The heat content asymptotics for variable geometries

Abstract

Let be a time-dependent family of Riemannian metrics on a manifold M with a smooth boundary. Let be the initial temperature of M and let be the specific heat of M. Impose Dirichlet or Neumann boundary conditions and let be the resulting total heat energy content of M. As , one can expand in an asymptotic series in half integer powers of the parameter t. We determine for in terms of geometric quantities; this extends previous results from the autonomous setting where the metric was independent of the parameter t to a dynamic setting where the metric is permitted to be time dependent.