Communications in Mathematical Physics

, Volume 209, Issue 3, pp 633–670

Albanese Maps and Off Diagonal Long Time Asymptotics for the Heat Kernel

  • Motoko Kotani
  • Toshikazu Sunada

DOI: 10.1007/s002200050033

Cite this article as:
Kotani, M. & Sunada, T. Comm Math Phys (2000) 209: 633. doi:10.1007/s002200050033

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 Schrödinger 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 [4]. 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.

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Motoko Kotani
    • 1
  • Toshikazu Sunada
    • 2
  1. 1.Department of Mathematics, Faculty of Science, Toho University, Miyama 2-2-1, Funabashi, Chiba 274, Japan. e-mail: kotani@toho-u.ac.jpJP
  2. 2.Mathematical Institute, Graduate School of Sciences, Tohoku University, Aoba, Sendai 980-77, Japan.¶e-mail: sunada@math.tohoku.ac.jpJP

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