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

我们得到了以下结果：(1)我们构造了紧辛流形上的Yang-Mills联络理论。(2)我们根据无限图的体积增长、进出度增长来估计其Cheeger常数、热核和格林核。(3)我们确定了平面区域上Dirichlet特征值问题的有限元方法的刚度矩阵和质量矩阵。(4)我们计算了半正则无限图的Cheeger常数、热核和Green核，并给出了每个无限图的显式比较定理。(5)我们将Yang-Mills理论推广到Weyl结构，建立了关于Weyl流形和仿射联络的Atiya-Hitchin-Singer理论。(6)建立了离散广义仿射曲面理论并给出了它的环群描述。(7)证明了仿射微分几何、Weyl几何、Yang-Mills理论之间的关系。(8)定义了从CR流形到Riemannan流形的伪调和映射的概念，给出了第一变分公式和第二变分公式。(9)阐明了每种Yang-Mills理论在Kaehler流形、CR流形和辛流形上的关系，刻画了紧辛流形上的Yang-Mills极小元。

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.