2003 Fiscal Year Final Research Report Summary
Global Analysis of the heat kernel and Green kernel of an Infinite Graph
| Project/Area Number |
13440051
|
|---|
| Research Category |
Grant-in-Aid for Scientific Research (B)
|
| Allocation Type | Single-year Grants |
|---|
| Section | 一般 |
|---|
| Research Field |
Global analysis
|
|---|
| Research Institution | Tohoku University |
|---|
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)
|
|---|
| Co-Investigator(Kenkyū-buntansha) |
ARISAWA Mariko Tohoku University, Graduate School of Information Sciences, Associate Professor, 大学院・情報科学研究科, 助教授 (50312632)
KANEKO Makoto Tohoku University, Graduate School of Information Sciences, Professor, 大学院・情報科学研究科, 教授 (10007172)
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)
OBATA Nobuaki Tohoku University, Graduate School of Information Sciences, Professor, 大学院・情報科学研究科, 教授 (10169360)
OHNO Yoshiki Tohoku University, Graduate School of Information Sciences, Associate Professor, 大学院・情報科学研究科, 助教授 (80005777)
|
|---|
| Project Period (FY) |
2001 – 2003
|
| Keywords | Yang-Mills connection / Dirichlet eigenvalue problem / finite element method / infinite graph / affine connection / oseudharmonic map / Green kernel / symplectic manifold |
|---|
| Research Abstract |
We have obtained the following results:
(1)We constructed the theory of Yang-Mills connections over compact symplectic manifolds.
(2)We estimated the Cheeger constant, the heat kernel and the Green kernel for an infinite graph in terms of the volume growth, growth of in and out degree.
(3)We determined the stiffness and mass matrices of the finite element method for the Dirichlet eigenvalue problem for a plane domain.
(4)We calculated the Cheeger constant, the heat kernel and Green kernel of semi-regular infinite graphs and gave the explicit comparison theorem for every infinite graph.
(5)We extended Yang-Mills theory to Weyl structure, and established Atiya-Hitchin-Singer theory to Weyl manifolds, and to affine connections.
(6)We formulated discrete improper affine surface theory and show its loop group description.
(7)We showed the relation in affine differential geometry, Weyl geometry, Yang-Mills theory.
(8)We defined the notion of pseudoharmonic maps from CR manifolds to a Riemanninan manifold, and showed the first variation formula and the second variation formula.
(9)We clarified the relation of each Yang-Mills theory on Kaehler manifolds, CR manifolds, and symplectic manifolds, and characterized the minimizers of the Yang-Mills functional over compact symplectic manifolds.
|
