Geometric structure and topology of manifolds and graphs

流形和图的几何结构和拓扑

基本信息

批准号：
10640078
负责人：
KATSUDA Atsushi
金额：
$ 2.05万
依托单位：
Okayama University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1998
资助国家：
日本
起止时间：
1998 至 1999
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-10640078/
关键词：
spectrum inverse problem stability the Hausdorff distance Laplacian isospectral graphs isoperimetric constants

项目摘要

The present project has been devoted to the study on the following subjects related to spectral geometry of manifolds and graphs.(1) Spectral geometry of graphs : We have proved the analog in graphs of the celebrated Faber-Krahn inequlity for domains in Euclidean spaces and find two methods of construction of isospectral pairs of graphs with respect to combinatorial Laplacian. These are contents of References. Moreover, I start to investigate aymptotic behavior of random walks defined on infinite cover of finite graphs by the Heisenberg group.(2) Stability of the Gel'fand inverse spectral problem : This project is joint work with Y. V. Kurylev and M. Lassas. The Gel'fand inverse spectral problem is to determine a Riemannian manifold with boundary from the spectrum and the boundary value of the Neuman Laplacian. This is solved by results Belishev and Kurylev combining the approximate controllabity results by Tataru. Then, one of next challenge is the stability. We first obtained stability results in the class of manifolds including the condition on the derivative of curvature and later, succeed to remove it. Moreover, we have obtained the existence results of harmonic coordinates in manifolds with boundary. These results are heavily depend on compactness arguments and thus, no effective estimate can not be given. However, we also have partial results foreffective estimates.

本项目致力于研究以下与流形和图的谱几何相关的主题。（1）图的谱几何：我们已经证明了著名的 Faber-Krahn 不等式在欧几里得空间中的域的图中的类比，以及找到关于组合拉普拉斯算子构建等谱图对的两种方法。这些是参考文献的内容。此外，我开始研究海森堡群在有限图的无限覆盖上定义的随机游走的渐近行为。(2) Gel'fand 逆谱问题的稳定性：该项目是与 Y. V. Kurylev 和 M. Lassas 的共同工作。 Gel'fand逆谱问题是根据谱和纽曼拉普拉斯算子的边界值来确定有边界的黎曼流形。 Belishev 和 Kurylev 结合 Tataru 的近似可控性结果解决了这个问题。那么，下一个挑战之一就是稳定性。我们首先得到了流形类的稳定性结果，包括曲率导数的条件，后来成功地去除了它。此外，我们还得到了有边界流形中调和坐标的存在性结果。这些结果在很大程度上取决于紧致性参数，因此无法给出有效的估计。然而，我们也有有效估计的部分结果。