CURVATURE AND STRUCURE OF SPACES

1.本研究项目的首席研究员T.Sakai一直致力于这一研究主题：黎曼流形的各种度量不变量之间的关系，以及它们与流形结构的关系。最近，他研究了含有一个函数f的黎曼流形的结构，如果f的拉普拉斯函数的绝对值由一个非负常数从下有界，则得到了一个关于f的拉普拉斯函数绝对值的不等式。此外，当上面的等式成立时，他确定了流形的黎曼结构。在本助学金的支持下，他调查了上述结果的扰动版本。最近，T.Colding和J.Cheeger发展了新的方法来研究Ricci曲率下界的黎曼流形的结构。将他们的方法应用于给定的黎曼流形M允许一个梯度为常范数1的函数f的情形，Sakai得到了f…更进一步：假设M的Ricci曲率由一个常数从下有界，且f的拉普拉斯算子的绝对值从上面的等式情形略有扰动，则M在Gromov-Hausdorff距离中接近模型黎曼流形(它表现为该不等式的等式情形，对应于欧几里得或双曲几何)，当限于距离球时。此外，他将这些结果推广到模型空间是一般翘曲乘积空间的情形。其他研究者的研究结果：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Lie群和H-空间的同伦性质，Shimakawa与Rabel构造了一般的位形空间的同调理论。Tanaka研究了拟线性发展方程，得到了经典解的存在唯一性的一些结果。Takeuchi研究了p-调和映射和p-拉普拉斯的第一本征值估计。较少

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