Geometric variational problems and submanifolds.

首先研究了球面S^n中具有2-参数大球族的三维极小子流形。在复射影空间中，（定向）大圆集与真实的（定向）2-平面Grassmannian和复二次曲面Q^相同<n-1>。然后将S^n中具有2-参数大球面族的子流形M构造为Q^n中二维曲面上的圆丛<n-1>。我们证明了：（1）如果Q^中的全纯曲线是复1维<n-1>的，则相应子流形M在S^n中的Gauss映射是退化的;（2）如果Q^中的全纯曲线<n-1>是一阶迷向的，则相应子流形M是极小的。和Reiko Miyaoka（上智大学），我们把以前的结果推广到球面中的高维子流形。特别地，如果Q^n中的复子流形M<n-1>是一阶迷向的，则S^n中相应的具有（dim_R）-参数族的大球面的子流形M（M上的圆丛）是严格的.利用Harvey和Lawson关于标定的结果，我们可以在复欧氏空间中构造特殊的拉格朗日子流形。证明了从秩为2，3，5的真实的Grassmannian中的某些齐次子流形中，可以构造S^n中的齐次严格子流形M，使得M的Gauss映射退化并满足Ferus等式.它们是E.Cartan等参超曲面的自然推广。

First we invetigated 3-dimensional minimal submanifolds with 2-parameter family of great spheres in a sphere S^n. Set of (oriened) great circles is identified with real (oriented) 2-plane Grassmannian and the complex quadric Q^<n-1> in a complex projective space. Then the submanifold M with 2-parameter family of great spheres in S^n is constructed as a circle bundle over a 2-dimensional surface Σ in Q^<n-1>. We showed that (1) Σ is a complex 1-dimensional holomorphic curve in Q^<n-1>, then the Gauss mapping of the corresponding submanifold M in S^n is degenerate, (2) the holomorphic curve Σ in Q^<n-1> is first order isotropic, then the corresponding M is minimal.Next, by a joint research with Goo Ishikawa (Hokkaido Univ.) and Reiko Miyaoka (Sophia Univ.), we generalized the former results to higher dimensional submanifolds in spheres. Especially, if a complex submanifold Σ in Q^<n-1> is first order isotropic, then the corresponding submnanifold M (circle bundle over Σ) with (dim_R Σ)-parameter family of great spheres in S^n is austere. Hence we can construct special Lagrangian submanifolds in complex Euclidean spaces by using the results with respect to the calibration by Harvey and Lawson. And we showed that from some homogeneous submanifolds in real Grassmannians of rank 2, 3, 5, one can construct homogeneous austere submanifolds M in S^n such that the Gauss mapping of M is degenerate and satisfying Ferus' equality. They are a natural generalization of E.Cartan's isoparametric hypersurfaces.