Asymptotic finite propagation speed for heat diffusion on certain Riemannian manifolds

Abstract

Using pointwise upper bounds recently obtained by the first author, we first show that the heat kernel h t ( x, y) on a Riemannian symmetric space of the noncompact type G K is asymptotically concentrated in an annulus centered at y and moving to infinity with finite speed 2 ¦ϱ¦, ϱ being as usual the half sum of all positive roots of G K . In the higher rank case we prove moreover that heat not only concentrates in an annulus but also along the ( K-orbit) of the ϱ-axis. By applying wave equation techniques developed by M. E. Taylor, we partially extend the above results to Riemannian manifolds with exponential volume growth and L 2 spectrum of the Laplacian bounded away from 0. Some extensions to the vector bundle case are also considered.