Bubbling phenomenons of harmonic maps and their compactification

调和映射的冒泡现象及其紧致化

• 批准号: 10640092
• 负责人: KOTANI Makoto
• 金额: $ 1.28万
• 依托单位: TOHOKU UNIVERSITY (1999)Toho University (1998)
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1998
• 资助国家: 日本
• 起止时间: 1998 至 1999
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-10640092/
• 关键词: crystal lattice; harmonic map; transition probability; random walk; central limit theorem; Albanese map; 調和写線; アルバネーゼ写線; フロッケ理論; 熱核; アルバネゼ写像

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.

讨论了非紧黎曼流形上热核的长时间渐近性态，其中流形上的交换等距群具有紧商的不连续自由作用。利用扭拉普拉斯算子的特征值和特征函数的摄动讨论，建立了热核的局部中心极限定理和当时间参数趋于无穷大时热核的渐近幂级数展开式.我们的想法和技术的动机部分类比与Floque-Bloch理论周期薛定谔算子。对于渐近展开，我们仔细使用经典的拉普拉斯方法。在讨论过程中，我们注意到与阿贝尔群作用有关的Albanese映射的概念与渐近性密切相关。一个类似的想法是可用于渐近的随机游走的转移概率的格子图。本文所得结果改进了我们以前的结果。在渐近性中，与格图的标准实现相关联的欧氏距离，我们称之为Albanese距离，起着至关重要的作用。

项目成果

M.Kotani,T.Shirai T.Sunada: "Asymptotic behavior of the transition probability of a random walk on an infinite graph"J. Funct. Anal.. 159. 664-689 (1998)

M.Kotani，T.Shirai T.Sunada：“无限图上随机游走的转移概率的渐近行为”J。

M.Kotani and T.Sunada: "Albanese maps and off diagonal long time asymptotics for the heat kernels"Comm. Math. Phys.. (発売予定).

M.Kotani 和 T.Sunada：“热核的阿尔巴尼亚映射和非对角线长时间渐近”物理数学..（待发布）。

H.Izeki and S.Nayatani: "Canonical metric on the domain of discontinuity of a Kleinian group"Semin. Theor. Spectre. Geom. Univ. Grenoble I. 16. 9-32 (1998)

H.Izeki 和 S.Nayatani：“克莱因群不连续域的规范度量”研讨会。

M. Kotani and T. Sunada: "Jacobian tori associated with a finite graph and its abelian covering graphs"Adv. in Appl. Math.. (in press).

M. Kotani 和 T. Sunada：“雅可比环面与有限图及其阿贝尔覆盖图相关”Adv。

T. Sunada: "Twisted group C* algebras applied to spectral geometry of noncompact manifolds"Surveys in Geometry. (1998)

T. Sunada：“扭曲 C* 群代数应用于非紧流形的谱几何”几何调查。