Geometry of the flat tori in the sphere and non- linear wave equations

球面平面环面的几何形状和非线性波动方程

• 批准号: 15540059
• 负责人: KITAGAWA Yoshihisa
• 金额: $ 1.92万
• 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: 2003
• 资助国家: 日本
• 起止时间: 2003 至 2005
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-15540059/
• 关键词: differential geometry; submanifold; flat torus; isometric deformation; mean curvature; 3-sphere; meromorphic mapping; dynamical system; 視覚化; ベックルンド変換; ベクトル場; 閉曲線; 体積

In this research, we studied geometry of flat tori in the 3-sphere, meromorphic mappings, surfaces of constant mean curvature and dynamical systems. The main results of this reseach are summarized as follows.1.Studies on flat tori in the 3-sphere. In this research, Y.Kitagawa studied the conjecture that any isometric deformation of compact surface in $S^3$ preserves the enclosed volume.As a result, he proved that the conjecture is ture for all flat tori in $S^3$.2.Studies on meromorphic mappings. In this research, Y.Aihara proved that for every hypersurface $D$ of degree $d$ in a complex projective space, there exists a holomorphic curve from the complex plane into the projective space whose deficiency for $D$ is positive and less than one.3.Studies on constant mean curvature surfaces and Backlund transformations. In this research, J.Inoguchi proved that Bianchi-Backlund transformation of a constant mean curvature surface is equivalent to the Darboux transformation and the simple type dressing.4.Studies on dynamical systems. In this research, K. Sakai proved that the $C^1$ interior of the set of expansive vector fields on a manifold is characterized as the set of vector fields without singularities satisfying both Axiom A and the quasi-transversality condition.

本文主要研究了三维球面中的平坦环面几何、亚纯映射、常平均曲率曲面和动力系统。本文的主要研究结果如下：1.三维球面中的平面环面的研究。在这一研究中，Y.Kitagawa研究了S^3 $中紧致曲面的等距变形保持封闭体积的猜想，并证明了该猜想对S^3 $中所有平坦环面都成立。2.亚纯映射的研究。Aihara证明了：对于复射影空间中的任意d次超曲面D，存在一条从复平面到射影空间的全纯曲线，其亏度D为正且小于1。3.常平均曲率曲面与Backlund变换的研究。在本研究中，J.Inoguchi证明了常平均曲率曲面的Bianchi-Backlund变换与Darboux变换和简单类型修饰等价。4.动力系统的研究。在本研究中，K. Sakai证明了流形上可扩张向量场集合的C^1 $内部被刻画为同时满足公理A和拟横截性条件的无奇点向量场集合。

项目成果

Uniqueness problems for meromorphic mappings under condition on the preimages of divisors

除数原像条件下亚纯映射的唯一性问题

• 发表时间: 2004
• 期刊: Adv.Studies in Pure Math. Vol.42
• 作者: K.Sakai;K.Moriyasu;W.Sun;Y.Aihara
• 通讯作者: Y.Aihara

Chracterizations of Bianchi-Backlund transformation of constant mean curvature surfaces

常平均曲率曲面的 Bianchi-Backlund 变换的表征

• 发表时间: 2005
• 期刊: International Journal of Mathematics 16
• 作者: J.Inoguchi;S.Kobayashi
• 通讯作者: S.Kobayashi

Algebraic dependence of meromorphic mappings in value distribution theory

• DOI: 10.1017/s0027763000008473
• 发表时间: 2003
• 期刊: Nagoya Mathematical Journal
• 影响因子: 0.8
• 作者: Yoshihiro Aihara

S.Yu.Pilyugin, K.Sakai, A.A.Rodionova: "Orbital and weak shadowing Properties"Discrete and Continuous Dynamical Systems. 9. 287-308 (2003)

S.Yu.Pilyugin、K.Sakai、A.A.Rodionova：“轨道和弱阴影特性”离散和连续动力系统。

RECENT TOPICS IN UNIQUENESS PROBLEM FOR MEROMORPHIC MAPPINGS

• DOI: 10.1007/1-4020-7951-6_13
• 发表时间: 2004
• 作者: Yoshihiro Aihara