Gromov-Hausdorff convergence and a theory of variational convergences
| Project/Area Number |
17540058
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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 | Tohoku University |
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Principal Investigator |
SHIOYA Takashi Tohoku University, Tohoku University, Graduate School of Science, Professor (90235507)
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| Co-Investigator(Kenkyū-buntansha) |
KUWAE Kazuhiro Faculty of Education, Kumamoto University, Associate Professor (80243814)
FUJIWARA Koji Graduate School of Information Sciences, 大学院・情報科学研究科, Professor (60229078)
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| Project Period (FY) |
2005 – 2007
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| Project Status |
Completed (Fiscal Year 2007)
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| Budget Amount *help |
¥3,600,000 (Direct Cost: ¥3,300,000、Indirect Cost: ¥300,000)
Fiscal Year 2007: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2006: ¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 2005: ¥1,300,000 (Direct Cost: ¥1,300,000)
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| Keywords | Geometry / Analysis / アレクサンドロフ空間 / Bishop-Gromov条件 / リッチ曲率 / ラプラシアン / 有界変動関数 / グリーンの公式 / 分割定理 / 比較定理 / エネルギー汎関数 / 測度距離空間 |
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| Research Abstract |
In these days, the study of geometric analysis on metric measure spaces is going around. The head investigator, Shioya, Studies such a subject and his main interest is curvature of metric measure spaces and convergence, especially Alexandrov spaces, Ricci curvature of metric measure spaces, and Gromov-Hausdorff convergence of metric measure spaces. On he other hand, Mosco studied variational convergences, which is a functional analytic theory of convergence of Dirichlet energy forms. We, Shioya and the investigator, Kuwae, thought that Mosco's theory is deeply related with the study of convergence of metric measure spaces, and have extended the theory in the geometric viewpoint. We have completed it in the period of this project. The concept of convergence in our theory is nowadays called the Mosco-Kuwae-Shioya convergence and is being widely applied to the finite dimensional method in probability theory and also to some homogenization problems.
Another study is on a Laplacian comparison theorem and a splitting theorem on Alexandrov spaces with some condition corresponding to a lower bound of Ricci curvature. This is still on going. For Riemannian manifods, the Ricci curvature being bounded below is equivalent to an infinitesimal version of the Bishop-Gromov inequality. Since it is impossible to define the Ricci curvature tensor on Alexandrov spaces, we consider the infinitesimal Bishop-Gromov inequality instead of the Ricci curvature bound. Different from Riemannian, the cut-locus is not necessarily a closed set in an Alexandrov space. That may even be a dense set. By this reason, the same proof as for Riemannian manifolds does not work and we develop a new method of proof.
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Report
(4 results)
Research Products
(38 results)
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[Presentation] Laplacian on Alexandrov spaces2007
Author(s)
Shioya, Takashi
Organizer
Stochastic Calculus on Manifolds, Graphs, and Rondom Structures
Place of Presentation
Hausdorff Research Institute for Mathematics, Bonn, Germany
Year and Date
2007-10-09
Description
「研究成果報告書概要(欧文)」より
Related Report
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