2004 Fiscal Year Final Research Report Summary
Theory of collapsing Riemannian manifolds and geometry of Alexandrov spaces
| Project/Area Number |
13440024
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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 | University of Tsukuba (2002-2004)
Kyushu University (2001) |
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Principal Investigator |
YAMAGUCHI Takao University of Tsukuba, Graduate School of Pure and Applied Sciences, Professor, 大学院・数理物質科学研究科, 教授 (00182444)
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| Co-Investigator(Kenkyū-buntansha) |
ITOH Mitsuhiro University of Tsukuba, Graduate School of Pure and Applied Sciences, Professor, 大学院・数理物質科学研究科, 教授 (40015912)
KAWAMURA Kazuhiro University of Tsukuba, Graduate School of Pure and Applied Sciences, Associate Professor, 大学院・数理物質科学研究科, 助教授 (40204771)
ISHIWATA Satoshi University of Tsukuba, Graduate School of Pure and Applied Sciences, Instructor, 大学院・数理物質科学研究科, 助手 (70375393)
OTSU Yukio Kyushu Univ, Grad.School of Math, Associate Professor, 大学院・数理物質科学研究科, 助教授 (80233170)
SHIOYA Takashi Tohoku Univ, Grad.Schol of Sa, Associate Professor, 大学院・理学研究科, 助教授 (90235507)
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| Project Period (FY) |
2001 – 2004
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| Keywords | Riemannian manfolds / collapsing / Gromov-Hausdorff / Alexandrov spare / 3-manifold / 4-manifold / graph manifold / energy form |
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| Research Abstract |
1.We have completed the study of collapsing 4-manifolds whose sectional curvature and diameter are uniformly bounded from below and above respectively, and established the geometry of 3-dimensional and 4-dimensional complete open spaces of nonnegative curvature (Yamaguchi).
2.We have proved that a 3-manifold with a lower curvature bound having a small volume is a graph manifold (Yamaguchi and Shioya).
3.We have determined the Gromov-Hausdorff convergence of surfaces with uniformly bounded total absolute curvature, and developed geometry of limit pearl spaces in detail such as singularities, homotopy types, number of pearls (Yamaguchi and Hori).
4.We have determined local geometric properties of a neighborhood of a singular point in an two-dimensional singular spaces with curvature bounded above proving that it is a gluing of several Lipschitz disks (Yamaguchi, Nagano and Shioya).
5.We have defined the notion of singular spaces with Ricci curvature bounded below, and introduced energy forms from such spaces to general metric spaces. We have proved the Poincare inequality and a compactness theorem using it (Kuwae and Shioya).
6.We have considered discrete approximations of spaces like Riemannian manifolds or Alexandrov spaces by graphs called nets, and proved that the convergence of Laplacians of nets to that of the space (Otsu). Using the idea of net-approximation above, we have studied the asymptotic behavior of heat operators on manifolds and obtained a central limit theorem for heat operator on nilpotent covering manifolds.
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Research Products
(14 results)
