Geometry of the flat tori in the 3-sphere and its higher dimensional generalization

3-球面平面环面的几何形状及其高维推广

• 批准号: 12640059
• 负责人: KITAGAWA Yoshihisa
• 金额: $ 2.3万
• 依托单位: Utsunomiya University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2000
• 资助国家: 日本
• 起止时间: 2000 至 2001
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-12640059/
• 关键词: differential geometry; submanifold; flat torus; isometric deformation; mean curvature; 3-sphere; meromorphic mapping; dynamical system

In this research, we studied geometry of flat tori in the 3-sphere, meromorphic mappings and dynamical systems. The main results of this research are summarized as follows.(1) Studies on isometric deformations of flat tori in the S-sphere.In this research, Y. Kitagawa studied isometric deformations of flat tori isometrically immersed in the 3-sphere S^3 with constant mean curvature. As a result, he obtained a classification of the flat tori isometrically immersed in S^3 which admit no isometric deformation.(2) Studies on algebraic dependence of meromorphic mappings.In this research, Y. Aihara proved some criteria for the propagation of algebraic dependence of dominant meromorphic mappings from an analytic finite covering space X over the complex m-space into a projective algebraic manifold. Moreover, applying these criteria, he obtained unicity theorems for meromorphic mappings, and gave conditions under which two holomorphic mappings from X into a smooth elliptic curve E are algebraically related.(3) Studies on vector fields with topological stability.In this research, K. Sakai (with K. Moriyasu and N. Sumi) gave a characterization of the structurally stable vector fields by making use of the notion of topological stability. More precisely, it was proved that the C^1 interior of the set of all topologically stable C^1 vector fields coincides with the set of all vector fields satisfying Axiom A and the strong transversality condition.

本文主要研究了三维球面上平坦环面的几何、亚纯映射和动力系统。本研究的主要成果概括如下。(1)研究了S-球面中平面环面的等距变形。北川研究了等距浸入具有常平均曲率的三维球面S^3中的平面环面的等距变形。结果，他得到了等距地浸入S^3中的不允许等距变形的平坦环面的分类。(2)研究了亚纯映射的代数依赖性。Aihara证明了支配亚纯映射的代数依赖从复m-空间上的解析有限覆盖空间X传播到射影代数流形的一些准则。此外，应用这些标准，他获得了唯一性定理的亚纯映射，并给予条件下，两个全纯映射从X到一个光滑的椭圆曲线E是代数相关的。(3)研究具有拓扑稳定性的向量场。Sakai（与K。Moriyasu和N. Sumi）利用拓扑稳定性的概念给出了结构稳定向量场的一个刻画。更确切地说，证明了所有拓扑稳定的C^1向量场的集合的C^1内部与满足公理A和强横截性条件的所有向量场的集合重合。

项目成果

Y.Aihara: "Uniqueness problem of meromorphic mappings on analytic covering spaces"Proc. of the Third ISAAC Congress(eds.H Begehr et al.). (to appear).

Y.Aihara：“解析覆盖空间上亚纯映射的唯一性问题”Proc。

H.Emori: "The Emergent Chain Model of Communication : Japanese Style of Exchanging Mathematical Idea and Sense"In B. Barton (Ed.), Language and Communication in Mathematics Education. 41-50 (2000)

H.Emori：“交流的涌现链模型：交换数学思想和意义的日本风格”，B. Barton（主编），《数学教育中的语言与交流》。

K.Sakai: "Shadowing properties of L-hyperbolic homeomorphisms"Topology and its Applications. 112. 229-243 (2001)

K.Sakai：“L-双曲同胚的遮蔽特性”拓扑及其应用。

Y. Aihara: "Propagation of algebraic dependence of meromorphic mappings"Taiwanese Math. J.. 5. 667-689 (2001)

Y. Aihara：“亚纯映射的代数依赖性的传播”台湾数学。

Y. Aihara: "Algebraic dependence of meromorphic mappings in value distribution theory"Nagoya Math. J.. (to appear).

Y. Aihara：“值分布理论中亚纯映射的代数依赖性”名古屋数学。