Differential Geometric Reserch on Manifolds

Kenmotsu发表了一篇论文，证明了部分正曲率流形中极小子流形的相交定理。Kenmotsu正在研究二维复空间形式中具有常高斯曲率的极小曲面的Kaehler角的局部行为：为了对这种极小曲面进行分类，我们首先得到了这种极小曲面的第二基本形式的微分几何刻画。利用它，我们得到了一个关于Kaehler角的超定系统。这归结为一个由两个常微分方程组成的系统，根据曲面的高斯曲率和环境空间的曲率的大小，这两个系统是不同的。我们对这些系统进行了广泛的分析，证明了它们即使在局部也不存在非平凡的公共解。它蕴含了极小曲面的局部分类定理。Fukaya在一般情况下证明了Arnold猜想。对于子流形几何的研究，R.Miyaoka广泛地研究了复射影空间中极小曲面与Toda方程之间的关系，并在Crelle期刊上发表了她的结果。山田在双曲空间中建立了常平均曲率曲面理论，在整体黎曼几何的研究中，Suyama给出了一种构造标准球的微分拓扑的新方法，并应用它证明了0.654个Pinch紧致Rimanan流形的一个可微Pinch定理.T.Sakai编写了一本关于全球黎曼几何的教科书，该教科书由美国数学学会出版。

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.