2001 Fiscal Year Final Research Report Summary
Analysis on Alexandrov Spaces
| Project/Area Number |
11440023
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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 |
Geometry
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| Research Institution | Mathematical Institute (2000-2001)
Kyushu University (1999) |
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Principal Investigator |
SHIOYA Takashi Mathematical Institute, Tohoku University, Associate Professor, 大学院・理学研究科, 助教授 (90235507)
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| Co-Investigator(Kenkyū-buntansha) |
KUWAE Kazuhiro Department of Mathematics, Yokohama City University, Associate Professor, 理学部, 助教授 (80243814)
OTSU Yukio Graduate School of Mathematics, Kyushu University, Associate Professor, 大学院・数理学研究院, 助教授 (80233170)
ITOH Junichi Faculty of Education, Kumamoto University, Professor, 教育学部, 教授 (20193493)
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| Project Period (FY) |
1999 – 2001
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| Keywords | Alexandrov spaces / Laplacian / Dirichlet forms / Gromov-Hausdorff convergence |
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| Research Abstract |
In this study, we construct foundation of analysis on Alexandrov spaces and develop it.
We prove that the embedding of the (1, 2)-Sobolev space W^<1,2>(X) of a compact Alexandrov space X into the L^2 space is compact. As a corollary, we obtain the discreteness of the spectrum of the Laplacian on a compact Alexandrov space. Here, the Laplacian is defined as an infinitesimal generator of the energy form, by using functional analysis. For a DC function on an Alexandrov space we have another concept of Laplacian depending on DC charts, called the DC-Laplacian. We investigate the relation between the DC-Laplacian and the functional analytic Laplacian. Moreover, we prove the continuity of the solutions of the eigen-equation and the heat equation on Alexaudrov spaces, and also the existence of the heat kernel. Let A(n) be the set of compact Atexandrov spaces of dimension n and curvature 【greater than or equal】 -1. We prove that on A(n) the spectral topology due to Kasue-Kumura coincides with the Gromov-Hausdorff topology. We introduce a concept of 'spaces of Ricci curvature bounded below' and energy of maps from such a space to a complete metric space. We prove the Poincare inequality and the existence of energy measure for it.
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