Studies on Bernstein type theorems for minimal submanifolds

设G是紧连通李群，Φ是G在R^n上具有余维2主轨道型的正交表示。这样的（G， Φ， R^n）被Hsiang-Lawson（1971）完全分类，它们正是2阶对称空间的各向同性表示。轨道空间R^n/G是一个R^2的域，它与对称空间G/K的Weyl室R^2/W （W=Weyl群）重合。Hsiang（1982）将构造R^n中常平均曲率的超曲面问题简化为求解R^2/W中曲线的常微分方程，并给出了许多例子。本文利用Hsiang的思想，将构造欧几里得空间中具有平坦法向连接的完全极小子流形的问题简化为求解R^2/W×R中曲线的常微分方程。重点是由旋转曲线生成的子流形总是具有平法向连接。定理1欧几里德空间中存在许多余维2不可约的具有平法向连接的完全极小子流形。这些例子与以下流形是微分同态的：S^P×S^q×R， SU (2)/T^2×R, G_2/T^2×R, F_4/Spin (8)×R,....在2000年用类似的思想我们得到了下面的定理。定理2双曲空间中存在许多具有平法向连接的余维2不可约完全极小子流形。这些例子是S^p×S^q×R的微分同态。

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.