广义Calabi猜想和典型仿射超曲面的分类问题研究

The investigations on the classification of typical hypersurfaces have always been the core subjects of differential geometric theory. Affine differential geometry is an important branch of differential geometry, among which the classification of the complete affine hyperspheres has attracted a large number of geometers and the related results, especially the proof of the Calabi conjecture on the qualitative classification of the hyperbolic affine hyperspheres, have got much attention. The existing related results（including those of ourselves) make us deeply aware that the centroaffine Tchebychev hypersurface plays the role of the affine hypersphere. In this project, relaying on our previous researches on affine hypersurfaces with typical properties and employing the algebra theory, analytical techniques and the techniques developed with reseaches on the classification of hypersurfaces, we will focus on the classification of affine hypersurfaces with typical properties, especially the following topics: the classification of the complete centroaffine Tchebychev hypersurfaces (especially the generalized Calabi conjecture on the hyperbolic case ), the classification of the centroaffine Tchebychev hypersurfaces with constant sectional curvature、Einstein affine hyperspheres、locally symmetric affine hyperspheres and the affine hypersurfaces with other typical properties.The researches and the relative results will enrich the theory of affine differential geometry especially those of centroaffine differential geometry.

典型超曲面的分类问题研究历来是微分几何理论的核心课题。仿射微分几何是微分几何的重要分支，其中完备仿射球的分类问题曾吸引大批几何学家，相关成果尤其是关于完备双曲型仿射球解析分类的Calabi猜想的证明受到人们的高度关注。已有成果使我们深刻认识到中心仿射Tchebychev超曲面扮演仿射球的角色。本项目将在前期研究基础上，综合运用代数理论、分析技巧及针对超曲面的分类问题而发展起来的特殊技巧，集中开展典型仿射超曲面的分类问题研究，特别专注于在下列课题上做出创新性研究成果：分类完备中心仿射Tchebychev超曲面，特别是将关于双曲型仿射球的Calabi猜想推广到双曲型中心仿射Tchebychev超曲面，分类常截面曲率中心仿射Tchebychev超曲面、Einstein仿射球、局部对称仿射球等典型仿射超曲面。这些研究及其成果将对丰富仿射微分几何理论尤其是中心仿射微分几何理论产生重要影响。

内容获取失败，请点击重试