Global Analysis of the heat kernel and Green kernel of an Infinite Graph

无限图热核和绿核的全局分析

• 批准号: 13440051
• 负责人: URAKAWA Hajime
• 金额: $ 9.41万
• 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: 2001
• 资助国家: 日本
• 起止时间: 2001 至 2003
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-13440051/
• 关键词: Yang-Mills connection; Dirichlet eigenvalue problem; finite element method; infinite graph; affine connection; oseudharmonic map; Green kernel; symplectic manifold; ヤング・ミズル接続; ワイル接続; モデュライ空間; チーガー定数; 熱核; 鋭角三角形分割; 境界値問題; 固有値; 離散計算幾何; 非退化CR

We have obtained the following results:(1)We constructed the theory of Yang-Mills connections over compact symplectic manifolds.(2)We estimated the Cheeger constant, the heat kernel and the Green kernel for an infinite graph in terms of the volume growth, growth of in and out degree.(3)We determined the stiffness and mass matrices of the finite element method for the Dirichlet eigenvalue problem for a plane domain.(4)We calculated the Cheeger constant, the heat kernel and Green kernel of semi-regular infinite graphs and gave the explicit comparison theorem for every infinite graph.(5)We extended Yang-Mills theory to Weyl structure, and established Atiya-Hitchin-Singer theory to Weyl manifolds, and to affine connections.(6)We formulated discrete improper affine surface theory and show its loop group description.(7)We showed the relation in affine differential geometry, Weyl geometry, Yang-Mills theory.(8)We defined the notion of pseudoharmonic maps from CR manifolds to a Riemanninan manifold, and showed the first variation formula and the second variation formula.(9)We clarified the relation of each Yang-Mills theory on Kaehler manifolds, CR manifolds, and symplectic manifolds, and characterized the minimizers of the Yang-Mills functional over compact symplectic manifolds.

我们已经获得了以下结果：（1）我们在紧凑的符号歧管上构建了杨木连接的理论。（2）我们估计了cheeger常数，热核，绿核和绿色内核，用于无限图，以体积的增长，in in and Out In和Out of In和Out of In和Out。 （4）我们计算了半规则无限图的花序常数，热内核和绿色内核，并给出了每个无限图的明确比较定理。（5）我们将Yang-Mills理论扩展到Weyl结构，并确定了Atiya-Hitchin-Hitchin-Hitchin-singer理论对Weyl-Hitchin和supper to super corportion confore confine Inder and cornection。描述。（7）我们显示了伴有差异几何形状，魏尔几何形状，阳米尔斯理论的关系。（8）我们定义了从cr歧管到riemanninan歧管的伪harmonic地图的概念，并显示了第一个变化公式和第二个变异式。歧管，并表征了阳米尔的最小化器在紧凑的符号歧管上的功能。

项目成果

H.Urakawa: "The Cheeger constant, the heat kernel and the Green kernel of an infinite graph"Monatshefte fur Mathematics. 138. 225-237 (2003)

H.Urakawa：“无限图的奇格常数、热核和格林核”《数学月刊》。

F.Ohtsuka: "Total excess on length surface"Mathematische Annalen. 319. 675-706 (2001)

F.Ohtsuka：“长度表面上的总过剩”数学年鉴。

H.Urakawa: "Yang-Mills theory and conjugate connections"Differential Geometry and Its Applications. (2002)

H.Urakawa：“杨-米尔斯理论和共轭连接”微分几何及其应用。

浦川肇: "数理システム科学"日本放送協会出版会. 230 (2002)

浦川肇：《数学系统科学》日本广播公司出版230（2002）。

J.Itoh: "Acute triangulations of the regular icosahedral surface"Discrete Computational Geometry. (発表予定). (2003)

J.Itoh：“正二十面体表面的锐角三角剖分”离散计算几何（待发表）。