Studies on Bernstein type theorems for minimal submanifolds

最小子流形的Bernstein型定理研究

基本信息

批准号：
11640068
负责人：
OKAYASU Takashi
金额：
$ 2.18万
依托单位：
IBARAKI UNIVERSITY (2000)University of Toyama (1999)
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1999
资助国家：
日本
起止时间：
1999 至 2000
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-11640068/
关键词：
minimal submanifold normal connection equivariant differential geometry hyperbolic space 同変微分幾何

项目摘要

Let G be a compact connected Lie group, Φ be an orthonormal representation of G on R^n with codimension 2 principal orbit type. Such (G, Φ, R^n) were classified completely by Hsiang-Lawson (1971) and they are exactly those isotropy representations of symmetric spaces of rank 2.The orbit space R^n/G is a domain of R^2, it coinsides with the Weyl chamber R^2/W (W=Weyl group) of some symmetric space G/K.Hsiang (1982) reduced the problem of constructing hypersurfaces of constant mean curvature in R^n to solving the ordinaly differential equations of curves in R^2/W, and get many examples.In this research, using Hsiang's idea, we reduced the problem of constructing complete minimal submanifolds with flat normal connection in the Euclidean spaces, to solving some ordinaly differential equations of curves in R^2/W×R.The point is that submanifolds generated from rotating curves have always flat normal connection.Theorem 1 There are many codimension 2 irreducible complete minimal submanifolds with flat normal connection in the Euclidean spaces. Those examples are diffeomorphic to the following manifolds :S^P×S^q×R, SU (2)/T^2×R, G_2/T^2×R, F_4/Spin (8)×R,....In 2000 using similar idea we get the following theorem.Theorem 2 There are many codimension 2 irreducible complete minimal submanifolds with flat normal connection in the hyperbolic spaces. Those examples are diffeomorphic to S^p×S^q×R.

设 G 为紧连通李群，Φ 为 G 在 R^n 上的正交表示，余维 2 主轨道类型，这样的 (G, Φ, R^n) 已被 Hsiang-Lawson (1971) 完全分类，它们是正是 2 阶对称空间的各向同性表示。轨道空间 R^n/G 是 R^2 的域，它与 Weyl 室 R^2/W 重合某对称空间的（W=Weyl群）G/K.Hsiang (1982)将R^n中构造常平均曲率超曲面的问题简化为求解R^2/W中曲线的常微分方程，得到许多在这项研究中，我们利用Hsiang的思想，将在欧几里得空间中构造具有平坦法线连接的完全最小子流形的问题简化为求解一些常微分方程R^2/W×R 中的曲线。要点是由旋转曲线生成的子流形总是具有平坦法向连接。定理 1 在欧几里得空间中存在许多具有平坦法向连接的余维 2 不可约完全最小子流形这些例子是微分同胚的。到以下流形：S^P×S^q×R，SU (2)/T^2×R，G_2/T^2×R， F_4/Spin (8)×R,....在2000年使用类似的想法我们得到以下定理。定理2 在双曲空间中存在许多具有平坦法向连接的余维2不可约完全最小子流形这些例子与S微分同胚。 ^p×S^q×R。