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表示公式及其概括研究了三维欧几里得空间中最小表面的性质。首先，我们为在三维欧几里得空间中最小表面的通量反向问题提供了肯定的结果。此外，作为（复杂的分析）通量的概括，我们定义了一种新的同源性不变式，也称为“通量”，用于双曲线三个空间中恒定平均曲率的表面。使用通量的平衡公式，我们为双曲线空间中的恒定平均曲率一个表面提供了一些不存在的结果。由于这种不存在的结果的延续，我们试图将完全曲率的完全曲率分类为完全曲率的柔软空间中的完整恒定平均曲率一表面，并且我们获得了与总均值相比的完整曲线，因为一般或相等的曲率是4 4的总体曲率。 WeierStrass型表示的最小表面的公式具有较高的欧几里得空间，我们在某些非紧凑型对称空间中定义了具有全态右高斯图的表面概念，并获得了Weierstrass-Bryant-Bryant类型表示。作为该公式的应用，我们获得了用于总绝对曲率的Osserman型不平等。

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。

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。

T.Kurose: "Conformal-projective geometry of statistical manifolds"the Interdisciplinary Information Science. (to appear).

T.Kurose：“统计流形的共形射影几何”跨学科信息科学。