A Modern Approach to Geometry of Curves and Surtaces and its Applications

曲线和曲面几何的现代方法及其应用

• 批准号: 09640106
• 负责人: UMEHARA Masaaki
• 金额: $ 2.24万
• 依托单位: HIROSHIMA UNIVERSITY (1998)Osaka University (1997)
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1997
• 资助国家: 日本
• 起止时间: 1997 至 1998
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09640106/
• 关键词: Cuvue; Suvface; Vertex; Constant mean cuvuature; Conlcal Singularity; 双曲型空間; 錐的特異点

We get the following results :1. The head investigator Masaaki Umehara gave a flux formula for surfaces of constant mean curvature 1 (i.e. CMC-1 surfaces) in the hyperbolic 3-space as an analogy for that of minimal surfaces in the Euclidean space, which is a joint work with Wayne Rossman (Kobe Univ.) and Kotaro Yamada (Kumamoto Univ.). As an application, he and K.Yamada classified all conformal metrics of constant curvature 1 with three conical singularities on 2-sphere.2. The head investigator Masaaki Umehara and Gudlaugur Thorbergs-son (Koln Univ.) gave a refinement of the classical four vertex theorem on simple closed curves, where they gave an abstract intrinsic approach regarding the theorem as a kind of homology theory on S^1 whose first Betti number is the number of vertices. As an application, they prove a four vertex theorem on some kind of space curves and also gave a new proof of the four vertex theorem for diffeo-morphisms on S^1. Moreover, they generalized the theory for much higher order geometry and gave sharp estimates for affine vertices on simple closed curves.3. The investigator Koiso investigated the elasticas in a Riemannian manifolds by analytic approach.

我们得到以下结果：1。负责人Masaaki Umehara为双曲线3空间中的恒定平均曲率1（即CMC-1表面）提供了通量公式，作为欧几里得空间中最小表面的表面，这是与Wayne Rossman（Kobe Univ。作为一种应用，他和K.Yamada将恒定曲率1的所有共形指标与2片圆锥形的三个圆锥形奇异性分类为2。调查员Masaaki Umehara和Gudlaugur Thorbergs-Son（Koln Univ。）对经典的四个顶点定理进行了简单的封闭曲线的改进，在此，他们给出了关于该定理作为S^1的一种同性恋理论的抽象固有方法，其第一个Betti的第一个betti编号是Vertices的数量。作为一个应用，他们在某种空间曲线上证明了四个顶点定理，还提供了S^1上的四个顶点定理的新证明。此外，他们对高阶几何形状的理论概括了，并在简单的封闭曲线上给出了仿射顶点的尖锐估计。3。研究者考索通过分析方法中的riemannian歧管调查了弹性。

项目成果

Wayne Rossman: "A new flux for mean curvature 1 surfaces in hyperbolic 3-space, and applications" Proc.Amer.Math.Soc.掲載予定.

Wayne Rossman：“双曲 3 空间中平均曲率 1 表面的新通量及其应用”Proc.Amer.Math.Soc 即将出版。

Wayne Rossman: "A new flux for mean curvature 1 surfaces in hy-perbolic 3-space, and applications" Proc.Amer.Math.Soc.(to appear).

Wayne Rossman：“双曲 3 空间中平均曲率 1 表面的新通量及其应用”Proc.Amer.Math.Soc.（即将出现）。

M.Umehara and K.Yamada: "A duality on CMC-1 surfaces in hyperbolic space,and a hypebolic anolog of the Ossermonn inequality" Tsukuba J.Math.21. 229-237 (1997)

M.Umehara 和 K.Yamada：“双曲空间中 CMC-1 表面的对偶性，以及 Ossermonn 不等式的双曲模拟”Tsukuba J.Math.21。

W.Rossman, M.Umehara and K.Yamada: "Irredrclble constant mear curoatue 1 surfaces in hyperbolic space with positive gerus" Tohoku Math.J.49. 449-484 (1997)

W.Rossman、M.Umehara 和 K.Yamada：“具有正 gerus 的双曲空间中的不可预测的常数 mear curoatue 1 曲面”Tohoku Math.J.49。

Masaaki Umehara: "Metric of constant curvature 1 with three conical singularities on 2-sphere" Illinois Journal of Mathematics. 掲載予定.

Masaaki Umehara：“2 球面上具有三个圆锥奇点的常曲率度量 1”，《伊利诺伊州数学杂志》即将出版。