1999 Fiscal Year Final Research Report Summary
Bubbling phenomenons of harmonic maps and their compactification
| Project/Area Number |
10640092
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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 (1999)
Toho University (1998) |
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Principal Investigator |
KOTANI Makoto Mathematical Institute, Tohoku University, Ass. Prof., 大学院・理学研究科, 助教授 (50230024)
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| Co-Investigator(Kenkyū-buntansha) |
NAKAGAWA Yasuhiro Mathematical Institute, Tohoku University, Ass. Prof., 大学院・理学研究科, 助教授 (90250662)
IZEKI Hiroyasu Mathematical Institute, Tohoku University, Ass. Prof., 大学院・理学研究科, 助教授 (90244409)
SUNADA Toshikazu Mathematical Institute, Tohoku University, Prof., 大学院・理学研究科, 教授 (20022741)
MIYAOKA Reiko Faculty of Science, Sophia University, Prof., 理工学部, 教授 (70108182)
OHNITA Yoshihiro Faculty of Science, Tokyo Metropolitan University, Prof., 理学部, 教授 (90183764)
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| Project Period (FY) |
1998 – 1999
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| Keywords | crystal lattice / harmonic map / transition probability / random walk / central limit theorem / Albanese map |
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| Research Abstract |
We discuss long time asymptotic behaviors of the heat kernel on a non-compact Riemannian manifold which admits a discontinuous free action of an abelian isometry group with a compact quotient. A local central limit theorem and the asymptotic power series expansion for the heat kernel as the time parameter goes to infinity are established by employing perturbation arguments on eigenvalues and eigenfunctions of twisted Laplacians. Our ideas and techniques are motivated partly by analogy with Floque-Bloch theory on periodic Schrodinger operators. For the asymptotic expansion, we make careful use of the classical Laplace method. In the course of a discussion, we observe that the notion of Albanese maps associated with the abelian group action is closely related to the asymptotics. A similar idea is available for asymptotics of the transition probability of a random walk on a lattice graph. The results obtained in the present paper refine our previous ones. In the asymptotics, the Euclidean distance associated with the standard realization of the lattice graph, which we call the Albanese distance, plays a crucial role.
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