Heat content and trace asymptotics for a time dependent family of Laplace operators
by
Peter B Gilkey
University of Oregon, USA
Coauthors: JH Park (Honam University, South Korea), Klaus Kirsten (University Manchester, United Kingdom)

Oral Communication

Let M be a compact Riemannian manifold with smooth boundary. We let D_t be a smooth time dependent family of operators of Laplace type; for example, we could take D_t:=\Delta_t to be the scalar Laplacian defined by a time dependent family of metrics g_t. We assume given a decomposition of the boundary \partialM=C_D \cup C_N as the disjoint union of closed (possibly empty) sets. We impose Dirichlet boundary conditions on C_D and Neumann boundary conditions on C_N. We extend previous work on the heat content asymptotics and the heat trace asymptotics to this setting.

The heat content asymptotics are defined as follows. Let \phi(x) be the initial temperature distribution let p(x;t) be a heat source on M, and let \psi be an auxiliary function defined only on the boundary. On the Neumann component C_N, we pump heat into M at a rate determined by \psi and on the Dirichlet component we keep the boundary at temperature \psi. Let u_{p, \phi, \psi} be the resulting temperature distribution and let p(x;t) be the specific heat. The heat content \beta(p, \phi, \psi, \rho)(t):=\int_Mu_{p, \phi, \psi}(x;t)\rho(x;t)dx has an asymptotic expansion as t\downarrow0 of the form \beta(p, \phi, \psi, \rho)(t) ~ \sum_{n >= 0}\beta_n(p, \phi, \psi, \rho) t^n/2 where the heat content asymptotics \beta_n are locally computable invariants. We give explicit combinatorial formulas for \beta_n for n=0, 1, 2, 3, 4.

Let K(t, x, y;D_t) be the fundamental solution for the homogeneous heat equation; u_{0, \phi, 0}=\int_MK(t, x, y;D_t)\phi(y)dy. Let a(t):=\int_MK(t, x, x;D_t); if D_t is independent of t, then a(t)=Tr(e^-tD). As t\downarrow0⁺ there is an asymptotic expansion of the form a(t) ~ \sum_{n >= 0}t^(n-m)/2a_n. The heat trace asymptotics a_n are locally computable and we give local formulae for a₀, a₂, and a₄ if \partialM is empty.

Both the a_n and \beta_n exhibit non-trivial dependence on the time dependent coefficients describing the symbol of the family D_t. It is this dependence upon the variable geometry involved that forms the focus of this report

Date received: May 22, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cadq-54.