2006 Fiscal Year Final Research Report Summary
Global analysis of the heat kernels on Riemannian manifolds and graphs
| Project/Area Number |
16340044
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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 | Tohoku University |
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Principal Investigator |
URAKAWA Hajime Tohoku University, Graduate School of Information Sciences, Professor, 大学院情報科学研究科, 教授 (50022679)
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| Co-Investigator(Kenkyū-buntansha) |
ASOH Tohl Tohoku University, Graduate School of Information Sciences, Associate Professor, 大学院情報科学研究科, 助教授 (00111352)
MUNEMASA Akihiro Tohoku University, Graduate School of Information Sciences, Professor, 大学院情報科学研究科, 教授 (50219862)
KANEKO Makoto Tohoku University, Graduate School of Information Sciences, Professor, 大学院情報科学研究科, 教授 (10007172)
OBATA Nobuaki Tohoku University, Graduate School of Information Sciences, Professor, 大学院情報科学研究科, 教授 (10169360)
ITOH Jin-ichi Kumamoto University, Faculty of Education, Professor, 教育学部, 教授 (20193493)
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| Project Period (FY) |
2004 – 2006
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| Keywords | heat kernel / convergence rate / Ricci curvature / Yang-Mills connection / bi-Yang-Mills connection / biharmonic map / Dirichlet eigenvalue problem / visualization |
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| Research Abstract |
The heat kernels of compact Riemannian manifolds converge to the equilibrium when time goes to infinity. We studied the rates of the heat kernels how do they reflect from the geometric structures of Riemannian manifolds. We showed the convergence rates are Lipshitz continuous on the deformation of Riemannian manifolds, we gave their upper estimation in terms of Ricci curvature and diameter, and also the upper estimation in terms of the non-zero first eigenvalue of the Laplacian. We gave their precise lower and upper estimations in the case of compact Riemannian symmetric spaces of rank one.
A Yang-Mills connection is a critical point of the Yang-Mills functional, and this is an analogue of harmonic map which is a critical point of the energy functional. Recently, the notion of biharmonic map was introduced which is a critical point of the 2-energy functional. We introduced the notion of 2-Yang-Mills connection which is a critical point of the 2-Yang-Mills functional. This notion is a natural generalization of Yang-Mills connection, and many further studies would be expected.
We introduced quite new method to visualize the Dirichlet or Neumann boundary eigenvalue problem of the Laplacian on plane domains. This method improved 20 percents fast comparing the known methods and reduced many steps input the data into computers. This new method made visualizations of the eigenvalue problems of the Laplacian on compact surfaces and bounded three dimensional domains. We applied to get patent of this method for programming of computer.
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[Book] 微積分の基礎2006
Author(s)
浦川 肇
Total Pages
232
Publisher
朝倉書店
Description
「研究成果報告書概要(和文)」より
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