ASYMPTOTIC EXPANSIONS FOR THE HEAT-CONTENT

Let M be a compact Riemannian manifold with smooth boundary partial de rivative M. We study the asymptotic expansions associated with the gen eralized heat operator Qe(-tPB) with suitable boundary conditions. A n ew invariant defined on the boundary of M is introduced, and a method is given that relates the heat content asymptotics for the generalized heat operator and the standard heat operator e-(tPB) with the new bou ndary asymptotics. As an application, we compute the boundary asymptot ics associated with an operator of Laplace type, and the asymptotics f or a generalized operator constructed from an operator of Dirac type.