Construction of submanifold with constant mean curvature, and its applications

常平均曲率子流形的构造及其应用

• 批准号: 10440024
• 负责人: YAMADA Kotaro
• 金额: $ 3.78万
• 依托单位: Kyushu University (2000)Kumamoto University (1998-1999)
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B).
• 财政年份: 1998
• 资助国家: 日本
• 起止时间: 1998 至 2000
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-10440024/
• 关键词: minimal surfaces; Weierstrass representation; CMC-1 surface; Gauss map; Osserman inequality; Total curvature; 非コンパクト型対称空間; キーワード7; キーワード8; 低平均曲率曲面; フラックス; モノドロミー問題; 定平均曲率曲面; 常微分方程式; 可積分系

We investigated properties of minimal surfaces in the three dimensional euclidean space using the Weierstrass representation formula, and generalizations of them. First, we gave an affirmative result for an inverse problem of flux for minimal surfaces in the three dimensional euclidean space. Moreover, as a generalization of (a complex analytic) flux, we defined a new homology invariant, which is also called as "flux", for surfaces of constant mean curvature one in the hyperbolic three space. Using the balancing formula of the flux, we proved some non-existence results for constant mean curvature one surface in hyperbolic space.As a continuation of this non-existence results, we tried to classify the complete constant mean curvature one surface in hyperbolic space with low total absolute curvature, and we obtained the complete classification for surfaces with total absolute curvature less than or equal to 4π.On the other hand, as a generalization of the Weierstrass-type representation formula for minimal surface with higher dimensional euclidean space, we defined a notion of surfaces with holomorphic right gauss map in some non-compact type symmetric space, and obtained the Weierstrass-Bryant type representation formula. As an application of this formula, we obtained an Osserman-type inequality for total absolute curvature.

我们使用Weierstrass表示公式研究了三维欧几里德空间中最小曲面的性质，并对其进行了推广。首先，我们对三维欧几里德空间中最小曲面的通量反问题给出了肯定的结果。 （复杂的解析）通量的推广，我们使用平衡为双曲三空间中的恒定平均曲率一的表面定义了一个新的同调不变量，也称为“通量”。根据通量公式，我们证明了双曲空间中恒定平均曲率一个曲面的一些不存在结果。作为这种不存在结果的延续，我们尝试对双曲空间中总绝对值较低的完整恒定平均曲率一个曲面进行分类曲率，并且我们获得了总绝对曲率小于或等于 4π 的曲面的完整分类。另一方面，作为具有更高的最小曲面的 Weierstrass 型表示公式的推广在维欧氏空间上，我们定义了某种非紧型对称空间中的全纯右高斯映射的曲面概念，并得到了Weierstrass-Bryant型表示公式，作为该公式的应用，我们得到了总的Osserman型不等式。绝对曲率。

项目成果

T.Kurose: "Conformal-projective geometry of statistical manifold"The Interdisciplinary Information Sciences. (to appear).

T.Kurose：“统计流形的共形射影几何”跨学科信息科学。

Y.Haraoka: "Quadratic relations for confluent hypergeometric functions on Z_<2,n+1>" Funkcialaj Ekvacioj. (To appear). (1999)

Y.Haraoka：“Z_<2,n 1> 上汇合超几何函数的二次关系”Funkcialaj Ekvacioj。

Shin Kato,Masaaki Umehara and Kotaro Yamada: "General existence of minimal surfaces of genus zero of catenoidal ends and prescribed flux"Communications in Analysis and Geometry. (to appear).

Shin Kato、Masaaki Umehara 和 Kotaro Yamada：“悬链末端零属极小曲面的一般存在性和指定通量”分析与几何通讯。

M.Umehara and K.Yamada: "Metrics of constant curvature 1 with three conical singularities on 2-sphere" Inidiana Journal of Mathematics. (To appear). (1999)

M.Umehara 和 K.Yamada：“2 球面上具有三个圆锥奇点的常曲率 1 的度量”Inidiana 数学杂志。

J.Itoh and M.Tanaka: "The dimension of a cut locus on a smooth riemannian manifold" Tohoku Mathematical Journal. 50. 571-575 (1998)

J.Itoh 和 M.Tanaka：“光滑黎曼流形上切割轨迹的维数”东北数学杂志。