Convergence of metric measure spaces and energy forms

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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