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.

在这项研究中，我们研究了在三个球体，mer态映射，恒定平均曲率和动力学系统的表面中的几何形状。该研究的主要结果总结如下。1。在3杆中对Flat Tori进行了研究。在这项研究中，Y.Kitagawa研究了以下猜想：$ s^3 $中的任何等距变形保留了封闭的体积。结果，他证明了猜想是所有平面托里的构想，均以$ s^3 $ .... meromormorphic映射研究。在这项研究中，Y.Aihara证明，对于复杂的投影空间中的每一个超出表情$ d $ d $ d $，都存在一个从复杂平面到投影空间的全体形态曲线，其不足$ d $的不足是积极的，小于1.3.在恒定的平均曲率表面和反向转换上的研究。在这项研究中，J.Inoguchi证明了恒定平均曲率表面的Bianchi-Backlund转换等效于Darboux转换和简单的型敷料。4。研究动态系统。在这项研究中，K。Sakai证明了一组宽阔的向量场的内部$ C^1 $的内部特征是一组矢量场，而没有满足Axiom a和Quasi-Transerversality条件的奇异性。

项目成果

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

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

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