Construction of submanifolds via integrable systems

本课题的主要研究成果如下：1.标记三次曲面模空间的标准射影浸入的构造及一致化方程的确定2。纯虚指数高斯微分方程定义的施瓦茨映射象的形状研究3.研究射影曲面的变换，如射影极小曲面和与线同余和微分不变量的关系; Appell微分系统定义的特殊曲面4。紧致辛流形的调和上同调的计算和在这样的流形上构造不同的辛结构5.通过DPW方法对H^3和R^3中的CMC曲面进行分类，并显式构造离散CMC曲面6。用平均曲率的积分确定旋转曲面的平均曲率为周期性的条件7.利用代数可积系统提出有限型多重调和映射8.重力作用下薄膜变分问题的表述及解的稳定性研究9.统一求解sinh-Gordon，Liouville，cosh-Gordon方程的方法及解的Backlund变换的构造10.小有限域上多项式理想素分解的实现11.由j（k（r））=（r）12定义的kappa函数的介绍和研究。定义三维双曲空间中具有奇点的平坦曲面的平坦前沿的概念及其理论的发展13。不一定不可约的Hermite对称空间的射影浸入的刚性证明

The result of the project is summarized as follows :1. Construction of a canonical projective immersion of the moduli space of marked cubic surfaces and determination of the uniformization equation2. Study of shape of the image of Schwarz map defined by the Gauss differential equation with purely imaginary exponents3. Study of transformations of projective surfaces such as projectively minimal surfaces and relation with line congruence and differential invariants; special surfaces defined by Appell's differential sytems4. Computation of the harmonic cohomology of compact symplectic nilmanifolds and construction of distinct symplectic structures on such manifolds5. Classification of CMC surfaces in H^3 and R^3 by the DPW method and explicit construction of discrete CMC surfaces6. Determination of the condition that the mean curvature of rotation surface is periodic in terms of the integral of the mean curvature7. Proposal of pluri-harmonic mapping of finite type making use of algebraic integrable systems8. Formulation of variational problem of thin films under the gravity and study of stability of solutions9. Unified method of solving sinh-Gordon, Liouville, and cosh-Gordon equation and construction of Backlund transformation of solutions10. Implementation of prime decomposition of polynomial ideals over small finite fields11. Introduction and study of the kappa function defined by j(k(r))=(r)12. Definition of the notion of flat fronts for flat surfaces with singular points in 3-dimensional hyperbolic space and development of its theory13. Proof of rigidity of projective immersions of Hermitian symmetric spaces not necessarily irreducible