Global analysis of the heat kernels on Riemannian manifolds and graphs

紧致黎曼流形的热核在时间趋于无穷时收敛于平衡态。研究了黎曼流形几何结构对热核的反射速率的影响。我们证明了黎曼流形变形的收敛率是Lipshitz连续的，我们给出了它们在里奇曲率和直径方面的上估计，以及在拉普拉斯的非零第一特征值方面的上估计。在秩为1的紧黎曼对称空间中给出了它们的精确上下估计。Yang-Mills连接是Yang-Mills泛函的一个临界点，这是调和映射的类似物，它是能量泛函的一个临界点。最近，引入了双调和映射的概念，它是二能泛函的一个临界点。我们引入了2-Yang-Mills连接的概念，它是2-Yang-Mills泛函的一个临界点。这一概念是对杨-米尔斯联系的自然推广，并期望有更多的进一步研究。介绍了一种新的方法来可视化平面上拉普拉斯算子的Dirichlet或Neumann边界特征值问题。该方法比已知方法提高了20%的速度，并减少了将数据输入计算机的许多步骤。该方法将拉普拉斯算子在紧曲面和有界三维域上的特征值问题可视化。我们申请了该方法的计算机编程专利。

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.