Large time behaviour of heat kernels on non-compact manifolds : fast and slow decays

Coulhon, Thierry

Journées équations aux dérivées partielles (1998), article no. 2, 12 p.

In this talk we shall present some joint work with A. Grigory’an. Upper and lower estimates on the rate of decay of the heat kernel on a complete non-compact riemannian manifold have recently been obtained in terms of the geometry at infinity of the manifold, more precisely in terms of a kind of $L^2$ isoperimetric profile. The main point is to connect the decay of the $L^1-L^{\infty }$ norm of the heat semigroup with some adapted Nash or Faber-Krahn inequalities, which is done by functional analytic methods. We shall give an outline of these results and show how they can give some 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?

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