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Differential Geometric Reserch on Manifolds

流形微分几何研究

基本信息

批准号：
07304006
负责人：
KENMOTSU Katsuei
金额：
$ 5.31万
依托单位：
Tohoku University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (A)
财政年份：
1995
资助国家：
日本
起止时间：
1995 至 1996
项目状态：
已结题

项目摘要

Kenmotsu has published a paper, in which he proved theorems for intersections of minimal submanifolds in manifolds with partially positive curvature. Kenmotsu is studying local behavior of the Kaehler angles of minimal surfaces with constant Gaussian curvature in two dimensional complex space forms : In order to classify such minimal surfaces, at first we obtained differential geometric characterization of the second fundamental forms of such minimal surfaces. By using it we obtained an overdetermined system for the Kaehler angle. This is reduced to a system of two ODE's. By the values of the Gaussian curvature of the surface and the curvature of ambiant space, these systems are different. We developed analysis to these systems extensively and proved that they have no non trivial common solution even locally. It implies local classification theorem of suchminimal surfaces.Fukaya has proved the Arnold conjecture in the general setting. This is really exciting.For reserch of submanifold geometry, R.Miyaoka has studied relations between minimal surfaces in complex projective spaces and the Toda equations extensively and published her results in the Crelle Journal. Yamada has contributed to construct the theory of constant mean curvature surfaces in the hyperbolic spaces.For the reserch of global Riemannan geometry, Suyama has given a new method to construct diffeotopy of standard spheres and applying it he proved a differentiable pinching theorem for 0.654 pinched compact riemannan manioflds. T.Sakai has written a textbook of global Riemannian geometry which was published by the American Math.Society.

Kenmotsu发表了一篇论文，其中他证明了具有部分正曲率的流形中极小子流形的交点定理。Kenmotsu研究了二维复空间形式中具有常高斯曲率的极小曲面的Kaehler角的局部性质：为了对这类极小曲面进行分类，首先得到了这类极小曲面的第二基本形式的微分几何特征。利用它得到了Kaehler角的超定系统。这被简化为两个ODE的系统。根据曲面的高斯曲率和周围空间的曲率的大小，这些系统是不同的。我们对这些系统进行了广泛的分析，证明了它们甚至在局部也没有非平凡的公共解。福谷在一般情况下证明了Arnold猜想。在子流形几何的研究方面，宫冈R.Miyaoka对复射影空间中极小曲面与户田方程之间的关系进行了广泛的研究，并在Crelle Journal上发表了她的结果。Yamada在双曲空间中建立了常中曲率曲面理论，Suyama在整体Riemannan几何的研究中，给出了一种构造标准球面拓扑的新方法，并应用该方法证明了0.654pinched紧致Riemannan流形的一个可微pinching定理. T.Sakai写了一本由美国数学学会出版的整体黎曼几何教科书。