CURVATURE AND STRUCURE OF SPACES

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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