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，其中他证明了在具有部分正弯曲的歧管中最小的亚曼福尔德相交的定理。 Kenmotsu正在研究最小表面的Kaehler角度的局部行为，其恒定高斯曲率以二维复杂空间形式形式：为了对这种最小表面进行分类，首先，我们获得了这种最小表面的第二个基本形式的不同几何表征。通过使用它，我们获得了Kaehler角度的过度确定系统。将其简化为两个Ode的系统。根据表面高斯曲率的值和Ambiant空间的曲率，这些系统是不同的。我们对这些系统进行了广泛的分析，并证明它们在本地甚至没有非微不足道的共同解决方案。它暗示了这种占地表面的局部分类定理。Fukaya已证明了一般环境中的Arnold猜想。这确实令人兴奋。对于Submanifold几何学的研究，R.Miyaoka研究了复杂的投影空间中最小表面与Toda方程式之间的关系，并在Crelle Journal中发布了她的结果。 Yamada为在双曲线空间中构建了恒定平均曲率表面的理论。对于Global Riemannan几何形状的研究，Suyama给出了一种新的方法来构建标准球的分散术，并应用了它，他证明了一个可靠的捏合理论，使Theorem对0.654 Pinched Compact Riemannanananan Maniioflds进行了捏合。 T.Sakai撰写了一本关于全球Riemannian几何形状的教科书，该文章由美国数学社会出版。