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Global Analysis of the Spectrum of an Infinite Graph

无限图谱的全局分析

基本信息

批准号：
10440056
负责人：
URAKAWA Hajime
金额：
$ 8.64万
依托单位：
Tohoku University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B).
财政年份：
1998
资助国家：
日本
起止时间：
1998 至 2000
项目状态：
已结题

项目摘要

We have obtained the following results :(1) We estimated the first eigenvalue and sectional curvature of compact symmetric spaces, and gave counter example of Elworthy-Rosenberg.(2) We obtained the Faber-Krahn type estimate for the first eigenvalue of the Dirichlet eigenvalue problem for a coonected finite graph with boundary.(3) We obtained the Barta type estimate of the Dirichlet eigenvalue problem for a graph, and sharp estimate of the infimum of the spectrum of an infinite graph.(4) We introduce the notion of surgery, and gave its application.(5) We characterized homogeneous spaces admitting affine projectively flat connections.(6) We gave the first variational formula of a natural functional on the space ofWeyl structures.(7) Harmonic morphisms play important roll in differential geometry oh harmonic maps. We consider the discrete harmonic morphisms, and obtained their complete characterization.(8) We obtained theory of solutions of the inhomogeneous Yang-Mills equation.(9) We gave a characterization of harmonic morphism between two graphs.(10) We obtain a general estimation formula of the spectrum of the discrete Laplacian for an infinite graph. Our method is quite new using our new notion of the incresing degree and decreasing one relative to distance from a fixed vertex.

得到以下结果：(1)我们估计了紧对称空间的第一特征值和截面曲率，并给出了Elworthy-Rosenberg的反例。(2)得到了一类带边界的连通有限图的Dirichlet特征值问题的第一特征值的Faber-Krahn型估计。(3)得到了图的Dirichlet特征值问题的Barta型估计，以及无限图谱的极小值的锐估计。(4)介绍了外科手术的概念，并给出了其应用。(5)刻画了具有仿射射影平面连接的齐次空间。(6)给出了weyl结构空间上的自然泛函的第一个变分公式。(7)调和态射在微分几何调和映射中占有重要地位。我们考虑了离散谐态射，并得到了它们的完整表征。(8)得到了非齐次Yang-Mills方程的解理论。(9)给出了两个图间调和态射的一个刻划。（10）得到了无限图离散拉普拉斯谱的一般估计公式。我们的方法是相当新的使用我们的新概念的增加和减少一个相对于距离一个固定的顶点。