Global analysis of the heat kernels on Riemannian manifolds and graphs

当时间趋于无穷大时，紧致黎曼流形的热核收敛于平衡点。我们研究了热核的速率是如何从黎曼流形的几何结构中反映出来的。证明了在黎曼流形的形变上收敛速度是Lipshitz连续的，给出了收敛速度关于Ricci曲率和直径的上估计，以及关于Laplacian的非零第一特征值的上估计.在秩为1的紧致Riemann对称空间中，给出了它们的精确上下估计. Yang-Mills联络是Yang-Mills泛函的一个临界点，它类似于调和映射，调和映射是能量泛函的临界点.最近，双调和映射的概念被引入，它是2-能量泛函的一个临界点。我们引入了2-Yang-Mills联络的概念，它是2-Yang-Mills泛函的一个临界点。这一概念是对Yang-Mills联络的自然推广，许多进一步的研究还有待于进一步的研究。我们引入了一种新的方法来可视化平面区域上Laplacian算子的Dirichlet或Neumann边界特征值问题。该方法比已知方法提高了20个百分点，减少了大量的数据输入计算机的步骤。这种新方法使紧致曲面和三维有界区域上的拉普拉斯算子的特征值问题可视化。该方法已申请专利用于计算机编程。

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.