The heat content asymptotics for variable geometries

Let g(t) be a time-dependent family of Riemannian metrics on a manifold M w ith a smooth boundary. Let phi be the initial temperature of M and let rho be the specific heat of M. Impose Dirichlet or Neumann boundary conditions and let beta(t) be the resulting total heat energy content of M. As t down arrow 0, one can expand beta similar to Sigma(n) beta(n)t(n/2) in an asympt otic series in half integer powers of the parameter t. We determine B-n for n less than or equal to 4 in terms of geometric quantities; this extends p revious results from the autonomous setting where the metric was independen t of the parameter t to a dynamic setting where the metric is permitted to be time dependent.