3 and case 5 of semi-homogenous cones or case 4.

AcknowledgementsThe second author would like to express the deepest gratitude for the support of the Hong Kong University of Science and Technology during the project，especially Min Yan，Yong-Chang Zhu，Bei-Fang Chen and Guo-Wu Meng.

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?March 31，2015;revised May 12，2015.The authors were supported by the NSF of China（10941002，11001262），and the Starting Fund for Distinguished Young Scholars of Wuhan Institute of Physics and Mathematics（O9S6031001）.

?Erxiao WANG.