Differential Geometric Reserch on Manifolds

点击翻译按钮获取中文摘要

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.