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.

本项目已专门研究与歧管和图形光谱几何相关的以下主题。（1）图的光谱几何形状：我们已经证明了著名的faber-krahn inequlity inequlity inequlity inequlity的类似物，用于欧几里得空间中的域名域，并找到了与等级层面相对于组合构造的两种方法。这些是参考文献的内容。此外，我开始研究Heisenberg Group的无限图封面上定义的随机步行的伴侣行为。（2）GEL'FAND逆光谱问题的稳定性：该项目是与Y. V. Kurylev和M. Lassas的联合工作。 GEL'FAND的反光谱问题是确定带有谱系边界的Riemannian歧管和Neuman Laplacian的边界值。结果是贝里舍夫和库里尔夫的结果，结合了塔塔鲁的近似控制能力结果。然后，下一个挑战之一是稳定性。我们首先在歧管类别中获得了稳定性结果，包括曲率衍生物的条件，然后成功地将其删除。此外，我们已经在具有边界的歧管中获得了谐波坐标的存在结果。这些结果在很大程度上取决于紧凑性论证，因此不能给出有效的估计。但是，我们也有部分结果的前置估计。