CURVATURE AND STRUCURE OF SPACES

1. 该研究项目的首席研究员T.Sakai一直致力于研究主题：黎曼流形的各种不变量之间的关系以及它们与流形结构的联系。最近，他研究了允许梯度为常范数的函数f的黎曼流形的结构，得到了在里奇曲率由一个非负常数从下有界的条件下f的拉普拉斯算子的绝对值的一个不等式。此外，他还确定了在上述不等式成立时流形的黎曼结构。在本补助金的支持下，他研究了上述结果的扰动版本。最近T.Colding和J.Cheeger发展了新的技术来研究Ricci曲率有界以下的黎曼流形的结构。将其方法应用于给定黎曼流形M允许一个梯度为常范数1的函数f的情况。更进一步：假设M的Ricci曲率从下以一个常数为界，且在上述不等式的相等情况下，f的拉普拉斯式的绝对值稍有扰动，则M在限定于距离球的Gromov-Hausdorff距离内接近于模型黎曼流形（表现为不等式的相等情况，对应于欧几里德几何或双曲几何）。进一步，他将这些结果推广到模型空间是一般翘曲积空间的情况。其他研究者的研究成果：Katsuda研究了图和黎曼流形的谱几何，并得到了Neumann边值问题反问题的稳定性结果。Mimura研究了李群和h空间的同调性质，Shimakawa构造了带标签的位形空间的一般同调理论。Tanaka研究了拟线性演化方程，得到了经典解的存在性和唯一性。Takeuchi研究了p-调和映射和p-拉普拉斯算子的第一特征值估计。少

1. T.Sakai, head investigator of this research program, has been working on the reseach theme : relationships between various metrical invariants of Riemannian manifolds, and their connection with the manifold structure. Recently, he studied the structure of Riemannian manifolds admitting a function f whose gradient is of constant norm, and obtained an inequality for the absolute value of the Laplacian of f provided that the Ricci curvatures are bounded from below by a nonnegative constant. Moreover, he determined the Riemannian structures of manifolds when equality holds in the inequality above. Under the support of the present Grant-in- Aid, he investigated the perturbed version of the above results. Recently T.Colding and J.Cheeger developed new techniques to study the structure of Riemannian manifolds with Ricci curvature bounded below. Applying their methods to the case where given Riemannian manifold M admits a function f whose gradient is of constant norm 1. Sakai obtained the f … More ollowing : Suppose that Ricci curvatures of M are bounded from below by a constant andthe absolute value of the Laplacian of f is slightly perturbed from the equality case of the inequality above, then M is close to a model Riemannian manifold (which appears as the equalty case of the inequality and corresponds to euclidean or hyperbolic geometry) in the Gromov-Hausdorff distance when restricted to distance balls. Furthermore, he generalized these results to the case where the model space is a general warped product space.2. Reseach results of other investigators : Katsuda studied spectral geometry of graph and Riemannian manifolds, and got stability results for the inverse problem of the Neumann boundary value problem. Mimura investigated homotopical properties of Lie groups and H-spaces, and Shimakawa constructed general homology theory for configuration spaces with rabel. Tanaka investigated quasi-linear evolution equation and got results on the existance and uniquness of classical solution. Takeuchi investigated p-harmonic maps and first eigenvalue estimates for the p-Laplacian. Less