Heat Content Asymptotics of a Riemannian Manifold with Boundary

Let M be a compact Riemannian manifold with a smooth C∞ boundary ∂M. Suppose M has initial temperature 1 at time t = 0 and suppose that ∂M is kept at temperature 0 for all t > 0. We study the asymptotic behavior of the amount of heat in M as t → 0+. We generalize these results to arbitrary operators of Laplace type with Dirichlet boundary conditions and to arbitrary smooth initial conditions.