Geometric variational problems and submanifolds.

首先，我们研究了球面S^n中具有两参数大球族的三维极小子流形.在复射影空间中，(定向的)大圆集与实(定向)2平面Grassman和复二次Q^&lt；n-1&gt；然后，将S^n中具有两参数大球族的子流形M构造为Q^&lt；n-1&gt；中二维曲面Σ上的圆丛。证明了(1)Σ是Q^&lt；n-1&gt；中的一维复全纯曲线，则对应的子流形M的高斯映射是退化的；(2)Q^&lt；n-1&gt；中的全纯曲线Σ是一阶迷向的，则对应的M是极小的。和Reiko Miyaoka(索菲亚大学)，我们把以前的结果推广到球面上的高维子流形上。特别地，如果Q^&lt；n-1&gt；中的复子流形Σ是一阶迷向的，则对应的子流形M(Σ上的圆丛)具有(dim_RΣ)-参数大球族在S^n中是严密的。因此，利用Harvey和Lawson的定标结果，我们可以在复欧氏空间中构造特殊的Lagrangian子流形。证明了从实Grassmannians中的一些2，3，5阶齐次子流形出发，可以构造出S^n中的齐次素子流形M，使得M的Gauss映射是退化的且满足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.