Global Analysis of the Spectrum of an Infinite Graph

无限图谱的全局分析

• 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
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-10440056/
• 关键词: first eigenvalue; Dirichlet eigenvalue problem; spectrum; infinite graph; affine connection; harmonic morphism; Green kernel; graph with boundary; 離散ラプラシアン; チーガー定数; 熱核; 離散曲面; 離散調和射; 次数; 射影平坦; ヤング・ミルズ接続; ワイル構造; ディリクレ問題; バルタの定理; ファーベル・クラー

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.

我们得到了如下结果：（1）估计了紧致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)本文给出了有边连通有限图Dirichlet特征值问题第一特征值的Faber-Krahn型估计。(3)我们得到了图的Dirichlet特征值问题的Barta型估计和无限图的谱的下确界的精确估计。(4)本文介绍了外科手术的概念，并给出了它的应用. (5)我们的特点是齐性空间允许仿射射影平坦的联系。(6)给出了Weyl结构空间上自然泛函的第一变分公式。(7)调和射在调和映射的微分几何中起着重要的作用。我们考虑了离散调和态射，得到了它们的完全刻画。(8)我们得到了非齐次Yang-Mills方程的解的理论。(9)给出了两个图之间的调和态射的一个刻画。(10)得到了无限图的离散Laplacian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incresing度和减少一个相对于距离固定顶点。

项目成果

浦川肇: "解析入門-級数/複素関数/ベクトル解析-" 裳華房, 152 (1998)

Hajime Urakawa：“分析导论 - 级数/复函数/向量分析” Shokabo，152 (1998)

J.Itoh: "The Lipschitz continuity of the distance function to the ait locus"Transactions of American Mathematical Society. 353. 21-40 (2001)

J.Itoh：“距离函数到 ait 轨迹的 Lipschitz 连续性”美国数学会汇刊。

H.Urakawa: "Eigenvalue pinching theorems on compact symmetric spaces" Proceedings of American Mathematical Society. 126. 3065-3069 (1998)

H.Urakawa：“紧对称空间上的特征值箍缩定理”美国数学会论文集。

T.Ichiyama: "A conformal gauge invariant functional for Weyl structures and the first variation formula"Tsukuba Journal of Mathematics. 23. 551-564 (1999)

T.Ichiyama：“Weyl 结构的共形规范不变量泛函和第一变分公式”筑波数学杂志。

H.Urakawa: "A discrete analogue of the harmonic morphism"Harmonic morphisms, harmonic maps and related topics. 413. 97-108 (2000)

H.Urakawa：“调和态射的离散模拟”调和态射、调和映射和相关主题。