Remark About Heat Diffusion on Periodic Spaces

TL;DR

This paper analyzes the asymptotic behavior of the heat kernel on certain Riemannian manifolds with periodic structures, revealing how heat diffusion is influenced by an effective Euclidean metric derived from topological properties.

Contribution

It provides a detailed computation of the heat kernel asymptotics on manifolds with a free cocompact Z^k-action, linking heat diffusion to the Hodge inner product on cohomology.

Findings

Heat kernel asymptotics are characterized for large t and d(x,y) ~ sqrt(t).

Heat diffusion is governed by an effective Euclidean metric from Hodge inner product.

Results connect geometric analysis with topological invariants.

Abstract

Let M be a complete Riemannian manifold with a free cocompact Z^k-action. Let k(t,x,y) be the heat kernel on M. We compute the asymptotics of k(t,x,y) in the limit in which t goes to infinity and d(x,y) is comparable to sqrt{t}. We show that in this limit, the heat diffusion is governed by an effective Euclidean metric on R^k coming from the Hodge inner product on H^1(M/Z^k; R).