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Remark about heat diffusion on periodic spaces

Author: John Lott
Journal: Proc. Amer. Math. Soc. 127 (1999), 1243-1249
MSC (1991): Primary 58G11
MathSciNet review: 1476376
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Abstract: Let $M$ be a complete Riemannian manifold with a free cocompact ${\Bbb Z}^k$-action. Let $k(t, m_1, m_2)$ be the heat kernel on $M$. We compute the asymptotics of $k(t, m_1, m_2)$ in the limit in which $t \rightarrow \infty$ and $d(m_1, m_2) \sim \sqrt{t}$. We show that in this limit, the heat diffusion is governed by an effective Euclidean metric on ${\Bbb R}^k$ coming from the Hodge inner product on $\mathrm{H}^1(M/{\Bbb Z}^k; {\Bbb R})$.

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Additional Information

John Lott
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109-1109

Received by editor(s): August 5, 1997
Additional Notes: Research supported by NSF grant DMS-9704633.
Communicated by: Jozef Dodziuk
Article copyright: © Copyright 1999 American Mathematical Society