Convergence of metric measure spaces and energy forms

对保守正则Dirichlet空间的收敛性进行了研究，并对极限空间进行了分析。所考虑的空间集特别包括黎曼流形、黎曼多面体、子黎曼流形。这种收敛是指能量形式和谱收敛的变分收敛，称为伽马收敛。主要结果如下：(1)考虑空间开子集的收敛序列，验证了核函数、格林函数、调和函数等的收敛性。此外，我们通过用极限空间的奇点来描述函数的能量集中现象或更一般的最小能量映射，展示了一种新的见解。(2)建立了度量图的收敛理论，即一维黎曼多面体。在我们的论证中，有效抵抗起着重要的作用。利用这个概念，我们讨论了能量形式的c…More收敛性，在一定变分意义上的Dirichlet能量形式，以及在Gromov-Hausdorff意义上的图的度规结构。该理论以一维空间集合上的一些特殊现象为特征。事实上，我们的极限空间中包含了一类重要的所谓分形集。我们提出了一种新的分形分析方法。(3)局部有限的无限网络可以看作是有限网络的极限。从这个角度出发，我们研究了无限网络的Royden紧化。证明了Royden边界在拟等距变换下的不变性，进一步证明了Royden紧化在有效阻力一致有界条件下的度量性。无限网络可以被认为是完备黎曼流形的良好近似。我们比较了完全黎曼流形和某些网络，重点讨论了有限狄利克雷能量的函数。少

We curried out the studies on convergence of conservative regular Dirichlet spaces and some analysis of the limit spaces. The set of spaces under consideration includes particularly Riemannian manifolds, Riemannian polyhedra, sub-Riemannian manifolds. The convergence is meant by a variational convergence, called the Gamma convergence, of energy forms and spectral convergence.The main results are described as follows :(1)Considering a convergent sequence of open subsets of the spaces, we verified the convergence of the kernel functions, the Green functions, harmonic functions and so on. Moreover we showed a new insight into phenomena of concentration of energies of functions or more generally maps of least energy by describing it in terms of singularities of the limit spaces.(2)We developed the convergence theory concerning metric graphs, that is, one dimensional Riemannian polyhedra. In our arguments, the effective resistance plays important roles. Using this notion, we discussed the c … More onvergence of energy forms, the Dirichlet energy forms, in a certain variatinal sense, and also the metric structure of the graphs in the Gromov-Hausdorff sense. The theory features some particular phenomena on the set of one dimensional spaces. An important class of so called fractal sets is in fact included in our limit spaces. We proposed a new approach to the analysis on fractals.(3)Locally finite, infinite networks may be viewed as limits of finite networks. From this point of view, we studied Royden's compactification of infinite networks. Among other things, we proved the invariance of the Royden boundaries under quasi-isometric transformations, and further the metrizability of the Royden compactification under the condition of the effective resistance being bounded uniformly. Infinite networks can be considered as good approximations of complete Riemannian manifolds. We compared complete Riemannian manifolds with certain networks, focusing the functions of finite Dirichlet energies. Less