2005 Fiscal Year Final Research Report Summary
Convergence of metric measure spaces and energy forms
| Project/Area Number |
15340053
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Global analysis
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| Research Institution | Kanazawa University |
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Principal Investigator |
KASUE Atsushi Kanazawa Univ., Graduate School of Natural Science, Professor, 自然科学研究科, 教授 (40152657)
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| Co-Investigator(Kenkyū-buntansha) |
NAKAO Shintaro Kanazawa Univ., Graduate School of Natural Science, Professor, 自然科学研究科, 教授 (90030783)
TAKANOBU Satoshi Kanazawa Univ., Graduate School of Natural Science, Professor, 自然科学研究科, 教授 (40197124)
USHIJIMA Akira Kanazawa Univ., Graduate School of Natural Science, Lecturer, 自然科学研究科, 講師 (50323803)
NAKAGAWA Yasuhiro Kanazawa Univ., Graduate School of Natural Science, Assistant Professor, 自然科学研究科, 助教授 (90250662)
KATO Shin Osaka City Univ., Graduate School of Science, Assistant Professor, 大学院・理学研究科, 助教授 (10243354)
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| Project Period (FY) |
2003 – 2005
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| Keywords | Dirichlet space / spectral convergence / kernel function / energy form / Riemannian distance / netwok / effective resistance / quasi-isometry |
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| Research Abstract |
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
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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
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