Representation formulas of Weierstrass type in submanifold theory

子流形理论中Weierstrass型的表示公式

• 批准号: 09640120
• 负责人: INOUE Hisao
• 金额: $ 2.18万
• 依托单位: Kumamoto University
• 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-09640120/
• 关键词: minimal surfaces; Weierstrass representation; monodoromoy; ODE; integrable system; モノドロミ-問題

We investigated a global properties of Weierstrass representation. First of all, we studied a fundamental problem related to global problems. In particular, as the monodromy problem for minimal surfaces in Euclidean geometry is considered as a period problem of certain integral of holomorphic forms, that of CMC-1 surfaces in hyperbolic space can be considerd as a monodromy problem of a ordinary differential equation on Riemann surfaces. In this context, the monodromy problem is the condition for SL(2, C)-monodromy group to be reduced to the unitary group. To find a criterion of such a condition is difficult. However, when a problem satisfies some symmetric properties, it can be solved explicitly. Using this fact, we have constructed a large amount of examples of CMC-1 surfaces. Related to this problem, we investigated metrics of constant positive curvature with conical singularities, and obtained a classification result.Related to classification of CMC-1 surfaces, we defined a homology invariant on CMC-1 surface, which is called flux, and proved some non-existence theorem using this invariant. Moreover, using Weierstrass representation for maximal surfaces in Minkowski 3-space, we have classified of surfaces with cone-like singularities with some finiteness.

我们研究了韦尔斯特拉斯表示的一个整体性质。首先，我们研究了一个与全球问题相关的根本问题。特别地，由于欧几里德几何中极小曲面的单调问题被看作是全纯形式的某些积分的周期问题，所以双曲空间中CMC-1曲面的单调问题可以看作是黎曼曲面上的常微分方程的单调问题。在本文中，单调问题是SL(2，C)-单调群归结为酉群的条件。要找到这样一个条件的标准是困难的。然而，当一个问题满足某些对称性质时，它可以显式求解。利用这一事实，我们构造了大量的CMC-1曲面的例子。结合CMC-1曲面的分类，定义了CMC-1曲面上的一个同调不变量，称为流量，并利用这个不变量证明了CMC-1曲面上的一些不存在定理。此外，利用三维Minkowski空间中极大曲面的Weierstrass表示，我们对具有一定有限性的锥形奇点曲面进行了分类。

项目成果

M.Umehara and K.Yamada: "Metrics of constant curvature 1 with three conical singularities on 2-sphere" Indiana 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”《印第安纳数学杂志》。

W.Rossman, M.Umehara and K.Yamada: "A new flux for mean curvature 1 surfaces in hyperbolic 3-space, and applications" Proceedings of American Math.Soc.(To appear). (1999)

W.Rossman、M.Umehara 和 K.Yamada：“双曲 3 空间中平均曲率 1 表面的新通量及其应用”美国数学会论文集（待发表）。

Y.Haraoka: "Monodromy of an Okubo system with non-semisimple exponents" Funkcialaj Ekvacioj. 40. 435-457 (1997)

Y.Haraoka：“具有非半简单指数的 Okubo 系统的 Monodromy”Funkcialaj Ekvacioj。