The Application Of Integrable System In The Construction Of Affine Spheres

The present Ph.D. dissertation is concerned with the affine spheres and their constructions in classical affine differential geometry. We will use the method of integrable systems. Although the completeness and classification of affine spheres has been well developed, we still lack the explicit formulas and images of nontrivial affine spheres. In recent years, Loftin-Yau-Zaslow’s important works about SYZ conjecture for mirror symmetry show that the construction of mirror symmetry structure need some special explicit formulas of definite affine spheres. However, the construction of this kind of affine sphere is still open.By the integrable system structure of affine spheres, the process of construction can be seen as loop group action. We compute the simple element of loop group, and prove the corresponding decomposition theorems. By these theorems, we give the explicit formula of new affine spheres and the concrete examples. We also compute the explicit formulas of Hildebrand’s complete hyperbolic affine spheres and the integrable system structure. The main contents are as follows:In chapter one and chapter two, we expatiate the development and the related research situa-tion of affine spheres, and introduce the main results of this Ph.D. dissertation briefly. We also re-view the basic concepts of affine differential geometry and the classification of semi-homogeneous cones.In chapter three, We first clarify the loop group formulations for both hyperbolic and elliptic affine spheres in R3, and then classify the rational elements with 3 poles or 6 poles in a real twisted loop group. We prove Iwasawa type decompositions for these two types rational elements respectively, and use them to obtain the dressing actions on hyperbolic and elliptic affine spheres, and get a formula which is similar to the classical Tzitzeica transformation. Some new examples with pictures will be produced. By these examples, we can find that the general global definite affine spheres is mixed by hyperbolic type and elliptic type. The corresponding results will publish in Asian J. Math..In chapter four, Hildebrand solved Monge-Ampere equation of semi-homogeneous cones and computed their corresponding complete hyperbolic affine spheres. We get a transformation formula by solving the Beltrami equation, which provide isothermal parameterizations for Hilde-brand’s new examples. With isothermal parameterizations, we can give the explicit formulas of Hildebrand’s examples by Weierstrass & function, ζ function and σ function, and obtain their affine metrics and affine cubic forms in the simplest expression. At last, we construct the whole associated family for each of Hildebrand’s examples by solving the structure equations. As an application, by analyzing the associated family, we can find in general any regular convex cone in R3 has a natural associated S1-family of such cones. This result provide a basis of further studies about the integrable structure in convex geometry. The corresponding results will publish in Acta Math. Sci. Ser. B Engl. Ed..