Differential equations and theory of submanifolds

我证明了具有6个重数为2的主曲率的等参超曲面的齐性，这是我几年来一直在研究的问题。我们还得到了Dorfmeister-Neher定理的一个新的证明，它统一地处理了重数为1的情形。研究了所得到的齐次超曲面，得到了如下结果：在重数为1的情况下，具有6个主曲率的超曲面是在具有3个主曲率的超曲面上的纤维，其中纤维不是局部测地球面。在重数2的情况下，纤维维度是6，而在重数1的情况下，这是3。纤维结构的发现是我们与G.Ishikawa和M.Kimura关于退化Gauss映射的结果的推广。此外，利用等参超曲面族填充环境空间的事实，我们得到了13维球面和7维球面之间有趣的关系。此外，利用这些超曲面作为例外群G_2的轨道，我们可以证明S^7-CP^2上存在一个完整群为G_2的度量。由此，我们将考虑一个实开版本的Calabi猜想，即当一个具有正Ricci曲率的紧致黎曼流形的某一部分被去掉时，它允许一个具有G_2完整的度量？这样，以G_2轨道表示的超曲面给我们提供了非常重要和有趣的问题。

I proved the homogeneity of isoparametric hypersurfaces with six principal curvatures with multiplicity two, which I had been tackling for several years. I also got a new proof of Dorfmeister-Neher's theorem which treats the multiplicity one case, in a unified manner.Investigating the resulted homogeneous hypersurfaces, I got the following As was known in the case of multiplicity one, the hypersurfaces with 6 principal curvatures are given as a fibration over those with 3 principal curvature, where the fibers aret otally geodesic spheres. In the case of multiplicity two, the fiber dimension is six, while in the case of multiplicity one, this is three. Discovery of the fibration structure is an extension of our former results on the degenerate Gauss mapping which was done with G. Ishikawa and M. Kimura.Moreover, using the fact that the family of isoparametric hypersurfaces fill the ambient space, we get an interesting relation between 13-dimensional sphere and 7-dimensional sphere. Furthermore, using that these hypersurfaces are given as orbits of the exceptional group G_2, we can show that there exists a metric on S^7-CP^2 of which holonomy group is G_2. From this, a real open version of Calabi conjecture will be considered, i.e., when a compact Riemannian manifolds with positive Ricci curvature from which a certain part removed, admits a metric with G_2 holonomy? In this way, hypersurfaces obtained as G_2 orbits suggest us very important and interesting problems.