Summary: Upper and lower estimates on the rate of decay of the heat kernel on a complete noncompact Riemannian manifold are recently obtained in terms of the geometry at infinity of the manifold, more precisely in terms of a kind of \(L^2\) isoperimetric profile. We give an outline of these results and show how they can give some partial answers to the following question: given the volume growth of a manifold, e.g. polynomial or exponential, how fast and how slow can the heat kernel decay be? The connection between the volume growth and the \(L^2\) isoperimetric profile will be made through Poincaré type inequalities. A large part of the material presented here is the result of a joint work with A. Grigor’yan.
For the entire collection see [Zbl 0990.00049].