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

得到了以下结果：(1)构造了紧辛流形上的Yang-Mills连接理论。(2)从体积增长、进出度增长等方面估计了无限图的Cheeger常数、热核和格林核。(3)确定了平面域Dirichlet特征值问题的有限元法的刚度矩阵和质量矩阵。(4)计算了半正则无限图的Cheeger常数、heat核和Green核，并给出了每个无限图的显式比较定理。(5)将Yang-Mills理论推广到Weyl结构，建立了适用于Weyl流形和仿射连接的Atiya-Hitchin-Singer理论。(6)建立了离散反常仿射曲面理论，给出了它的环群描述。(7)在仿射微分几何、Weyl几何、Yang-Mills理论中给出了这一关系。(8)定义了从CR流形到riemananninan流形的伪调和映射的概念，并给出了第一变分公式和第二变分公式。(9)阐明了Kaehler流形、CR流形和辛流形上各Yang-Mills理论的关系，并刻画了紧致辛流形上Yang-Mills泛函的极小值。

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.