1999 Fiscal Year Final Research Report Summary
Geometric structure and topology of manifolds and graphs
| Project/Area Number |
10640078
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Geometry
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| Research Institution | Okayama University |
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Principal Investigator |
KATSUDA Atsushi Okayama University, Science, Associate Prof., 理学部, 助教授 (60183779)
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| Co-Investigator(Kenkyū-buntansha) |
TAMURA Hideo Okayama University, Science, Prof., 理学部, 教授 (30022734)
SHIMOKAWA Kazuhisa Okayama University, Science, Prof., 理学部, 教授 (70109081)
SAKAI Takashi Okayama University, Science, Prof., 理学部, 教授 (70005809)
TAKEUCHI Hiroshi Shikoku University, Management and Information, Prof., 経営情報学部, 教授 (20197271)
IKEDA Akira Okayama University, Education, Prof., 教育学部, 教授 (30093363)
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| Project Period (FY) |
1998 – 1999
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| Keywords | spectrum / inverse problem / stability / the Hausdorff distance / Laplacian / isospectral graphs / isoperimetric constants |
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| Research Abstract |
The present project has been devoted to the study on the following subjects related to spectral geometry of manifolds and graphs.
(1) Spectral geometry of graphs : We have proved the analog in graphs of the celebrated Faber-Krahn inequlity for domains in Euclidean spaces and find two methods of construction of isospectral pairs of graphs with respect to combinatorial Laplacian. These are contents of References. Moreover, I start to investigate aymptotic behavior of random walks defined on infinite cover of finite graphs by the Heisenberg group.
(2) Stability of the Gel'fand inverse spectral problem : This project is joint work with Y. V. Kurylev and M. Lassas. The Gel'fand inverse spectral problem is to determine a Riemannian manifold with boundary from the spectrum and the boundary value of the Neuman Laplacian. This is solved by results Belishev and Kurylev combining the approximate controllabity results by Tataru. Then, one of next challenge is the stability. We first obtained stability results in the class of manifolds including the condition on the derivative of curvature and later, succeed to remove it. Moreover, we have obtained the existence results of harmonic coordinates in manifolds with boundary. These results are heavily depend on compactness arguments and thus, no effective estimate can not be given. However, we also have partial results foreffective estimates.
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