Global analysis of the heat kernels on Riemannian manifolds and graphs

黎曼流形和图上热核的全局分析

基本信息

批准号：
16340044
负责人：
URAKAWA Hajime
金额：
$ 10.43万
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：
2004
资助国家：
日本
起止时间：
2004 至 2006
项目状态：
已结题

项目摘要

The heat kernels of compact Riemannian manifolds converge to the equilibrium when time goes to infinity. We studied the rates of the heat kernels how do they reflect from the geometric structures of Riemannian manifolds. We showed the convergence rates are Lipshitz continuous on the deformation of Riemannian manifolds, we gave their upper estimation in terms of Ricci curvature and diameter, and also the upper estimation in terms of the non-zero first eigenvalue of the Laplacian. We gave their precise lower and upper estimations in the case of compact Riemannian symmetric spaces of rank one.A Yang-Mills connection is a critical point of the Yang-Mills functional, and this is an analogue of harmonic map which is a critical point of the energy functional. Recently, the notion of biharmonic map was introduced which is a critical point of the 2-energy functional. We introduced the notion of 2-Yang-Mills connection which is a critical point of the 2-Yang-Mills functional. This notion is a natural generalization of Yang-Mills connection, and many further studies would be expected.We introduced quite new method to visualize the Dirichlet or Neumann boundary eigenvalue problem of the Laplacian on plane domains. This method improved 20 percents fast comparing the known methods and reduced many steps input the data into computers. This new method made visualizations of the eigenvalue problems of the Laplacian on compact surfaces and bounded three dimensional domains. We applied to get patent of this method for programming of computer.

当时间到达无穷大时，紧凑的riemannian歧管的热核会趋于平衡。我们研究了热核的速率，它们如何反映出黎曼流形的几何结构。我们表明收敛速率是lipshitz在黎曼歧管的变形上的连续，我们在ricci曲率和直径方面给出了它们的上限，也给出了laplacian的非零第一本特征值的上限。在紧凑的riemannian对称空间的情况下，我们给出了它们的精确和上层估计。杨米尔斯连接是Yang-Mills功能的关键点，这是谐波映射的类似物，这是能量功能的关键点。最近，引入了Biharmonic图的概念，这是2能功能的关键点。我们介绍了2-yang-mills连接的概念，这是2-yang-mills功能的关键点。这一概念是杨利尔斯连接的自然概括，预计会有许多进一步的研究。我们引入了相当新的方法，以可视化laplacian在平面域上的dirichlet或neumann边界特征值问题。该方法改进了20个易位，比较已知方法并将许多步骤减少到计算机中。这种新方法使Laplacian在紧凑型表面和有界的三维域上的特征值问题进行了可视化。我们申请了该方法用于编程计算机的专利。